Numerical optimization of hybrid dielectric/HTS resonators for surface impedance evaluation of HTS films

1956 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 9, NO. 2, JUNE 1999 Numerical Optimization of Hybrid Dielectric/HTS Resonators for Surface Impedance Evaluation of HTS Films C. Collado, D. Gonzalo, E. Rozan, J.M. O’Callaghan. Universitat Politecnica de Catalunya. Campus Nord UPC-D3. Barcelona 08034, Spain. C. Sans Universitat de Vic. C/ Miramarges 4. Vic 08500, Spain AbstractThis work describes an alternative to the traditional dielectric resonator topology used for measuring surface impedance in High Temperature Superconducting (HTS) films. A gap is introduced above the dielectric so that only the lower film is in direct contact with it. This arrangement has been used extensively for mechanical tuning of dielectric resonators and, when used for surface impedance measurement, it can be designed to make the losses in the upper film small relative to the overall resonator losses. Then, measured results are mostly due to one of the films and not the average of two. The specifics of a resonator design for measuring 2-inch wafers are presented. An analysis and optimization of the resonator is done using a numerically efficient mode-matching algorithm. I. INTRODUCTION Commonly used techniques for evaluating surface impedance in HTS thin films are based on measurements of resonators on either patterned, planar structures or in a dielectric resonator cavity in which the upper and lower walls are made of HTS films and are in contact with the dielectric [l]. Only the latter approach is non-destructive, but it yields the average surface resistance of two films. Thus, to measure a sample with unknown properties, it has to be paired with a reference sample to determine its surface impedance. The degree of accuracy to which the surface resistance of the reference is known can be a significant source of error in these measurements. Therefore, other non-destructive methods need to be explored further. 11. HYBRID DIELECTRICIHTS RESONATOR The dielectric resonator topology under study is a modification of the traditional dielectric-HTS one [ 11, where a gap is introduced between the upper endplate and the dielectric cylinder (Fig. 1). The purpose of introducing this gap is to lower the RF losses in the upper film and Manuscript received September 15, 1998 This work has been funded by the Spanish Ministry of Education and Culture through CICYT grant MAT95-1038-C02-02 and by the Catalan ClRlT through grant SGROOO43. Coupling prDba1 Sapphire/ crystol .GI , U‘ U Fig. 1. Dielectric resonator structure. Definitions of dielectric height (hd), dielectric radius (rd) and gap size (h,) concentrate them in the lower one, so that its R,v is the largest contributing factor to the total losses of the cavity. Experimental use of this type of cavity for measurement of R,v is mentioned in [2], but no design details are given. Am analytical study of this topology, based on an a.xia1 modematching method with a single mode is presented in [3]. No description is given in any of these works of the procedure to optimize the cavity for R, determination purposes. In [3] it is pointed out that, as the upper gap is increased, the electromagnetic fields on the normal metal side walls also increase, so the contribution of the RF losses of the upper film are reduced at the expense of increasing those of the side walls. For R, determination, an optimal height exists for .a given resonator size and a given diameter of the superconducting endplates, for which the ratio of the RI3 losses in the lower plate to the overall losses in the cavity is maximum. This optimum may be found using an efficien.t mode-matching code [4], which calculates the electromagnetic field distribution in structures with azimutall symmetry. The algorithm is not restricted to small gap sizes and can be used to optimize the whole structure (gap size:, dielectric height and dielectric diameter) for a given diameter of the supeconducting wafers to be measured. Reference [2] proposes a surface resistance derermination procedure based in two consecutive measurements of quality factors in which the top and bottom samples are ‘exchangecl. The optimization proposed here might make this unnecessary, so that a single measurement and a rough estimate of the surface resistance of the top film and side walls might lead to an accurate determination of the surface resistance of the lower film. 1051-8223/99$10.00 0 1999 IEEE 1957 Fig. 2. Radial regions (A,B,C) and axial layers (1,2,3) considered by the mode-matching algorithm. Dielectric permitivity can be defined independently for each of the 9 cells (AI..C3). A. Full-wave radial mode-matching method The software developed [4] can analyze up to three different radial regions, each one divided into three different layers (Fig.2). The