Kinetic inductance is the manifestation of the inertial mass of mobile
charge carriers
In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term is used ...
in alternating electric fields as an equivalent series
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
. Kinetic inductance is observed in high carrier mobility conductors (e.g.
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 ...
) and at very high frequencies.
Explanation
A change in
electromotive force
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
(emf) will be opposed by the
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
of the charge carriers since, like all objects with mass, they prefer to be traveling at constant velocity and therefore it takes a finite time to accelerate the particle. This is similar to how a change in emf is opposed by the finite rate of change of magnetic flux in an inductor. The resulting phase lag in voltage is identical for both energy storage mechanisms, making them indistinguishable in a normal circuit.
Kinetic inductance (
) arises naturally in the
Drude model
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltage ...
of
electrical conduction
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
considering not only the DC conductivity but also the finite relaxation time (collision time)
of the mobile charge carriers when it is not tiny compared to the wave period 1/f. This model defines a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
conductance at radian frequency ω=2πf given by
. The imaginary part, -σ
2, represents the kinetic inductance. The Drude complex conductivity can be expanded into its real and imaginary components:
where
is the mass of the charge carrier (i.e. the effective
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
mass in metallic
conductors) and
is the carrier number density. In normal metals the collision time is typically
s, so for frequencies < 100 GHz
is very small and can be ignored; then this equation reduces to the DC conductance
. Kinetic inductance is therefore only significant at optical frequencies, and in superconductors whose
.
For a superconducting wire of cross-sectional area
, the kinetic inductance of a segment of length
can be calculated by equating the total kinetic energy of the
Cooper pairs
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 ...
in that region with an equivalent inductive energy due to the wire's current
:
where
is the electron mass (
is the mass of a Cooper pair),
is the average Cooper pair velocity,
is the density of Cooper pairs,
is the length of the wire,
is the wire cross-sectional area, and
is the current. Using the fact that the current
, where
is the electron charge, this yields:
The same procedure can be used to calculate the kinetic inductance of a normal (i.e. non-superconducting) wire, except with
replaced by
,
replaced by
, and
replaced by the normal carrier density
. This yields:
The kinetic inductance increases as the carrier density decreases. Physically, this is because a smaller number of carriers must have a proportionally greater velocity than a larger number of carriers in order to produce the same current, whereas their energy increases according to the ''square'' of velocity. The
resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
also increases as the carrier density
decreases, thereby maintaining a constant ratio (and thus phase angle) between the (kinetic) inductive and resistive components of a wire's
impedance for a given frequency. That ratio,
, is tiny in normal metals up to
terahertz frequencies.
Applications
Kinetic inductance is the principle of operation of the highly sensitive
photodetector
Photodetectors, also called photosensors, are sensors of light or other electromagnetic radiation. There is a wide variety of photodetectors which may be classified by mechanism of detection, such as Photoelectric effect, photoelectric or photoc ...
s known as
kinetic inductance detectors (KIDs). The change in the
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 Cooper ...
density brought about by the absorption of a single
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
in a strip of superconducting material produces a measurable change in its kinetic inductance.
Kinetic inductance is also used in a design parameter for superconducting
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 ...
s:
is the ratio of the
kinetic inductance of the
Josephson junctions
In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mech ...
in the qubit to the geometrical inductance of the flux qubit. A design with a low beta behaves more like a simple inductive loop, while a design with a high beta is dominated by the Josephson junctions and has more
hysteretic
Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...
behavior.
[https://books.google.com/books?id=yOA8rUo5N4oC&pg=PA157 or ]
See also
*
Drude model
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltage ...
*
Electrical conduction
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
*
Electron mobility
In solid-state physics, the electron mobility characterises how quickly an electron can move through a metal or semiconductor when pulled by an electric field. There is an analogous quantity for holes, called hole mobility. The term carrier mobili ...
*
Inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
*
Superconductivity
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 ...
References
{{reflist
External links
YouTube video on kinetic inductance from MIT
Electrodynamics
Superconductivity