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Josephson effect

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In physics, the Josephson effect is a phenomenon that occurs when two <a href="Superconductors" class="mw-redirect" title="Superconductors">superconductors</a> are placed in proximity, with some barrier or restriction between them. The effect is named after the British physicist <a href="Brian_Josephson" title="Brian Josephson">Brian Josephson</a>, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.<a href="#cite_note-possibleNewEffects-1">[1]</a><a href="#cite_note-Joe-2">[2]</a> It is an example of a <a href="Macroscopic_quantum_phenomenon" class="mw-redirect" title="Macroscopic quantum phenomenon">macroscopic quantum phenomenon</a>, where the effects of quantum mechanics are observable at ordinary, rather than atomic, scale. The Josephson effect has many practical applications because it exhibits a precise relationship between different physical measures, such as voltage and frequency, facilitating highly accurate measurements.

The Josephson effect produces a current, known as a <a href="Supercurrent" title="Supercurrent">supercurrent</a>, that flows continuously without any voltage applied, across a device known as a Josephson junction (JJ). These consist of two or more superconductors coupled by a weak link. The weak link can be a thin insulating barrier (known as a <a href="Superconducting_tunnel_junction" title="Superconducting tunnel junction">superconductor–insulator–superconductor junction</a>, 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-c-S).

Josephson junctions have important applications in <a href="Quantum_circuit" title="Quantum circuit">quantum-mechanical circuits</a>, such as <a href="SQUID" title="SQUID">SQUIDs</a>, <a href="Superconducting_quantum_computing" title="Superconducting quantum computing">superconducting qubits</a>, and <a href="RSFQ" class="mw-redirect" title="RSFQ">RSFQ</a> digital electronics. The <a href="NIST" class="mw-redirect" title="NIST">NIST</a> standard for one <a href="Volt" title="Volt">volt</a> is achieved by <a href="Josephson_voltage_standard" title="Josephson voltage standard">an array of 20,208 Josephson junctions in series</a>.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The DC Josephson effect had been seen in experiments prior to 1962,<a href="#cite_note-5">[5]</a> 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 <a href="Brian_Pippard" title="Brian Pippard">Brian Pippard</a> at the <a href="New_Museums_Site#Mond_Laboratory" title="New Museums Site">Mond Laboratory</a> of the <a href="University_of_Cambridge" title="University of Cambridge">University of Cambridge</a>. That year, Josephson took a many-body theory course with <a href="Philip_W._Anderson" title="Philip W. Anderson">Philip W. Anderson</a>, a <a href="Bell_Labs" title="Bell Labs">Bell Labs</a> 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 <a href="Ivar_Giaever" title="Ivar Giaever">Ivar Giaever</a> 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 <a href="Marvin_L._Cohen" title="Marvin L. Cohen">Cohen</a>, <a href="Leopoldo_M%C3%A1ximo_Falicov" title="Leopoldo Máximo Falicov">Falicov</a>, and Phillips about the superconductor-barrier-normal metal system.<a href="#cite_note-6">[6]</a><a href="#cite_note-trueGenius-7">[7]</a>: 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 <a href="New_Museums_Site#Mond_Laboratory" title="New Museums Site">Mond</a> 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.

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After further review, they concluded that Josephson's results were valid. Josephson then submitted "Possible new effects in superconductive tunnelling" to <a href="Physics_Letters" title="Physics Letters">Physics Letters</a> in June 1962<a href="#cite_note-possibleNewEffects-1">[1]</a>. The newer journal Physics Letters was chosen instead of the better established <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a> due to their uncertainty about the results. <a href="John_Bardeen" title="John Bardeen">John Bardeen</a>, 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.<a href="#cite_note-trueGenius-7">[7]</a>: 222–227 See also: <a href="John_Bardeen#Josephson_effect_controversy" title="John Bardeen">John Bardeen § Josephson effect controversy</a>.

In January 1963, Anderson and his <a href="Bell_Labs" title="Bell Labs">Bell Labs</a> 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".<a href="#cite_note-8">[8]</a> These authors were awarded patents<a href="#cite_note-9">[9]</a> on the effects that were never enforced, but never challenged.

Before Josephson's prediction, it was only known that single (i.e., non-paired) electrons can flow through an insulating barrier, by means of <a href="Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">quantum tunneling</a>. Josephson was the first to predict the tunneling of superconducting <a href="Cooper_pair" title="Cooper pair">Cooper pairs</a>. For this work, Josephson received the <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> in 1973.<a href="#cite_note-10">[10]</a> John Bardeen was one of the nominators.<a href="#cite_note-trueGenius-7">[7]</a>: 230

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Types of Josephson junction include the <a href="Phi_Josephson_junction" title="Phi Josephson junction">φ Josephson junction</a> (of which <a href="Pi_Josephson_junction" title="Pi Josephson junction">π Josephson junction</a> is a special example), <a href="Long_Josephson_junction" title="Long Josephson junction">long Josephson junction</a>, and <a href="Superconducting_tunnel_junction" title="Superconducting tunnel junction">superconducting tunnel junction</a>. Other uses include:

<ul><li>A "Dayem bridge" is a <a href="Thin-film" class="mw-redirect" title="Thin-film">thin-film</a> Josephson junction where the weak link comprises a superconducting wire measuring a few <a href="Micrometre" title="Micrometre">micrometres</a> or less.<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a></li>

<li>The <a href="Josephson_junction_count" title="Josephson junction count">Josephson junction count</a> is a <a href="Proxy_variable" class="mw-redirect" title="Proxy variable">proxy variable</a> for a device's complexity</li>

<li><a href="SQUID" title="SQUID">SQUIDs</a>, or superconducting quantum interference devices, are very sensitive <a href="Magnetometer" title="Magnetometer">magnetometers</a> that operate via the Josephson effect</li>

<li>Superfluid helium quantum interference devices (SHeQUIDs) are the <a href="Superfluid" class="mw-redirect" title="Superfluid">superfluid</a> helium analog of a dc-SQUID<a href="#cite_note-13">[13]</a></li>

