Josephson effect#The AC Josephson effect

The DC Josephson effect had been seen in experiments prior to 1962,{{Cite web |last=Josephson |first=Brian D. |date=December 12, 1973 |title=The Discovery of Tunneling Supercurrents (Nobel Lecture) |url=https://www.nobelprize.org/prizes/physics/1973/josephson/lecture/}} but had been attributed to "super-shorts" or breaches in the insulating barrier leading to the direct conduction of electrons between the superconductors.

In 1962, Brian Josephson became interested in superconducting tunneling. He was then 23 years old and a second-year graduate student of Brian Pippard at the Mond Laboratory of the University of Cambridge. That year, Josephson took a many-body theory course with Philip W. Anderson, a Bell Labs employee on sabbatical leave for the 1961–1962 academic year. The course introduced

Josephson to the idea of broken symmetry in superconductors, and he "was fascinated by the idea of broken symmetry, and wondered whether there could be any way of observing it experimentally". Josephson studied the experiments by Ivar Giaever and Hans Meissner, and theoretical work by Robert Parmenter. Pippard initially believed that the tunneling effect was possible but that it would be too small to be noticeable, but Josephson did not agree, especially after Anderson introduced him to a preprint of "Superconductive Tunneling" by Cohen, Falicov, and Phillips about the superconductor-barrier-normal metal system.{{cite journal |title=Superconductive Tunneling |first1=M. H. |last1=Cohen |first2=L. M. |last2=Falicov |first3=J. C. |last3=Phillips |date=15 April 1962 |journal=Physical Review Letters |volume=8 |issue=8 |pages=316–318 |doi=10.1103/PhysRevLett.8.316 |bibcode=1962PhRvL...8..316C |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.8.316}}{{r|trueGenius|pp=223-224}}

Josephson and his colleagues were initially unsure about the validity of Josephson's calculations. Anderson later remembered:

We were all—Josephson, Pippard and myself, as well as various other people who also habitually sat at the Mond tea and participated in the discussions of the next few weeks—very much puzzled by the meaning of the fact that the current depends on the phase.

After further review, they concluded that Josephson's results were valid. Josephson then submitted "Possible new effects in superconductive tunnelling" to Physics Letters in June 1962{{r|possibleNewEffects}}. The newer journal Physics Letters was chosen instead of the better established Physical Review Letters due to their uncertainty about the results. John Bardeen, by then already Nobel Prize winner, was initially publicly skeptical of Josephson's theory in 1962, but came to accept it after further experiments and theoretical clarifications.{{cite book |url=https://archive.org/details/truegeniuslifesc0000hodd |url-access=registration |title=True Genius: The Life and Science of John Bardeen |publisher=Joseph Henry Press |page=[https://archive.org/details/truegeniuslifesc0000hodd/page/117 117] |isbn=9780309084086 |last1=Daitch |first1=Vicki |last2=Hoddeson |first2=Lillian |year= 2002 }}{{rp|pp=222–227}} See also: {{section link|John Bardeen|Josephson effect controversy}}.

In January 1963, Anderson and his Bell Labs colleague John Rowell submitted the first paper to Physical Review Letters to claim the experimental observation of Josephson's effect "Probable Observation of the Josephson Superconducting Tunneling Effect".{{cite journal |last1=Anderson |first1=P. W. |last2=Rowell |first2=J. M. |date=15 March 1963 |title=Probable Observation of the Josephson Tunnel Effect |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.10.230 |journal=Physical Review Letters |volume=10 |issue=6 |pages=230 |bibcode=1963PhRvL..10..230A |doi=10.1103/PhysRevLett.10.230}}

These authors were awarded patents{{Cite patent|number=US3335363A|title=Superconductive device of varying dimension having a minimum dimension intermediate its electrodes|gdate=1967-08-08|invent1=Anderson|invent2=Dayem|inventor1-first=Philip W.|inventor2-first=Aly H.|url=https://patents.google.com/patent/US3335363A/en}} on the effects that were never enforced, but never challenged.{{Citation needed|date=March 2023}}

