Josephson junction

In physics, a Josephson junction is a nonlinear, dissipationless circuit element that uses the Josephson effect in superconductors to form a nonlinear inductor.^[1] It consists of a weak barrier (such as an insulator) separating two superconductors.^[2][3] Pairs of electrons tunnel through the barrier, producing an effective inductance that is dependent on the phase difference between the superconductors.

The Josephson junction is used as a circuit element in superconducting loops. In a circuit, the Josephson inductance and the junction capacitance can be used to form a nonlinear oscillator, similar to a LC circuit, with a current-dependent inductance.^[4] Circuits with Josephson junctions are used to make superconducting magnetometers known as SQUIDs, in classical logic gates for ultrafast computing, and in circuit quantum electrodynamics to create superconducting qubits.^[5]

In 1962, Brian Josephson predicted the Josephson effect: that pairs of superconducting electrons known as Cooper pairs could tunnel through the gap between two superconducting layers, if they were weakly separated.^[2] In 1963, Phillip Anderson and John Rowell at Bell Labs constructed the first Josephson junction, using tin-oxide-lead junctions, to verify this effect. A thin film of tin was oxidized to form the barrier, and then a thin-film cross strip of lead formed the counter electrode.^[6]

In the 1960s, James Zimmerman, Arnold Silver, and colleagues working at a Ford lab on magnetic resonance realized that Josephson junction nonlinearity, combined with the flux quantization in superconducting loops, could be used to measure small changes in magnetic flux, and create a magnetometer with high levels of magnetic field sensitivity precision.^[7][8] They created the first such device using films of tin separated by plastic,^[8] with two Josephson junctions separated by a macroscopic gap. This was the first SQUID magnetometer.

In 1966, at IBM, Juri Matisoo demonstrated sub-nanosecond switching in a Josephson junction. This and subsequent results indicated that Josephson Junctions could be faster than any available transistor at the time. This began a superconducting supercomputer project at IBM, known as the Josephson signal processor, with intentions to use (Pb-In-Au)-oxide-(Pb-Bi) Josephson junctions as a basis for logic circuits to enable ultrafast computing.^[9] This project lasted from 1967 to September 1983.^[10] The initial attempt was shut down due to insufficient speed advantage compared to rapidly developing silicon technology, but some Japanese companies remained interested, since JJ-based technology avoided the problem of large semiconductor mainframes and power dissipation.^[3]

In 1980, Tony Leggett proposed using the phase of a Josephson junction as a macroscopic quantum coordinate.^[11] In 1987, John Clarke, Michel H. Devoret, and John M. Martinis used a Josephson junction-based circuit to demonstrating that a macroscopic system could have quantum mechanical properties (specifically, tunnelling and energy quantization).^[12] They won the 2025 Nobel Prize in physics for their work.

In 1999, Yasunobu Nakamura and colleagues used Josephson junctions connected in a superconducting loop to show coherent quantum oscillations in a Josephson circuit, making this the first superconducting qubit.^[13] Superconducting qubits require the nonlinearity provided by Josephson junctions, since it allows the first two energy levels of the circuit to be addressed independently of all the others. All major superconducting qubit architectures use Josephson junctions.

The Josephson effect arises when two superconductors are weakly coupled through a thin insulating barrier. Without any voltage applied, the two superconducting electrodes' electrical charge carriers, which are Cooper pairs, will cross the barrier as a supercurrent, ${\textstyle I_{J}}$.^[14] This supercurrent is given by:

$${\displaystyle I_{J}=I_{0}\sin(\varphi )}$$

Where ${\textstyle I_{0}}$ is the critical current and ${\displaystyle \varphi }$ is the phase difference between the two superconductor's wavefunctions. Because Cooper pairs tunnel phase-coherently, the supercurrent depends only on the phase difference. This is the first Josephson equation, known as the current-phase relation.

When a voltage is applied, the phase difference evolves in time. This is described by the second Josephson equation, the voltage-phase relation:

$${\displaystyle {\frac {d\varphi (t)}{dt}}={\frac {2e}{\hbar }}V={\frac {2\pi }{\Phi _{0}}}V}$$

The two Josephson equations can be used to derive the effective inductance.

$${\displaystyle V(t)={\frac {\hbar }{2e}}{\frac {d\delta }{dt}}={\frac {\hbar }{2e}}\left({\frac {dI}{d\delta }}\right)^{-1}{\frac {dI}{dt}}}$$

By analogy with an inductor, ${\textstyle V(t)=L_{J}{\frac {dI}{dt}}}$, so the effective inductance is:

$${\displaystyle L_{J}={\frac {\hbar }{2e}}\left({\frac {dI}{d\delta }}\right)^{-1}={\frac {\Phi _{0}}{2\pi }}\left({\frac {dI}{d\delta }}\right)^{-1}}$$

