# Josephson effect

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Josephson junction array chip developed by the National Bureau of Standards as a standard volt

The Josephson effect is the phenomenon of supercurrent, a current that flows indefinitely long 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.[1][2] The DC Josephson effect had been seen in experiments prior to 1962,[3] 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.[4] These authors were awarded patents on the effects that were never enforced, but never challenged.

Before Josephson's prediction, it was only known that normal (i.e. non-superconducting) electrons can flow through an insulating barrier, by means of quantum tunneling. Josephson was the first to predict the tunneling of superconducting Cooper pairs. For this work, Josephson received the Nobel Prize in Physics in 1973.[5] Josephson junctions have important applications in quantum-mechanical circuits, such as SQUIDs, superconducting qubits, and RSFQ digital electronics. The NIST standard for one volt is achieved by an array of 20,208 Josephson junctions in series.[6]

## Applications

The electrical symbol for a Josephson junction

Types of Josephson junction include the pi Josephson junction, varphi Josephson junction, long Josephson junction, and Superconducting tunnel junction. A "Dayem bridge" is a thin-film variant of the Josephson junction in which the weak link consists of a superconducting wire with dimensions on the scale of a few micrometres or less.[7][8] The Josephson junction count of a device is used as a benchmark for its complexity. The Josephson effect has found wide usage, for example in the following areas.

SQUIDs, or superconducting quantum interference devices, are very sensitive magnetometers that operate via the Josephson effect. They are widely used in science and engineering.

In precision metrology, the Josephson effect provides an exactly reproducible conversion between frequency and voltage. Since the frequency is already defined precisely and practically by the caesium standard, the Josephson effect is used, for most practical purposes, to give the standard representation of a volt, the Josephson voltage standard. However, the International Bureau of Weights and Measures has not changed the official SI unit definition.[9][10]

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".[11]

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 ${\displaystyle \scriptstyle {\frac {1}{2e}}h}$  that carries the digital information: the absence of switching is equivalent to 0, while one switching event carries a 1.

Josephson junctions are integral in superconducting quantum computing as qubits such as in a flux qubit or others schemes where the phase and charge act as the conjugate variables.[12]

Superconducting tunnel junction detectors (STJs) may become a viable replacement for CCDs (charge-coupled devices) for use in astronomy and astrophysics in a few years. These devices are effective across a wide spectrum from ultraviolet to infrared, and also in x-rays. The technology has been tried out on the William Herschel Telescope in the SCAM instrument.

Quiterons and similar superconducting switching devices.

Josephson effect has also been observed in SHeQUIDs, the superfluid helium analog of a dc-SQUID.[13]

## The effect

Diagram of a single Josephson junction. A and B represent superconductors, and C the weak link between them.

The basic equations governing the dynamics of the Josephson effect are[14]

${\displaystyle U(t)={\frac {\hbar }{2e}}{\frac {\partial \varphi }{\partial t}}}$  (superconducting phase evolution equation)
${\displaystyle I(t)=I_{c}\sin(\varphi (t))}$  (Josephson or weak-link current-phase relation)

where ${\displaystyle U(t)}$  and ${\displaystyle I(t)}$  are the voltage across and the current through the Josephson junction, ${\displaystyle \varphi (t)}$  is the "phase difference" across the junction (i.e., the difference in phase factor, or equivalently, argument, between the Ginzburg–Landau complex order parameter of the two superconductors composing the junction), and ${\displaystyle I_{c}}$  is a constant, the "critical current" of the junction. The critical current is an important phenomenological parameter of the device that can be affected by temperature as well as by an applied magnetic field. The physical constant ${\displaystyle {\frac {h}{2e}}}$  is the magnetic flux quantum ${\displaystyle \Phi _{0}}$ , the inverse of which is the Josephson constant.

Typical I-V characteristic of a superconducting tunnel junction, a common kind of Josephson junction. The scale of the vertical axis is 50 μA and that of the horizontal one is 1 mV. The bar at ${\displaystyle U=0}$  represents the DC Josephson effect, while the current at large values of ${\displaystyle \left|U\right|}$  is due to the finite value of the superconductor bandgap and not reproduced by the above equations.

The three main effects predicted by Josephson follow from these relations:

### 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 phase difference across the insulator, and may take values between ${\displaystyle -I_{c}}$  and ${\displaystyle I_{c}}$ .