dielectric constant in each region and layer can be set independently, and the whole volume analyzed is assumed to be enclosed by conducting walls. Fields in every radial region are expressed as an infinite series of particular solutions of the wave equation in cylindrical coordinates. Azimutal and axial eigenfunctions are harmonic functions while radial eigenfunctions are Bessel functions, which must be chosen appropriately to avoid singularity in the axis (region A), fulfill boundary conditions on the external walls (region C), or provide continuity between adjacent regions (region B). The eigenvalues of each radial region are found by solving a pair of transcendental equations related to the three-layer problem. A system of equations is then found by forcing continuity conditions of the tangential fields between radial regions A and B, and B and C, from which the coefficients of the mode expansion are found. In our case 12 modes are sufficient. Fig. 3 illustrates the field and current distributions for a sapphire dielectric with a 12 mm diameter and a 6 mm height. B. Optimization. Basics and alternatives For R, determination, the optimal dimensions of the cavity in Fig. 1 are those that make the losses of the lower plate dominant with respect to the other type of losses. This is equivalent to maximizing Qo/Qlc,w, where: U is the total stored energy in the cavity, Pd the total dissipated power, Pd-/ow the power dissipated in the lower wall, and Qo the unloaded quality factor. Both the total energy U and the dissipated powers Pd and Prl-lOw can be calculated from the field distribution inside the cavity with properly defined surface and volume integrals of the electromagnetic fields [5]. Definitions analogous to that of Qlclw in (1) can be used for Qu/,, Q.,d and Qd which account for the losses in the upper wall, side wall, and dielectric respectively. Thus, the unloaded quality factor can be written as: ...._.......... .... , I,,................ I#,,.-......... . ... Iltrr. ............. It,.- ]I!..-::::::::' :::: 'id, ........ .. -),,,.......... /j/i\ ,,,........ . \$:j:::::;:;;:;;: hzj::: : :: : : : : : : : I \-.%.-A " .. . . . . . . . . . . . . -+c+ 4- - . . . . . . . . . . . Fig. 3. Field and current distributions in the cavity with a TEOI mode and a sapphire dielectric. Bottom plate diameter is 48mm, dielectric diameter is 12mm and dielectric height is 6mm. The dashed line represents the sapphire outline. (Top) Radial cut of the resonator showing magnetic field lines. (Bottom) Currents at the surface of the lower HTS film. The relevance of maximizing Q0/Qluw is due to the fact that this defines the sensitivity of the system: (3) which can. be proven from (1) and (2), and from the proportionality between and R,. Also, as Qo/Qlc,w is increased, the accuracy in estimating Q,,, QSd and Qd has a lesser effect on the R,y predicted from the measured Qo and (2). A software code has been developed, which includes the electromagnetic mode-matching algorithm described previously, and is capable of calculating Qrclw , Q,, QSd and Qd as a function of dielectric height (hJ, dielectric radius (rd) and gap size (h,) (Fig. 1). To test it, the cavity dimensions given in [2] were used, and the code results were compared with those in Table I in that reference. The agreement is excellent, as shown in Table I of this work. The software developed can use several strategies to vary hd, rd and h, in the search for a maximum of Qo/QlcIw : 1) variation of the gap size for a given (fixed) dimension of the dielectric; 2) exhaustive search varying hd, rd and h, over a specified range, with a specified incremental step size; and 3) 1958 -DQ-Side wall (Qsd) ~ "i j_ + .... 4 - Q-Upper wall (Qup) ' r.i.. .... ~. I. :. i.. ........ : - . Q-Dielectric (Qd) ~ - .. . --+-.Q~/QIow~ . ,. F ,< j .TABLE I COMPARISONS BETWEEN 121 AND THIS WORK Glow is defined as the product of the surface resistance of the lower plate and Plow; Cup and Gsd are defined similarly. f (GHz) Glow GUP Gsd PI 9.5 546 4.90e3 1.15e6 This work 9.510 541 4.81e3 l.lle6 Error % 0.8% 1.7% 3.3% ,. ,. -' \ I .. -. ....... + .....; ........................................... I +, L.- -- ............... & .................................. \.> +-I--- t __ x- -. 1 *. -\-&--*----+-; ;A' \ I - /I y; .. ,, . ,- ~, . I- \; r :\ ....... ~ ................................ gradient search using Matlab [6]. In this process it is assumed that the upper and lower walls are made of YBa2Cu3O7.8 (YBCO), and the side walls of a normal metal (usually OFHC copper, but it can be changed by the user). Their R,T is scaled as the calculated resonant frequency of the cavity changes due to variations in hd, rd and h,. The scaling law is taken as f * for YBCO and f for the metal, and the values of R,y at lOGHz and 77K are taken as 300@ (YBCO) and 9mQ (Cu) [7] respectively. The frequency dependence of the loss tangent of sapphire is neglected, taking tank4 [8] throughout the optimization. C. Optimum gap forfixed dielectric and wafer sizes 10 i ;\ [1 *ixi.