Kinetic inductance

Kinetic inductance is the manifestation of the inertial mass of mobile charge carriers in alternating electric fields as an equivalent series inductance. Kinetic inductance is observed in high carrier mobility conductors (e.g. superconductors) and at very high frequencies.
Explanation.
A change in electromotive force (emf) will be opposed by the inertia 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 (formula_1) arises naturally in the Drude model of electrical conduction considering not only the DC conductivity but also the finite relaxation time (collision time) formula_2 of the mobile charge carriers when it is not tiny compared to the wave period 1/f. This model defines a complex conductance at radian frequency ω=2πf given by formula_3. The imaginary part, -σ2, represents the kinetic inductance. The Drude complex conductivity can be expanded into its real and imaginary components:
formula_4
where formula_5 is the mass of the charge carrier (i.e. the effective electron mass in metallic conductors) and formula_6 is the carrier number density. In normal metals the collision time is typically formula_7 s, so for frequencies < 100 GHz formula_8 is very small and can be ignored; then this equation reduces to the DC conductance formula_9. Kinetic inductance is therefore only significant at optical frequencies, and in superconductors whose formula_10.
For a superconducting wire of cross-sectional area formula_11, the kinetic inductance of a segment of length formula_12 can be calculated by equating the total kinetic energy of the Cooper pairs in that region with an equivalent inductive energy due to the wire's current formula_13:
formula_14
where formula_15 is the electron mass (formula_16 is the mass of a Cooper pair), formula_17 is the average Cooper pair velocity, formula_18 is the density of Cooper pairs, formula_12 is the length of the wire, formula_11 is the wire cross-sectional area, and formula_13 is the current. Using the fact that the current formula_22, where formula_23 is the electron charge, this yields:
formula_24
The same procedure can be used to calculate the kinetic inductance of a normal (i.e. non-superconducting) wire, except with formula_16 replaced by formula_15, formula_27 replaced by formula_23, and formula_18 replaced by the normal carrier density formula_6. This yields:
formula_31
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 also increases as the carrier density formula_6 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, formula_33, is tiny in normal metals up to terahertz frequencies.
Applications.
Kinetic inductance is the principle of operation of the highly sensitive photodetectors known as kinetic inductance detectors (KIDs). The change in the Cooper pair density brought about by the absorption of a single photon 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 qubits: formula_34 is the ratio of the kinetic inductance of the Josephson junctions 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 behavior.
Graphene inductors were predicted to allow significant miniaturization in radio-frequency electronics applications.
History.
In January 2018, a team led by Kaustav Banerjee at University of California, Santa Barbara demonstrated on-chip spiral inductors based on intercalated multilayer graphene which exploit kinetic inductance at room temperature, intended for frequencies in the 10-50 GHz range. In these microscopic coils, kinetic inductance increases the net inductance by up to 50%. Since this does not increase the coil's resistance, its Q is similarly increased, achieving Q factors typically of 12.