Josephson effect

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In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. The effect is named after the British physicist Brian Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.^[1][2] It is an example of a macroscopic quantum phenomenon, 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 supercurrent, 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 superconductor–insulator–superconductor junction, or S-I-S), a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-c-S).

Josephson junctions have important applications in quantum-mechanical circuits, such as SQUIDs, superconducting qubits, and RSFQ digital electronics. The NIST standard for one volt is achieved by an array of 20,208 Josephson junctions in series.^[3]

The DC Josephson effect had been seen in experiments prior to 1962,^[4] 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 into 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.^{[5][6]: 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^[1]. 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.^{[6]: 222–227} See also: 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".^[7] These authors were awarded patents^[8] on the effects that were never enforced, but never challenged.^{[citation needed]}

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.^[9] John Bardeen was one of the nominators.^[6]: 230

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.^[10][11]
• 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-SQUID^[12]
• 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".^[13]
• 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 ${\displaystyle \scriptstyle {\frac {1}{2e}}h}$. 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.^[14]
• Superconducting tunnel junction detectors are used in superconducting cameras

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 ${\displaystyle \psi _{A}={\sqrt {n_{A}}}e^{i\phi _{A}}}$, and superconductor B ${\displaystyle \psi _{B}={\sqrt {n_{B}}}e^{i\phi _{B}}}$, which can be interpreted as the wave functions of Cooper pairs in the two superconductors. If the electric potential difference across the junction is ${\displaystyle V}$, then the energy difference between the two superconductors is ${\displaystyle 2eV}$, since each Cooper pair has twice the charge of one electron. The Schrödinger equation for this two-state quantum system is therefore:^[15]

$${\displaystyle i\hbar {\frac {\partial }{\partial t}}{\begin{pmatrix}{\sqrt {n_{A}}}e^{i\phi _{A}}\\{\sqrt {n_{B}}}e^{i\phi _{B}}\end{pmatrix}}={\begin{pmatrix}eV&K\\K&-eV\end{pmatrix}}{\begin{pmatrix}{\sqrt {n_{A}}}e^{i\phi _{A}}\\{\sqrt {n_{B}}}e^{i\phi _{B}}\end{pmatrix}},}$$

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

$${\displaystyle {\frac {\partial }{\partial t}}({\sqrt {n_{A}}}e^{i\phi _{A}})={\dot {\sqrt {n_{A}}}}e^{i\phi _{A}}+{\sqrt {n_{A}}}(i{\dot {\phi }}_{A}e^{i\phi _{A}})=({\dot {\sqrt {n_{A}}}}+i{\sqrt {n_{A}}}{\dot {\phi }}_{A})e^{i\phi _{A}},}$$

and therefore the Schrödinger equation gives:

$${\displaystyle ({\dot {\sqrt {n_{A}}}}+i{\sqrt {n_{A}}}{\dot {\phi }}_{A})e^{i\phi _{A}}={\frac {1}{i\hbar }}(eV{\sqrt {n_{A}}}e^{i\phi _{A}}+K{\sqrt {n_{B}}}e^{i\phi _{B}}).}$$

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

$${\displaystyle \varphi =\phi _{B}-\phi _{A}.}$$The Schrödinger equation can therefore be rewritten as:

$${\displaystyle {\dot {\sqrt {n_{A}}}}+i{\sqrt {n_{A}}}{\dot {\phi }}_{A}={\frac {1}{i\hbar }}(eV{\sqrt {n_{A}}}+K{\sqrt {n_{B}}}e^{i\varphi }),}$$

and its complex conjugate equation is:

$${\displaystyle {\dot {\sqrt {n_{A}}}}-i{\sqrt {n_{A}}}{\dot {\phi }}_{A}={\frac {1}{-i\hbar }}(eV{\sqrt {n_{A}}}+K{\sqrt {n_{B}}}e^{-i\varphi }).}$$

Add the two conjugate equations together to eliminate ${\displaystyle {\dot {\phi }}_{A}}$:

$${\displaystyle 2{\dot {\sqrt {n_{A}}}}={\frac {1}{i\hbar }}(K{\sqrt {n_{B}}}e^{i\varphi }-K{\sqrt {n_{B}}}e^{-i\varphi })={\frac {K{\sqrt {n_{B}}}}{\hbar }}\cdot 2\sin \varphi .}$$

Since ${\displaystyle {\dot {\sqrt {n_{A}}}}={\frac {{\dot {n}}_{A}}{2{\sqrt {n_{A}}}}}}$, we have:

$${\displaystyle {\dot {n}}_{A}={\frac {2K{\sqrt {n_{A}n_{B}}}}{\hbar }}\sin \varphi .}$$

Now, subtract the two conjugate equations to eliminate ${\displaystyle {\dot {\sqrt {n_{A}}}}}$:

$${\displaystyle 2i{\sqrt {n_{A}}}{\dot {\phi }}_{A}={\frac {1}{i\hbar }}(2eV{\sqrt {n_{A}}}+K{\sqrt {n_{B}}}e^{i\varphi }+K{\sqrt {n_{B}}}e^{-i\varphi }),}$$

which gives:

$${\displaystyle {\dot {\phi }}_{A}=-{\frac {1}{\hbar }}(eV+K{\sqrt {\frac {n_{B}}{n_{A}}}}\cos \varphi ).}$$

Similarly, for superconductor B we can derive that:

$${\displaystyle {\dot {n}}_{B}=-{\frac {2K{\sqrt {n_{A}n_{B}}}}{\hbar }}\sin \varphi ,\,{\dot {\phi }}_{B}={\frac {1}{\hbar }}(eV-K{\sqrt {\frac {n_{A}}{n_{B}}}}\cos \varphi ).}$$