<li>In precision <a href="Metrology" title="Metrology">metrology</a>, the Josephson effect is a <a href="Reproducibility" title="Reproducibility">reproducible</a> conversion between <a href="Frequency" title="Frequency">frequency</a> and <a href="Voltage" title="Voltage">voltage</a>. The <a href="Josephson_voltage_standard" title="Josephson voltage standard">Josephson voltage standard</a> takes the <a href="Caesium_standard" title="Caesium standard">caesium standard</a> definition of frequency and gives the standard representation of a <a href="Volt" title="Volt">volt</a></li>

<li><a href="Single-electron_transistor" title="Single-electron transistor">Single-electron transistors</a> are often made from <a href="Superconducting" class="mw-redirect" title="Superconducting">superconducting</a> materials and called "superconducting single-electron transistors".<a href="#cite_note-14">[14]</a></li>

<li><a href="Elementary_charge" title="Elementary charge">Elementary charge</a> is most precisely measured in terms of the Josephson constant and the von Klitzing constant which is related to the <a href="Quantum_Hall_effect" title="Quantum Hall effect">quantum Hall effect</a></li>

<li><a href="RSFQ" class="mw-redirect" title="RSFQ">RSFQ</a> digital electronics are based on shunted Josephson junctions. Junction switching emits one <a href="Magnetic_flux_quantum" title="Magnetic flux quantum">magnetic flux quantum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{2e}}h}">

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<li><a href="Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting quantum computing</a> uses Josephon junctions as <a href="Qubits" class="mw-redirect" title="Qubits">qubits</a> such as in a <a href="Flux_qubit" title="Flux qubit">flux qubit</a> or other schemes where the phase and charge are <a href="Conjugate_variables" title="Conjugate variables">conjugate variables</a>.<a href="#cite_note-15">[15]</a></li>

<li><a href="Superconducting_tunnel_junction" title="Superconducting tunnel junction">Superconducting tunnel junction</a> detectors are used in <a href="Superconducting_camera" title="Superconducting camera">superconducting cameras</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="The_Josephson_equations">The Josephson equations</h2></div>

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 <a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau order parameter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{A}={\sqrt {n_{A}}}e^{i\phi _{A}}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{A}={\sqrt {n_{A}}}e^{i\phi _{A}}}</annotation>

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</math><img src="./60e81f059ef10f7ab612183984e637e11c4df8da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.9ex; height:3.509ex;" alt="{\displaystyle \psi _{A}={\sqrt {n_{A}}}e^{i\phi _{A}}}" loading="lazy">, and superconductor B <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{B}={\sqrt {n_{B}}}e^{i\phi _{B}}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{B}={\sqrt {n_{B}}}e^{i\phi _{B}}}</annotation>

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</math><img src="./be853c0753a73b086f314cc4f234177ff8f85c81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.941ex; height:3.509ex;" alt="{\displaystyle \psi _{B}={\sqrt {n_{B}}}e^{i\phi _{B}}}" loading="lazy">, which can be interpreted as the <a href="Wave_function" title="Wave function">wave functions</a> of <a href="Cooper_pair" title="Cooper pair">Cooper pairs</a> in the two superconductors. If the electric potential difference across the junction is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle 2eV}</annotation>

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</math><img src="./5e922c6ea18e256b8c0d424eb62f1032ff192572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.033ex; height:2.176ex;" alt="{\displaystyle 2eV}" loading="lazy">, since each Cooper pair has twice the charge of one electron. The <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for this <a href="Two-state_quantum_system" title="Two-state quantum system">two-state quantum system</a> is therefore:<a href="#cite_note-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\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}},}</annotation>

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where the constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is a characteristic of the junction. To solve the above equation, first calculate the time derivative of the order parameter in superconductor A:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}({\sqrt {n_{A}}}e^{i\phi _{A}})={\dot {\sqrt {n_{A}}}}e^{i\phi _{A}}+{\sqrt {n_{A}}}(i{\dot {\phi }}_{A}e^{i\phi _{A}})=({\dot {\sqrt {n_{A}}}}+i{\sqrt {n_{A}}}{\dot {\phi }}_{A})e^{i\phi _{A}},}</annotation>

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and therefore the Schrödinger equation gives:

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<annotation encoding="application/x-tex">{\displaystyle ({\dot {\sqrt {n_{A}}}}+i{\sqrt {n_{A}}}{\dot {\phi }}_{A})e^{i\phi _{A}}={\frac {1}{i\hbar }}(eV{\sqrt {n_{A}}}e^{i\phi _{A}}+K{\sqrt {n_{B}}}e^{i\phi _{B}}).}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle \varphi =\phi _{B}-\phi _{A}.}</annotation>

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</math>The Schrödinger equation can therefore be rewritten as:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\sqrt {n_{A}}}}+i{\sqrt {n_{A}}}{\dot {\phi }}_{A}={\frac {1}{i\hbar }}(eV{\sqrt {n_{A}}}+K{\sqrt {n_{B}}}e^{i\varphi }),}</annotation>

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and its <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> equation is:

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\sqrt {n_{A}}}}-i{\sqrt {n_{A}}}{\dot {\phi }}_{A}={\frac {1}{-i\hbar }}(eV{\sqrt {n_{A}}}+K{\sqrt {n_{B}}}e^{-i\varphi }).}</annotation>

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Add the two conjugate equations together to eliminate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\phi }}_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\phi }}_{A}}</annotation>

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</math><img src="./cfddb8c66af0ec382e667cf956495f6e7bf9b1e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.932ex; height:3.176ex;" alt="{\displaystyle {\dot {\phi }}_{A}}" loading="lazy">:

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2{\dot {\sqrt {n_{A}}}}={\frac {1}{i\hbar }}(K{\sqrt {n_{B}}}e^{i\varphi }-K{\sqrt {n_{B}}}e^{-i\varphi })={\frac {K{\sqrt {n_{B}}}}{\hbar }}\cdot 2\sin \varphi .}</annotation>