Then, using the first Josephson equation:

$${\displaystyle {\frac {dI}{d\delta }}=I_{0}\cos(\delta )}$$

So the Josephson inductance is

${\displaystyle L_{J}={\frac {\Phi _{0}}{2\pi I_{0}\cos(\delta )}}}$

The Josephson junction is both non-linear and non-dissipative. It is non-linear because its effective inductance depends on the superconducting phase difference ${\displaystyle \delta }$, and it is (ideally) non-dissipative because it operates without resistance as long as the current remains below the critical value ${\displaystyle I_{0}}$. However, at non-zero voltage, there is dissipation introduced by additional channels of loss. This is modeled as a shunt resistor in the RCSJ model.

The potential energy of the junction is:

$${\displaystyle U(\varphi )=-E_{J}\cos \varphi }$$

The Hamiltonian for this system is:

$${\displaystyle H={\frac {Q^{2}}{2C}}-E_{J}\cos \phi }$$

Or equivalently:

$${\displaystyle H=4E_{C}n^{2}-E_{J}\cos \phi }$$

where ${\textstyle E_{C}={\frac {e^{2}}{2C}}}$ is the charging energy and ${\textstyle n={\frac {Q}{2e}}}$is the number of Cooper pairs.

To study the dynamics of a Josephson junction, the model can be simplified by assuming that the normal conductance is constant. In this model, the Josephson junction is characterized by the Josephson inductance and the normal resistance R of the junction.^[15]

Similarly to a parallel plate capacitor, the Josephson junction design must have some capacitance because it consists of two electrodes separated by an insulator or weak link.^[16] It also may have some resistance, due to the dissipative current at finite voltage. Thus, the Josephson junction can be modeled as an ideal non-linear Josephson element in parallel with a normal resistor and capacitor. This is known as the resistively and capacitively-shunted junction (RCSJ) model.^[17] Alternatively, in the limit where capacitance is small, the resistively-shunted junction (RSJ) model is used.

In both models, the normal resistance will depend on bias voltage and temperature, ${\displaystyle R_{N}=R(V,T)}$.^[18] It is thought that lower temperatures will make the resistance go to zero, because there will be fewer quasiparticles in the circuit. However, this in practice difficult, as there exist a non-negligible population of non-equilibrium quasiparticles, due to cosmic rays or other phenomena, that are not accounted for by temperature alone.^[18]

The main equation of the RCSJ model is:

$${\displaystyle {\frac {\hbar C}{2e}}\,{\frac {\partial ^{2}\varphi }{\partial t^{2}}}\;+\;{\frac {\hbar }{2eR}}\,{\frac {\partial \varphi }{\partial t}}\;+\;I_{c0}\sin(\varphi )\;=\;I}$$

This differential equation describes the dynamics of the Josephson junction when a bias current ${\displaystyle I}$ is applied.^[19]

Rewriting the RCSJ model equation in dimensionless coordinates gives:

$${\displaystyle {\frac {\partial ^{2}\varphi }{\partial \tau ^{2}}}+{\frac {1}{Q}}\,{\frac {\partial \varphi }{\partial \tau }}+\sin(\varphi )={\frac {I}{I_{c0}}}\,.}$$

Where the dimensionless time variable ${\displaystyle \tau =\omega _{p}t}$, with

${\displaystyle \omega _{p}={\sqrt {\frac {2eI_{0}}{\hbar C}}}\,.}$

This is known as the Josephson plasma frequency of the junction. It represents the untitled frequency of the Josephson junction oscillations.

When a current is applied to a Josephson junction, the RCSJ model can be used to describe its behavior. A current-biased Josephson junction exhibits an effective tilted washboard of potential energy.^[20] Although the bare Josephson junction resembles periodic wells with no tilt, the effective potential energy takes on an extra term in a current-biased circuit:

$${\displaystyle U(\varphi )=-E_{J}\cos \varphi -{\frac {\hbar }{2e}}I\varphi }$$

where ${\displaystyle I}$ is the bias current.

This is of the form of a general tilted washboard potential, given by:

$${\displaystyle U(x)=-Ax-B\cos(x)}$$

Their graphs are tilted cosine functions, with the degree of tilt given by the constant A. The Josephson junction acts like a damped nonlinear oscillator.