Before Josephson's prediction, it was only known that single (i.e., non-paired) 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.{{Cite web |title=The Nobel Prize in Physics 1973 |url=https://www.nobelprize.org/prizes/physics/1973/summary/ |access-date=2023-03-01 |website=The Nobel Prize}} John Bardeen was one of the nominators.{{r|trueGenius|p=230}}

==Applications==

File:Josephson junction symbol.svg for a Josephson junction]] Types of Josephson junction include the φ Josephson junction (of which π Josephson junction is a special example), long Josephson junction, and superconducting tunnel junction. Other uses include:

• A "Dayem bridge" is a thin-film Josephson junction where the weak link comprises a superconducting wire measuring a few micrometres or less.{{cite journal |last1=Anderson |first1=P. W. |last2=Dayem |first2=A. H. |year=1964 |title=Radio-frequency effects in superconducting thin film bridges |journal=Physical Review Letters |volume=13 |issue=6 |page=195 |bibcode=1964PhRvL..13..195A |doi=10.1103/PhysRevLett.13.195}}{{cite web |last=Dawe |first=Richard |title=SQUIDs: A Technical Report – Part 3: SQUIDs |website=rich.phekda.org |date=28 October 1998 |url=http://rich.phekda.org/squid/technical/part3.html |format=website |access-date=2011-04-21 |archive-url=https://web.archive.org/web/20110727172927/http://rich.phekda.org/squid/technical/part3.html |archive-date=27 July 2011 |url-status=dead}}
• The Josephson junction count is a proxy variable for a device's complexity
• SQUIDs, or superconducting quantum interference devices, are very sensitive magnetometers that operate via the Josephson effect
• Superfluid helium quantum interference devices (SHeQUIDs) are the superfluid helium analog of a dc-SQUIDSato, Y.; Packard, R. (October 2012), Superfluid helium interferometers, Physics Today, p. 31.
• In precision metrology, the Josephson effect is a reproducible conversion between frequency and voltage. The Josephson voltage standard takes the caesium standard definition of frequency and gives the standard representation of a volt
• Single-electron transistors are often made from superconducting materials and called "superconducting single-electron transistors".{{cite journal |last1=Fulton |first1=T. A. |last2=Gammel |first2=P. L. |last3=Bishop |first3=D. J. |last4=Dunkleberger |first4=L. N. |last5=Dolan |first5=G. J. |year=1989 |title=Observation of Combined Josephson and Charging Effects in Small Tunnel Junction Circuits |journal=Physical Review Letters |volume=63 |issue=12 |pages=1307–1310 |bibcode=1989PhRvL..63.1307F |doi=10.1103/PhysRevLett.63.1307 |pmid=10040529}}
• Elementary charge is most precisely measured in terms of the Josephson constant and the von Klitzing constant which is related to the quantum Hall effect
• RSFQ digital electronics are based on shunted Josephson junctions. Junction switching emits one magnetic flux quantum $\backslash scriptstyle\backslash frac\{1\}\{2\; e\}h$. Its presence and absence represents binary 1 and 0.
• Superconducting quantum computing uses Josephon junctions as qubits such as in a flux qubit or other schemes where the phase and charge are conjugate variables.{{cite journal |last1=Bouchiat |first1=V. |last2=Vion |first2=D. |last3=Joyez |first3=P. |last4=Esteve |first4=D. |last5=Devoret |first5=M. H. |year=1998 |title=Quantum coherence with a single Cooper pair |journal=Physica Scripta |volume=T76 |page=165 |bibcode=1998PhST...76..165B |doi=10.1238/Physica.Topical.076a00165 |s2cid=250887469 |doi-access=}}
• Superconducting tunnel junction detectors are used in superconducting cameras