### The AC Josephson effect

With a fixed voltage ${\displaystyle U_{DC}}$  across the junction, the phase will vary linearly with time and the current will be an AC current with amplitude ${\displaystyle I_{c}}$  and frequency ${\displaystyle {\frac {2e}{h}}U_{DC}}$ . The complete expression for the current drive ${\displaystyle I_{\text{ext}}}$  becomes:

${\displaystyle I_{\text{ext}}=C_{J}{\frac {\operatorname {d} \!v}{\operatorname {d} \!t}}+I_{c}\sin \varphi +{\frac {V}{R}}.}$

This means a Josephson junction can act as a perfect voltage-to-frequency converter.

### The inverse AC Josephson effect

If the phase takes the form ${\displaystyle \varphi (t)=\varphi _{0}+n\omega t+a\sin(\omega t)}$ , the voltage and current will be

${\displaystyle U(t)={\frac {\hbar }{2e}}\omega (n+a\cos(\omega t)),{\text{ and }}I(t)=I_{c}\sum _{m=-\infty }^{\infty }J_{m}(a)\sin(\varphi _{0}+(n+m)\omega t),}$

The DC components will then be

${\displaystyle U_{DC}=n{\frac {\hbar }{2e}}\omega ,{\text{ and }}I(t)=I_{c}J_{-n}(a)\sin \varphi _{0}.}$

Hence, for distinct AC voltages, the junction may carry a DC current and the junction acts like a perfect frequency-to-voltage converter.

## Josephson phase

The Josephson phase is the difference of the phases of the quantum mechanical wave function in two superconducting electrodes forming a Josephson junction.[15]

If the macroscopic wave functions ${\displaystyle \Psi _{1}}$  and ${\displaystyle \Psi _{2}}$  in superconductors 1 and 2 are given by

${\displaystyle \Psi _{j}={\sqrt {n_{s}}}e^{i\theta _{j}},\forall j\in \left\{1,2\right\},}$

then the Josephson phase is defined by

${\displaystyle \varphi \;{\overset {\underset {\mathrm {def} }{}}{=}}\;\theta _{2}-\theta _{1}.}$

## Josephson energy

The Josephson energy is the potential energy accumulated in a Josephson junction when a supercurrent flows through it. One can think of a Josephson junction as a non-linear inductance which accumulates (magnetic field) energy when a current passes through it. In contrast to real inductance, no magnetic field is created by a supercurrent in a Josephson junction — the accumulated energy is the Josephson energy.[16]

For the simplest case the current-phase relation (CPR) is given by the first Josephson relation:

${\displaystyle I_{s}=I_{c}\sin(\varphi ),}$

where ${\displaystyle I_{s}}$ , is the supercurrent flowing through the junction, ${\displaystyle I_{c}}$ , is the critical current, and ${\displaystyle \varphi }$ , is the Josephson phase. Imagine that initially at time ${\displaystyle t=0}$  the junction was in the ground state ${\displaystyle \varphi =0}$  and finally at time ${\displaystyle t}$  the junction has the phase ${\displaystyle \varphi }$ . The work done on the junction (so the junction energy is increased by)

${\displaystyle U=\int _{0}^{t}I_{s}V\operatorname {d} \!t={\frac {\Phi _{0}}{2\pi }}\int _{0}^{t}I_{s}{\frac {\operatorname {d} \!\varphi }{\operatorname {d} \!t}}\operatorname {d} \!t={\frac {\Phi _{0}}{2\pi }}\int _{0}^{\varphi }I_{c}\sin \varphi \operatorname {d} \!\varphi ={\frac {\Phi _{0}I_{c}}{2\pi }}(1-\cos \varphi ).}$

Here ${\displaystyle E_{J}={\frac {\Phi _{0}I_{c}}{2\pi }}}$  sets the characteristic scale of the Josephson energy, and ${\displaystyle 1-\cos \varphi }$  sets its dependence on the phase ${\displaystyle \varphi }$ . The energy ${\displaystyle U(\varphi )}$  accumulated inside the junction depends only on the current state of the junction, but not on history or velocities, i.e. it is a potential energy. Note, that ${\displaystyle U(\varphi )}$  has a minimum equal to zero for the ground state ${\displaystyle \varphi =2\pi n}$ , ${\displaystyle n}$  is any integer.

## Josephson inductance

Imagine that the Josephson phase across the junction is ${\displaystyle \varphi _{0}}$ , and the supercurrent flowing through the junction is

${\displaystyle I_{0}=I_{c}\sin \varphi _{0}.}$

(This is the same equation as above, except now we will look at small variations in ${\displaystyle I_{s}}$  and ${\displaystyle \varphi }$  around the values ${\displaystyle I_{0}}$  and ${\displaystyle \varphi _{0}}$ .)

