Kinetic Inductance: The Inertia of Electrons in Circuits

Most people who have studied electronics know inductance as a property of coils and loops: when current flows, it creates a magnetic field, and that field stores energy and resists changes in the current. There is a second, less familiar kind of inductance that has nothing to do with magnetic fields. It comes from the fact that the electrons carrying a current have mass. Accelerating them takes energy and takes time, and that behavior looks, to the rest of the circuit, exactly like an inductor.

This effect is called kinetic inductance. In ordinary copper wiring at everyday frequencies it is so small that it can be ignored. In superconductors, very thin films, two-dimensional materials such as graphene, and at optical frequencies, it can dominate. Kinetic inductance is the working principle of some of the most sensitive light detectors ever built, including detectors used in astronomical cameras and single-photon counters used in quantum communication. It is also a design tool for superconducting quantum circuits and a newer class of low-noise amplifiers.

This article explains what kinetic inductance is, where it comes from physically, how to calculate it, how it was discovered and measured, the technologies that depend on it, its limits, and some common points of confusion.

The basic idea

Wikipedia defines kinetic inductance as "the manifestation of the inertial mass of mobile charge carriers in alternating electric fields as an equivalent series inductance." Unpacking that definition:

• Inertial mass. Electrons, and the electron pairs that carry current in superconductors, have mass. An object with mass resists changes in its velocity.
• Mobile charge carriers. Current is the motion of these charged particles. To change the current, you must change their velocity.
• Alternating fields. When a voltage changes, it takes a finite time for the carriers to speed up or slow down. The current therefore lags the voltage.
• Equivalent series inductance. A conventional inductor also makes current lag voltage, because changing the current requires changing the magnetic flux, which takes time. The two effects produce the same phase lag and are indistinguishable from outside the circuit.

The total inductance of a conductor is the sum of the two contributions:

L_total = L_geometric + L_kinetic

The geometric (or magnetic) inductance depends on the shape of the conductor and its surroundings. The kinetic inductance depends on the properties of the charge carriers: how many there are, how heavy they are, and how they scatter.

An analogy

Imagine water flowing through a pipe. If you suddenly increase the pressure at one end, the water does not instantly flow faster, because the water itself has mass and must be accelerated. A long, narrow pipe holding a large mass of fast-moving water resists changes more than a short, wide one. In this analogy, kinetic inductance is the inertia of the water itself, while geometric inductance is more like the energy stored in a flywheel attached to the flow. Both resist changes in flow, but for different reasons.

Why it is usually invisible: the Drude model

In normal metals, electrons do not accelerate freely for long. They constantly collide with lattice vibrations, impurities and defects, losing their momentum. The average time between collisions is called the scattering time, τ.

The Drude model, a classical description of electrons in metals, gives the conductivity at angular frequency ω (where ω = 2πf) as a complex number:

σ(ω) = ne²τ / [m(1 + iωτ)]

Here n is the number of charge carriers per unit volume, e is the electron charge and m is the carrier mass (the effective mass in a metal). Separating real and imaginary parts:

σ = (ne²τ/m) × [1/(1 + ω²τ²) − i ωτ/(1 + ω²τ²)]

The real part describes ordinary resistive conduction. The imaginary part describes the kinetic inductance. The ratio between them is ωτ.

In normal metals, Wikipedia gives a scattering time of about 10⁻¹⁴ seconds. At frequencies below 100 GHz, ωτ is therefore very small, and the conductivity reduces to the familiar direct-current value σ₀ = ne²τ/m. The kinetic inductance becomes significant only when ωτ approaches 1, which happens at optical frequencies, or when τ becomes effectively infinite, which is what happens in superconductors.

A worked check (hypothetical numbers)

At 100 GHz, ω = 2π × 10¹¹ ≈ 6.28 × 10¹¹ radians per second. With τ = 10⁻¹⁴ s:

ωτ = 6.28 × 10¹¹ × 10⁻¹⁴ ≈ 0.0063

The kinetic inductive impedance is less than 1% of the resistance. At an optical frequency of 500 THz, ω ≈ 3.14 × 10¹⁵ rad/s and ωτ ≈ 31, so the inductive term dominates. This is one reason metals behave so differently in visible light than in a radio circuit.

