Modeling the nonlinearity of superconducting strip transmissionlines

The use of the Ginzburg-Landau (GL) theory to predict the nonlinear behavior in a superconducting stripline resonator as a function of input current is reported. A method for calculating the nonlinear inductance and the fractional change in the resonant frequency (Δf/f) of a stripline resonator is presented. By solving the GL equations inside the superconducting strip, the spatial variation of the number density of superconducting electrons and, hence, the spatial variation of the magnetic penetration depth are obtained for different values of input current. First, an infinite parallel plate transmission line is considered where the one-dimensional GL equations are solved. The two-dimensional case of a stripline is then considered. Nonlinear inductances are calculated as functions of input current for different superconducting striplines. Comparisons of the calculated Δf/f with measurements for YBa₂Cu₃O_7-x stripline resonators show excellent agreement     

Current distribution in superconducting thin-film strips

A closed-form expression is presented for the current distribution in an isolated strip made from a superconducting thin film with thickness less than a few penetration depths. This equation gives a good approximation within certain limits for the resistance and inductance per unit length of superconducting striplines and microstrip lines. Both the ac resistance of an isolated superconducting strip and the resistance per unit length of a superconducting stripline calculated using this closed-form expression agree well with the numerical results generated by the modified Weeks' method. The possibility of using this expression along with the integral equation method to develop an efficient full-wave method for analyzing superconducting planar transmission lines is discussed     

Investigation of high-temperature superconductors with terahertzbandwidth electrical pulses

The authors summarize recent attempts to propagate ultrashort electrical pulses containing frequency components up to 1 THz on transmission lines made from thin films of the high-temperature superconductor YBa₂Cu₃O_7-δ. Optoelectronic generation of extremely short electrical transients and jitter-free electrooptic detection of the pulse shape and amplitude as they propagate along the superconducting line make it possible to investigate the high-frequency properties of the high-T_c material with high accuracy because of the the long interaction lengths that are possible on transmission lines. The authors discuss the influence of epitaxy, surface roughness, and magnetic field on the pulse propagation. Important parameters such as the magnetic penetration depth and the superconducting energy gap are directly determined from the experimental data. Finally, the authors compare their results with weak-coupling BCS (Bardeen-Cooper-Schrieffer) theory     

Stripline resonator measurements of Zs versus Hrf in YBa2Cu3O7-x thin films

A report is presented on measurements of the surface impedance, Z_S, of YBa₂Cu₃O_7-x thin films using a stripline resonator. The films were deposited on LaAlO₃ substrates by off-axis magnetron sputtering. The authors obtained Z_S as a function of frequency from 1.5 to 20 GHz, as a function of temperature from 4 K to the transition temperature (~90 K), and as a function of the RF magnetic field from zero to 300 Oe. At low temperatures the surface resistance, R _S, of the films shows a very weak dependence on the magnetic field up to 225 to 250 Oe. At 77 K, R_S is proportional to the square of the field. The penetration depth shows a much weaker dependence on the field than does R_S. The origins of the magnetic field dependence of Z_S are also discussed     

Wideband Nonlinear Response of High-Temperature Superconducting Thin Films From Transmission-Line Measurements

We report on a technique for extracting an accurate value of the nonlinear inductance in superconducting transmission lines. This novel technique assesses the frequency dependence of the transmission line's nonlinear response. A wideband nonlinear measurement system was used to simultaneously measure the third-order spurious signals at 2f₁ - f_2, 2f₂ - f₁ + f₂, 2f₂ + f₁, 3f₁, and 3f₂ frequencies. Measurements for different values of the fundamental frequencies f₁ and f₂ allow us to study the spurious signal generation from 1 to 21 GHz. We demonstrate this technique by measuring several superconducting YBa₂Cu₃O_7-x coplanar waveguide transmission line geometries patterned in a single chip at 80 K. The results show a linear frequency dependence of the nonlinear response, indicating a dominant contribution of the nonlinear inductance over the nonlinear resistance omegaDeltaL(i) Gt DeltaR(i). The experimentally obtained nonlinear inductances are then used to determine device-independent measures of the linearity of the thin-film material in order to provide the foundation for modeling the nonlinear response of specific devices.     

