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The Josephson effect is the phenomenon of supercurrent, a current that flows continuously without any voltage applied, across a device known as a Josephson junction (JJ), which consists of two or more superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier (known as a superconductor-insulator-superconductor junction, or S-I-S), a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-s-S).
The Josephson effect is an example of a macroscopic quantum phenomenon. It is named after the British physicist Brian David Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link. The DC Josephson effect had been seen in experiments prior to 1962, but had been attributed to "super-shorts" or breaches in the insulating barrier leading to the direct conduction of electrons between the superconductors. The first paper to claim the discovery of Josephson's effect, and to make the requisite experimental checks, was that of Philip Anderson and John Rowell. These authors were awarded patents on the effects that were never enforced, but never challenged.
Single-electron transistors are often constructed of superconducting materials, allowing use to be made of the Josephson effect to achieve novel effects. The resulting device is called a "superconducting single-electron transistor".
The Josephson effect is also used for the most precise measurements of elementary charge in terms of the Josephson constant and von Klitzing constant which is related to the quantum Hall effect.
RSFQ digital electronics is based on shunted Josephson junctions. In this case, the junction switching event is associated to the emission of one magnetic flux quantum that carries the digital information: the absence of switching is equivalent to 0, while one switching event carries a 1.
Add the two conjugate equations together to eliminate :
Since , we have:
Now, subtract the two conjugate equations to eliminate :
Similarly, for superconductor B we can derive that:
Noting that the evolution of Josephson phase is and the time derivative of charge carrier density is proportional to current , the above solution yields the Josephson equations:
(1st Josephson relation, or weak-link current-phase relation)
(2nd Josephson relation, or superconducting phase evolution equation)
where and are the voltage across and the current through the Josephson junction, and is a parameter of the junction named the critical current. The critical current of the Josephson junction depends on the properties of the superconductors, and can also be affected by environmental factors like temperature and externally applied magnetic field.
Typical I-V characteristic of a superconducting tunnel junction, a common kind of Josephson junction. The scale of the vertical axis is 50 ?A and that of the horizontal one is 1 mV. The bar at represents the DC Josephson effect, while the current at large values of is due to the finite value of the superconductor bandgap and not reproduced by the above equations.
There are three main effects predicted by Josephson that follow directly from the Josephson equations:
The DC Josephson effect
The DC Josephson effect is a direct current crossing the insulator in the absence of any external electromagnetic field, owing to tunneling. This DC Josephson current is proportional to the sine of the Josephson phase (phase difference across the insulator, which stays constant over time), and may take values between and .
The AC Josephson effect
With a fixed voltage across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC (Alternating Current) with amplitude and frequency . This means a Josephson junction can act as a perfect voltage-to-frequency converter.
The inverse AC Josephson effect
Microwave radiation of a single (angular) frequency can induce quantized DC voltages across the Josephson junction, in which case the Josephson phase takes the form , and the voltage and current across the junction will be:
The DC components are:
This means a Josephson junction can act like a perfect frequency-to-voltage converter, which is the theoretical basis for the Josephson voltage standard.
When the current and Josephson phase varies over time, the voltage drop across the junction will also vary accordingly; As shown in derivation below, the Josephson relations determine that this behavior can be modeled by a kinetic inductance named Josephson Inductance.
Rewrite the Josephson relations as:
Now, apply the chain rule to calculate the time derivative of the current:
This gives the expression for the kinetic inductance as a function of the Josephson Phase:
Here, is a characteristic parameter of the Josephson junction, named the Josephson Inductance.
Note that although the kinetic behavior of the Josephson junction is similar to that of an inductor, there is no associated magnetic field. This behaviour is derived from the kinetic energy of the charge carriers, instead of the energy in a magnetic field.
Based on the similarity of the Josephson junction to a non-linear inductor, the energy stored in a Josephson junction when a supercurrent flows through it can be calculated.
The supercurrent flowing through the junction is related to the Josephson phase by the current-phase relation (CPR):
The superconducting phase evolution equation is analogous to Faraday's law:
Assume that at time , the Josephson phase is ; At a later time , the Josephson phase evolved to . The energy increase in the junction is equal to the work done on the junction:
This shows that the change of energy in the Josephson junction depends only on the initial and final state of the junction and not the path. Therefore the energy stored in a Josephson junction is a state function, which can be defined as:
Here is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by . An alternative but equivalent definition is also often used.
Again, note that a non-linear magnetic coil inductor accumulates potential energy in its magnetic field when a current passes through it; However, in the case of Josephson junction, no magnetic field is created by a supercurrent -- the stored energy comes from the kinetic energy of the charge carriers instead.
The RCSJ model
The Resistively Capacitance Shunted Junction (RCSJ) model, or simply shunted junction model, includes the effect of AC impedance of an actual Josephson junction on top of the two basic Josephson relations stated above.
As per Thévenin's theorem, the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel to the ideal Josephson Junction. The complete expression for the current drive becomes:
where the first term is displacement current with - effective capacitance, and the third is normal current with - effective resistance of the junction.
The Josephson penetration depth characterizes the typical length on which an externally applied magnetic field penetrates into the long Josephson junction. It is usually denoted as and is given by the following expression (in SI):
where is the thickness of the Josephson barrier (usually insulator), and are the thicknesses of superconducting electrodes, and and are their London penetration depths. The Josephson penetration depth usually ranges from a few µm to several mm if the critical supercurrent density is very low.
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^McCumber, D. E. (1968-06-01). "Effect of ac Impedance on dc Voltage-Current Characteristics of Superconductor Weak-Link Junctions". Journal of Applied Physics. 39 (7): 3113-3118. doi:10.1063/1.1656743. ISSN0021-8979.
^Chakravarty, Sudip; Ingold, Gert-Ludwig; Kivelson, Steven; Zimanyi, Gergely (1988-03-01). "Quantum statistical mechanics of an array of resistively shunted Josephson junctions". Physical Review B. 37 (7): 3283-3294. doi:10.1103/PhysRevB.37.3283. PMID9944915.