A neat result for normal wires

For a normal wire of length l and cross-sectional area A, the kinetic inductance is:

L_K = (m / ne²) × (l / A)

The direct-current resistance of the same wire, using σ₀ = ne²τ/m, is:

R = (m / ne²τ) × (l / A)

Dividing one by the other gives a simple relationship:

L_K = R × τ

In other words, the kinetic inductance of a normal wire equals its resistance multiplied by the scattering time. A hypothetical 1-ohm copper wire with τ = 10⁻¹⁴ s has a kinetic inductance of about 10⁻¹⁴ henries, or 10 femtohenries, which is negligible next to the geometric inductance of any practical wire. This also explains a point Wikipedia makes: reducing the carrier density increases both kinetic inductance and resistance, so their ratio, ωτ, stays the same, and in normal metals it stays tiny up to terahertz frequencies.

Kinetic inductance in superconductors

In a superconductor, the picture changes completely. Below its critical temperature, a superconductor carries current through Cooper pairs: pairs of electrons bound together that move through the lattice without scattering. With no scattering, τ is effectively infinite, there is no resistance, and nothing stops the kinetic inductance from mattering.

Deriving the formula

Wikipedia derives the superconducting kinetic inductance by equating the kinetic energy of the moving Cooper pairs with the energy stored in an inductor. For a wire of length l and cross-section A, containing Cooper pairs of mass 2m_e at density n_s moving with average velocity v:

½ × (2m_e v²) × (n_s l A) = ½ L_K I²

The current is I = 2e v n_s A, since each pair carries charge 2e. Solving gives:

L_K = (m_e / 2n_s e²) × (l / A)

The structure of this formula reveals the key physics:

• Fewer carriers means more kinetic inductance. To carry the same current, a smaller number of Cooper pairs must move faster. Kinetic energy goes up with the square of velocity, so the stored energy, and hence the inductance, rises.
• Long, thin conductors have more kinetic inductance. The l/A factor is the same geometric factor that governs resistance.
• Anything that reduces the Cooper-pair density increases L_K. This includes raising the temperature toward the critical temperature, applying a large current, or breaking pairs by absorbing photons. This sensitivity is what makes kinetic inductance useful for detectors.

Connection to the London penetration depth

The same physics appears in the London theory of superconductivity. Wikipedia credits Geertruida de Haas-Lorentz with first deriving the magnetic penetration depth in 1925, with Fritz and Heinz London later deriving it within their London equations in 1935. The London penetration depth is:

λ_L = √(m / μ₀ n q²)

where μ₀ is the magnetic constant. Typical values range from 50 to 500 nanometres. Since λ_L² is proportional to m/n, it is directly related to the kinetic inductance. For a thin film much thinner than the penetration depth, the kinetic inductance per square of film scales as λ_L²/t, where t is the thickness. In practical units, the sheet kinetic inductance is often written as:

L□ ≈ μ₀ λ² / t

Sheet inductance: "per square"

Thin-film engineers describe kinetic inductance per square (written pH/□ or "picohenries per square"), just as they describe sheet resistance in ohms per square. A strip of film that is ten times longer than it is wide contains ten squares and has ten times the sheet inductance. This makes it easy to design inductors by drawing long, narrow meanders.

Worked example (hypothetical). A film with λ = 300 nm and thickness t = 10 nm has:

L□ = (1.257 × 10⁻⁶ H/m) × (300 × 10⁻⁹ m)² / (10 × 10⁻⁹ m) ≈ 1.13 × 10⁻¹¹ H ≈ 11 pH/□

A meander that is 1,000 squares long would have a kinetic inductance of about 11 nH.

Checking a published case. A study of epitaxial niobium nitride films described a 4 nm film with a penetration depth of about 500 nm. For a stripline 100 nm wide with an effective length of 500 µm, the authors calculated a kinetic inductance of about 392 nH. Using the formula: λ²/t = (500 × 10⁻⁹)² / (4 × 10⁻⁹) = 6.25 × 10⁻⁵ m. Multiplying by μ₀ gives about 78.5 pH per square. The strip contains 500 µm ÷ 100 nm = 5,000 squares, so the total is 5,000 × 78.5 pH ≈ 393 nH, matching the published figure.

The Mattis–Bardeen route: from resistance to inductance

There is a convenient practical formula that links kinetic inductance to quantities that are easy to measure. It comes from the microscopic theory of superconductors. In 1958, D. C. Mattis and John Bardeen published a theory of the electrodynamic response of normal and superconducting metals, describing how superconductors absorb and respond to electromagnetic radiation. At low temperatures and low frequencies, it leads to a widely used relation for thin films:

L□ = ħ R□ / (π Δ₀)

Here ħ is the reduced Planck constant, R□ is the sheet resistance just above the critical temperature, and Δ₀ is the superconducting energy gap at zero temperature. In the BCS theory of superconductivity, Δ₀ = 1.76 k_B T_c, where k_B is the Boltzmann constant and T_c is the critical temperature.