Imagine that we add little extra current (direct or alternating) ${\displaystyle \delta _{I(t)}\ll I_{c}}$  through the junction, and want to see how it reacts. The phase across the junction changes to become ${\displaystyle \varphi =\varphi _{0}+\delta _{\varphi }}$ . One can write:

${\displaystyle I_{0}+\delta _{I}=I_{c}\sin(\varphi _{0}+\delta _{\varphi }).\,}$

Assuming that ${\displaystyle \delta _{\varphi }}$  is small, we make a Taylor expansion in the right hand side to arrive at

${\displaystyle \delta _{I}=I_{c}\cos(\varphi _{0})\delta _{\varphi }.}$

The voltage across the junction (we use the 2nd Josephson relation) is

${\displaystyle V={\frac {\Phi _{0}}{2\pi }}{\dot {\varphi }}={\frac {\Phi _{0}}{2\pi }}(\underbrace {\dot {\varphi _{0}}} _{=0}+{\dot {\delta _{\varphi }}})={\frac {\Phi _{0}}{2\pi }}{\frac {\dot {\delta _{I}}}{I_{c}\cos(\varphi _{0})}}.}$

If we compare this expression with the expression for voltage across the conventional inductance

${\displaystyle V=L{\frac {\partial I}{\partial t}},}$

we can define the so-called Josephson inductance

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

One can see that this inductance is not constant, but depends on the phase ${\displaystyle \varphi _{0}}$  across the junction. The typical value is given by ${\displaystyle L_{J}(0)}$  and is determined only by the critical current ${\displaystyle I_{c}}$ . Note that, according to definition, the Josephson inductance can even become infinite or negative, if ${\displaystyle \cos \varphi _{0}\leq 0}$ .

One can also calculate the change in Josephson energy

${\displaystyle \delta _{U(\varphi _{0})}=U(\varphi )-U(\varphi _{0})=E_{J}(\cos(\varphi _{0})-\cos(\varphi _{0}+\delta _{\varphi }).}$

Making Taylor expansion for small ${\displaystyle \delta _{\varphi }}$ , we get

${\displaystyle \approx E_{J}\sin(\varphi _{0})\delta _{\varphi }={\frac {E_{J}\sin(\varphi _{0})}{I_{c}\cos \varphi _{0}}}\delta _{I}.}$

If we now compare this with the expression for increase of the inductance energy ${\displaystyle \delta _{E_{L}}=LI\delta _{I}}$ , we again get the same expression for ${\displaystyle L}$ .

Note, that although Josephson junction behaves like an inductance, there is no associated magnetic field. The corresponding energy is hidden inside the junction. The Josephson Inductance is also known as a Kinetic Inductance – the behaviour is derived from the kinetic energy of the charge carriers, not energy in a magnetic field.[17]

As an alternative to the above approach to finding the Josephson Inductance, the equation for voltage across an inductor can be used (given by ${\displaystyle V=L{\frac {\partial I}{\partial t}}}$ ). By finding the derivative of the current with respect to time, and rearranging in the form of the inductance equation, inductance can be found.

Firstly, using the chain rule

${\displaystyle {\frac {\partial I}{\partial t}}={\frac {\partial I}{\partial \varphi }}{\frac {\partial \varphi }{\partial t}},}$

and from the Josephson junction equations

${\displaystyle {\frac {\partial I}{\partial \varphi }}=I_{C}\cos(\varphi ),}$
${\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {2eV}{\hbar }}={\frac {2\pi V}{\Phi _{0}}},}$

Combining these three equations gives

${\displaystyle {\frac {\partial I}{\partial t}}={\frac {I_{c}\cos(\varphi )2eV}{\hbar }}={\frac {I_{c}\cos(\varphi )2\pi V}{\Phi _{0}}},}$

and by rearranging to find in the form of ${\displaystyle V=L{\frac {\partial I}{\partial t}}}$

${\displaystyle V={\frac {\hbar }{2eI_{c}\cos(\varphi )}}{\frac {\partial I}{\partial t}}.}$

## Josephson penetration depth

The Josephson penetration depth characterizes the typical length on which an externally applied magnetic field penetrates into the long Josephson junction. Josephson penetration depth is usually denoted as ${\displaystyle \lambda _{J}}$  and is given by the following expression (in SI):

