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IEEE International Symposium on Electromagnetic Compatibility

Modeling linear and nonlinear inductive effects in superconducting planar circuits

James C. Booth, Christopher L. Holloway, Edward F. Kuester

Published Nov 13, 2002• 1 citation
Publisher

Abstract

Superconducting planar circuits possess properties that can be very different from their normal-metal counterparts, which makes modeling efforts more complicated for superconducting components. In this paper we present closed-form expressions for the resistance and internal inductance per unit length of superconducting planar transmission lines in terms of the relevant material parameters. We also include a simple calculation of the nonlinear response of superconducting devices based on a current-dependent superconducting penetration depth. Taken together, these two descriptions can be used to predict the linear and nonlinear response of superconducting planar devices in a manner that is conducive to use in circuit design. The results of these calculations compare well with 76 K measurements of coplanar waveguide transmission lines and resonators fabricated from thin films of the high T/sub c/ superconductor YBa/sub 2/Cu/sub 3/O/sub 7-/spl delta//.

DOI
10.1109/isemc.2001.950711

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Superconducting planar circuits possess properties that can be very different from their normal-metal counterparts, which makes modeling efforts more complicated for superconducting components. In this paper we present closed-form expressions for the resistance and internal inductance per unit length of superconducting planar transmission lines in terms of the relevant material parameters. We also include a simple calculation of the nonlinear response of superconducting devices based on a current-dependent superconducting penetration depth. Taken together, these two descriptions can be used to predict the linear and nonlinear response of superconducting planar devices in a manner that is conducive to use in circuit design. The results of these calculations compare well with 76 K measurements of coplanar waveguide transmission lines and resonators fabricated from thin films of the high T/sub c/ superconductor YBa/sub 2/Cu/sub 3/O/sub 7-/spl delta//.
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- f2,2f2- f1+ f2, 2f2+ f1, 3f1, and 3f2frequencies. Measurements for different values of the fundamental frequencies f1and f2allow us to study the spurious signal generation from 1 to 21 GHz. We demonstrate this technique by measuring several superconducting YBa2Cu3O7-xcoplanar 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.
Superconducting planar resonators of thin molybdenum carbide (MoC) films were fabricated by optical lithography. The films were deposited via reactive magnetron sputtering and by changing the deposition parameters, the sheet resistance of the films was varied. The microwave properties of the resonators were studied by broadband coplanar waveguide spectroscopy in flip-chip configuration. The kinetic inductance of MoC films governs the high frequency response of the resonators and is determined by the complex conductivity of the superconducting film. Well below the critical temperature of transition, kinetic inductance is proportional to film sheet resistance in normal state. The temperature-dependent resonances in the broadband spectra below critical temperature are described by a lumped LC model, where the inductance consists of geometric and kinetic inductance of the resonator. The temperature and frequency dependent complex conductivity of the superconducting film is calculated by the Dynes model for dirty superconductors. Furthermore, a numerical model of the kinetic planar resonator in Sonnet software is presented, which can be utilized to design resonators with desired properties, based on highly disordered superconductors. An on-chip design of a kinetic planar resonator filter is presented. The high kinetic inductance of the superconducting films allows us to minimize microwave filter dimensions.
Inductance of superconducting thin-film inductors and structures with linewidth down to 250 nm has been experimentally evaluated. The inductors include various striplines and microstrips, their 90° bends and meanders, interlayer vias, etc., typically used in superconducting digital circuits. The circuits have been fabricated by a fully planarized process with 8 niobium layers, developed at MIT Lincoln Laboratory for very-large-scale superconducting integrated circuits. Excellent run-to-run reproducibility and inductance uniformity of better than 1% across 200-mm wafers have been found. It has been found that the inductance per unit length of stripline and microstrip line inductors continues to grow as the inductor linewidth is reduced deep into the submicron range to the widths comparable to the film thickness and magnetic field penetration depth. It is shown that the linewidth reduction does not lead to widening of the parameter spread due to diminishing sensitivity of the inductance to the linewidth and dielectric thickness. The experimental results were compared with numeric inductance extraction using commercial software and freeware, and a good agreement was found for 3-D inductance extractors. Methods of further miniaturization of circuit inductors for achieving circuit densities> 106Josephson junctions per cm2are discussed.
We present calculations of the properties of the first coplanar waveguide transmission line resonators patterned from high-Tc superconducting films at 8.8 and 6.5 GHz [1, 2]. Based on field calculations of the corresponding coplanar waveguides with infinite conductivity, performed by a partial wave synthesis (mode-matching method) [3], we obtain the superconductor losses subsequently using London's theory. Due to the high conductivity of the superconductive material both the electric and magnetic fields within the superconductor may be assumed to be parallel to the surface planes. For the small conductor thickness the tangential electrical field on each surface plane depends on the tangential magnetic fields on both surface planes. This relation is modelled by the surface impedance matrix Zs. The substrate losses are treated in the field calculations and are also taken into account. The calculated overall Q values are compared with the measured values and it has to be pointed out, that the superconducting thin films are due to the available material datas of the YBaCuO superconductor and the used substrate materials of a very good quality, because they reach nearly their upper limit given by the known material datas.
The authors evaluate the frequency dependence of the nonlinear response in high-temperature superconductor YBa2Cu3O7−x thin films by simultaneously measuring the nonlinear intermodulation products and harmonic generation in broadband superconducting transmission lines at 76K. The frequencies of the two-tone incident signal are set to produce spurious signals from 1to21GHz. They extract a nonlinear term ∣ΔR2+jωΔL2∣ by applying a model of spurious signal generation in superconducting transmission lines to their measurement results. They found that this nonlinear term follows a linear dependence on frequency, indicating a dominant contribution of the nonlinear inductance ΔL2 over the nonlinear resistance ΔR2, (ωΔL2⪢ΔR2), for the superconducting nonlinear response.
We have investigated the electrical properties of high-T c dc-SQUID's formed in a coplanar waveguide, and discussed their applicability to single flux quantum (SFQ) circuits. The planar type Josephson junctions were fabricated in NdBa2Cu3O y (NBCO) superconducting thin films by a narrow-focused ion beam (FIB) irradiation technique. The critical current and normal resistance values of the Josephson junction at 4.2 K were 115 µA and 8 Ω, respectively. The inductance was estimated by applying a direct injection of the control current into the dc-SQUID loop and measuring the periodic voltage response. A loop inductance of 47 pH was obtained for a 25 µm-wide and 20 µm-long dc-SQUID at 4.2 K. It has been clarified that for a stable operation of SFQ logic circuits, a 10 µm-square dc-SQUID's are required.