This formula tells engineers something useful: films with high normal-state resistance and low critical temperature have high kinetic inductance. That is why disordered, thin films of materials such as titanium nitride, niobium nitride and niobium titanium nitride are favored when large kinetic inductance is wanted.

Worked check using published data. A 2018 study of titanium nitride (TiN) films grown by atomic layer deposition reported a film 8.9 nm thick with a resistivity of 449 µΩ·cm and a critical temperature of 3.01 K, and a measured sheet inductance of 234 pH/□. Using the formula:

• Sheet resistance: R□ = ρ/t = (4.49 × 10⁻⁶ Ω·m) / (8.9 × 10⁻⁹ m) ≈ 504 Ω
• Energy gap: Δ₀ = 1.76 × (1.381 × 10⁻²³ J/K) × 3.01 K ≈ 7.31 × 10⁻²³ J
• Sheet inductance: L□ = (1.055 × 10⁻³⁴ J·s × 504 Ω) / (π × 7.31 × 10⁻²³ J) ≈ 2.32 × 10⁻¹⁰ H ≈ 232 pH/□

The result agrees with the reported 234 pH/□ to within about 1%.

How film thickness changes things

The same TiN study reported how sheet inductance rose as the films got thinner:

From the thickest to the thinnest film, the sheet inductance increased more than a hundredfold. Three things change together: the films get thinner (fewer squares of cross-section), their resistivity rises (more disorder), and their critical temperature drops (smaller gap). All three push kinetic inductance up. Films around 5 nm thick did not become superconducting at all, which marks a practical floor. The study also found that the thinnest films had internal quality factors around 100,000, while films thicker than 14 nm exceeded 1 million at single-photon power, showing a trade-off between high inductance and low loss.

Two-dimensional conductors and graphene

Kinetic inductance is especially pronounced in two-dimensional conductors, where electrons are confined to a sheet. For a 2D conductor with electron density n per unit area, width W and length l, Wikipedia gives:

L_K = (m_e / ne²) × (l / W)

Graphene is a special case. Its electrons behave as if they have no individual mass, because their energy rises linearly with momentum rather than quadratically. Even so, the collective electron system still has inertia. Wikipedia gives the kinetic inductance of graphene as:

L_K = (πħ² / e² ε_F) × (l / W)

where ε_F is the Fermi energy. A 2014 study by Yoon and colleagues measured this "collective dynamical mass" of the Dirac fermions in graphene. Because the Fermi energy can be tuned by applying a gate voltage, graphene's kinetic inductance can be adjusted electrically, which is of interest for compact high-frequency circuits.

A brief history

Hertz's 1880 experiment is a nice illustration of how small the effect is in ordinary metals. He was looking for evidence that electric current had inertia, and found only an upper bound. Kinetic inductance became measurable and useful only once superconductors and, later, thin-film fabrication made it large.

Kinetic inductance detectors

The most important application of kinetic inductance is in detecting light, from millimetre waves to X-rays.

How a kinetic inductance detector works

A kinetic inductance detector (KID), also called a microwave kinetic inductance detector (MKID), consists of a superconducting strip that forms part of a microwave resonator together with a capacitor. The sequence of events is:

1. A photon is absorbed in the superconducting strip.
2. If its energy exceeds twice the superconducting gap, it breaks Cooper pairs, creating excess unpaired electrons called quasiparticles.
3. With fewer Cooper pairs, the kinetic inductance increases, following L_K ∝ 1/n_s.
4. The resonator's frequency, f = 1/(2π√(LC)), shifts downward, and its losses rise.
5. A microwave probe tone at the resonant frequency picks up a change in phase and amplitude, which is measured.

Wikipedia's article on kinetic inductance detectors notes that the phase shift scales with the energy of the absorbed photon, which lets the detector measure photon energies as well as count photons. These detectors operate at cryogenic temperatures, typically below 1 kelvin.

