Correction factor for determining the London penetration depth from strip resonators

The strip resonator technique is a popular way to measure the temperature (T)-dependent London penetration depth /spl lambda//sub L/(T) in superconducting thin films. The temperature dependence can provide fundamental information about the superconducting energy gap and hence insight into the pairing mechanism. Since /spl lambda//sub L/(T) characterizes the film's response to a magnetic field near the surface, it qualifies the suitability of the superconducting film for microwave device applications. There has been much controversy regarding the actual form of the temperature dependency, with some researchers reporting a weak-coupled Bardeen-Cooper-Schriefer (BCS)-like behavior and others favoring a Gorter-Casimir type fit. This paper shows that the disagreement can be at least partially attributed to a temperature sensitive term traceable to stray susceptance coupled into the resonator. The effect is inherent to the technique, but a simple procedure to compensate for it can be used and is presented here as a correction factor (1+/spl xi/).


INTRODUCTION
The resonant frequency of a high°Q microwave transmission line is inversely proportional to the square root of the sum of kinetic and magnetic inductanceper unit length.The kinetic inductance (I._(T)),associatedwith the inertial mass of the charge carriers, is strongly dependent on the penetration depth. Hence, the shift in resonant frequency with temperature can, in principle, yield a sensitive measure of _(T). However, extracting the zero temperature penetration depth Q.(0)) generally requires the assumption of a particular theoretical model to which the data is curve fit. The situationis exasperated by the complex interdependency among variables such as film thickness (t), circuit geometry including strip width (W) and substrate thickness (h), critical temperature (Tc), and Z(0). The penetration depth is also sensitiveto the quality of the film,especially near its surface, as well as the transition width A(T), which is an indicator of phase purity. Some studies have focussed only on extremely low impedancelines [1] or strictly low temperature (i.e. T < Tc/2) Z(T) dependence [2]. For most practicalmicrowave applications, line impedanceswill be in the neighborhood of 50 f_, and film thickness will be of the same order as the penetration depth. Experimental investigations using strip transmission lines near Tc have invariably revealed a strong deviation from theory [3][4][5] when t = _,.This short paper shows that the disagreement can be attributed, at least in part, to the susceptance coupled into the resonator from the gap discontinuity as well as the feed line of electrical length [31. The coupled susceptance is modifiedby the temperature dependent characteristic impedance of the resonator. When the effect is taken into account, the natural resonant frequency of the resonator is shown to increase as T approaches To, and the _.(T)profile changes accordingly. The situation when the strip characteristic impedance is not matched to the generator is included.

DERIVATION OF THE PERTURBATION OF THE NATURAL RESONANT FREQUENCY DUE TO LOADING
A lumped equivalent circuit model representing the excited resonator is shown in figure 1. A transmission line • gap is more often depicted as a capacitive pi network [6]. But it is mathematicallyconvenient to model it as shown here, and the transformation is straightforward. It is well known that the measuredresonant frequency (too')of an inductively or capacitivelycoupled resonator is pulled from the actual resonant frequency (too)of the isolated circuit because of the reactance or susceptance associated with the coupling mechanism. A good estimate of the unperturbed resonant frequency can be obtained by considering the coupled susceptance in the calculation of too-The total susceptance of the loaded resonator is where B is the susceptanceof the network left of the transformer. Since to2LC= 1at resonance and Qo= RtooC,it follows that too= too'(I + nZRB/(2Qo)) (2) and finally, using the approximation R=2ZoQo/IIfrom [7], shallow field penetration. Here, the inductance was derived from the imaginary part of the impedance calculated from the phenomenological loss equivalence method [10].This method has been shown to provide accurate results for both attenuation and phase velocity for quasi-TEM, normal and superconductor, transmission lines.
Determiningc(T) is not so straightforward but it can be estimated from [3] where resonant frequency versus temperaturedata was provided for a metallic conductor on LaAIO3. Ae(T) was taken as -550 ppm/K which is an order of magnitude more severe than results disclosed in [4]. Still, the effect is subtle and the correction factor of (3) [11]. A series network can be made equivalent to a parallel network, and vice versa, at one frequency. Since we are interested in the behavior of the circuit of figure 1  noted that the strip transmissionline gap parameters are assumed to be static. It will merely be mentioned that following the above approach and solving for the conductance, one can show that the Q of the resonator will depend on the feedline characteristics.

SAMPLE CALCULATION
In order to illustratethe impact of the correction factor on a practical resonator, an example is presented here.

CONCLUSIONS
A corrective term has been presented which slightly modifies the shape of the resonant frequency versus temperaturecurve of strip resonators, used to evaluate the London penetration depth. The term is temperature sensitive primarilybecause of the kineticinductance associated with the superconducting resonator, and is not expected to be as significantwhen t is thick compared to _..But, some earlierresonator based measurements may have overestimated_ (0) because of this effect. This may help explain some of the disa_eement in measurements of 3tdetermined by other methods. An implication of the approach presented herein is that the observed resonant frequency will depend on the characteristics of the feed line.