Maximum heat-sink temperature for CW operation of adouble-heterostructure semiconductor injection laser

Assuming that the temperature dependence of the threshold current for pulsed operation is known, an analytical expression for the maximum heat-sink temperature, T_hm, for CW operation of the laser can be derived. The maximum heat-sink temperature is expressed in terms of the characteristic temperature T₀, the room-temperature threshold current for pulsed operation I₀ , the equivalent effective thermal resistance &thetas;, and the equivalent effective series electrical resistance r of the device. It is shown that the values of T_hm can be enhanced by increasing the value of T₀ or by decreasing the values of I₀, &thetas;, and r     

Phenomenological loss equivalence method for planar quasi-TEMtransmission lines with a thin normal conductor or superconductor

The incremental inductance rule for conductor loss calculations is not valid if conductor thickness decreases and becomes comparable to the penetration depth. A simple approach, referred to as the phenomenological loss equivalence method is proposed for characterizing a planar quasi-TEM transmission line with a thin normal conductor or superconductor over a wide range of field penetrations. For microstrip lines with a thin copper or high-T_c superconductor, the conductor losses calculated by this method agree very well with the published data calculated by the finite-element method and the Monte Carlo method, respectively. Because of the simplicity of the calculation, the method should be very useful for the computer-aided design of monolithic microwave circuits     

Comparison of radio-frequency and microwave superconductingproperties of YBaCuO dedicated to magnetic resonance imaging

Properties of YBaCuO thin films are evaluated in two distinct frequency ranges using different patterns made during the same process on LaAlO₃ substrate. Microwave superconducting properties in the range 1-45 GHz are determined by S-parameters measurement of a superconducting coplanar waveguide in the range 53-95 K. We obtain a surface resistance of 0.4 mΩ at 10.8 GHz and 77 K. Radio-frequency properties are obtained by measuring the Q-factor of a superconducting resonator (YBCO multiturn transmission lines separated by a sapphire sheet) dedicated to surface magnetic resonance imaging. At 52 MHz and 77 K we measure a Q-factor of 33180. The extraction of the radio-frequency surface resistance from Q-factor measurements in the 64-95 K range takes into account external loss mechanisms and nonuniform normal current distribution and leads to a 0.0093-μΩ surface resistance at 52 MHz and 77 K, in good agreement with the value extrapolated from microwave measurements assuming an ω² frequency dependence. The evaluation of λ₀ is carried out by using several models for X_L(t). Least squares fits to data in the microwave and radio-frequency domain are performed using the Gorter-Casimir expression for X_L(t) and give the same λ₀ value for both devices     

The coplanar resonator technique for determining the surfaceimpedance of YBa2Cu3O7-δ thinfilms

We describe how coplanar microwave resonators fabricated from patterned thin films of YBa₂Cu₃O₇(-δ) (YBCO) can he used to measure the ab-plane microwave surface impedance Z_s=R_s+jX_s of the films, in particular the absolute value and temperature dependence of the magnetic penetration depth λ. The current distribution of the resonator is calculated by modelling the resonator as a network of coupled transmission lines of rectangular cross-sections; this is then used to estimate the ab-plane λ(T) from the measurements of resonators of different geometries patterned onto the same film. We obtain values of λ(0) in the range 150-220 nm. The unloaded quality factors of the linear resonators at 7.95 GHz are around 45000 at 15 K and around 6500 at 77 K. We estimate the corresponding values of the intrinsic R_s at 7.95 GHz to be 23 μΩ and 110 μΩ at 15 K and 77 K, respectively. These values are comparable with those of other high quality unpatterned YBCO films reported in the literature. Z_s for the best optimised films appears to be insensitive to the effects of patterning     

Microwave measurement of temperature and current dependences ofsurface impedance for high-Tc superconductors

