# Josephson Effect

The Josephson effect is a physical effect that describes the tunnel current between two superconductors . It was theoretically predicted by Brian D. Josephson in 1962 and later verified in numerous experiments, first in 1962 by John Rowell at Bell Laboratories (partly with Philip Warren Anderson ). Josephson received the Nobel Prize in Physics in 1973 .

Although the Josephson effect was the first to be measured in superconductors, the term was generalized: One speaks of the Josephson effect if two macroscopic wave functions are weakly coupled to one another (coupling via a tunnel barrier).

## Clear description

The electric current in superconductors is not carried by individual electrons , as is the case in normal conductors , but by electron pairs, the so-called Cooper pairs , as postulated by the BCS theory .

If two superconductors are separated by a non-superconducting barrier a few nanometers thick, Cooper pairs from one superconductor can tunnel through the barrier into the other . Such a superconductor-normal conductor -superconductor (SNS) or superconductor- insulator- superconductor (SIS) arrangement is called a Josephson contact . If a power source is now connected to the Josephson junction and a small electrical current is passed through the contact, it continues to behave like a superconductor without interruption, as the Cooper pairs tunnel through the barrier.

## The Josephson equations

In a superconductor, all Cooper pairs are in the same quantum mechanical state , so they can be described by one and the same wave function (see BCS theory ). In a Josephson junction, the wave functions of the two superconductors are coupled through the thin, non-superconducting layer. The size of the coupling is essentially determined by the thickness of the layer. The supercurrent (current carried by Cooper pairs) through this layer has the size ${\ displaystyle I _ {\ mathrm {J}}}$

${\ displaystyle I _ {\ mathrm {J}} = I _ {\ mathrm {c}} \ sin \ Delta \ varphi \ quad}$ (1st Josephson equation),

where represents the phase difference of the superconducting wave functions on both sides of the barrier and is the critical current of the barrier. ${\ displaystyle \ Delta \ varphi}$${\ displaystyle I _ {\ mathrm {c}}}$

The following applies:

${\ displaystyle {\ frac {\ partial \ Delta \ varphi} {\ partial t}} = {\ frac {2 \ pi} {\ Phi _ {0}}} U \ quad}$with (2nd Josephson equation),${\ displaystyle \ quad \ Phi _ {0} = {\ frac {h} {2e}} \ quad}$

where is the magnetic flux quantum . This equation is called the 2nd Josephson equation. Due to the changing phase difference, a constantly changing supercurrent occurs. By inserting the 2nd Josephson equation into the 1st Josephson equation, the corresponding frequency ( Josephson frequency ) is obtained ${\ displaystyle \ Phi _ {0}}$

${\ displaystyle f _ {\ mathrm {J}} = {\ frac {2eU} {h}} = K _ {\ mathrm {J}} \ cdot U \; \; \;}$or .${\ displaystyle \; \; \ omega _ {\ mathrm {J}} = {\ frac {2eU} {\ hbar}} = 2 \ pi K _ {\ mathrm {J}} \ cdot U}$

Here is

${\ displaystyle K _ {\ mathrm {J}} = {\ frac {1} {\ Phi _ {0}}} = {\ frac {2 \, e} {h}} = 4 {,} 835 \; 978 \; 484 \ ldots \ cdot 10 ^ {14} \, {\ frac {\ mathrm {Hz}} {\ mathrm {V}}}}$

the Josephson constant . It is known exactly because e ( elementary charge ) and h ( Planck's quantum of action ) have been used to define the SI units since 2019 and a fixed value has been assigned to them. Since frequencies can be measured very precisely, the Josephson effect is suitable as a voltage standard for the precise representation of the unit volt . For the (very low) voltage of 1 µV, for example, the frequency is 483.5978… MHz.

It remains to be mentioned that the current through a Josephson junction is dependent on external magnetic fields. Strictly speaking, the 1st Josephson equation reads:

${\ displaystyle I _ {\ mathrm {J}} = I _ {\ mathrm {c}} \ sin \ left (\ Delta \ varphi - {\ frac {2 \ pi} {\ Phi _ {0}}} \ int _ {\ mathrm {superconductor} 1} ^ {\ mathrm {superconductor} 2} {\ textbf {A}} d {\ textbf {l}} \ right)}$

Here, the magnetic vector potential and the integral is a line integral that extends from the first superconductor over the barrier to the second superconductor. ${\ displaystyle {\ textbf {A}}}$

## Characteristic curve of a Josephson contact

Current-voltage characteristic of a non- hysteretic Josephson contact

Several processes take place at a Josephson junction (non-hysteresis case without magnetic field):

• Tunneling Cooper Pairs through the Barrier (Josephson Effect)
• Individual electrons tunnel through the barrier
• Breaking of Cooper pairs and tunneling of the resulting single electrons through the barrier (at high voltages compared to the band gap of the superconductor)
• Ohmic conduction through the barrier with SNS contacts (not with SIS contacts)
• The two superconductors with the barrier in between behave like the plates of a plate capacitor.

Since all these processes take place in parallel, the following applies

• The voltage measured is the voltage of the Josephson effect; it can be calculated directly from it using the 2nd Josephson equation .${\ textstyle {\ frac {\ partial \ Delta \ varphi} {\ partial t}}}$
• The measured current is the total current of all processes taking place, i.e. essentially the sum of the current of the Cooper pairs and the current of the individual electrons, smoothed over time by the capacitor effect of the contact.