${\displaystyle \lambda _{J}={\sqrt {\frac {\Phi _{0}}{2\pi \mu _{0}d'j_{c}}}},}$

where ${\displaystyle \Phi _{0}}$  is the magnetic flux quantum, ${\displaystyle j_{c}}$  is the critical current density (A/m²), and ${\displaystyle d'}$  characterizes the inductance of the superconducting electrodes[18]

${\displaystyle d'=d_{I}+\lambda _{1}\tanh \left({\frac {d_{1}}{2\lambda _{1}}}\right)+\lambda _{2}\tanh \left({\frac {d_{2}}{2\lambda _{2}}}\right),}$

where ${\displaystyle d_{I}}$  is the thickness of the Josephson barrier (usually insulator), ${\displaystyle d_{1}}$  and ${\displaystyle d_{2}}$  are the thicknesses of superconducting electrodes, and ${\displaystyle \lambda _{1}}$  and ${\displaystyle \lambda _{2}}$  are their London penetration depths.

## References

1. ^ B. D. Josephson (1962). "Possible new effects in superconductive tunnelling". Phys. Lett. 1 (7): 251–253. Bibcode:1962PhL.....1..251J. doi:10.1016/0031-9163(62)91369-0.
2. ^ B. D. Josephson (1974). "The discovery of tunnelling supercurrents". Rev. Mod. Phys. 46 (2): 251–254. Bibcode:1974RvMP...46..251J. doi:10.1103/RevModPhys.46.251.
3. ^ Josephson, Brian D. (December 12, 1973). "The Discovery of Tunneling Supercurrents (Nobel Lecture)" (PDF).
4. ^ P. W. Anderson; J. M. Rowell (1963). "Probable Observation of the Josephson Tunnel Effect". Phys. Rev. Lett. 10 (6): 230. Bibcode:1963PhRvL..10..230A. doi:10.1103/PhysRevLett.10.230.
5. ^ The Nobel prize in physics 1973, accessed 8-18-11
6. ^ Steven Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003.
7. ^ P. W. Anderson; A. H. Dayem (1964). "Radio-frequency effects in superconducting thin film bridges". Phys. Rev. Lett. 13 (6): 195. Bibcode:1964PhRvL..13..195A. doi:10.1103/PhysRevLett.13.195.
8. ^ Dawe, Richard (28 October 1998). "SQUIDs: A Technical Report – Part 3: SQUIDs" (website). http://rich.phekda.org. Retrieved 2011-04-21. External link in |publisher= (help)
9. ^ International Bureau of Weights and Measures (BIPM), SI brochure, section 2.1.: SI base units, section 2.1.1: Definitions, accessed 22 June 2015
10. ^ Practical realization of units for electrical quantities (SI brochure, Appendix 2). BIPM, [last updated: 20 February 2007], accessed 22 June 2015.
11. ^ T. A. Fulton; P. L. Gammel; D. J. Bishop; L. N. Dunkleberger; G. J. Dolan (1989). "Observation of Combined Josephson and Charging Effects in Small Tunnel Junction Circuits". Phys. Rev. Lett. 63 (12): 1307–1310. Bibcode:1989PhRvL..63.1307F. doi:10.1103/PhysRevLett.63.1307. PMID 10040529.
12. ^ V. Bouchiat; D. Vion; P. Joyez; D. Esteve; M. H. Devoret (1998). "Quantum coherence with a single Cooper pair". Physica Scripta. T76: 165. Bibcode:1998PhST...76..165B. doi:10.1238/Physica.Topical.076a00165.
13. ^ Physics Today, Superfluid helium interferometers, Y. Sato and R. Packard, October 2012, page 31
14. ^ Barone, A.; Paterno, G. (1982). Physics and Applications of the Josephson Effect. New York: John Wiley & Sons. ISBN 978-0-471-01469-0.
15. ^ R. Feynman, R. Leighton, and M. Sands, The Feynman lectures on physics, Addison-Wesley, Reading, Mass, Volume III (1965).
16. ^ Michael Tinkham, Introduction to superconductivity, Courier Corporation, 1986
17. ^ Devoret, M; Wallraff, A; Martinis, J (2004). "Superconducting Qubits: A Short Review". arXiv:cond-mat/0411174.
18. ^ Weihnacht, M (1969). "Influence of Film Thickness on D. C. Josephson Current". Physica Status Solidi B. 32 (2): 169. Bibcode:1969PSSBR..32..169W. doi:10.1002/pssb.19690320259.