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Applied Physics Letters

Theory of microwave intermodulation in a high-T c superconducting microstrip resonator

T. Dahm, Douglas J. Scalapino

Published Dec 30, 1996• 62 citations
Publisher

Abstract

We calculate the intermodulation power for a superconducting microstrip resonator arising from the intrinsic microscopic nonlinearities. We focus on high-Tc superconductors where it is believed that the order parameter has dx2−y2 symmetry. A characteristic upturn of the intermodulation power is found at low temperatures. We discuss different ways to reduce intermodulations by changing the geometry of the microstrip.

DOI
10.1063/1.116960

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We calculate the intermodulation power for a superconducting microstrip resonator arising from the intrinsic microscopic nonlinearities. We focus on high-Tc superconductors where it is believed that the order parameter has dx2−y2 symmetry. A characteristic upturn of the intermodulation power is found at low temperatures. We discuss different ways to reduce intermodulations by changing the geometry of the microstrip.
The penetration depth and surface resistance of a superconductor depend upon the superfluid current density. This dependence gives rise to nonlinear mixing in a superconducting microstrip resonator. Here we discuss the problem of intermodulation in which two signals at ω1 and ω2, laying within the pass band of a microstrip cavity resonance, mix and generate a signal at 2ω1−ω2. An expression relating the power generated at 2ω1−ω2 to the power transmitted at ω1 and ω2 is given. We focus on the high-Tc superconductors where it is believed that the order parameter has dx2−y2 symmetry. We find for a resonator with a large unloaded Q that intermodulation arises dominantly from the reactive nonlinear inductance of the superconducting film.
The issue of intermodulation distortion in superconducting devices for rf and microwave applications is discussed. The classical frequency conversion theory for nonlinear two-port devices in the small signal limit is shown to apply to microstrip resonators regardless of their geometry and material. Two tone, third order intermodulation measurements allow more sensitive detection of nonlinearity compared to surface impedance measurements, provided a low noise, spurious free experimental setup is adopted. Measurements carried out on high quality Nb meanderline microstrip resonators show that at very low power level nonlinearity is dominated by intrinsic device properties. Film defects start to play a predominant role above a critical power level that strongly depends on the defect’s nature.
We study the power dependence of two-tone intermodulation arising in high-Tc superconducting microwave resonators as a function of each tone separately. In the regime where the intermodulation power varies as the square of the input power for equally strong tones, we observe unexpected behavior of the power dependence on the individual tones. We show that this behavior can be understood in terms of a nonlinear inductance which varies linearly with the magnitude of the input current. These findings provide a consistent phenomenological picture of the unusual power dependence and should place constraints on a microscopic description of nonlinear behavior in high-Tc films where the intermodulation products vary like the square of the input power.
Nonlinearities give rise to intermodulation distortion in superconducting microwave devices and currently limit their use to low power applications. We have developed a cryogenic imaging technique to spatially resolve intermodulation distortion and used it to image an 8.2 GHz high temperature superconducting Tl2Ba2CaCu2O8 microwave resonator. The images reveal that the fundamental and intermodulation electric fields obey a fixed relation throughout the device. We note that further refinements of intermodulation theory in resonant devices may be required to fully describe the data.
The simplest model of nonlinear response of a superconducting thin film is used for modeling the nonlinear phenomena in a superconducting transmission line and a microstrip resonator. The specified characteristic power of the transmission line is suggested to use as a fitting parameter for numerical description of the microstrip line nonlinearity at microwaves. Quantitative agreement of simulated and experimental data has been obtained for the incident power dependent transmission coefficient of a microstrip line section and a high quality microstrip resonator. Numerical results have also been obtained for the power of the third harmonic radiated from the nonlinear resonator.
This paper presents a comprehensive study of microwave nonlinearities in superconductors, with an emphasis on intermodulation distortion and third-harmonic generation. It contains the analysis of various resonant and nonresonant test devices and its validation using numerical simulations based on harmonic balance (HB). The HB simulations made on test devices show that the closed-form equations for intermodulation and third-harmonic generation are only valid at low power levels. The paper also contains examples of application of HB to illustrate that this technique is useful to simulate superconductive devices other than simple test devices, and that the validity of the simulations is not restricted to low drive power levels. Most of the analyses and simulations of this paper are based on electrical parameters that describe the nonlinearities in the superconducting material. These parameters are compatible with many existing models of microwave nonlinearities in superconductors. We discuss the particulars on how to relate these electrical parameters with one of the existing models that postulates that the nonlinear effects are due to a dependence of the penetration depth on the current density in the superconductor.