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arXiv:1210.5929 · Journal of Applied Physics

Inductances and attenuation constant for a thin-film superconducting coplanar waveguide resonator

John R. Clem

Published Jan 4, 2013• 30 citations
arXivHTMLPDF

Abstract

The geometric, kinetic, and total inductances and the attenuation constant are theoretically analyzed for a thin-film superconducting coplanar waveguide resonator consisting of a current-carrying central conductor, adjacent slots, and ground planes that return the current. The analysis focuses on films of thickness d obeying d<2λ (λ is the London penetration depth), for which the material properties are characterized by the two-dimensional screening length Λ=2λ2/d. Introducing a cut-off procedure that guarantees that the magnitudes of the currents in the central conductor and the ground planes are equal, new and simpler results are obtained for the kinetic inductance and the attenuation constant for small Λ. Exact results for arbitrary Λ are presented for the geometric, kinetic, and total inductances in the limit of tiny slot widths, and approximate results are presented for arbitrary slot widths.

DOI
10.1063/1.4773070

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The geometric, kinetic, and total inductances and the attenuation constant are theoretically analyzed for a thin-film superconducting coplanar waveguide resonator consisting of a current-carrying central conductor, adjacent slots, and ground planes that return the current. The analysis focuses on films of thickness d obeying d<2λ (λ is the London penetration depth), for which the material properties are characterized by the two-dimensional screening length Λ=2λ2/d. Introducing a cut-off procedure that guarantees that the magnitudes of the currents in the central conductor and the ground planes are equal, new and simpler results are obtained for the kinetic inductance and the attenuation constant for small Λ. Exact results for arbitrary Λ are presented for the geometric, kinetic, and total inductances in the limit of tiny slot widths, and approximate results are presented for arbitrary slot widths.
An analytical expression for the kinetic inductance of the superconducting coplanar waveguide is given using the conformal mapping technique. This expression is shown to be in good agreement with experimental results obtained from the temperature dependence of the resonant frequency of the NbN coplanar waveguide resonators, especially in the case of a film thickness smaller than the magnetic penetration depth. It is also shown that the magnetic penetration depth of the superconducting thin films can be evaluated by this method.
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 objective of this thesis is to evaluate a generic process for fabrication and characterization of the Superconductive coplanar waveguide (CPW) resonators. Superconductive CPW resonators with various lengths and shapes are designed to investigate their electrical and magnetic properties as well as resonance properties and sensitivities. In the first part of thesis, two different models are introduced in order to estimate the nonlinear kinetic inductance of a superconducting CPW resonator. The first model is based on Bean criticalstate model and the second one is based on current dependence of London penetration depth, λL(I). The existence of a shift in resonant frequency, f0, of Superconductive CPW resonator caused by a non-linear kinetic inductance is also shown experimentally. Simulations were carried out to estimate the nonlinear kinetic inductance due to the selfinduced magnetic field penetration. The rest of the thesis is concerned with development of very smooth Aluminum (Al) thin films with RMS (Root Mean Square) roughness ∼ 1 nm and CAD (Computer Aid Design) of superconductive CPW resonators. Experimental investigation of a generic fabrication technique for superconductive CPW resonator is carried out. Many resonators are fabricated with different design parameters, such as centerline or gap width, film thickness and gap capacitors length. The fabrication process is described in detail. Electron Beam Lithography is used to fabricate Nb and Al CPW resonators which are coupled to outer conductors via gap capacitors. We have fabricated GHz frequency CPW resonators with quality factors, Q up to 5× 10.
The objective of this thesis is to evaluate a generic process for fabrication and characterization of the Superconductive coplanar waveguide (CPW) resonators. Superconductive CPW resonators with various lengths and shapes are designed to investigate their electrical and magnetic properties as well as resonance properties and sensitivities. In the first part of thesis, two different models are introduced in order to estimate the nonlinear kinetic inductance of a superconducting CPW resonator. The first model is based on Bean critical-state model and the second one is based on current dependence of London penetration depth. The existence of a shift in resonant frequency of Superconductive CPW resonator caused by a non-linear kinetic inductance is also shown experimentally. Simulations were carried out to estimate the nonlinear kinetic inductance due to the self- induced magnetic field penetration. The rest of the thesis is concerned with development of very smooth Aluminum (Al) thin films with RMS (Root Mean Square) roughness 1~nm and CAD (Computer Aid Design) of superconductive CPW resonators. Experimental investigation of a generic fabrication technique for superconductive CPW resonator is carried out. Many resonators are fabricated with different design parameters, such as centerline or gap width, film thickness and gap capacitors length. The fabrication process is described in detail. Electron Beam Lithography is used to fabricate Nb and Al CPW resonators which are coupled to outer conductors via gap capacitors. We have fabricated GHz frequency CPW resonators with quality factors, Q up 5X10^5.
Photon detectors based on the change of kinetic inductance of a thin superconducting film have a number of applications, particularly in astronomy, owing to their high sensitivity and ease of integration into large arrays. Here we discuss in detail the analysis of kinetic inductance detectors that use thin film microwave coplanar resonators. Photon absorption decreases the electron pair density, increasing the magnetic penetration depth /spl lambda/, which causes a decrease in the resonant frequency (or phase) of an irradiated resonator. To quantify this effect, we first compute the resonator current distribution, from which the /spl lambda/-dependent parameters (such as kinetic inductance) are calculated. Optimum responsivity for phase measurement is achieved by using the thinnest film with the narrowest center conductor width at the lowest possible temperature. However, the responsivity is compromised by extrinsic microwave losses, in particular due to residual surface resistance, which is likely to be significant in the thinnest films.
Analysis of superconducting circuits requires knowledge of current density distribution in the film cross section. This article aims to investigate the current density distribution in superconducting coplanar waveguides (SCPWs), a basic component with increasing use in microwave integrated circuits. In this regard, first, the CPW is mapped to a pair of parallel plates and the vector potential is described through the Green's function concept. Then, based on the Green's function and utilizing London's second equation, an integral equation is formed which yields the current density distribution. The procedure is verified by COMSOL Multiphysics software. Examining the current density for several SCPWs shows that as the gap is reduced, the SCPW experiences much more current density especially at the edges. Based on the proposed procedure, the kinetic inductance of SCPW is examined, which is another focus of the article. In this article, for several CPWs with different aspect ratios, the kinetic inductance is evaluated and discussed. Results show that as the ratio is increased, the kinetic inductance is decreased.