FIG. 2: The magnetic field dependance of the quality factor of the resonator. The measured quality factor Q$_{m}$ is shown as open circles, and the internal quality factor Q$_{i}$ is shown as filled circles. This residual field is likely related to the field leaking out of the superconducting magnet in the ADR. Once cold, the devices showed little response to applied magnetic fields of over several hundred mG, where the resonator Q$_{i}$ starts degrading. This degradation remains when the field is ramped down to zero. When the device was warmed past T$_{c}$ and recooled the original Q$_{i}$ was recovered, likely indicating trapped magnetic flux. Using the measured thickness of the a-Si:H film of 200 nm from Figure 1 and 50 nm as the penetration depth of aluminum, Equation 1, and the known lengths and resonant frequencies of the resonators it is possible to derive the dielectric constant of a-Si:H. This calculation yields ϵ$_{r}$ = 11 . 4. The error in the determination of ϵ$_{r}$ scales with the error in the measurement of the dielectric thickness. Most of the computed quantities in this paper like NEP vary only weakly with ϵ$_{r}$ . Previous work has shown that the dielectric loss tangent is strongly dependent on the electric field in the dielectric [6, 15–17]. Figure 3 shows the loss tangent as a function of electric field in one of the a-Si:H microstrip resonators. The electric field at the open ends of the resonator can be calculated from [14]: where m = 1 / 2 for a half wave resonator, P is the microwave power on the feedline, Q$_{m}$ is the measured quality factor, Q$_{c}$ is the coupling quality factor, and Z$_{0}$ is the characteristic impedance of the microstrip transmission line Z$_{0}$ = $^{√}$L/C = 1 /v$_{p}$C . Since d ≪ w we use a E$_{0}$ = 1 d$^{√}$ 4 Z$_{0}$ πm Q 2 m Q$_{c}$ P (5) 6 − 100 − 50 0 50 100 150 Magnetic Field (mG) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Q / 10 5