</semantics>

</math>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\sqrt {n_{A}}}}={\frac {{\dot {n}}_{A}}{2{\sqrt {n_{A}}}}}}">

<mover>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mover>

</mrow>

<mfrac>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\sqrt {n_{A}}}}={\frac {{\dot {n}}_{A}}{2{\sqrt {n_{A}}}}}}</annotation>

</semantics>

</math><img src="./30c3780e82eb3f3ddbb1efeceaedb937f31b9d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.688ex; height:6.176ex;" alt="{\displaystyle {\dot {\sqrt {n_{A}}}}={\frac {{\dot {n}}_{A}}{2{\sqrt {n_{A}}}}}}" loading="lazy">, we have:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {n}}_{A}={\frac {2K{\sqrt {n_{A}n_{B}}}}{\hbar }}\sin \varphi .}</annotation>

</semantics>

</math>

Now, subtract the two conjugate equations to eliminate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\sqrt {n_{A}}}}}">

<mover>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\sqrt {n_{A}}}}}</annotation>

</semantics>

</math><img src="./689adb3d1d1d7ba5dc5da01e8d7af55478930b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.795ex; height:3.509ex;" alt="{\displaystyle {\dot {\sqrt {n_{A}}}}}" loading="lazy">:

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

<msqrt>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2i{\sqrt {n_{A}}}{\dot {\phi }}_{A}={\frac {1}{i\hbar }}(2eV{\sqrt {n_{A}}}+K{\sqrt {n_{B}}}e^{i\varphi }+K{\sqrt {n_{B}}}e^{-i\varphi }),}</annotation>

</semantics>

</math>

which gives:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\phi }}_{A}=-{\frac {1}{\hbar }}(eV+K{\sqrt {\frac {n_{B}}{n_{A}}}}\cos \varphi ).}</annotation>

</semantics>

</math>

Similarly, for superconductor B we can derive that:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mfrac>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {n}}_{B}=-{\frac {2K{\sqrt {n_{A}n_{B}}}}{\hbar }}\sin \varphi ,\,{\dot {\phi }}_{B}={\frac {1}{\hbar }}(eV-K{\sqrt {\frac {n_{A}}{n_{B}}}}\cos \varphi ).}</annotation>

</semantics>

</math>

Noting that the evolution of Josephson phase is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\varphi }}={\dot {\phi }}_{B}-{\dot {\phi }}_{A}}">

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\varphi }}={\dot {\phi }}_{B}-{\dot {\phi }}_{A}}</annotation>

</semantics>

</math><img src="./02e92200e2a6a785b3f2eba69e217389975d3722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.353ex; height:3.176ex;" alt="{\displaystyle {\dot {\varphi }}={\dot {\phi }}_{B}-{\dot {\phi }}_{A}}" loading="lazy"> and the time derivative of <a href="Charge_carrier_density" title="Charge carrier density">charge carrier density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {n}}_{A}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {n}}_{A}}</annotation>

</semantics>

</math><img src="./0d853a929176100bee537383b4e3b578e9c501e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.86ex; height:2.509ex;" alt="{\displaystyle {\dot {n}}_{A}}" loading="lazy"> is proportional to current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{A}\approx n_{B}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{A}\approx n_{B}}</annotation>

</semantics>

</math><img src="./07640e8cffbc13372a410aa20e8055a1d51a7051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.832ex; height:2.009ex;" alt="{\displaystyle n_{A}\approx n_{B}}" loading="lazy">, the above solution yields the Josephson equations:<a href="#cite_note-barone-17">[17]</a>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(t)=I_{c}\sin(\varphi (t))}</annotation>

</semantics>

</math><img src="./745bfd3aa11306a69821e1afa0471acea6573e16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.107ex; height:2.843ex;" alt="{\displaystyle I(t)=I_{c}\sin(\varphi (t))}" loading="lazy">

(1)

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {2eV(t)}{\hbar }}}</annotation>

</semantics>

</math><img src="./085c6054c876051341aa8e1722c3cd821047c7e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.291ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {2eV(t)}{\hbar }}}" loading="lazy">

(2)

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(t)}</annotation>

</semantics>

</math><img src="./383b47023708ac9df0e198448491048ec7acb2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:2.843ex;" alt="{\displaystyle V(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(t)}</annotation>

</semantics>

</math><img src="./e2434c9d80c34c95e25cc81ba6700f756a29dac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.821ex; height:2.843ex;" alt="{\displaystyle I(t)}" loading="lazy"> are the voltage across and the current through the Josephson junction, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{c}}</annotation>

</semantics>

</math><img src="./7463de5fff5fee0c540ac1d8cd1aca43f4852c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.967ex; height:2.509ex;" alt="{\displaystyle I_{c}}" loading="lazy"> 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 <a href="Josephson_constant" class="mw-redirect" title="Josephson constant">Josephson constant</a> is defined as:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{J}={\frac {2e}{h}}\,,}</annotation>

</semantics>

</math>

and its inverse is the <a href="Magnetic_flux_quantum" title="Magnetic flux quantum">magnetic flux quantum</a>:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{0}={\frac {h}{2e}}=2\pi {\frac {\hbar }{2e}}\,.}</annotation>

</semantics>

</math>

The superconducting phase evolution equation can be reexpressed as:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=2\pi [K_{J}V(t)]={\frac {2\pi }{\Phi _{0}}}V(t)\,.}</annotation>

</semantics>

</math>

If we define:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi =\Phi _{0}{\frac {\varphi }{2\pi }}\,,}</annotation>

</semantics>

</math>

then the voltage across the junction is:

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\frac {\Phi _{0}}{2\pi }}{\frac {\partial \varphi }{\partial t}}={\frac {d\Phi }{dt}}\,,}</annotation>

</semantics>

</math>

which is very similar to <a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>. But note that this voltage does not come from magnetic energy, since there is <a href="Meissner_effect" title="Meissner effect">no magnetic field in the superconductors</a>; Instead, this voltage comes from the kinetic energy of the carriers (i.e. the Cooper pairs). This phenomenon is also known as <a href="Kinetic_inductance" title="Kinetic inductance">kinetic inductance</a>.

