# Kinetic Inductance

## Origin of the Kinetic Inductance

Kinetic inductance originates in the kinetic energy required by each electron that is contributing to a flow of current. In general the electrons in a solid are moving around continuously, evenly distributed amongst all the possible directions in the crystal. Thus they all posess kinetic energy even when no current flows.
When a current flows the electric field adds a small drift velocity component to the whole electon distribution which requries the electron system to acquire kinetic energy: (click for full size image)

In a normal or superconducting material this kinetic energy is equvalent mathematically to the energy invested in creating a magnetic field - the energy is effectively stored until the electrons decelerate again. This is often neglected in normal materials as their resistance requires that energy continually be applied to sustain the current due to charge carrier scattering - ohmic resistance.

### Derivation of kinetic inductance formula

To derive the kinetic inductance for a conducting material is fairly simple The averge excess kinetic energy of the charge carriers energy is given by: Where m is the charge carrier mass and v the drift velocity and n the number of charge carriers per unit volume. The transport current in general can be described in terms of the cross sectional area of the conductor (A), the drift velocity and the fundamental charge q: The total kinetic energy can be expressed in terms of the transport current I and equivalence to inductance used: Thus Lis the kinetic inductance of a conductor with n carriers per unit volume, carrier mass m and carrier charge q, carrying a current I, of cross sectional area A is: The voltage generated by this extra, intrinsic inductance, when the currrent and the carrier density can change is given by: or, in the complex frequency domain: This actually means that all conductors have some inductance which is independent of the exact shape into which the material is formed. In general, for normal materials, the freqeuncy at which the impedance contribution of this inductance becomes significant compared to the resistance is very large (~10THz). Hence one can generally safely neglect kinetic inductance for normal electronics.

Recently kinetic inductance has been rediscovered by researchers trying to develop metamaterial systems for extremely high frequencies (ie optical metamaterials). In these cases typical conductor sizes are ~ 200nm or less and kinetic inductance starts to become very large when compared to the geometrical inductance which generalyl scales with size. This has limited efforts to increase the operation freqeuncy of metamaterials by simple scaling and has required the development of novel structures.

In superconductors, the kinetic inductance is far more significant at lower frequencies and hence does play an important role in the behaviour of superconducting devices and systems.

If a material is excited by light (or otherwise) such that the number of charge carriers varies with time, in the presence of a constant transport current, a voltage pulse is generated according to the differenctial of carrier density: We have used this process to develop novel superconducting optical devices such as photomixers and ultrafast samplers