A key parameter is the kinetic inductance fraction, often written α: the ratio of kinetic inductance to total inductance. Jonas Zmuidzinas's 2012 review of superconducting microresonators in the Annual Review of Condensed Matter Physics describes how α, together with the volume of the inductor and the resonator quality factor, sets the magnitude of the detector's response. A large α means a given number of broken pairs produces a larger frequency shift.

The multiplexing advantage

The breakthrough that made KIDs attractive was not sensitivity alone but frequency-division multiplexing. The 2003 Nature paper by Day and colleagues was titled "A broadband superconducting detector suitable for use in large arrays." Because each resonator can be designed with a slightly different resonant frequency, hundreds or thousands of detectors can be connected to a single microwave transmission line. One broadband microwave signal containing many tones reads them all at once.

A 2002 conference paper by the same group estimated that passive frequency multiplexing could allow roughly 10,000 detectors to be read out with a single cryogenic transistor amplifier. Competing detector technologies, such as transition-edge sensors, require more complex readout electronics for each pixel. Wikipedia's KID article describes KIDs as a more compact, lower-cost and less complex alternative.

Worked example: tuning a resonator array (hypothetical)

Suppose a resonator uses a TiN meander like the 8.9 nm film above, 1,000 squares long, giving a kinetic inductance of about 234 nH, paired with a 1 pF capacitor. Ignoring geometric inductance:

f = 1 / (2π × √(234 × 10⁻⁹ H × 1 × 10⁻¹² F)) = 1 / (2π × 4.84 × 10⁻¹⁰ s) ≈ 329 MHz

If photon absorption increased the inductance by 0.1%, the frequency would drop by about 0.05%, because frequency scales as one over the square root of inductance. That is about 0.16 MHz. Designers place neighboring resonators far enough apart in frequency that such shifts do not cause them to overlap, then pack as many as the available bandwidth allows.

Astronomical instruments

KIDs have been deployed at many observatories. Wikipedia's article lists, among others, the Caltech Submillimeter Observatory, the Atacama Pathfinder Experiment (APEX), the IRAM 30-metre telescope, the Large Millimeter Telescope, Palomar Observatory for optical and near-infrared work, and the balloon-borne experiments OLIMPO (which flew in 2018) and BLAST-TNG (which flew in 2020). They are also planned for future instruments, including the PRobe far-Infrared Mission for Astrophysics (PRIMA) concept.

In optical and near-infrared astronomy, MKIDs can record the arrival time and approximate energy of each photon, producing a kind of color video with no filters. Wikipedia notes that the spectral resolution in this mode is fundamentally limited to roughly R ≈ 100, meaning the detector can distinguish wavelengths differing by about 1%, which is coarse compared with a spectrograph but remarkable for an imaging detector.

Superconducting nanowire single-photon detectors

A second major detector technology also depends on kinetic inductance, though in a different way. Superconducting nanowire single-photon detectors (SNSPDs) were developed in 2001 by scientists at Moscow State Pedagogical University and the University of Rochester. The first fully operational prototype system was demonstrated in 2005 by the National Institute of Standards and Technology in Boulder and BBN Technologies, as part of the DARPA Quantum Network.

How they work

An SNSPD is a long, narrow superconducting wire, usually made of niobium nitride (NbN), biased with a current just below the maximum it can carry while superconducting. When a photon is absorbed, it creates a small resistive hotspot. Current diverts around it, exceeding the critical current in the surrounding region, and a resistive section forms across the wire. This produces a voltage pulse that signals a detection. The wire then cools and returns to the superconducting state.

According to Wikipedia, NbN has a critical temperature of about 10 K, and SNSPDs typically operate between 1 K and 4 K. Lower-critical-temperature materials such as tungsten silicide (WSi), molybdenum silicide (MoSi) and niobium titanium nitride (NbTiN) are also used.

The role of kinetic inductance

In an SNSPD, kinetic inductance controls how quickly the detector recovers. After a detection, current must flow back into the nanowire. Because the long, thin wire has a large kinetic inductance L_k, and the readout circuit has a resistance R (often 50 ohms), the current returns with a time constant:

τ = L_k / R

Wikipedia notes that the detector's dead time is on the order of a few nanoseconds. The time constant must not be too short, however. If current returns faster than the hotspot can cool, the wire stays resistive and the detector "latches" and stops responding. Wikipedia states that the NbN hotspot cools in under 100 picoseconds, so the inductive time constant must be slower than that. Kinetic inductance therefore acts both as a speed limit and as a safety brake.