<div class="mw-heading mw-heading2"><h2 id="Three_main_effects">Three main effects</h2></div>

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

<div class="mw-heading mw-heading3"><h3 id="The_DC_Josephson_effect">The DC Josephson effect</h3></div>

The DC Josephson effect is a direct current crossing the insulator in the absence of any external electromagnetic field, owing to <a href="Tunnel_(quantum_mechanics)" class="mw-redirect" title="Tunnel (quantum mechanics)">tunneling</a>. 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -I_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -I_{c}}</annotation>

</semantics>

</math><img src="./fe616011ce7d7eedd3d70e97b95d2d62021ca85d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.775ex; height:2.509ex;" alt="{\displaystyle -I_{c}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{c}}</annotation>

</semantics>

</math><img src="./7463de5fff5fee0c540ac1d8cd1aca43f4852c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.967ex; height:2.509ex;" alt="{\displaystyle I_{c}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="The_AC_Josephson_effect">The AC Josephson effect</h3></div>

With a fixed voltage <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{DC}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{DC}}</annotation>

</semantics>

</math><img src="./f3ddc032d07ad729ddd28d58b7ce8d470ac2bf88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.197ex; height:2.509ex;" alt="{\displaystyle V_{DC}}" loading="lazy"> across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC (<a href="Alternating_current" title="Alternating current">alternating current</a>) with amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{c}}</annotation>

</semantics>

</math><img src="./7463de5fff5fee0c540ac1d8cd1aca43f4852c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.967ex; height:2.509ex;" alt="{\displaystyle I_{c}}" loading="lazy"> and frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{J}V_{DC}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{J}V_{DC}}</annotation>

</semantics>

</math><img src="./09fbb8bf3ceb023e39a9e06c044b41bf97150599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.443ex; height:2.509ex;" alt="{\displaystyle K_{J}V_{DC}}" loading="lazy">. This means a Josephson junction can act as a perfect voltage-to-frequency converter.

<div class="mw-heading mw-heading3"><h3 id="The_inverse_AC_Josephson_effect">The inverse AC Josephson effect</h3></div>

Microwave radiation of a single <a href="Angular_frequency" title="Angular frequency">(angular) frequency</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> can induce quantized DC voltages<a href="#cite_note-18">[18]</a> across the Josephson junction, in which case the Josephson phase takes the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)=\varphi _{0}+n\omega t+a\sin(\omega t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)=\varphi _{0}+n\omega t+a\sin(\omega t)}</annotation>

</semantics>

</math><img src="./401dbeb24ef30b5e4681ffc2a54fb6eb9b2f5b1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.77ex; height:2.843ex;" alt="{\displaystyle \varphi (t)=\varphi _{0}+n\omega t+a\sin(\omega t)}" loading="lazy">, and the voltage and current across the junction will be:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(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),}</annotation>

</semantics>

</math>

The DC components are:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\text{DC}}=n{\frac {\hbar }{2e}}\omega ,{\text{ and }}I_{\text{DC}}=I_{c}J_{-n}(a)\sin \varphi _{0}.}</annotation>

</semantics>

</math>

This means a Josephson junction can act like a perfect frequency-to-voltage converter,<a href="#cite_note-19">[19]</a> which is the theoretical basis for the Josephson voltage standard.

<div class="mw-heading mw-heading2"><h2 id="Josephson_inductance">Josephson inductance</h2></div>

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 <a href="Kinetic_inductance" title="Kinetic inductance">kinetic inductance</a> named Josephson Inductance.<a href="#cite_note-20">[20]</a>

Rewrite the Josephson relations as:

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial I}{\partial \varphi }}&=I_{c}\cos \varphi ,\\{\frac {\partial \varphi }{\partial t}}&={\frac {2\pi }{\Phi _{0}}}V.\end{aligned}}}</annotation>

</semantics>

</math><img src="./730953b770c35eb2c65213a7116f67a8709758c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:15.544ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial I}{\partial \varphi }}&=I_{c}\cos \varphi ,\\{\frac {\partial \varphi }{\partial t}}&={\frac {2\pi }{\Phi _{0}}}V.\end{aligned}}}" loading="lazy"></dd></dl>

Now, apply the <a href="Chain_rule" title="Chain rule">chain rule</a> to calculate the time derivative of the current:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\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,}</annotation>

</semantics>

</math><img src="./ccca17644018662c04f7910e56d534a027c2d1bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.926ex; height:6.176ex;" alt="{\displaystyle {\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,}" loading="lazy"></dd></dl>

Rearrange the above result in the form of the <a href="Current%E2%80%93voltage_characteristic" title="Current–voltage characteristic">current–voltage characteristic</a> of an inductor:

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {\Phi _{0}}{2\pi I_{c}\cos \varphi }}{\frac {\partial I}{\partial t}}=L(\varphi ){\frac {\partial I}{\partial t}}.}</annotation>

</semantics>

</math><img src="./3a88a9efbd731b1d2abec903e98eb33dfb8990a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.899ex; height:5.843ex;" alt="{\displaystyle V={\frac {\Phi _{0}}{2\pi I_{c}\cos \varphi }}{\frac {\partial I}{\partial t}}=L(\varphi ){\frac {\partial I}{\partial t}}.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle L(\varphi )={\frac {\Phi _{0}}{2\pi I_{c}\cos \varphi }}={\frac {L_{J}}{\cos \varphi }}.}</annotation>

</semantics>

</math><img src="./9d49e09a642fa3647bb10d6fade5d346fc02e307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.314ex; height:5.843ex;" alt="{\displaystyle L(\varphi )={\frac {\Phi _{0}}{2\pi I_{c}\cos \varphi }}={\frac {L_{J}}{\cos \varphi }}.}" loading="lazy"></dd></dl>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{J}=L(0)={\frac {\Phi _{0}}{2\pi I_{c}}}}">

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<annotation encoding="application/x-tex">{\displaystyle L_{J}=L(0)={\frac {\Phi _{0}}{2\pi I_{c}}}}</annotation>

</semantics>

</math><img src="./7e863ab0ca27fbaeac737a655fd31341fcb2969e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.905ex; height:5.843ex;" alt="{\displaystyle L_{J}=L(0)={\frac {\Phi _{0}}{2\pi I_{c}}}}" loading="lazy"> 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.