File:Single josephson junction.svg

The Josephson effect can be calculated using the laws of quantum mechanics. A diagram of a single Josephson junction is shown at right. Assume that superconductor A has Ginzburg–Landau order parameter $\backslash psi\_A=\backslash sqrt\{n\_A\}e^\{i\backslash phi\_A\}$, and superconductor B $\backslash psi\_B=\backslash sqrt\{n\_B\}e^\{i\backslash phi\_B\}$, which can be interpreted as the wave functions of Cooper pairs in the two superconductors. If the electric potential difference across the junction is $V$, then the energy difference between the two superconductors is $2eV$, since each Cooper pair has twice the charge of one electron. The Schrödinger equation for this two-state quantum system is therefore:{{Cite web |url=https://feynmanlectures.caltech.edu/III_21.html |title=The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, Section 21-9: The Josephson junction |website=feynmanlectures.caltech.edu |access-date=2020-01-03}}

$$i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}$$

\begin{pmatrix}

\sqrt{n_A}e^{i\phi_A} \\

\sqrt{n_B}e^{i\phi_B}

\end{pmatrix}

=

\begin{pmatrix}

eV & K \\

K & -eV

\end{pmatrix}

\begin{pmatrix}

\sqrt{n_A}e^{i\phi_A} \\

\sqrt{n_B}e^{i\phi_B}

\end{pmatrix},

where the constant $K$ is a characteristic of the junction. To solve the above equation, first calculate the time derivative of the order parameter in superconductor A:

$$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; (\backslash sqrt\{n\_A\}e^\{i\backslash phi\_A\})=\backslash dot\; \backslash sqrt\{n\_A\}e^\{i\backslash phi\_A\}+\; \backslash sqrt\{n\_A\}\; (i\; \backslash dot\; \backslash phi\_A\; e^\{i\backslash phi\_A\})=(\backslash dot\; \backslash sqrt\{n\_A\}+\; i\; \backslash sqrt\{n\_A\}\; \backslash dot\; \backslash phi\_A)\; e^\{i\backslash phi\_A\}\; ,$$

and therefore the Schrödinger equation gives:

$$(\backslash dot\; \backslash sqrt\{n\_A\}+\; i\; \backslash sqrt\{n\_A\}\; \backslash dot\; \backslash phi\_A)\; e^\{i\backslash phi\_A\}$$

=\frac{1}{i\hbar}(eV\sqrt{n_A}e^{i\phi_A}+K\sqrt{n_B}e^{i\phi_B}).

The phase difference of Ginzburg–Landau order parameters across the junction is called the Josephson phase:

$$\backslash varphi=\backslash phi\_B-\backslash phi\_A.$$The Schrödinger equation can therefore be rewritten as:

$$\backslash dot\; \backslash sqrt\{n\_A\}+\; i\; \backslash sqrt\{n\_A\}\; \backslash dot\; \backslash phi\_A$$

=\frac{1}{i\hbar}(eV\sqrt{n_A}+K\sqrt{n_B}e^{i\varphi}),

and its complex conjugate equation is:

$$\backslash dot\; \backslash sqrt\{n\_A\}-\; i\; \backslash sqrt\{n\_A\}\; \backslash dot\; \backslash phi\_A$$

=\frac{1}{-i\hbar}(eV\sqrt{n_A}+K\sqrt{n_B}e^{-i\varphi}).

Add the two conjugate equations together to eliminate $\backslash dot\; \backslash phi\_A$:

$$2\backslash dot\; \backslash sqrt\{n\_A\}=\backslash frac\{1\}\{i\backslash hbar\}(K\backslash sqrt\{n\_B\}e^\{i\backslash varphi\}-K\backslash sqrt\{n\_B\}e^\{-i\backslash varphi\})=\backslash frac\{K\backslash sqrt\{n\_B\}\}\{\backslash hbar\}\; \backslash cdot\; 2\backslash sin\; \backslash varphi.$$

Since $\backslash dot\; \backslash sqrt\{n\_A\}=\backslash frac\{\backslash dot\; n\_A\}\{2\backslash sqrt\{n\_A\}\}$, we have:

$$\backslash dot\; n\_A=\backslash frac\{2K\backslash sqrt\{n\_An\_B\}\}\{\backslash hbar\}\backslash sin\; \backslash varphi.$$

Now, subtract the two conjugate equations to eliminate $\backslash dot\; \backslash sqrt\{n\_A\}$:

$$2i\; \backslash sqrt\{n\_A\}\; \backslash dot\; \backslash phi\_A$$

=\frac{1}{i\hbar}(2eV\sqrt{n_A}+K\sqrt{n_B}e^{i\varphi}+K\sqrt{n_B}e^{-i\varphi}),

which gives:

$$\backslash dot\; \backslash phi\_A$$

=-\frac{1}{\hbar}(eV+K\sqrt{\frac{n_B}{n_A}}\cos \varphi).

