Radio-Frequency Losses in the Superconducting Penetration Depth

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Abstract
Trapped flux has been shown to be responsible for a large part of the residual ac losses in both types I and II superconductors. The authors have made theoretical and experimental investigations of losses in the 40–400 MHz range to establish a more detailed low‐field model. The surface resistance rs, as given by Pippard, is exceedingly small for Sn, Pb, and other pure metals below about 0.95 Tc at these frequencies. As a consequence, trapped‐flux effects provide the dominant losses in rolled‐foil resonant circuits. The theoretical model is simply Ohmic losses in the normal regions of trapped fluxoids bounded by the penetration depth and the surface. This gives an added resistance rh=rh(0)V(t), where the temperature function is V(t) = (1−t4)−1/2(1−t2)−1. The magnitude of the loss is predicted to be proportional to the density of fluxoids trapped, which in turn is assumed proportional to the background magnetic field for low fields. The experimental technique consisted of pulse determinations of circuit Q in the superconducting and normal state as a function of temperature and weak background magnetic field on cooling below Tc. The most detailed effort was made on pure tin foil from 60–350 MHz with background H fields up to 6 G. Data on these and on Sn–In, Pb–Sn, Ta, and Nb samples showed the function V(t) gave the best fit. The flux‐trapping loss was linearly proportional to the cooling field. The Pippard surface‐resistance term was separated out along with a very small residual loss, r0. The latter was independent of field or temperature and could be annealed out. The Pippard term had a frequency dependence between ν4/3 and ν3/2 as expected. These results differed from Haden et al. in that no break in the decay rate was observed in the pure Sn circuits. This could be explained by the absence of an observable fluxoid core transition. The absence of a strong frequency dependence is consistent with Ohmic loss and ruled out a dominant hysteresis behavior. The nature of the flux‐trapping loss is seen to make it the principle dissipating mechanism for most circuit applications since it resists annealing, does not depend upon impurities, and even though greatly reduced by nulling out external fields, may be partially self‐induced due to thermoelectric effects on cooling.

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