Perturbation formulas for TE₀₁₁-mode dielectric rod resonator and for a TE₀₁₁-mode circular cavity resonator are derived to determine the surface impedance Z_s of superconductors from measured values of resonant frequencies and unloaded Q. The relation between the maximum surface current density of a superconductor, J_s, and output power from a signal generator is derived. On the basis of these analytical results, a measurement technique is proposed to evaluate the temperature and J_s dependencies of Z_s for superconductors. The measured results of the temperature dependence of Z_s for YBCO and copper plates are presented. From these results, it is verified that the dielectric resonator is suitable for measuring the surface reactance for YBCO. From these Z_s values the temperature dependences of the skin depth and the penetration depth and those of the complex conductivity are obtained on the basis of the two-fluid model. These measured values agree well with the theoretical curves     

Correction factor for determining the London penetration depth fromstrip resonators

The strip resonator technique is a popular way to measure the temperature (T)-dependent London penetration depth λ_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 λ _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+ξ)     

The propagation characteristics of wave-guiding structures withvery thin superconductors; application to coplanar waveguide YBa2Cu3O7-x resonators

The propagation characteristics of waveguiding structures with superconductors which are thin compared to the magnetic penetration depth are analyzed. The complex propagation constant is evaluated within the framework of the modified spectral domain method without the need for numerical calculations in the complex plane. Good agreement is found with the results of other methods. The numerical analysis is instrumental in deducing results for the penetration depth and the surface resistance of YBa₂Cu₃O_7-x thin films on sapphire with a PrBa₂Cu₃O_7-x buffer layer. Recent observations of a non-single-gap BCS temperature dependence are confirmed     

Field response of ultra-thin type II superconducting transmissionlines

This work describes the effect of a tangential magnetic field on the superconducting penetration depth, λ, as modeled by the theory of Ginzburg and Landau. In particular, an increasing magnetic field decreases the magnitude of the order parameter Ψ. Consequently, the London equations have been modified to include field dependent values of the penetration depth, λ(T,H) and complex conductivity, σ(T,H). The analysis assumes that the superconducting films are free of vortices. For Type II films, this criteria can only be met when the thickness of the conductors is less than 1.8ξ_GL. The G-L theory is used to find closed form solutions that describe the electromagnetic characteristics of a kinetic inductance transmission line including phase velocity, impedance, and energy loss, as functions of power, temperature and magnetic fields. Limitations of power are also discussed. To validate these concepts, experiments verifying the effects of temperature and bias field were implemented using niobium superconductors 500 Å thick separated by 1000 Å of Al₂O₃. Potential device applications include variable phase-shifters, tunable filters, and extremely sensitive bolometers     

Microstrip ring resonator technique for measuring microwaveattenuation in high-Tc superconducting thin films

Microwave attenuation of high-T_c superconducting (HTS) films sputtered on MgO and ZrO₂ were measured using a microstrip ring resonator circuit. The results of Y-Ba-Cu-O and Bi-Sr-Ca-Cu-O resonators were compared to those for gold-plated resonators of identical design. The losses of superconducting and gold-plated films were determined from unloaded Q-factor measurements. The attenuation of Y-Ba-Cu-O film on an MgO substrate is approximately 31% lower than that of gold films at 6.6 GHz and 33% lower at 19.2 GHz for temperatures below 50 K. The approach of using microstrips to characterize microwave losses shows the usefulness of HTS films in integrated circuit technology     

Absorption at high microwave power by large-area Tl-basedsuperconducting films on metallic substrates

Microwave surface resistance measurements have been made on large-area Ti-Ba-Ca-Cu-O thick films magnetron-sputtered onto oriented Ag alloy substrates by replacing the end wall of an 18 GHz TE₀₁₁ mode Cu cavity with the superconducting film. The best surface resistance values obtained are 4 and 14 mΩ at 10 K and 77 K, respectively; corresponding Cu values are 8 and 21 mΩ. The dependence of the surface resistance on microwave power was measured in a similar way except that a Nb cavity was used instead of a Cu cavity. Typically, the surface resistance of the film begins to rise in 1-10 Oe of microwave field and saturates in 20-60 Oe. A model is presented relating the observed saturation to critical penetration of Josephson junctions. Films exhibiting the highest degree of c-axis texturing show the weakest dependence of surface resistance on power and also exhibit the sharpest transition to the superconducting state as measured at high frequency     