Similarly, for superconductor B we can derive that:

$$\backslash dot\; n\_B=-\backslash frac\{2K\backslash sqrt\{n\_An\_B\}\}\{\backslash hbar\}\backslash sin\; \backslash varphi\; ,\; \backslash ,\; \backslash dot\; \backslash phi\_B$$

=\frac{1}{\hbar}(eV-K\sqrt{\frac{n_A}{n_B}}\cos \varphi).

Noting that the evolution of Josephson phase is $\backslash dot\; \backslash varphi=\backslash dot\; \backslash phi\_B-\backslash dot\; \backslash phi\_A$ and the time derivative of charge carrier density $\backslash dot\; n\_A$ is proportional to current $I$, when $n\_A\; \backslash approx\; n\_B$, the above solution yields the Josephson equations:{{Cite book |title=Physics and Applications of the Josephson Effect |last1=Barone |first1=A. |last2=Paterno |first2=G. |publisher=John Wiley & Sons |year=1982 |isbn=978-0-471-01469-0 |location=New York}}

{{equation|1=I(t) = I_c \sin (\varphi (t)) |2=1}}

{{equation|1=\frac{\partial \varphi}{\partial t} = \frac{2 e V(t)}{\hbar} |2=2}}

where $V(t)$ and $I(t)$ are the voltage across and the current through the Josephson junction, and $I\_c$ is a parameter of the junction named the critical current. Equation (1) is called the first Josephson relation or weak-link current-phase relation, and equation (2) is called the second Josephson relation or superconducting phase evolution equation. 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.

The Josephson constant is defined as:

$$K\_J=\backslash frac\{2\; e\}\{h\}\backslash ,,$$

and its inverse is the magnetic flux quantum:

$$\backslash Phi\_0=\backslash frac\{h\}\{2\; e\}=2\; \backslash pi\; \backslash frac\{\backslash hbar\}\{2\; e\}\backslash ,.$$

The superconducting phase evolution equation can be reexpressed as:

$$\backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}\; =\; 2\; \backslash pi\; [K\_JV(t)]\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash Phi\_0\}V(t)\; \backslash ,.$$

If we define:

$$\backslash Phi=\backslash Phi\_0\backslash frac\{\backslash varphi\}\{2\; \backslash pi\}\backslash ,,$$

then the voltage across the junction is:

$$V=\backslash frac\{\backslash Phi\_0\}\{2\; \backslash pi\}\backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}=\backslash frac\{d\backslash Phi\}\{dt\}\backslash ,,$$

which is very similar to Faraday's law of induction. But note that this voltage does not come from magnetic energy, since there is no magnetic field in the superconductors; Instead, this voltage comes from the kinetic energy of the carriers (i.e. the Cooper pairs). This phenomenon is also known as kinetic inductance.

File:I-V characteristics of Josephson Junction.JPG

There are three main effects predicted by Josephson that follow directly from the Josephson equations:

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 $-I\_c$ and $I\_c$.

With a fixed voltage $V\_\{DC\}$ across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC (alternating current) with amplitude $I\_c$ and frequency $K\_J\; V\_\{DC\}$. This means a Josephson junction can act as a perfect voltage-to-frequency converter.

Microwave radiation of a single (angular) frequency $\backslash omega$ can induce quantized DC voltages{{Cite journal |last1=Langenberg |first1=D. N. |last2=Scalapino |first2=D. J. |last3=Taylor |first3=B. N. |last4=Eck |first4=R. E. |date=1966-04-01 |title=Microwave-induced D.C. voltages across Josephson junctions |journal=Physics Letters |volume=20 |issue=6 |pages=563–565 |doi=10.1016/0031-9163(66)91114-0 |bibcode=1966PhL....20..563L |issn=0031-9163}} across the Josephson junction, in which case the Josephson phase takes the form $\backslash varphi\; (t)\; =\; \backslash varphi\_0\; +\; n\; \backslash omega\; t\; +\; a\; \backslash sin(\backslash omega\; t)$, and the voltage and current across the junction will be:

$$V(t)\; =\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash omega\; (\; n\; +\; a\; \backslash cos(\; \backslash omega\; t)\; ),\; \backslash text\{\; and\; \}\; I(t)\; =\; I\_c\; \backslash sum\_\{m\; =\; -\backslash infty\}^\backslash infty\; J\_m\; (a)\; \backslash sin\; (\backslash varphi\_0\; +\; (n\; +\; m)\; \backslash omega\; t),$$