Worked example (hypothetical). A nanowire with 500 nH of kinetic inductance read out through 50 Ω has τ = 500 × 10⁻⁹ / 50 = 10 ns. Reducing the wire length to cut the inductance to 100 nH gives τ = 2 ns, a faster detector, but with less margin against latching.

Performance

Wikipedia reports that fibre-coupled SNSPDs in optical cavities have reached detection efficiencies greater than 98% at 1,550 nm, the telecommunications wavelength. Timing jitter can be as low as 3 picoseconds at visible wavelengths, count rates reach the tens of megahertz, and dark count rates are extremely low. A 400,000-pixel array has been reported. Applications include quantum key distribution, optical quantum computing, LIDAR, deep-space optical communication, dark matter searches and exoplanet detection.

Kinetic inductance in quantum circuits

Josephson junctions as nonlinear inductors

Superconducting qubits are built from Josephson junctions: two superconductors separated by a very thin barrier. Brian Josephson predicted in 1962 that a supercurrent can tunnel through such a barrier. A Josephson junction behaves like a nonlinear inductor with a characteristic Josephson inductance:

L_J = Φ₀ / (2π I_c)

where Φ₀ = h/2e is the magnetic flux quantum (about 2.07 × 10⁻¹⁵ webers) and I_c is the junction's critical current. This inductance is kinetic in origin: it reflects the energy stored in the motion of the superconducting condensate across the junction, not a magnetic field.

Worked example (hypothetical). A junction with a critical current of 10 nA has L_J = 2.07 × 10⁻¹⁵ / (2π × 10 × 10⁻⁹) ≈ 33 nH. Small junctions can provide large inductances in a tiny footprint.

The β parameter in flux qubits

In superconducting flux qubits, a design parameter called β compares the Josephson (kinetic) inductance of the junctions to the geometrical inductance of the qubit loop, as Wikipedia describes. A qubit with low β behaves much like a simple inductive loop, while one with high β is dominated by the junctions and shows more hysteretic behavior.

Superinductors and fluxonium

Some qubit designs need an inductance so large that its impedance exceeds the superconducting resistance quantum h/4e² (about 6.45 kΩ) at the operating frequency, while losing almost no energy. Such elements are called superinductors. Geometric coils cannot achieve this in a small space; kinetic inductance can.

The fluxonium qubit, proposed in 2009 by a Yale group led by Michel Devoret, uses a superinductor made from an array of Josephson junctions. Fluxonium operates at low frequencies of roughly 0.1 to 1 GHz, compared with 4 to 5 GHz for the more common transmon qubit, and has achieved relaxation times above 1 millisecond and single-qubit gate fidelities up to 99.998%. Thin, disordered films such as the high-inductance TiN described above are another route to superinductors; the TiN study listed superinductors in protected qubits among its target applications, noting characteristic impedances up to 28 kΩ in its thinnest film.

Nonlinear kinetic inductance and parametric amplifiers

Kinetic inductance is not perfectly constant. When a large current flows, it reduces the density of Cooper pairs slightly, which increases the inductance. To a first approximation, the inductance depends on the square of the current:

L(I) ≈ L₀ [1 + (I/I*)²]

where I* is a scale current set by the material and geometry. This nonlinearity is usually a nuisance in detectors, because strong readout signals shift and distort the resonance. Zmuidzinas's review identifies the build-up of nonlinear kinetic inductance in higher-resistivity superconductors as one of the factors limiting readout power.

The same nonlinearity can also be exploited. In 2012, Byeong Ho Eom, Peter Day, Henry LeDuc and Jonas Zmuidzinas described a superconducting parametric amplifier that uses the nonlinear kinetic inductance of a superconducting transmission line to transfer power from a strong pump tone to a weak signal. The device is simply a patterned metal film on a dielectric substrate. Their paper reported gain extending over 2 GHz on either side of an 11.56 GHz pump tone and an upper limit on added noise of 3.4 photons at 9.4 GHz, with a dynamic range comparable to microwave transistor amplifiers.

This addressed a long-standing trade-off: transistor amplifiers have wide bandwidth and high dynamic range but noise far above the quantum limit, while Josephson parametric amplifiers approach the quantum limit but are usually narrowband with limited dynamic range. Kinetic inductance travelling-wave amplifiers aim for the best of both and are of interest for reading out large detector arrays and many qubits at once.