<div class="mw-heading mw-heading2"><h2 id="Josephson_energy">Josephson energy</h2></div>

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.<a href="#cite_note-21">[21]</a>

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

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<annotation encoding="application/x-tex">{\displaystyle I=I_{c}\sin \varphi .}</annotation>

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</math><img src="./4b98490a4716300b125222446dfd1f7601cbda83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.034ex; height:2.676ex;" alt="{\displaystyle I=I_{c}\sin \varphi .}" loading="lazy"></dd></dl>

The superconducting phase evolution equation is analogous to <a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law</a>:

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<annotation encoding="application/x-tex">{\displaystyle V=\operatorname {d} \!\Phi /\operatorname {d} \!t\,.}</annotation>

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</math><img src="./18c8b137c21a546f54e4c66a572a78e8c9802275.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.572ex; height:2.843ex;" alt="{\displaystyle V=\operatorname {d} \!\Phi /\operatorname {d} \!t\,.}" loading="lazy"></dd></dl>

Assume that at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation>

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</math><img src="./cb0768c0bd659f2f84fb5ef9f4b74f336123d915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}" loading="lazy">, the Josephson phase is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{1}}</annotation>

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</math><img src="./d7daf493c8f6ef669c04c7b9715532fc35d12d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{1}}" loading="lazy">; At a later time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation>

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</math><img src="./749fee708b41e7079eabd50d61c8bf3e965db16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}" loading="lazy">, the Josephson phase evolved to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{2}}</annotation>

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</math><img src="./c08631714273b6c8edaa9573ef3d8c548314a930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.574ex; height:2.176ex;" alt="{\displaystyle \varphi _{2}}" loading="lazy">. The energy increase in the junction is equal to the work done on the junction:

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<annotation encoding="application/x-tex">{\displaystyle \Delta E=\int _{1}^{2}IV\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 \,.}</annotation>

</semantics>

</math><img src="./a0fa6754ea92363c677d82d6aa5ab2e2ff790707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:72.268ex; height:6.676ex;" alt="{\displaystyle \Delta E=\int _{1}^{2}IV\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 \,.}" loading="lazy"></dd></dl>

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 <a href="Thermodynamic_process_path" class="mw-redirect" title="Thermodynamic process path">path</a>. Therefore, the energy stored in a Josephson junction is a <a href="State_function" title="State function">state function</a>, which can be defined as:

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<annotation encoding="application/x-tex">{\displaystyle E(\varphi )=-{\frac {\Phi _{0}I_{c}}{2\pi }}\cos \varphi =-E_{J}\cos \varphi \,.}</annotation>

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</math><img src="./69ba89adf598ee683f5ea47d9d2ec48c40d00e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.287ex; height:5.343ex;" alt="{\displaystyle E(\varphi )=-{\frac {\Phi _{0}I_{c}}{2\pi }}\cos \varphi =-E_{J}\cos \varphi \,.}" loading="lazy"></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{J}=|E(0)|={\frac {\Phi _{0}I_{c}}{2\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{J}=|E(0)|={\frac {\Phi _{0}I_{c}}{2\pi }}}</annotation>

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</math><img src="./6a0de48b55702326a4fa37a9a1aba49cc9eb998e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.762ex; height:5.343ex;" alt="{\displaystyle E_{J}=|E(0)|={\frac {\Phi _{0}I_{c}}{2\pi }}}" loading="lazy"> is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{J}=L_{J}I_{c}^{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle E_{J}=L_{J}I_{c}^{2}}</annotation>

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</math><img src="./5ae28780e03c70edcb3a973888baeb35d44aa209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.213ex; height:2.843ex;" alt="{\displaystyle E_{J}=L_{J}I_{c}^{2}}" loading="lazy">. An alternative but equivalent definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(\varphi )=E_{J}(1-\cos \varphi )}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle E(\varphi )=E_{J}(1-\cos \varphi )}</annotation>

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</math><img src="./e5edde856b2071c3896334168059bdd59d47d743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.022ex; height:2.843ex;" alt="{\displaystyle E(\varphi )=E_{J}(1-\cos \varphi )}" loading="lazy"> is also often used.

Again, note that a non-linear <a href="Inductor" title="Inductor">magnetic coil inductor</a> accumulates <a href="Potential_energy" title="Potential energy">potential energy</a> 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.

<div class="mw-heading mw-heading2"><h2 id="The_RCSJ_model">The RCSJ model</h2></div>

The Resistively Capacitance Shunted Junction (RCSJ) model,<a href="#cite_note-22">[22]</a><a href="#cite_note-23">[23]</a> 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 <a href="Th%C3%A9venin's_theorem" title="Thévenin's theorem">Thévenin's theorem</a>,<a href="#cite_note-24">[24]</a> the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel<a href="#cite_note-25">[25]</a> to the ideal Josephson Junction. The complete expression for the current drive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\text{ext}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle I_{\text{ext}}}</annotation>

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</math><img src="./bf62c49b39114886332e79bc0b95a86c73d75919.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:2.509ex;" alt="{\displaystyle I_{\text{ext}}}" loading="lazy"> becomes:

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<annotation encoding="application/x-tex">{\displaystyle I_{\text{ext}}=C_{J}{\frac {\operatorname {d} \!V}{\operatorname {d} \!t}}+I_{c}\sin \varphi +{\frac {V}{R}},}</annotation>

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</math><img src="./5e5750606aa0991d3bfbcd4aa224a14d70285d91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.51ex; height:5.509ex;" alt="{\displaystyle I_{\text{ext}}=C_{J}{\frac {\operatorname {d} \!V}{\operatorname {d} \!t}}+I_{c}\sin \varphi +{\frac {V}{R}},}" loading="lazy"></dd></dl>

where the first term is displacement current with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{J}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{J}}</annotation>

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</math><img src="./3d3f1eb1e505eb7a31d2fc30c71f9211f5b6fae4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle C_{J}}" loading="lazy"> – effective capacitance, and the third is normal current with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> – effective resistance of the junction.