The DC components are:

$$V\_\backslash text\{DC\}\; =\; n\; \backslash frac\{\backslash hbar\}\{2\; e\}\; \backslash omega,\; \backslash text\{\; and\; \}\; I\_\backslash text\{DC\}\; =\; I\_c\; J\_\{-n\}\; (a)\; \backslash sin\; \backslash varphi\_0.$$

This means a Josephson junction can act like a perfect frequency-to-voltage converter,{{Cite journal |last1=Levinsen |first1=M. T. |last2=Chiao |first2=R. Y. |last3=Feldman |first3=M. J. |last4=Tucker |first4=B. A. |date=1977-12-01 |title=An inverse ac Josephson effect voltage standard |journal=Applied Physics Letters |volume=31 |issue=11 |pages=776–778 |doi=10.1063/1.89520 |bibcode=1977ApPhL..31..776L |issn=0003-6951}} 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.{{cite arXiv |eprint=cond-mat/0411174 |first1=M. |last1=Devoret |first2=A. |last2=Wallraff |title=Superconducting Qubits: A Short Review |date=2004 |last3=Martinis |first3=J.}}

Rewrite the Josephson relations as:

:$$

\begin{align}

\frac{\partial I}{\partial \varphi} &= I_c\cos\varphi,\\

\frac{\partial \varphi}{\partial t} &= \frac{2\pi}{\Phi_0}V.

\end{align}

Now, apply the chain rule to calculate the time derivative of the current:

:$$

\frac{\partial I}{\partial t} = \frac{\partial I}{\partial \varphi}\frac{\partial \varphi}{\partial t}=I_c\cos\varphi\cdot\frac{2\pi}{\Phi_0}V,

Rearrange the above result in the form of the current–voltage characteristic of an inductor:

:$$

V = \frac{\Phi_0}{2\pi I_c\cos\varphi} \frac{\partial I}{\partial t}=L(\varphi)\frac{\partial I}{\partial t}.

This gives the expression for the kinetic inductance as a function of the Josephson Phase:

:$$

L(\varphi) = \frac{\Phi_0}{2\pi I_c\cos\varphi} = \frac{L_J}{\cos\varphi}.

Here, $L\_J=L(0)=\backslash frac\{\backslash Phi\_0\}\{2\backslash pi\; I\_c\}$ 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.Michael Tinkham, Introduction to superconductivity, Courier Corporation, 1986.

The supercurrent flowing through the junction is related to the Josephson phase by the current-phase relation (CPR):

:$I\; =\; I\_c\; \backslash sin\backslash varphi.$

The superconducting phase evolution equation is analogous to Faraday's law:

:$V=\backslash operatorname\{d\}\backslash !\backslash Phi/\backslash operatorname\{d\}\backslash !t\backslash ,.$

Assume that at time $t\_1$, the Josephson phase is $\backslash varphi\_1$; At a later time $t\_2$, the Josephson phase evolved to $\backslash varphi\_2$. The energy increase in the junction is equal to the work done on the junction:

:$$

\Delta E = \int_1^2 I V\operatorname{d}\!{t}

= \int_{1}^{2} I\operatorname{d}\!\Phi

= \int_{\varphi_1}^{\varphi_2} I_c\sin \varphi \operatorname{d}\!\left(\Phi_0\frac{\varphi}{2\pi}\right)

= -\frac{\Phi_0 I_c}{2\pi} \Delta\cos\varphi\,.

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:

:$E(\backslash varphi)=-\backslash frac\{\backslash Phi\_0\; I\_c\}\{2\backslash pi\}\backslash cos\backslash varphi=-E\_J\backslash cos\backslash varphi$

\,.

