9.4 Josephson junctions

Josephson Junctions

Josephson junctions are one of the most important structures in superconducting electronics and quantum computing. They consist of two superconductors separated by a thin insulating barrier, and they allow Cooper pairs to tunnel quantum mechanically between the two sides. This tunneling gives rise to remarkable phenomena, including the DC and AC Josephson effects, flux quantization, and Shapiro steps, all of which have direct applications in voltage metrology, SQUIDs, and superconducting qubits.

Superconductor-Insulator-Superconductor Structure

A Josephson junction has a sandwich geometry: two superconducting electrodes with a thin insulating barrier between them. The barrier is typically only a few nanometers thick and is often made of aluminum oxide ($\text{Al}_2\text{O}_3$) or magnesium oxide ($\text{MgO}$). The electrodes are usually low-temperature superconductors such as niobium, aluminum, or lead.

The barrier must be thin enough that the superconducting wavefunctions on each side overlap, enabling quantum tunneling of Cooper pairs across the insulator. If the barrier is too thick, the tunneling probability drops exponentially and the junction stops functioning.

Tunneling of Cooper Pairs

In a superconductor, electrons pair up into Cooper pairs through electron-phonon interactions. These pairs share a single macroscopic quantum wavefunction characterized by a well-defined phase.

Cooper pairs can tunnel coherently through the insulating barrier of a Josephson junction even with zero applied voltage. This is a macroscopic quantum effect: quantum behavior that manifests at a scale you can measure with lab instruments. The tunneling current depends on the phase difference $\delta$ between the superconducting wavefunctions on either side of the barrier, which is the central variable in all Josephson junction physics.

DC Josephson Effect

The DC Josephson effect is the flow of a dissipationless supercurrent through the junction with no voltage applied. The supercurrent is governed by:

$I = I_c \sin(\delta)$

where $I_c$ is the critical current (the maximum supercurrent the junction can carry) and $\delta$ is the phase difference between the two superconductors.

The critical current $I_c$ depends on junction properties: barrier thickness, junction area, and the superconducting energy gap $\Delta$. A thinner barrier or larger area gives a higher $I_c$.

The physical meaning here is striking: the superconducting wavefunctions on both sides of the barrier are phase-locked, and the current flows purely because of their phase relationship, not because of any voltage driving it.

AC Josephson Effect

When you apply a constant DC voltage $V$ across the junction, the phase difference evolves in time, and the supercurrent oscillates at a frequency:

$f = \frac{2eV}{h}$

This is the AC Josephson effect. The numerical value of the proportionality constant is:

$f = 483.597 \text{ GHz/mV} \times V$

The factor of $2e$ (not $e$) appears because the tunneling carriers are Cooper pairs, each carrying charge $2e$. This precise frequency-voltage relationship is what makes Josephson junctions so valuable for voltage metrology: if you know the frequency, you know the voltage to extraordinary precision, since it depends only on fundamental constants.

Current-Voltage Characteristics

The I-V curve of a Josephson junction is highly nonlinear and has two distinct regimes:

1. Below $I_c$: The junction carries a supercurrent with zero voltage drop. This is the DC Josephson regime. The junction behaves as a perfect short circuit for DC.
2. Above $I_c$: A finite voltage develops across the junction, and it enters the resistive state. In this regime the AC Josephson effect operates, with the supercurrent oscillating at the Josephson frequency while a DC voltage appears across the junction.

The transition from the zero-voltage state to the resistive state is abrupt and hysteretic in underdamped junctions (those with large capacitance relative to dissipation), which is important for switching applications and qubit design.

Shapiro Steps

When you apply an external AC signal (at frequency $f$) to a DC-biased Josephson junction, the I-V curve develops flat voltage plateaus called Shapiro steps. These steps appear at quantized voltages:

$V_n = \frac{nhf}{2e}$

where $n$ is an integer ($n = 0, 1, 2, \ldots$), $h$ is Planck's constant, and $e$ is the electron charge.

Shapiro steps arise from phase-locking between the junction's internal Josephson oscillation and the external AC drive. When the two frequencies are commensurate, the time-averaged voltage locks onto one of these discrete values. The step heights depend on the amplitude of the applied AC signal and follow Bessel function behavior.

Shapiro steps are the basis of the modern Josephson voltage standard, which defines the volt in terms of frequency and fundamental constants.

Josephson Penetration Depth

The Josephson penetration depth $\lambda_J$ sets the length scale over which magnetic fields and phase gradients vary along the plane of the junction:

$\lambda_J = \sqrt{\frac{\hbar}{2e\mu_0 J_c d}}$

Here $J_c$ is the critical current density, $d$ is the effective magnetic thickness of the junction (accounting for London penetration depths on both sides plus the barrier thickness), and $\mu_0$ is the vacuum permeability.