<div class="mw-heading mw-heading2"><h2 id="Josephson_penetration_depth">Josephson penetration depth</h2></div>

The Josephson penetration depth characterizes the typical length on which an externally applied <a href="Magnetic_field" title="Magnetic field">magnetic field</a> penetrates into the <a href="Long_Josephson_junction" title="Long Josephson junction">long Josephson junction</a>. It is usually denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{J}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{J}}</annotation>

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</math><img src="./308700e7898e22630a74d5f543dea1596c6cd6e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.628ex; height:2.509ex;" alt="{\displaystyle \lambda _{J}}" loading="lazy"> and is given by the following expression (in SI):

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{J}={\sqrt {\frac {\Phi _{0}}{2\pi \mu _{0}d'j_{c}}}},}</annotation>

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</math><img src="./4adf80616b3402c8a242d5339a3f5e049075e898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.289ex; height:7.676ex;" alt="{\displaystyle \lambda _{J}={\sqrt {\frac {\Phi _{0}}{2\pi \mu _{0}d'j_{c}}}},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{0}}</annotation>

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</math><img src="./32f33de1903508a18f10c5fc11d788de19e043dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Phi _{0}}" loading="lazy"> is the magnetic flux quantum, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{c}}">

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<annotation encoding="application/x-tex">{\displaystyle j_{c}}</annotation>

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</math><img src="./f1c1152b8226d25f2c54d33922fdbbf5d92d16ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.929ex; height:2.509ex;" alt="{\displaystyle j_{c}}" loading="lazy"> is the critical supercurrent density (A/m2), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'}">

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<annotation encoding="application/x-tex">{\displaystyle d'}</annotation>

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</math><img src="./1f310a68106a9e308bdaf887ff8f7171c4cb9d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d'}" loading="lazy"> characterizes the inductance of the superconducting electrodes<a href="#cite_note-26">[26]</a>

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<annotation encoding="application/x-tex">{\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),}</annotation>

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</math><img src="./1b98fcf686e2c966fd8ed86f01ec9739ec0e70ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.155ex; height:6.176ex;" alt="{\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),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{I}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle d_{I}}</annotation>

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</math><img src="./e05e5521b300f096191936a901599a02fce36db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.27ex; height:2.509ex;" alt="{\displaystyle d_{I}}" loading="lazy"> is the thickness of the Josephson barrier (usually insulator), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation>

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</math><img src="./4cccb5a6a2f1acab4ca255e0be86c224ed82282a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation>

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</math><img src="./9276f8f68c5c23329de74ad76e69f6801358fb1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}" loading="lazy"> are the thicknesses of superconducting electrodes, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}}">

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</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}}</annotation>

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</math><img src="./571a423bece8f29bcd1b48572f18dd4f6213dce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{2}}</annotation>

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</math><img src="./6b668a1bd1e8ab9452ca975b7497546e7c1ba187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \lambda _{2}}" loading="lazy"> are their <a href="London_penetration_depth" title="London penetration depth">London penetration depths</a>. The Josephson penetration depth usually ranges from a few <a href="Micrometre" title="Micrometre">μm</a> to several mm if the critical current density is very low.<a href="#cite_note-27">[27]</a>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Josephson_effect" class="extiw external" title="commons:Category:Josephson effect">Josephson effect</a>.</div></div>

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<ul><li><a href="Pi_Josephson_junction" title="Pi Josephson junction">Pi Josephson junction</a></li>

<li><a href="Varphi_Josephson_junction" class="mw-redirect" title="Varphi Josephson junction">φ Josephson junction</a></li>

<li><a href="Josephson_diode" title="Josephson diode">Josephson diode</a></li>

<li><a href="Andreev_reflection" title="Andreev reflection">Andreev reflection</a></li>

<li><a href="Fractional_vortices" title="Fractional vortices">Fractional vortices</a></li>

<li><a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a></li>

<li><a href="Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a></li>

<li><a href="Macroscopic_quantum_self-trapping" title="Macroscopic quantum self-trapping">Macroscopic quantum self-trapping</a></li>

<li><a href="Quantum_computer" class="mw-redirect" title="Quantum computer">Quantum computer</a></li>

<li><a href="Quantum_gyroscope" title="Quantum gyroscope">Quantum gyroscope</a></li>

<li><a href="Rapid_single_flux_quantum" title="Rapid single flux quantum">Rapid single flux quantum</a> (RSFQ)</li>

<li><a href="Semifluxon" title="Semifluxon">Semifluxon</a></li>

<li><a href="Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li>

<li><a href="Josephson_vortex" title="Josephson vortex">Josephson vortex</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFJosephson1962" class="citation journal cs1">Josephson, B. D. (1962). "Possible new effects in superconductive tunnelling". Physics Letters. 1 (7): 251–253. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1962PhL.....1..251J">1962PhL.....1..251J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0031-9163%2862%2991369-0">10.1016/0031-9163(62)91369-0</a>.</cite>

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<dl><dd>Also in <cite id="CITEREFJosephson1974" class="citation journal cs1">Josephson, B. D. (1974). "The Discovery of Tunnelling Supercurrents". Europhysics News. 5 (3): 1–5. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974ENews...5c...1J">1974ENews...5c...1J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1051%2Fepn%2F19740503001">10.1051/epn/19740503001</a>.</cite></dd></dl>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> Steven Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003.