Here $E\_J\; =\; |E(0)|=\backslash frac\{\backslash Phi\_0\; I\_c\}\{2\backslash pi\}$ is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by $E\_J\; =\; L\_JI^2\_c$. An alternative but equivalent definition $E(\backslash varphi)=E\_J(1-\backslash cos\backslash varphi)$ 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 Resistively Capacitance Shunted Junction (RCSJ) model,{{Cite journal |last=McCumber |first=D. E. |date=1968-06-01 |title=Effect of ac Impedance on dc Voltage-Current Characteristics of Superconductor Weak-Link Junctions |journal=Journal of Applied Physics |volume=39 |issue=7 |pages=3113–3118 |doi=10.1063/1.1656743 |bibcode=1968JAP....39.3113M |issn=0021-8979}}{{Cite journal |last1=Chakravarty |first1=Sudip |last2=Ingold |first2=Gert-Ludwig |last3=Kivelson |first3=Steven |last4=Zimanyi |first4=Gergely |date=1988-03-01 |title=Quantum statistical mechanics of an array of resistively shunted Josephson junctions |journal=Physical Review B |volume=37 |issue=7 |pages=3283–3294 |doi=10.1103/PhysRevB.37.3283 |pmid=9944915 |bibcode=1988PhRvB..37.3283C|url=https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/40643 }} 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,{{Cite web |title=AC Thevenin's Theorem |url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/acthev.html |website=hyperphysics.phy-astr.gsu.edu |access-date=2020-01-03}} the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel{{Cite web |title=Dynamics of RF SQUID |url=http://phelafel.technion.ac.il/~orensuch/theory/Theory.html |website=phelafel.technion.ac.il |access-date=2020-01-11 |archive-date=2021-06-13 |archive-url=https://web.archive.org/web/20210613133720/http://phelafel.technion.ac.il/~orensuch/theory/Theory.html |url-status=dead}} to the ideal Josephson Junction. The complete expression for the current drive $I\_\backslash text\{ext\}$ becomes:

:$I\_\backslash text\{ext\}\; =\; C\_J\; \backslash frac\{\backslash operatorname\{d\}\backslash !V\}\{\backslash operatorname\{d\}\backslash !t\}\; +\; I\_c\; \backslash sin\; \backslash varphi\; +\; \backslash frac\{V\}\{R\},$

where the first term is displacement current with $C\_J$ – effective capacitance, and the third is normal current with $R$ – 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 $\backslash lambda\_J$ and is given by the following expression (in SI):

:$\backslash lambda\_J=\backslash sqrt\{\backslash frac\{\backslash Phi\_0\}\{2\backslash pi\backslash mu\_0\; d\text{'}\; j\_c\}\},$

where $\backslash Phi\_0$ is the magnetic flux quantum, $j\_c$ is the critical supercurrent density (A/m²), and $d\text{'}$ characterizes the inductance of the superconducting electrodes{{cite journal |last1=Weihnacht |first1=M. |year=1969 |title=Influence of Film Thickness on D. C. Josephson Current |journal=Physica Status Solidi B |volume=32 |issue=2 |page=169 |bibcode=1969PSSBR..32..169W |doi=10.1002/pssb.19690320259}}

:$$

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 $d\_I$ is the thickness of the Josephson barrier (usually insulator), $d\_1$ and $d\_2$ are the thicknesses of superconducting electrodes, and $\backslash lambda\_1$ and $\backslash lambda\_2$ are their London penetration depths. The Josephson penetration depth usually ranges from a few μm to several mm if the critical current density is very low.{{cite book |last1=Buckel |first1=Werner |last2=Kleiner |first2=Reinhold |title=Supraleitung |date=2004 |publisher=Wiley-VCH Verlag GmbH&Co.KGaA |location=Tübingen |isbn=3527403485 |page=67 |edition=6.}}

{{commons category|Josephson effect}}

{{div col|colwidth=30em}}

• Pi Josephson junction
• φ Josephson junction
• Josephson diode
• Andreev reflection
• Fractional vortices
• Ginzburg–Landau theory
• Macroscopic quantum phenomena
• Macroscopic quantum self-trapping
• Quantum computer
• Quantum gyroscope
• Rapid single flux quantum (RSFQ)
• Semifluxon
• Zero-point energy
• Josephson vortex

{{div col end}}

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