In circuit quantization, flux and charge are treated as generalized coordinates. The quantization step promotes ${\displaystyle \phi }$ (dimensionless phase) and ${\displaystyle n}$ (Cooper pair number) to operators satisfying the commutation relations. This gives the quantum Hamiltonian for the Josephson junction, used in circuit quantum electrodynamics:

$${\displaystyle {\hat {H}}=4E_{C}{\hat {n}}^{2}-E_{J}\cos {\hat {\phi }}}$$

This forms the basic nonlinear element used in circuit quantum electrodynamics and in superconducting qubits.^[21]

The superconductor-insulator-superconductor design, also called a superconducting tunnel junction, has a thin insulating material as the barrier, typically around 1-2 nanometers thick. They are the most widely used form of Josephson junction, and are used for superconducting qubits, SQUIDs, and voltage standard devices.

Aluminum is often used to fabricate SIS junctions because it can form an oxide that is the correct thickness to act as the insulating barrier.

The barrier is a thin normal metal, rather than an insulator. The normal metal may not be superconducting, or it may be a superconductor with a smaller critical temperature. Although for a SIS Josephson junction, the current-phase relationship is often close to being perfectly sinusoidal, the SNS junction current-phase relation has a ramp-like dependence.^[22]

SNS designs have higher critical currents and lower impedance than traditional SIS designs. Implementations of SNS junctions include SNS sandwiches, variable-thickness bridges, and ramp junctions.^[23] They are used in rapid single flux quantum logic circuits which encode digital information based on magnetic flux quanta since they have a non-hysteretic current–voltage dependence.^[24]

The barrier is a constriction or weak link, such as a microbridge of superconducting material.^[25] These junctions can carry larger critical currents than SIS junctions. ScS junctions may be used in cases where an all-superconducting junction would be preferable to introducing an insulator or non-superconducting metal to the device.^[26] ScS junctions have a sub-millimeter frequency range of around 500–1000 GHz.^[27]

Because of the non-negligible presence of non-equilibrium electrons (often called quasiparticles in this context), the dynamic behavior of ScS junctions is much more complicated than traditional resistive models.^[28]

In traditional Josephson junctions, the ground state phase is zero. However, phi Josephson junctions instead have a non-zero Josephson phase in the ground state. In this case, the first Josephson equation (the current-phase relation) is modified, from ${\textstyle I=I_{0}\cos(\delta )}$ to:

$${\displaystyle I=I_{0}\cos(\delta +\phi )}$$

Phi-junctions can be physically realized by breaking symmetries, such as by using an altermagnet.^[29]

Pi Josephson junctions are a kind of Phi Josephson junction with a 180 degree phase difference between them. The first Josephson equation (the current-phase relation) is modified in this case, from ${\textstyle I=I_{0}\cos(\delta )}$ to:

$${\displaystyle I=I_{0}\cos(\delta +\pi )=-I_{0}\sin(\delta )}$$

Pi-junctions can be physically realized using a ferromagnetic barrier, which is usually magnetic but may be insulated.^[30] This architecture is often called a superconductor-ferromagnetic-superconductor (SFS) junction.^[31] In an SFS junction, the ground state energy oscillates between 0 and π as a function of the ferromagnet's thickness. Applications of the SFS Pi-junction include digital superconducting logic and memory.^[32]

Long josephson junctions are josephson junctions where at least one parameter exceeds the Josephson penetration depth ${\displaystyle \lambda _{J}}$. The Josephoson penetration depth describes how far an external magnetic field penetrates into a Josephson junction, and is given by:

$${\displaystyle \lambda _{J}={\sqrt {\frac {\Phi _{0}}{2\pi \mu _{0}j_{c}\left(L+2\lambda _{L}\right)}}}}$$

where L is the length of the junction. In Josephson junctions with dimensions that exceed the penetration depth, the superconducting phase varies spatially, described by the sine-Gordon equation.^[33] These long Josephson junctions can support propagating flux quanta known as fluxons.^[33][34] They may be used as a Josephson transmission line or as a flux flow oscillator.^[34]

The majority of Josephson junctions are fabricated in a cleanroom using thin-film deposition and patterning. There are two major methods used for fabrication. In shadow evaporation, patterning occurs during deposition using a resist mask. In whole-wafer or trilayer methods, patterning occurs after deposition by lithography and etching.

In both methods, a superconducting metal is deposited onto an insulating substrate. The most common substrates are Bare Si and oxidized Si, usually with wafers of diameters 50 mm to 150 mm.^[35]

In shadow evaporation, shadow masks, made from highly intricate machined masks, are used to pattern the Josephson junction. This was the first method used for fabricating Josephson junctions. In shadow mask techniques, the junction is defined during deposition rather than by post-deposition patterning by using a physical max transferred to an underlying film using wet or dry etching.^[35]

In research settings and for superconducting qubits, shadow evaporation is the dominant method because it preserves interface cleanliness and is highly precise.

