A Josephson junction is a nonlinear, dissipationless circuit element that uses the Josephson effect in superconductors to form a nonlinear inductor. It consists of a weak barrier (such as an insulator) separating two superconductors. 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. 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.
History In 1962, British physicist 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. 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.
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. They created the first such device using films of tin separated by plastic, 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. This project lasted from 1967 to September 1983. 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.
In 1980, Tony Leggett proposed using the phase of a Josephson junction as a macroscopic quantum coordinate. 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). 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. 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.
Theory
Josephson equations
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, . This supercurrent is given by:
Where is the critical current and 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:
The two Josephson equations can be used to derive the effective inductance.
By analogy with an inductor, , so the effective inductance is:
Then, using the first Josephson equation:
So the Josephson inductance is
The Josephson junction is both non-linear and non-dissipative. It is non-linear because its effective inductance depends on the superconducting phase difference , and it is (ideally) non-dissipative because it operates without resistance as long as the current remains below the critical value . 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.
Energy stored in the junction The potential energy of the junction is:
The Hamiltonian for this system is:
Or equivalently:
where is the charging energy and is the number of Cooper pairs.
RCSJ equivalent circuit model 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.
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. 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. 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, . It is thought that lower temperatures will make the resistance go to zero, because there will be fewer quasiparticles in the circuit. However, in practice this is difficult, as there exists a non-negligible population of non-equilibrium quasiparticles, due to cosmic rays or other phenomena, that are not accounted for by temperature alone.
The main equation of the RCSJ model is:
This differential equation describes the dynamics of the Josephson junction when a bias current is applied.
Tilted washboard potential Rewriting the RCSJ model equation in dimensionless coordinates gives:
Where the dimensionless time variable , with
This is known as the Josephson plasma frequency of the junction. It represents the frequency of the Josephson junction oscillations in the untilted washboard potential.
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. 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:
where is the bias current.
This is of the form of a general tilted washboard potential, given by:
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.
Circuit quantization In circuit quantization, flux and charge are treated as generalized coordinates. The quantization step promotes (dimensionless phase) and (Cooper pair number) to operators satisfying the commutation relations. This gives the quantum Hamiltonian for the Josephson junction, used in circuit quantum electrodynamics:
This forms the basic nonlinear element used in circuit quantum electrodynamics and in superconducting qubits.
Types of Josephson junctions
Superconductor-Insulator-Superconductor (SIS)
The superconductor-insulator-superconductor (SIS) tunnel junction, also called a superconducting tunnel junction, has two superconducting electrodes (S) separated by a thin insulating barrier (I). If the barrier is made of an insulating oxide, for example, aluminum oxide (Al2O3), it would be about 1 nanometer thick; a thicker oxide barrier would block quantum tunneling. If a barrier is made of other, less perfect insulators, for example, from amorphous silicon, it may be several nanometers thick. SIS structures are widely used as Josephson junctions, but sometimes, when a supercurrent (zero-voltage current) isn't required, they are used for their nonlinear current-voltage characteristics. They are used in SQUIDs, Josephson voltage standards (in both cases, the application is of an analog type), and in superconducting digital circuits, for example in A/D and D/A converters, or in superconducting processors (in those cases, an application is digital, tunnel junctions assisting in creating logical 1 and 0 states), as well as in Quantum Computing, where they help form superconducting qubits.
In the vast majority of applications, the two electrodes (designated as "S" in the SIS abbreviation) are made of niobium (Nb), which has the highest superconducting transition temperature among elements in the Periodic Table. Clean Niobium superconducts below 9.2 K, which allows operation of a superconducting circuit containing SIS Nb-based junctions in liquid Helium, which has a boiling temperature of 4.2 K at normal pressure. Occasionally, when operational temperature must be much lower, as is the case in Quantum Computing (qubits), "S" layers can be made of aluminum (Al), which has a transition temperature into the superconducting state of 1.2 K. Aluminum is also used as a thin layer deposited over a niobium base electrode in order to create high-quality refractory (Nb-based) tunnel junctions (see below in the section "Fabrication"). The reason for using Al is that it forms a remarkably stable, uniform, pinhole-free oxide layer, which is ideal for creating a reliable tunnel barrier. It is not an accident that the first tunnel junctions created by Ivar Giaever were made of aluminum. Control of Al oxide thickness can be easily achieved by timing its oxidation in air or at a partial pressure of oxygen within a deposition system (see below in the "Fabrication" section).
Superconductor-Normal-Superconductor (SNS) 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.
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. 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.
Superconductor-constriction-Superconductor (ScS) The barrier is a constriction or weak link, such as a microbridge of superconducting material. 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. ScS junctions have a sub-millimeter frequency range of around 500–1000 GHz.
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.
Phi Josephson junction