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFJosephson1973" class="citation web cs1">Josephson, Brian D. (December 12, 1973). <a rel="nofollow" class="external text" href="https://www.nobelprize.org/prizes/physics/1973/josephson/lecture/">"The Discovery of Tunneling Supercurrents (Nobel Lecture)"</a>.</cite>

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<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFMcCumber1968" class="citation journal cs1">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. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1968JAP....39.3113M">1968JAP....39.3113M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1656743">10.1063/1.1656743</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-8979">0021-8979</a>.</cite>

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<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFChakravartyIngoldKivelsonZimanyi1988" class="citation journal cs1">Chakravarty, Sudip; Ingold, Gert-Ludwig; Kivelson, Steven; Zimanyi, Gergely (1988-03-01). <a rel="nofollow" class="external text" href="https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/40643">"Quantum statistical mechanics of an array of resistively shunted Josephson junctions"</a>. Physical Review B. 37 (7): 3283–3294. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988PhRvB..37.3283C">1988PhRvB..37.3283C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.37.3283">10.1103/PhysRevB.37.3283</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9944915">9944915</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/acthev.html">"AC Thevenin's Theorem"</a>. hyperphysics.phy-astr.gsu.edu. Retrieved 2020-01-03.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210613133720/http://phelafel.technion.ac.il/~orensuch/theory/Theory.html">"Dynamics of RF SQUID"</a>. phelafel.technion.ac.il. Archived from <a rel="nofollow" class="external text" href="http://phelafel.technion.ac.il/~orensuch/theory/Theory.html">the original</a> on 2021-06-13. Retrieved 2020-01-11.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFWeihnacht1969" class="citation journal cs1">Weihnacht, M. (1969). "Influence of Film Thickness on D. C. Josephson Current". Physica Status Solidi B. 32 (2): 169. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969PSSBR..32..169W">1969PSSBR..32..169W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fpssb.19690320259">10.1002/pssb.19690320259</a>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFBuckelKleiner2004" class="citation book cs1">Buckel, Werner; Kleiner, Reinhold (2004). Supraleitung (6. ed.). Tübingen: Wiley-VCH Verlag GmbH&Co.KGaA. p. 67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3527403485</bdi>.</cite>

</li>

</ol></div></div>

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</style><div id="Condensed_matter_physics656" style="font-size:114%;margin:0 4em"><a href="Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="State_of_matter" title="State of matter">States of matter</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Solid" title="Solid">Solid</a></li>

<li><a href="Liquid" title="Liquid">Liquid</a></li>

<li><a href="Plasma_(physics)" title="Plasma (physics)">Plasma</a></li>

<li><a href="Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li>

<li><a href="Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li>

<li><a href="Fermi_gas" title="Fermi gas">Fermi gas</a></li>

<li><a href="Supersolid" title="Supersolid">Supersolid</a></li>

<li><a href="Superfluidity" title="Superfluidity">Superfluid</a></li>

<li><a href="Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li>

<li><a href="Time_crystal" title="Time crystal">Time crystal</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phase phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li>

<li><a href="Phase_transition" title="Phase transition">Phase transition</a></li>

<li><a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li>

<li><a href="Critical_phenomena" title="Critical phenomena">Critical phenomena</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electrons in solids</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hall_effect" title="Hall effect">Hall effect</a></li>

<li><a href="Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li>

<li><a href="Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li>

<li><a href="Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li>

<li><a href="Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li>

<li><a href="Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li>

<li><a href="Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Drude_model" title="Drude model">Drude model</a></li>

<li><a href="Free_electron_model" title="Free electron model">Free electron model</a></li>

<li><a href="Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a></li>

<li><a href="Bloch's_theorem" title="Bloch's theorem">Bloch's theorem</a></li>

<li><a href="Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid theory</a></li>

<li><a href="Electronic_band_structure" title="Electronic band structure">electronic band structure</a></li>

<li><a href="Anderson_localization" title="Anderson localization">Anderson localization</a></li>

<li><a href="BCS_theory" title="BCS theory">BCS theory</a></li>

<li><a href="Tight_binding_model" class="mw-redirect" title="Tight binding model">tight binding model</a></li>

<li><a href="Hubbard_model" title="Hubbard model">Hubbard model</a></li>

<li><a href="Density_functional_theory" title="Density functional theory">Density functional theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li>

<li><a href="Mott_insulator" title="Mott insulator">Mott insulator</a></li>

<li><a href="Semiconductor" title="Semiconductor">Semiconductor</a></li>

<li><a href="Semimetal" title="Semimetal">Semimetal</a></li>

<li><a href="Electrical_conductor" title="Electrical conductor">Conductor</a></li>

<li><a href="Superconductivity" title="Superconductivity">Superconductor</a></li>

<li><a href="Topological_insulator" title="Topological insulator">Topological insulator</a></li>

<li><a href="Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Couplings</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Thermoelectric_effect" title="Thermoelectric effect">Thermoelectricity</a></li>

<li><a href="Piezoelectricity" title="Piezoelectricity">Piezoelectricity</a></li>

<li><a href="Ferroelectricity" title="Ferroelectricity">Ferroelectricity</a></li>

<li><a href="Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li>

<li><a href="Electrostriction" title="Electrostriction">Electrostriction</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Magnetic phases</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amorphous_magnet" title="Amorphous magnet">Amorphous magnet</a></li>

<li><a href="Diamagnetism" title="Diamagnetism">Diamagnet</a></li>

<li><a href="Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a></li>

<li><a href="Paramagnetism" title="Paramagnetism">Paramagnet</a></li>

<li><a href="Superparamagnetism" title="Superparamagnetism">Superparamagnet</a></li>

<li><a href="Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li>

<li><a href="Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li>

<li><a href="Metamagnetism" title="Metamagnetism">Metamagnet</a></li>

<li><a href="Spin_glass" title="Spin glass">Spin glass</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anyon" title="Anyon">Anyon</a></li>

<li><a href="Bogoliubov_quasiparticle" title="Bogoliubov quasiparticle">Bogoliubov quasiparticle</a></li>