A general analysis of propagation along multiple-layersuperconducting stripline and microstrip transmission lines

A rigorous spectral-domain formulation for a superconducting stripline or microstrip transmission line with a multiple-layer dielectric substrate is presented. The formulation models the strip conductor as a surface current with an equivalent surface impedance, where the surface impedance is approximated in closed form when the strip is either much thinner or much thicker than a penetration depth. In either case the surface impedance is related to the complex conductivity of the material, which is calculated from a two-fluid model. Results are presented to show the slow-wave propagation and attenuation along both microstrip and stripline packages in a realistic multiple-layer configuration, which accounts for the field penetration into the superconducting ground planes     

Hybrid high temperature superconductor/GaAs 10 GHz microwaveoscillator: temperature and bias effects

Hybrid YBa₂Cu₃O_7-x superconductor/GaAs microwave oscillators have been designed, fabricated and characterized. The planar oscillators were built on a single 10 mm×10 mm LaAlO₃ substrate. The active elements in the hybrid oscillators were GaAs MESFETs. A ring resonator was used to select and stabilize the frequency. A superconducting ring resonator had a loaded Q at 77 Kg which was 8 times larger than the loaded Q of a ring resonator fabricated out of copper. S-parameters of the GaAs FET were measured at cryogenic temperatures and used to design the oscillator which had a reflection mode configuration. The transmission lines, RF chokes and bias lines were all fabricated from YBa₂Cu₃ O_7-x superconducting thin films. The performance of the oscillators was measured as a function of temperature. The rate of change of the frequency as a function of temperature was smaller for an oscillator patterned from a pulsed laser deposited film than for an oscillator patterned from a sputtered film. As a function of bias at 77 K, the best circuit had an output power of 11.5 dBm and a maximum efficiency of 11.7% The power of the second harmonic was 25 dB to 35 dB below that of the fundamental, for every circuit. At 77 K, the best phase noise of the superconducting oscillators was 68 dBc/Hz at an offset frequency of 10 kHz and less than -93 dBc/Hz at an offset frequency of 100 kHz. At an offset frequency of 10 kHz, the superconducting oscillator had 12 dB less phase noise than the copper oscillator at 77 K     

Double-active-layer index-guided InGaAsP-InP laser diode

A buried crescent InGaAsP-InP laser with two active layers was fabricated to study the temperature behavior of the double-carrier-confinement structure. An anomalously high characteristic temperature T₀ was measured, and optical switching behavior was observed. A mode analysis and numerical calculation using a rate equation approach explained qualitatively very well the experimental results. It was revealed that both the Auger recombination in this special double-active-layer configuration and the temperature-dependent leakage current, which leads to uniform carrier distribution in both active regions, are essential to increase T₀     

Microwave characteristics of high Tcsuperconducting coplanar waveguide resonator

A theoretical formulation has been developed to calculate the coupling coefficient, London penetration depth, and surface resistance of a coplanar waveguide resonator fabricated from films of superconducting YBCO material. Experimental data of the reflection coefficient as a function of temperature and frequency agree reasonably well with calculations. The formulation is of sufficient generality to be applicable to other guided structures     

Attenuation and dispersion for high-Tc superconductingmicrostrip lines

The attenuation and dispersion of microstrip lines of the high-T_c superconductor YBa₂Cu₃O₇ (YBCO) on yttria-stabilized zirconia substrates as a function of frequency and temperature are calculated. The effect on pulse propagation of superconducting and model dispersion in addition to the attenuation is demonstrated. At 60 K, microstrip lines of YBCO are significantly less attenuating at frequencies below 500 GHz than microstrip lines of copper at the same temperature. This advantage is particularly significant at the higher attenuations that result as the substrate thickness is made smaller for miniaturization or to improve the microstrip line bandwidth. The application of YBCO for microstrip lines appears to be most useful at frequencies above 100 GHz and dielectric thicknesses less than 100 μm, where the attenuation of cooled copper is prohibitively large. Cooled to temperatures below 20 K, YBCO may make possible a new generation of extremely high bandwidth (~5 THz), small-feature-size (~5 μm) circuits and devices