In physics, the Josephson effect is a phenomenon that occurs when two
superconductors
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
are placed in proximity, with some barrier or restriction between them. It is an example of a
macroscopic quantum phenomenon
Macroscopic quantum phenomena are processes showing quantum behavior at the macroscopic scale, rather than at the atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are superfluidity and sup ...
, 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 physics quantities, 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
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defin ...
is achieved by
an array of 20,208 Josephson junctions in series.
History
The Josephson effect is named after the British physicist
Brian David Josephson
Brian David Josephson (born 4 January 1940) is a Welsh theoretical physicist and professor emeritus of physics at the University of Cambridge. Best known for his pioneering work on superconductivity and quantum tunnelling, he was awarded the Nob ...
, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.
[
] The DC Josephson effect had been seen in experiments prior to 1962, but had been attributed to "super-shorts" or breaches in the insulating barrier leading to the direct conduction of electrons between the superconductors. The first paper to claim the discovery of Josephson's effect, and to make the requisite experimental checks, was that of
Philip Anderson and John Rowell. These authors were awarded patents 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
quantum tunneling
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
. Josephson was the first to predict the tunneling of superconducting
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
s. For this work, Josephson received the
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 1973.
Applications
Types of Josephson junction include the
φ Josephson junction (of which
π Josephson junction is a special example),
long Josephson junction
In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict.
In terms of underlying model a ''short Josephson ...
, and
superconducting tunnel junction
The superconducting tunnel junction (STJ) — also known as a superconductor–insulator–superconductor tunnel junction (SIS) — is an electronic device consisting of two superconductors separated by a very thin layer of insulating material. Cu ...
. A "Dayem bridge" is a
thin-film
A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ap ...
variant of the Josephson junction in which the weak link consists of a superconducting wire with dimensions on the scale of a few
micrometre
The micrometre ( international spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American spelling), also commonly known as a micron, is a unit of length in the International System of Unit ...
s or less. The
Josephson junction count
The Josephson junction count is the number of Josephson junctions on a superconducting integrated circuit chip. Josephson junctions are active circuit elements in superconducting circuits. The Josephson junction count is a measure of circuit or d ...
of a device is used as a benchmark for its complexity. The Josephson effect has found wide usage, for example in the following areas.
SQUIDs, or superconducting quantum interference devices, are very sensitive
magnetometers that operate via the Josephson effect. They are widely used in science and engineering.
In precision
metrology, the Josephson effect provides an exactly reproducible conversion between
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
. Since the frequency is already defined precisely and practically by the
caesium standard, the Josephson effect is used, for most practical purposes, to give the standard representation of a
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defin ...
, the
Josephson voltage standard.
Single-electron transistor
A single-electron transistor (SET) is a sensitive electronic device based on the Coulomb blockade effect. In this device the electrons flow through a tunnel junction between source/drain to a quantum dot (conductive island). Moreover, the electric ...
s are often constructed of
superconducting
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
materials, allowing use to be made of the Josephson effect to achieve novel effects. The resulting device is called a "superconducting single-electron transistor".
The Josephson effect is also used for the most precise measurements of
elementary charge in terms of the Josephson constant and von Klitzing constant which is related to the
quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exh ...
.
RSFQ digital electronics is based on shunted Josephson junctions. In this case, the junction switching event is associated to the emission of one
magnetic flux quantum
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
that carries the digital information: the absence of switching is equivalent to 0, while one switching event carries a 1.
Josephson junctions are integral in
superconducting quantum computing
Superconducting quantum computing is a branch of solid state quantum computing which implements superconducting electronic circuits using superconducting qubits as artificial atoms, or quantum dots. For superconducting qubits, the two logic sta ...
as
qubits
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
such as in a
flux qubit
In quantum computing, more specifically in superconducting quantum computing, flux qubits (also known as persistent current qubits) are micrometer sized loops of superconducting metal that is interrupted by a number of Josephson junctions. These d ...
or others schemes where the phase and charge act as the
conjugate variables.
Superconducting tunnel junction detectors (STJs) may become a viable replacement for CCDs (
charge-coupled devices
A charge-coupled device (CCD) is an integrated circuit containing an array of linked, or coupled, capacitors. Under the control of an external circuit, each capacitor can transfer its electric charge to a neighboring capacitor. CCD sensors are ...
) for use in
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
and
astrophysics in a few years. These devices are effective across a wide spectrum from ultraviolet to infrared, and also in x-rays. The technology has been tried out on the
William Herschel Telescope
The William Herschel Telescope (WHT) is a optical/near-infrared reflecting telescope located at the Roque de los Muchachos Observatory on the island of La Palma in the Canary Islands, Spain. The telescope, which is named after William Hersc ...
in the
SCAM
A confidence trick is an attempt to defraud a person or group after first gaining their trust. Confidence tricks exploit victims using their credulity, naïveté, compassion, vanity, confidence, irresponsibility, and greed. Researchers hav ...
instrument.
Quiterons and similar superconducting switching devices.
Josephson effect has also been observed in superfluid helium quantum interference devices (
SHeQUIDs), the
superfluid
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
helium analog of a dc-SQUID.