<li><a href="Exciton" title="Exciton">Exciton</a></li>

<li><a href="Magnon" title="Magnon">Magnon</a></li>

<li><a href="Phonon" title="Phonon">Phonon</a></li>

<li><a href="Pines'_demon" title="Pines' demon">Pines' demon</a></li>

<li><a href="Plasmon" title="Plasmon">Plasmon</a></li>

<li><a href="Polariton" title="Polariton">Polariton</a>

<ul><li><a href="Exciton-polariton" title="Exciton-polariton">Exciton-polariton</a></li>

<li><a href="Phonon_polariton" title="Phonon polariton">Phonon polariton</a></li></ul></li>

<li><a href="Polaron" title="Polaron">Polaron</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Soft_matter" title="Soft matter">Soft matter</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amorphous_solid" title="Amorphous solid">Amorphous solid</a></li>

<li><a href="Granular_material" title="Granular material">Granular matter</a></li>

<li><a href="Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li>

<li><a href="Polymer" title="Polymer">Polymer</a></li>

<li><a href="Interface_and_colloid_science" title="Interface and colloid science">Colloids</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li> Category</li>

<li> <a href="https://commons.wikimedia.org/wiki/Category:Condensed_matter_physics" class="extiw external" title="commons:Category:Condensed matter physics">Commons</a></li>

<li><a href="Portal%3APhysics" title="Portal:Physics">Physics Portal</a></li>

<li> Physics WikiProject</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Superconductivity212" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Superconductivity212" style="font-size:114%;margin:0 4em"><a href="Superconductivity" title="Superconductivity">Superconductivity</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Bean's_critical_state_model" title="Bean's critical state model">Bean's critical state</a></li>

<li><a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li>

<li><a href="Kohn%E2%80%93Luttinger_superconductivity" title="Kohn–Luttinger superconductivity">Kohn–Luttinger</a></li>

<li><a href="London_equations" title="London equations">London</a></li>

<li><a href="Matthias_rules" title="Matthias rules">Matthias</a></li>

<li><a href="Mattis%E2%80%93Bardeen_theory" title="Mattis–Bardeen theory">Mattis–Bardeen</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Characteristic parameters</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Superconducting_coherence_length" title="Superconducting coherence length">coherence length</a></li>

<li><a href="Critical_field" title="Critical field">critical field</a></li>

<li><a href="London_penetration_depth" title="London penetration depth">London penetration depth</a></li>

<li><a href="Silsbee_effect" title="Silsbee effect">Silsbee current</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abrikosov_vortex" title="Abrikosov vortex">Abrikosov vortices</a></li>

<li><a href="Andreev_reflection" title="Andreev reflection">Andreev reflection</a></li>

<li><a href="Cooper_pair" title="Cooper pair">Cooper pairs</a></li>

<li><a href="Flux_pinning" title="Flux pinning">flux pinning</a></li>

<li><a href="Flux_pumping" title="Flux pumping">flux pumping</a></li>

<li><a href="Little%E2%80%93Parks_effect" title="Little–Parks effect">Little–Parks effect</a></li>

<li><a href="Meissner_effect" title="Meissner effect">Meissner effect</a></li>

<li><a href="Homes's_law" title="Homes's law">Homes's law</a></li>

<li><a href="Persistent_current" title="Persistent current">persistent currents</a></li>

<li><a href="Proximity_effect_(superconductivity)" title="Proximity effect (superconductivity)">proximity effect</a></li>

<li><a href="Reentrant_superconductivity" title="Reentrant superconductivity">reentrance</a></li>

<li><a href="SU(2)_color_superconductivity" title="SU(2) color superconductivity">SU(2) color</a></li>

<li><a href="Supercurrent" title="Supercurrent">supercurrents</a></li>

<li><a href="Superdiamagnetism" title="Superdiamagnetism">superdiamagnetism</a></li>

<li><a href="Superstripes" title="Superstripes">superstripes</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Superconductor_classification" title="Superconductor classification">Classification</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">By magnetic response</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Types

<li><a href="Ferromagnetic_superconductor" title="Ferromagnetic superconductor">ferromagnetic</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By explanation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Conventional_superconductor" title="Conventional superconductor">conventional</a></li>

<li><a href="Unconventional_superconductor" title="Unconventional superconductor">unconventional</a></li>

<li><a href="Topological_superconductor" title="Topological superconductor">topological</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By critical temperature</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>low temperature</li>

<li><a href="High-temperature_superconductivity" title="High-temperature superconductivity">high temperature</a></li>

<li><a href="Room-temperature_superconductor" title="Room-temperature superconductor">room temperature</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By composition</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Covalent_superconductor" title="Covalent superconductor">covalent</a></li>

<li><a href="Cuprate_superconductor" title="Cuprate superconductor">cuprates</a></li>

<li><a href="Heavy_fermion_superconductor" title="Heavy fermion superconductor">heavy fermion</a></li>

<li><a href="Iron-based_superconductor" title="Iron-based superconductor">iron-based</a></li>

<li><a href="Oxypnictide" title="Oxypnictide">oxypnictides</a></li>

<li><a href="Organic_superconductor" title="Organic superconductor">organic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Technological_applications_of_superconductivity" title="Technological applications of superconductivity">Technological applications</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cryotron" title="Cryotron">cryotron</a></li>

<li><a href="Superconducting_magnet" title="Superconducting magnet">electromagnets</a></li>

<li><a href="Superconducting_quantum_computing" title="Superconducting quantum computing">quantum computing</a></li>

<li><a href="Rutherford_cable" title="Rutherford cable">Rutherford cable</a></li>

<li><a href="SQUID" title="SQUID">SQUID</a></li>

<li><a href="Superconducting_wire" title="Superconducting wire">wires</a></li>

<li><a href="SCMaglev" title="SCMaglev">SCMaglev</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">List of superconductors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bilayer_graphene" title="Bilayer graphene">bilayer graphene</a></li>

<li><a href="BSCCO" class="mw-redirect" title="BSCCO">BSCCO</a></li>

<li><a href="Thallium_barium_calcium_copper_oxide" title="Thallium barium calcium copper oxide">TBCCO</a></li>

</div></td></tr></tbody></table></div>

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