- A -/ Y U. 11'1,,,,111,,,,,111,1,, Fig. 4 shows the results of the gap height optimiza,tion for a dielectric with hp6mm, rp6mm and a 2 inch wafer where only an area of 48mm in diameter is exposed to the electromagnetic fields in the cavity. This figure indicates that, while the losses in the side walls are strongly dependent on h,, this dependence is moderate in the upper wall, and weak in the dielectric and lower wall. Therefore, the optimum height represents a trade-off between the losses in the upper wall and those in the side wall. This results in an optimum gap of h,=6mm and Qo/Q1,,,,=O.85, which essentially coincides with the results reported in [2]. As shown later, substantial improvements can be made by adjusting h,l, rd and h, jointly. Finally, Fig. 5 shows that these results degrade !strongly if the diameter of metal side wall is reduced. D. Substitution of the upper HTSplate by a cooper one Unsuccessful attempts have been made at rep lacing the YBCO in the upper wall by copper. Even when the gap height optimization process was performed taking into account the new R,, of the upper plate, the sensitivity (Qo/Ql(,,,) values were too low to allow an (acceptable accuracy in the determination of the R,, of the bottoin plate. Inl other words, the gap can never be high enough to replace the: upper film by a normal metal without making it a dominani. source of loss in the cavity and rendering it useless €or surface: resistance determination. Fig. 6 shows the effects of this substitution in the sensitivity (Qo/Q,,J. When dimensions of the wafer and the dielectric are known and fixed, only h, needs to be varied to find a maximum for Qo/Qro,,,. This is done for two reasons: to explore a possible improved design of the cavity in [2] (while maintaining the sapphire used); and to find a good initial set of values of hd, rd and h, from which a second optimization can be done using a gradient search in which all three variables are allowed to change. E. Joint optimization of hd, rd and h, The values of hd, rd and h, have been optimized using two parallel approaches: a gradient optimization and an IO" 1 O'O 1 0' 1 o6 1 0.8 1 o8 0.6 Q . gi 4 0.4 0.2 0 Fig. 4. Dependence of Qw, elow, Qsd with gap height (h,). The optimum height (max. QO /Q& represents a trade-off between the losses in the upper and side walls (elq, Qsd), whose dependence on h, is much stronger than that of the lower wall (elow) and dielectric (Qd). The maximum of Qo/QlOw is at h,=6mm. 0.8 0.6 8 . 0.4 0.2 Cavity radius (rnrn) Fig. 5. Dependence of Qw, elow, Q,Td with cavity radius. As the radius is decreased, losses in the side walls increase, become dominant, and degrade QO /el(,,,,. Gap height is hg=6mm. As in Fig. 4. dielectric height is h~6mm 1959 Fig. 6. Deterioration of the sensitivity (Qo/QlOw) as the value of R, of the upper sample is increased. A strong degradation occurs for the range of R,T values corresponding to cooper at 77K at the resonant frequencies of the cavity (about 10 mn). For these values of Rs in the upper wall, the Qo/Qim is too low for determination of the R,y of the lower HTS film. exhaustive search over a broad range of hd, rd and h, to ensure that the gradient optimization did not stop at a local maximum of Qo/Qbw. The exhaustive search was done from 2mm< hdlOmm, 3mm<r614mm and 1mm<hg<15mm. A maximum was found at hd=2mm, r~5mm and h,=6mm with Qo/Ql,,,~0.94. The search showed that this optimum value would degrade sharply when hd was decreased beyond its optimum value. However, the dependence with hd and rd was much weaker, Gradient optimizations were run starting from several values, including the ones from Sect. 11-C. They all reached similar values of Qo/Qro,,,. The best one was the one started from hs2mm, rd=5mm and h,=6mm (the optimum of the exhaustive search), which was used to fine-tune the previous search resulting in hd=l.7mm, r&.Omm and hg=4.6mm with Qo/Qlow =0.95. In other words, if no other losses are considered in a cavity with these dimensions, only a 5% overestimation of the surface resistance of the lower sample will be made, and this may be greatly with estimates of the surface resistance of the side walls, upper film and dielectric loss tangent. 111. CONCLUSION An optimization of the image dielectric resonator cavity has been done by numerical techniques. The software developed for the optimization has been cross-checked with previously published results. The optimized cavity should be allow the determination of surface resistance of YBCO with an error lower, than 5% if no other sources of RF loss in the cavity are taken into account. REFERENCES Z-Y Shen, C. Wilker, P. Pang, W. L. Holstein, D. Face, D.J. 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