Other applications

The TiN study and the wider literature list several more uses for high kinetic inductance:

• Superconducting phase shifters, using the current dependence of kinetic inductance to tune signal delay.
• Ultra-sensitive current sensors, exploiting the same nonlinearity.
• Tunable nanoinductors, demonstrated by Annunziata and colleagues in 2010, which use current to adjust inductance in very small structures.
• Hybrid quantum systems, in which high-impedance resonators couple strongly to individual electron spins or other quantum objects, because a high-inductance resonator concentrates large voltage fluctuations.
• Compact resonators and filters, since large kinetic inductance allows long transmission line lengths to be replaced by short ones.

Materials comparison

Design considerations and limits

Loss and noise

High kinetic inductance usually comes from thin, disordered films, and such films tend to have more loss. The TiN data show quality factors falling from more than 1 million in thicker films to about 100,000 in the thinnest. Loss reduces detector sensitivity and qubit coherence.

Temperature sensitivity

Because kinetic inductance depends on the Cooper-pair density, it changes with temperature as thermally excited quasiparticles appear. This is useful for thermometry but means that devices must be held at very stable cryogenic temperatures. KIDs typically run below 1 kelvin.

Quasiparticle lifetime

In a KID, the response to a photon lasts as long as the excess quasiparticles survive before recombining into Cooper pairs. This quasiparticle lifetime sets how long the signal persists and influences sensitivity. It is a material property and is affected by impurities and stray radiation.

Readout power limits

Stronger readout tones improve the signal-to-noise ratio but eventually cause problems: they can break Cooper pairs (generating quasiparticles in materials like aluminum) or drive the resonator into its nonlinear regime (in high-resistivity films). Choosing the readout power is a balancing act.

Thin-film variability

The thinnest films are near the limit of superconductivity. Small variations in thickness or composition can shift the critical temperature and kinetic inductance across a wafer, causing resonator frequencies to land in the wrong place in a multiplexed array. In SNSPDs, Wikipedia notes that local constrictions in the wire can cause efficiency to vary widely across a device.

Surface oxidation

Thin nitride films can change over time. The TiN study observed slow growth of a 5 to 8 nm oxide layer over months, which matters when the entire film is only about 9 nm thick.

Kinetic and geometric inductance side by side

Because the two kinds of inductance add together and look the same to a circuit, it helps to compare them directly.

The last two rows explain why kinetic inductance was a curiosity for most of the history of electrical engineering and became a central design parameter only with the rise of thin-film superconducting devices.

How kinetic inductance is measured in practice

Because kinetic and geometric inductance are indistinguishable at a single frequency and temperature, measurements rely on changing something that affects only the kinetic part.

Temperature sweeps. The most common approach is to build a superconducting resonator and measure its resonant frequency as the temperature rises toward the critical temperature. Thermally excited quasiparticles reduce the Cooper-pair density, raising the kinetic inductance and lowering the frequency. The geometric inductance does not change. Fitting the frequency shift against temperature, using Mattis–Bardeen theory, gives both the kinetic inductance fraction and the energy gap. Later papers citing Zmuidzinas's review describe obtaining α and the zero-temperature gap Δ₀ by fitting measured resonator data in this way.

Comparison with simulation. Electromagnetic simulation software can calculate the geometric inductance of a structure accurately. Comparing the simulated resonant frequency with the measured one reveals how much extra inductance the superconductor contributes.

From normal-state resistance. For a quick estimate, engineers measure the sheet resistance just above the critical temperature and the critical temperature itself, then apply L□ = ħR□/(πΔ₀) with Δ₀ = 1.76 k_B T_c. The TiN worked example earlier shows how well this can match a direct measurement.

Current dependence. Measuring how the resonant frequency shifts as a direct current is applied reveals the nonlinear part of the kinetic inductance, which matters for amplifiers and for setting readout power in detectors.

These methods give a practical rule for anyone designing superconducting circuits: always budget for kinetic inductance when predicting resonant frequencies in thin films, because in a high-inductance film the kinetic part can be many times the geometric part, and ignoring it would put the resonance far from where a purely geometric model predicts.

Common misconceptions

"Kinetic inductance is a magnetic effect"

It is not. Kinetic inductance stores energy in the motion of charge carriers, not in a magnetic field. It appears in the same place in circuit equations as magnetic inductance and causes the same phase lag, which is why the two can be added together, but the physics is different. A straight superconducting nanowire with almost no geometric inductance can have hundreds of nanohenries of kinetic inductance.