Noting that the evolution of Josephson phase is ${\displaystyle {\dot {\varphi }}={\dot {\phi }}_{B}-{\dot {\phi }}_{A}}$ and the time derivative of charge carrier density ${\displaystyle {\dot {n}}_{A}}$ is proportional to current ${\displaystyle I}$, when ${\displaystyle n_{A}\approx n_{B}}$, the above solution yields the Josephson equations:^[16]

${\displaystyle I(t)=I_{c}\sin(\varphi (t))}$ (1)

${\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {2eV(t)}{\hbar }}}$ (2)

where ${\displaystyle V(t)}$ and ${\displaystyle I(t)}$ are the voltage across and the current through the Josephson junction, and ${\displaystyle I_{c}}$ 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:

$${\displaystyle K_{J}={\frac {2e}{h}}\,,}$$

and its inverse is the magnetic flux quantum:

$${\displaystyle \Phi _{0}={\frac {h}{2e}}=2\pi {\frac {\hbar }{2e}}\,.}$$

The superconducting phase evolution equation can be reexpressed as:

$${\displaystyle {\frac {\partial \varphi }{\partial t}}=2\pi [K_{J}V(t)]={\frac {2\pi }{\Phi _{0}}}V(t)\,.}$$

If we define:

$${\displaystyle \Phi =\Phi _{0}{\frac {\varphi }{2\pi }}\,,}$$

then the voltage across the junction is:

$${\displaystyle V={\frac {\Phi _{0}}{2\pi }}{\frac {\partial \varphi }{\partial t}}={\frac {d\Phi }{dt}}\,,}$$

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.

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 ${\displaystyle -I_{c}}$ and ${\displaystyle I_{c}}$.

With a fixed voltage ${\displaystyle V_{DC}}$ across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC (alternating current) with amplitude ${\displaystyle I_{c}}$ and frequency ${\displaystyle K_{J}V_{DC}}$. This means a Josephson junction can act as a perfect voltage-to-frequency converter.

Microwave radiation of a single (angular) frequency ${\displaystyle \omega }$ can induce quantized DC voltages^[17] across the Josephson junction, in which case the Josephson phase takes the form ${\displaystyle \varphi (t)=\varphi _{0}+n\omega t+a\sin(\omega t)}$, and the voltage and current across the junction will be: $${\displaystyle V(t)={\frac {\hbar }{2e}}\omega (n+a\cos(\omega t)),{\text{ and }}I(t)=I_{c}\sum _{m=-\infty }^{\infty }J_{m}(a)\sin(\varphi _{0}+(n+m)\omega t),}$$

The DC components are: $${\displaystyle V_{\text{DC}}=n{\frac {\hbar }{2e}}\omega ,{\text{ and }}I_{\text{DC}}=I_{c}J_{-n}(a)\sin \varphi _{0}.}$$

This means a Josephson junction can act like a perfect frequency-to-voltage converter,^[18] 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.^[19]

Rewrite the Josephson relations as:

${\displaystyle {\begin{aligned}{\frac {\partial I}{\partial \varphi }}&=I_{c}\cos \varphi ,\\{\frac {\partial \varphi }{\partial t}}&={\frac {2\pi }{\Phi _{0}}}V.\end{aligned}}}$

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

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

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

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

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

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

Here, ${\displaystyle L_{J}=L(0)={\frac {\Phi _{0}}{2\pi I_{c}}}}$ 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.^[20]

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

${\displaystyle I=I_{c}\sin \varphi .}$

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

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

Assume that at time ${\displaystyle t_{1}}$, the Josephson phase is ${\displaystyle \varphi _{1}}$; At a later time ${\displaystyle t_{2}}$, the Josephson phase evolved to ${\displaystyle \varphi _{2}}$. The energy increase in the junction is equal to the work done on the junction:

${\displaystyle \Delta E=\int _{1}^{2}IV\operatorname {d} \!{t}=\int _{1}^{2}I\operatorname {d} \!\Phi =\int _{\varphi _{1}}^{\varphi _{2}}I_{c}\sin \varphi \operatorname {d} \!\left(\Phi _{0}{\frac {\varphi }{2\pi }}\right)=-{\frac {\Phi _{0}I_{c}}{2\pi }}\Delta \cos \varphi \,.}$

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:

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

Here ${\displaystyle E_{J}=|E(0)|={\frac {\Phi _{0}I_{c}}{2\pi }}}$ is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by ${\displaystyle E_{J}=L_{J}I_{c}^{2}}$. An alternative but equivalent definition ${\displaystyle E(\varphi )=E_{J}(1-\cos \varphi )}$ 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,^[21][22] 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,^[23] the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel^[24] to the ideal Josephson Junction. The complete expression for the current drive ${\displaystyle I_{\text{ext}}}$ becomes:

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

where the first term is displacement current with ${\displaystyle C_{J}}$ – effective capacitance, and the third is normal current with ${\displaystyle R}$ – 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 ${\displaystyle \lambda _{J}}$ and is given by the following expression (in SI):

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

where ${\displaystyle \Phi _{0}}$ is the magnetic flux quantum, ${\displaystyle j_{c}}$ is the critical supercurrent density (A/m²), and ${\displaystyle d'}$ characterizes the inductance of the superconducting electrodes^[25]

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

where ${\displaystyle d_{I}}$ is the thickness of the Josephson barrier (usually insulator), ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ are the thicknesses of superconducting electrodes, and ${\displaystyle \lambda _{1}}$ and ${\displaystyle \lambda _{2}}$ are their London penetration depths. The Josephson penetration depth usually ranges from a few μm to several mm if the critical current density is very low.^[26]

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