DC Josephson Effect: If the voltage is small, i. H. the energy of the Cooper pairs in the electric field over negligible, results in the minimization of the free energy of the system that is in equilibrium with sets. Ie ${\ displaystyle k_ {B} T}$${\ textstyle \ Delta \ varphi = (2z + 1) {\ frac {\ pi} {2}}}$${\ textstyle z \ in \ mathbb {Z}}$

${\ displaystyle I _ {\ mathrm {J}} = \ pm I _ {\ mathrm {c}}}$

If no external voltage source is connected, this Josephson current is compensated for by tunneling individual electrons in the opposite direction so that no voltage builds up. If a (low) voltage is applied, the Josephson current flows away due to the electric field through the voltage source, so that essentially applies to the measured current: ${\ displaystyle I _ {\ mathrm {total}} = 0}$

${\ displaystyle I _ {\ mathrm {total}} = I _ {\ mathrm {J}} = \ pm I _ {\ mathrm {c}}}$

AC Josephson effect : If the voltage is so high that the thermal effects are negligible, the Resistively and Capacitively Shunted Junction (RCSJ) model is usually used to set up a DGL for . The total current is used as the sum of the Josephson current, ohmic current, and the current of a capacitor. If a constant voltage is forced by a voltage source, the result is a phase of ${\ displaystyle \ Delta \ varphi}$

${\ displaystyle \ Delta \ varphi = \ Delta \ varphi _ {\ mathrm {0}} + \ omega _ {\ mathrm {J}} t}$

so the Josephson current is then an alternating current with a circular frequency . ${\ displaystyle \ omega _ {\ mathrm {J}}}$

## Technical implementation of Josephson contacts

Schematic structure of a Josephson contact

The barrier separating the two superconductors may only be a few nanometers thick so that quantum mechanical tunneling processes can take place. This can be done in different ways:

• Construction of an SNS or SIS arrangement using thin-film technology by sputtering or laser ablation
• A thin tip made of superconducting material, which is pressed onto a superconductor (point contact / tip contact), has a similar effect, since tunnel effects occur on the sides of the tip (in the normally conducting state, a current that is so high that the thinnest point may be sent through the arrangement the constriction is oxidized by the heat and a thin insulating oxide layer is created)
• A very narrow constriction in a superconducting film (the effects are the same as with tip contact)
• In strongly anisotropic high-temperature superconductors such as Bi2212 or the Pnictid LaO 0.9 F 0.1 FeAs, superconductivity only takes place in planes, with insulating layers between the planes. Structuring can therefore be used to produce intrinsic Josephson junctions from single crystals.

## Applications

Josephson contacts are used as extremely fast switching elements and very precise voltage stabilizers . They are also used in systems for measuring extremely small magnetic fluxes ( SQUIDs ).

Josephson junctions are very accurate frequency-to-voltage converters. With the inverse Josephson effect , the Josephson contact is operated with a voltage of the form

${\ displaystyle U (t) = {\ frac {\ hbar} {2e}} \ omega (n + a \ cos (\ omega t))}$

One can show that I c is then constant. This arrangement is used in calibration offices as a very precise frequency-to-voltage converter for the calibration of voltages and is then called the Josephson standard or Josephson quantum standard.

## Limitations

Since Josephson effects only occur in connection with superconductors, they have to be cooled to very low temperatures, which makes their operation technically complex and under certain circumstances very costly. A frequently used superconducting material for the production of such contacts is niobium , which becomes superconducting at 9.2 Kelvin . Liquid helium (with a temperature of 4.2 Kelvin) is used for cooling to these temperatures . Josephson junctions made from high-temperature superconductor materials can also be cooled at liquid nitrogen temperatures (77.4 Kelvin). Liquid nitrogen is significantly cheaper and easier to manufacture than liquid helium, but the manufacturing process for Josephson junctions from high-temperature superconductors is significantly more expensive.

## literature

• Brian D. Josephson: Possible New Effects in Superconducting Tunneling. In: Physics Letters. Vol. 1, No. 7, 1962, pp. 251-253, doi : 10.1016 / 0031-9163 (62) 91369-0 .
• John M. Rowell, Philip Warren Anderson , Donald E. Thomas: Image of the Phonon Spectrum in the Tunneling Characteristic between Superconductors. In: Physical Review Letters. Vol. 10, No. 8, 1963, pp. 334-336, doi : 10.1103 / PhysRevLett.10.334 .
• Sidney Shapiro: Josephson Current in Superconducting Tunneling: The Effect of Microwaves and other Observations. In: Physical Review Letters. Vol. 11, No. 2, 1963, pp. 80-82, doi : 10.1103 / PhysRevLett.11.80 .
• Brian D. Josephson: The Discovery of Tunneling Supercurrents . (PDF; 188 kB), Nobel Prize Speech 1973.
• Antonio Barone, Gianfranco Paternò: Physics and Applications of the Josephson effect. John Wiley & Sons, New York NY 1982, ISBN 0-471-01469-9 .
• Konstantin K. Likharev: Dynamics of Josephson Junctions and Circuits. 3rd printing. Gordon and Breach Science Publishers, New York NY et al. 1986, ISBN 2-88124-042-9 .
• Charles Kittel : Introduction to Solid State Physics. 13th, corrected edition. Oldenbourg, Munich et al. 2002, ISBN 3-486-27219-5 .
• Werner Buckel , Reinhold Kleiner: Superconductivity. Basics and Applications. 6th, completely revised and expanded edition. Wiley-VCH, Weinheim 2004, ISBN 3-527-40348-5 .

## Individual evidence

1. ^ Josephson effect in Bose-Einstein Condensates in "Nature 449, 579 (2007)"
2. CODATA Recommended Values. National Institute of Standards and Technology, accessed June 6, 2019 . Value for .${\ displaystyle K _ {\ mathrm {J}}}$
3. ^ Josephson tunnel junctions with ferromagnetic interlayer. Self-published by Forschungszentrum Jülich, Jülich, 2006, accessed on March 25, 2014 .