"Superconductors have zero impedance"

Superconductors have zero resistance to direct current. At any nonzero frequency, their kinetic inductance gives them a reactive impedance, and at finite temperature a small amount of loss as well. This is exactly what KIDs measure.

"Kinetic inductance only matters in superconductors"

It matters wherever ωτ is not small. In normal metals at optical frequencies, kinetic inductance contributes to how metals reflect and absorb light, and it is significant in nanoscale optical structures. It is also prominent in two-dimensional conductors.

"More carriers mean more inductance"

The opposite is true. Fewer carriers must move faster to carry the same current, storing more kinetic energy. Materials with low carrier density, such as disordered nitride films, have high kinetic inductance.

Frequently asked questions

How big is kinetic inductance in a normal copper wire?

Extremely small. Using L_K = R × τ, a 1-ohm wire with a scattering time of 10⁻¹⁴ seconds has a kinetic inductance of about 10 femtohenries. That is negligible next to the geometric inductance of any practical wire.

Why do detectors use thin films?

Kinetic inductance per square scales as λ²/t, so thinner films have more of it. Thin films also have higher sheet resistance and often lower critical temperatures, both of which raise kinetic inductance through the relation L□ = ħR□/(πΔ₀). A larger kinetic inductance fraction makes the detector's frequency more sensitive to broken Cooper pairs.

What is the difference between a KID and an SNSPD?

A KID is a resonator whose frequency shifts slightly when photons break Cooper pairs; the superconductor stays superconducting throughout. An SNSPD is a current-biased nanowire that briefly switches to a resistive state when a photon arrives, producing a voltage pulse. In KIDs, kinetic inductance is the sensing mechanism. In SNSPDs, it mainly sets the recovery time.

Can kinetic inductance be measured directly?

It is usually measured by building a resonator and comparing its resonant frequency with what the geometric inductance alone would predict, or by tracking how the resonant frequency changes with temperature. The difference reveals the kinetic contribution.

Is kinetic inductance useful outside cryogenics?

The superconducting applications all require cryogenic temperatures. At room temperature, kinetic inductance is relevant in plasmonics and terahertz devices, and in two-dimensional materials such as graphene, where it can be tuned electrically.

Why does the kinetic inductance increase with current?

A large current reduces the density of Cooper pairs, as some of the condensate's energy is used up by the flow. Fewer pairs mean higher kinetic inductance. This nonlinearity is the basis of kinetic inductance parametric amplifiers.

Summary

Kinetic inductance is the electrical signature of the mass of moving charge. In a normal wire at ordinary frequencies, collisions erase it almost entirely, leaving a contribution of femtohenries. Remove the collisions, as superconductivity does, or squeeze the carriers into very thin films and two-dimensional sheets, and kinetic inductance can become the dominant inductance in a circuit.

That transformation has made kinetic inductance a working tool across modern physics. It powers the resonators in kinetic inductance detectors used on telescopes around the world, sets the recovery speed of superconducting nanowire single-photon detectors that reach efficiencies above 98%, provides the superinductors in long-lived fluxonium qubits, and enables wideband parametric amplifiers. The same few equations, L_K ∝ (m/ne²)(l/A) and L□ = ħR□/(πΔ₀), explain all of them, and they show why designers seek out thin, disordered superconductors with few carriers and small energy gaps.

Sources

• Kinetic inductance (Wikipedia)
• Kinetic inductance detector (Wikipedia)
• Superconducting nanowire single-photon detector (Wikipedia)
• London penetration depth (Wikipedia)
• Mattis–Bardeen theory (Wikipedia)
• Josephson effect (Wikipedia)
• Fluxonium (Wikipedia)
• Shearrow et al. (2018), Atomic layer deposition of titanium nitride for quantum circuits (arXiv:1808.06009)
• Penetration depth and tunneling studies in very thin epitaxial NbN films (arXiv:0911.4181)
• Eom, Day, LeDuc and Zmuidzinas (2012), A Wideband, Low-Noise Superconducting Amplifier with High Dynamic Range (arXiv:1201.2392)
• Zmuidzinas (2012), Superconducting Microresonators: Physics and Applications, Annual Review of Condensed Matter Physics (Caltech record)
• Day et al. (2003), A broadband superconducting detector suitable for use in large arrays, Nature 425, 817–821
• Mazin (2004), Microwave Kinetic Inductance Detectors, PhD thesis, Caltech
• Day et al. (2002), Superconducting kinetic inductance photon detectors, Proc. SPIE 4849