Studies of the Extrinsic Sources of Nonlinearity

 

I  Background and motivation.

            The microwave nonlinearity of superconductors was well known before the discovery of HTS, and shortly after its discovery, nonlinearity was already a subject of research[1].  In 1994, the observation of nonlinear signal distortion in superconducting filters was first reported[2].  Ten years later, nonlinear signal distortion still plagues the superconducting devices in commercial use[3].  I propose to systematically study the extrinsic causes of nonlinearity.

            Superconductivity is an ideal research area for physics undergraduates.  When I first began studying this subject as a graduate student, I was impressed by its use of all of the elements of the physics curriculum: electromagnetics in the London model; thermodynamics and statistical mechanics in Ginzburg-Landau theory; quantum mechanics in BCS theory and flux quantization; even classical mechanics in the Drude-like two-fluid model equation of motion.  Superconductivity is a research topic which is both accessible to undergraduates and relies on this broad use of physics.  The field is accessible due to the small-scale nature of its experiments and the phenomenological descriptions of superconducting properties. 

 

 

II  Experimental setup

            Distortion is measured by introducing a clean sinusoidal signal at frequency, f, to a device sample and examining the harmonic content of the electromagnetic fields in the presence of the sample.  This measurement is called harmonic distortion, and the emitted components will be at integer multiples of the incident frequency: 2f, 3f    We sometimes turn to measurement of 3^rd order intermodulation distortion (IMD)[4], as shown in Fig. 1.  Here we measure the mixing between the 2^nd and 3^rd harmonics of two closely spaced input signals, f₁ and f₂, at 3f₂-2f₁ and 3f₁-2f₂. 

            The various values of the 2^nd and 3^rd harmonic and the IMD will be dominated by different extrinsic mechanisms.  Especially at low temperatures, fluxon nucleation in the grain boundaries dominates the 2^nd harmonic[5].  The nonlinear conductivity of the grain boundaries dominates the 3^rd harmonic[6], while both are expected to dominate the IMD.  At higher temperatures, nucleation of fluxons at the edge of the film often dominates all components of the nonlinearity[7].  We will do measurements on resonators of various geometries from about 25 Kelvin to T_C, giving all of these mechanisms the opportunity to cause observable nonlinearity in these experiments.

            A test sample and its fixture are depicted in Fig. 2.  The disk resonator is excited in the TM₀₁₀ mode which has no current at the film edge[8].  In contrast to this TEM mode resonators are characterized by current crowding at the film edge.  So by using both types of sample, the two dominant extrinsic causes of nonlinearity will be viewed. 

            We have already fabricated thin film HTS devices and measured their nonlinearity in a different laboratory[9].  The experimental samples in Fig. 2 will allow us to consider the nonlinearity of YBa₂Cu₃O₇ and Tl₂Ba₂CaCu₂O₈ with so many of the dominant variables kept out of the picture.  Different extrinsic mechanisms dominate the nonlinearity in the low and the high temperature regions, with Josephson fluxons dominant at the low temperatures and edge critical state flux dominant at high temperature⁸. 

 

 

Studies of the Intrinsic Sources of Nonlinearity

 

            The fluxon generated nonlinearity described above is considered to be extrinsic in nature.  The proposed experiments do not consider intrinsic nonlinearity from Gorkov excitations and the nonlinear Meissner effect.  These intrinsic nonlinearities, in particular the nonlinear Meissner effect, are currently the subject of study in an NSF funded program headed by Prof. Steven Anlage at the University of Maryland in which I am a co-PI[10].  The above study of extrinsic nonlinearity will serve as a compliment to that investigation.  However, ultimately, the local probe method used in the NSF work can permit the direct observation of extrinsic nonlinearities due to Josephson vortices.

 

 

 

Making Nonlinearity Useful

 

Radio frequency (RF) circuit designers are increasingly confronted with the need to account for and also harness nonlinear behavior of circuit elements.  The harnessing of nonlinearity dates back to frequency modulation and the birth of the heterodyne technique[11].  Accounting for nonlinearity is growing in importance in step with the use of digital modulation of radio signals.  For example, spectral regrowth around a loaded CDMA frequency assignment reduces the network capacity by forcing transmitters to go to higher power.  Accurate knowledge of device nonlinearity and predictive modeling of nonlinear behavior, have become essential tools of RF design.

Earlier this year Maury Microwave Corporation (U.S.A) and NMDG Engineering, bvba (Belgium) announced the release of a large signal network analyzer (LSNA) under license from Agilent Technologies[12]. 

A calibrated diode would provide the user with a known magnitude and phase of both the fundamental and the higher harmonics for use in LSNA calibration.  But the lack of a low noise diode has forced LSNA users to follow a more complicated calibration procedure which includes the calibration approach for a classic VNA plus the added correction of the amplitude and phase errors of the harmonics relative to the fundamental[13].

This calibration approach is not too much more detailed than that of a conventional VNA.  However, an affordable nonlinear device could serve calibration purposes, and could be used as a primary reference for the phase and amplitude of the fundamental and higher harmonics.  I intend to pursue the development of a superconducting microstrip nonlinearity reference, fully integrated in a user friendly portable cryogenic package. 

A 50 Ohm transmission line will be designed and patterned onto a YBCO thin film.  Nonlinear transmission lines were modeled in the Ginzburg-Landau formalism by Megahed[14].  Coutts, et al.[15] went on to show the predictability of the amplitude attenuation and the phase of the signal as it propagates down a nonlinear transmission line.  In a later work, Coutts, et al.[16] demonstrated a nonlinear HTS transmission line with power controlled phase shift.  Our design will build on an understanding of nonlinear transmission lines derived from the various results mentioned above, to design an HTS-based nonlinear transmission line patterned onto a wafer (or a chip) of YBCO on LaAlO₃.