The Josephson equations
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 , and superconductor B
, which can be interpreted as the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s of
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
s in the two superconductors. If the electric potential difference across the junction is
, then the energy difference between the two superconductors is
, since each Cooper pair has twice the charge of one electron. The
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
for this
two-state quantum system
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a sys ...
is therefore:
where the constant
is a characteristic of the junction. To solve the above equation, first calculate the time derivative of the order parameter in superconductor A:
and therefore the Schrödinger equation gives:
The phase difference of Ginzburg-Landau order parameters across the junction is called the Josephson phase:
The Schrödinger equation can therefore be rewritten as:
and its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
equation is:
Add the two conjugate equations together to eliminate
:
Since
, we have:
Now, subtract the two conjugate equations to eliminate
:
which gives:
Similarly, for superconductor B we can derive that:
Noting that the evolution of Josephson phase is
and the time derivative of
charge carrier density is proportional to current
, the above solution yields the Josephson equations:
where
and
are the voltage across and the current through the Josephson junction, and
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
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
is defined as:
and its inverse is the
magnetic flux quantum
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
:
The superconducting phase evolution equation can be reexpressed as:
If we define:
then the voltage across the junction is:
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.
Three main effects
There are three main effects predicted by Josephson that follow directly from the Josephson equations:
The DC Josephson effect
The DC Josephson effect is a direct current crossing the insulator in the absence of any external electromagnetic field, owing to
tunneling. This DC Josephson current is proportional to the sine of the Josephson phase (phase difference across the insulator, which stays constant over time), and may take values between
and
.
The AC Josephson effect
With a fixed voltage
across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC (
Alternating Current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
) with amplitude
and frequency
. This means a Josephson junction can act as a perfect voltage-to-frequency converter.
The inverse AC Josephson effect
Microwave radiation of a single
(angular) frequency can induce quantized DC voltages across the Josephson junction, in which case the Josephson phase takes the form
, and the voltage and current across the junction will be:
The DC components are:
This means a Josephson junction can act like a perfect frequency-to-voltage converter, which is the theoretical basis for the
Josephson voltage standard.
Josephson inductance
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.
Rewrite the Josephson relations as:
:
Now, apply the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
to calculate the time derivative of the current:
:
Rearrange the above result in the form of the
current–voltage characteristic
A current–voltage characteristic or I–V curve (current–voltage curve) is a relationship, typically represented as a chart or graph, between the electric current through a circuit, device, or material, and the corresponding voltage, or ...
of an inductor:
:
This gives the expression for the kinetic inductance as a function of the Josephson Phase:
:
Here,
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.
Josephson energy
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.
The supercurrent flowing through the junction is related to the Josephson phase by the current-phase relation (CPR):
:
The superconducting phase evolution equation is analogous to
Faraday's law:
:
Assume that at time
, the Josephson phase is
; At a later time
, the Josephson phase evolved to
. The energy increase in the junction is equal to the work done on the junction:
:
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
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
. Therefore the energy stored in a Josephson junction is a
state function, which can be defined as:
:
Here
is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by
. An alternative but equivalent definition
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 RCSJ model
The Resistively Capacitance Shunted Junction (RCSJ) model, 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
As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that ''"For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B ...
, the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel to the ideal Josephson Junction. The complete expression for the current drive
becomes:
:
where the first term is displacement current with
- effective capacitance, and the third is normal current with
- effective resistance of the junction.
Josephson penetration depth
The Josephson penetration depth characterizes the typical length on which an externally applied
magnetic field penetrates into the
long Josephson junction
In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth \lambda_J. This definition is not strict.
In terms of underlying model a ''short Josephson ...
. It is usually denoted as
and is given by the following expression (in SI):
:
where
is the
magnetic flux quantum
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
,
is the
critical supercurrent density
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
* Critical Software, a company specializing in ...
(A/m
2), and
characterizes the inductance of the superconducting electrodes
:
where
is the thickness of the Josephson barrier (usually insulator),
and
are the thicknesses of superconducting electrodes, and
and
are their
London penetration depth
In superconductors, the London penetration depth (usually denoted as \lambda or \lambda_L) characterizes the distance to which a magnetic field penetrates into a superconductor and becomes equal to e^ times that of the magnetic field at the surface ...
s. The Josephson penetration depth usually ranges from a few
μm to several mm if the critical supercurrent density is very low.
See also
*
Pi Josephson junction
*
φ Josephson junction
*
Josephson diode
A Josephson diode is an electronic device that superconducts electrical current in one direction and is resistive in the other direction. The device is a Josephson junction exhibiting a superconducting diode effect (SDE). It is an example of a ...
*
Andreev reflection
Andreev reflection (AR), named after the Russian physicist Alexander F. Andreev, is a type of particle scattering which
occurs at interfaces between a superconductor (S) and a normal state material (N). It is a charge-transfer process by which ...
*
Fractional vortices
*
Ginzburg–Landau theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenol ...
*
Macroscopic quantum phenomena
Macroscopic quantum phenomena are processes showing quantum behavior at the macroscopic scale, rather than at the atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are superfluidity and su ...
*
Macroscopic quantum self-trapping In quantum mechanics, macroscopic quantum self-trapping is when two Bose-Einstein condensates weakly linked by an energy barrier which particles can tunnel through, nevertheless end up with a higher average number of bosons on one side of the junc ...
*
Quantum computer
*
Quantum gyroscope
A quantum gyroscope is a very sensitive device to measure angular rotation based on quantum mechanical principles. The first of these was built by Richard Packard and his colleagues at the University of California, Berkeley. The extreme sensit ...
*
Rapid single flux quantum
In electronics, rapid single flux quantum (RSFQ) is a digital electronic device that uses superconducting devices, namely Josephson junctions, to process digital signals. In RSFQ logic, information is stored in the form of magnetic flux quanta ...
(RSFQ)
*
Semifluxon
*
Zero-point energy
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty pri ...
*
Josephson vortex
In superconductivity, a Josephson vortex (after Brian Josephson from Cambridge University) is a quantum vortex of supercurrents in a Josephson junction (see Josephson effect). The supercurrents circulate around the vortex center which is situated ...
References
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Condensed matter physics
Superconductivity
Sensors
Mesoscopic physics
Energy (physics)