3. Kinetic Inductance Thermometry

The idea of kinetic inductance dates from the time of Faraday. Kinetic inductance is essentially the inertial mass of the charge carriers. The energy associated with a current I is

$${{1}\over {2}} {L_MI^2+\int_{\rm conductor}{1\over 2}nm_qv^2\,d\tau} \equiv {1\over 2} {L_MI^2+{1\over 2}L_kI^2}$$ (75)

For a uniform current in a homogeneous conductor the kinetic inductance is

$$\biggl( {m_q\over nq^2} \biggr) \biggl( {l\over \sigma} \biggr)$$ (76)

L_k is the kinetic inductance of the circuit. L_k depends on the geometry of the conductor through the length l and the cross-sectional area $\sigma$ of the conductor. It also depends on the material through the number density n of the current carriers, their mass m_q, and their charge q (Meservey and Tedrow (1969)).

In most situations L_k is negligible compared to L_M. This is because in a normal metal the time between collisions for the charge carriers $\tau_{c}^{}$ is short so that the kinetic reactance will not be comparable to the ohmic resistance for frequencies less than $\sim$ 10¹³ Hz where $\omega$$\tau_{c}^{}$ $\sim$ 1. In a superconductor the current is carried without dissipation by Cooper pairs so $\tau_{c}^{}$$\to$$\infty$. Thus the kinetic reactance dominates the ohmic resistance for all frequencies. The geometric inductance L_M is controlled by the geometry of the circuit. For a long thin superconductor with a ground plane to reduce L_M, L_k may dominate.

The kinetic inductance arises naturally from the first London equation:

$$\bf E {\partial\over{\partial t}} \Lambda \bf J_{s}^{}$$ (77)

Substituting into Poynting's theorem and neglecting the normal channel yields (Orlando and Delin (1991))

$$\oint_{\Sigma}^{} \bf S \bf s {d\over{dt}} \int_{V}^{} \biggl( {1\over 2} \epsilon \bf E^{2}_{} {1\over 2} \mu_{0}^{} \bf H^{2}_{} {1\over 2} \Lambda \bf J_{s}^{2} \biggr)$$ (78)

The third term in the integral may be rewritten

$${1\over 2} \Lambda \bf J_{s}^{2} \biggl( {1\over 2} \bf v_{s}^{2} \biggr)$$ (79)

This term represents the kinetic energy of the Cooper pairs whose number density and effective mass are n^* and m^*. So the kinetic inductance depends on the penetration depth

$$\lambda \sqrt{\Lambda(T)\over \mu_0}$$ (80)

and hence on the temperature.

The inductance of a superconducting stripline is given by (Meyers (1961))

$$\mu_{0}^{} {l\over w} \biggl[ \lambda_{s}^{} {d_s\over \lambda_s} \lambda_{g}^{} {d_g\over \lambda_g} \biggr]$$ (81)

where

$$\lambda_{g,s}^{} {\lambda_{g,s}(0)\over\sqrt{1-\bigl({T\over T_c}\bigr)^4}}$$ (82)

is the penetration depth in the superconducting film. The subscripts g and s refer to the ground plane and meander strip, respectively. The length of the meander strip is l, w is its width, t is the thickness of the dielectric layer, and d_g and d_s are the thicknesses of the ground plane and meander strip as shown in Figure 3.3. The critical temperature is T_c and $\lambda$(0) refers to the penetration depth at absolute zero. This function varies very rapidly with temperature close to T_c. There are two limiting cases which are of interest here. For thick films d > > $\lambda$ and L_k $\propto$ $\lambda$. For thin films d < < $\lambda$ and the temperature dependence is stronger: L_k $\propto$ $\lambda^{2}_{}$. Clearly, to maximize sensitivity, a kinetic inductance thermometer should be designed to operate in the latter limit. For a superconductor such as Al ( $\lambda$(0) $\approx$ 500 Å) this would require films less than 100 Å thick.

The elimination of Johnson noise and the use of a SQUID preamplifier could improve the resolution by a factor of $\sim$50 over that of a resistive calorimeter with the same total heat capacity.

Figure 6: Cross-section of a kinetic inductor. The supercurrents flow approximately in the shaded regions.