The superconducting films, such as aluminum, have low melting points, so thermal evaporation is used for deposition.^[35] The evaporation source is usually a wire, and temeratures range from 1000 to 2000 degrees Celsius for film condensation during the evaporation step.

A method developed by Kroger and colleagues in 1981 has patterning occurring after deposition. Instead of one layer at a time, a continuous trilayer stack is deposited across the entire wafer. In this method, the entire superconductor-barrier-superconductor sandwich is formed before patterning.^[36]

This method is more reporducible and scalable than traditional shadow evaporation techniques. Niobium trilayer processes are used for commercial production of SQUID sensors and voltage standard devices.^[35]

Whole wafer processes use sputtering deposition. This is an alternative deposition technique, to reduce potential interactions between the evaporant and supporting material. In this method, the target material is bombarded with energetic ions. The ions can eject the target atoms into the gas phase. The sputter deposition method allows sputter more complex alloys to be deposited, but sputter deposition rates are slow.^[35]

This is the preferred method of deposition for Nb/Al-AlOx/Nb Josephson junctions.

Usually a high-resolution lithography using an electron beam or focused ion beam etches the pattern on the device. Dry or wet etching can be used. Dry etching processes use physical etchants or reactive gasses, while wet etching process submerge the substrate in liquid etchants.^[5]

Superconducting qubits using Josephson junctions are popular candidates for a quantum processor because superconductors inherently have very low dissipation, making long coherence times possible.^[37] These kinds of qubits are used by Google, IBM, Rigetti, and others to engineer quantum processors.

A superconducting qubit is essentially a nonlinear resonator formed from the Josephson inductance and its junction capacitance, which acts like a non-linear LC circuit.^[37] Since this oscillator is nonlinear, it is anharmonic, and allows the first two energy levels of the circuit to be addressed independently of the others. This is necessary because the system in principle has many energy levels, but the qubit's operating space includes only the two lowest states.^[37] In fact, In order to detect energy quantization at all, a circuit must include a non-linear element.^[5]

Almost all superconducting qubit architectures use Josephson junctions (with exceptions being topological qubits). The standard implementation is a Josephson tunnel junction, but other architectures use constriction junctions.

A SQUID, or superconducting quantum interference device, consists of two Josephson junctions arranged in a loop, with magnetic flux threading through it. SQUIDs effectively act as a single flux-tunable Josephson junction, because the applied flux biases the phase across the junctions. Thus, the SQUID's critical current, and hence its effective inductance, can be tuned by the external magnetic flux.

Because Josephson junctions have a critical current, they can be fashioned into a binary logic circuit: when the current through the junction is less than the critical current, the voltage is zero, but once it exceeds the critical current, the voltage oscillates in time.^[2] Thus, Josephson junction switching between zero-voltage and finite-voltage states can be used in logic circuits.An example of applications in logic circuits is the rapid single flux quantum technology, which stores information in the form of magnetic flux quanta, transferred by single flux quantum voltage pulses produced by Josephson junctions.

Logic devices using Josephson junctions have ultra-high switching speeds, of around 10 picoseconds, and low power dissipation of around 1 µW.^[38] It is an active area of research because it may be able to overcome scaling limits in CMOS electronics. However, its level of integration remains significantly less than that of CMOS circuits.^[39]

Highly integrated arrays of Josephson junctions, containing more than 10,000 or even 100,000 junctions, can be used as a voltage standard.^[41] This voltage standard has achieved accuracy of 1 parts per billion or better, allowing reference of the volt up to just to physical constants.^[41] The junctions used are typically SIS or SNS.^[42]

A centimeter-wide superconducting circuit containing a Josephson junction was used by John Clarke, Michel H. Devoret, and John M. Martinis to demonstrate macroscopic quantum phenomena. In the device, billions of superconducting electrons in the circuit formed a collective macroscopic system described by a single quantum phase.^[43] By showing evidence of tunneling behavior in the junction, they were able to demonstrate quantum tunneling in a macroscopic object. This was the subject of the 2025 Nobel Prize in Physics.

Josephson junctions are used in ultra-low noise amplifiers, like Josephson parametric amplifiers (JPAs) and traveling-wave parametric amplifiers (TWPAs), because they are largely dissipationless. Other non-linear elements such as semiconductors contribute to dissipation, even in low noise amplifiers like HEMTs. However, because there is no added noise when using a Josephson junction, superconducting amplifiers using Josephson junctions are able to operate near the quantum noise limit. This makes them useful for experiments that require extremely low noise, such as axion detection^[44] and readout of solid-state qubits.^[45]

• Josephson effect
• Superconducting tunnel junction
• SQUID
• Josephson voltage standard
• Superconducting quantum computing