This parameter determines whether a junction is classified as:

• Short junction (dimensions $\ll \lambda_J$): The phase difference is approximately uniform across the junction. Most junctions used in qubits and SQUIDs fall in this category.
• Long junction (dimensions $\gg \lambda_J$): The phase can vary significantly along the junction, and the junction can support Josephson vortices (fluxons) that propagate along the barrier.

Flux Quantization in Josephson Junctions

In a superconducting loop containing one or more Josephson junctions, the total magnetic flux through the loop is quantized in units of the flux quantum:

$\Phi_0 = \frac{h}{2e} \approx 2.068 \times 10^{-15} \text{ Wb}$

The quantization condition for a loop with a single junction is:

$\oint \nabla\varphi \cdot d\mathbf{l} = 2\pi n - \frac{2\pi}{\Phi_0}\Phi$

where $\varphi$ is the phase of the superconducting order parameter, $n$ is an integer, and $\Phi$ is the total enclosed flux.

This flux quantization causes the critical current of a junction in a loop to be periodically modulated by the applied magnetic field. That periodic modulation is the operating principle behind SQUIDs.

SQUID: Superconducting Quantum Interference Device

A SQUID exploits the interference between supercurrents in a loop containing Josephson junctions to achieve extraordinary magnetic field sensitivity.

• DC SQUID: A superconducting loop with two Josephson junctions. It's biased with a constant current just above the combined critical current. The voltage across the SQUID oscillates as a function of the applied flux with period $\Phi_0$, allowing flux changes far smaller than $\Phi_0$ to be detected.
• RF SQUID: A superconducting loop with a single Josephson junction, inductively coupled to an RF tank circuit. Changes in the enclosed flux shift the resonant characteristics of the tank circuit.

SQUIDs can detect magnetic fields as small as $\sim 10^{-15}$ T, making them the most sensitive magnetometers available. Applications include biomagnetism (measuring magnetic fields from brain or heart activity), geophysical surveying, and materials characterization.

RCSJ Model

The resistively and capacitively shunted junction (RCSJ) model is the standard dynamical model for a Josephson junction. It represents the junction as three parallel elements:

1. An ideal Josephson element carrying supercurrent $I_c \sin(\delta)$
2. A resistor $R$ representing quasiparticle tunneling and dissipation
3. A capacitor $C$ representing the geometric capacitance of the junction electrodes

Applying Kirchhoff's current law gives the equation of motion for the phase:

$\frac{\hbar C}{2e}\ddot{\delta} + \frac{\hbar}{2eR}\dot{\delta} + I_c \sin(\delta) = I$

where $I$ is the external bias current. This equation is mathematically equivalent to a driven, damped pendulum, which provides useful physical intuition: $\delta$ plays the role of the pendulum angle, the capacitance acts as inertia, the resistance provides damping, and the bias current is the driving torque.

The Stewart-McCumber parameter $\beta_c = 2eI_cR^2C/\hbar$ determines the damping regime:

• $\beta_c \gg 1$: Underdamped (hysteretic I-V curve)
• $\beta_c \ll 1$: Overdamped (non-hysteretic I-V curve)

Josephson Junction Applications

Josephson junctions underpin several important technologies:

• Voltage standards: Arrays of Josephson junctions driven by microwaves produce Shapiro steps at precisely known voltages, forming the basis of the international definition of the volt.
• SQUIDs: Used for ultra-sensitive magnetic measurements in fields ranging from neuroscience to mineral exploration.
• Superconducting digital electronics: Rapid single flux quantum (RSFQ) logic uses the motion of individual flux quanta through Josephson junction circuits to perform high-speed, low-power computation.
• Superconducting qubits: The nonlinearity of the Josephson junction is what makes superconducting qubits possible (see below).

Superconducting Qubits

Josephson junctions provide the essential nonlinear, dissipationless inductance needed to create an anharmonic oscillator, which is what distinguishes a qubit (with addressable $|0\rangle$ and $|1\rangle$ states) from a simple harmonic LC circuit (where all energy levels are equally spaced).

The main qubit types are:

• Charge qubit (Cooper pair box): A small superconducting island coupled to a reservoir through a Josephson junction. The qubit states correspond to different numbers of excess Cooper pairs on the island. Sensitive to charge noise.
• Flux qubit: A superconducting loop interrupted by one or more Josephson junctions. The two qubit states correspond to persistent currents circulating clockwise and counterclockwise. Operates near half a flux quantum of applied flux.
• Phase qubit: A current-biased Josephson junction where the qubit states are the lowest two energy levels in a tilted washboard potential.
• Transmon qubit: An evolution of the charge qubit designed to suppress charge noise by operating in the regime $E_J \gg E_C$, where $E_J$ is the Josephson energy and $E_C$ is the charging energy. The transmon is currently the most widely used qubit architecture in superconducting quantum processors.