Abstract:
Based on its superconductive properties relating to “nonlineanty,” a conventional HTS strip is divisible into three “domains,” namely, a medial domain and two lateral domains. The nonlinearity associated with the conventional strip&#39;s medial domain is considerably greater than that which is associated with its lateral domains. Similarly divisible into a medial domain and two lateral domains, the present invention&#39;s HTS strip uniquely exploits these physical distinctions by causing more (e.g., most) of the current that it conducts to be conducted by its lateral domains. Various inventive designs accomplish this through narrowing or interruption/punctuation (e.g., via holes and/or trenches) or degradation, or some combination thereof, of the medial domain. By thus “re-proportioning” current conduction as compared with a conventional strip, an inventive strip succeeds in “re-proportioning” the associated nonlinearities. Consequently, the total nonlinearity associated with an inventive strip is significantly lower than that which is associated with a conventional strip.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to high temperature superconductors, more particularly to the use thereof in filters suitable for electronic applications such as those involving radar or communications. 
     High temperature superconductor (abbreviated “HTS”) filters have already seen some commercial success in the capacity of operation in or with receiver antennae in cellular phone base stations. Nevertheless, the potential exists for expanded use of HTS filters. The main impediment to extension of HTS filters to emitter antennae applications is the observed “surface impedance nonlinearity” that is associated with the dependence of the HTS filters on the microwave field power level. This nonlinearity occurrence generates undesirable “intermodulation” products that are particularly detrimental to radar and secure communication applications. Recent work asserts the origin of this phenomenon as an intrinsic property of HTS, and hence concludes that this phenomenon cannot be eliminated. 
     At the front end of any microwave receiver-antenna is a filter that cuts off (“excises”) all frequencies outside a predetermined frequency window (“bandpass”) to prevent the totality of airways signals from overwhelming the device. The operational principle of such filters may be described in terms of a “resonator cavity,” according to which the cavity is designed to resonate at a predetermined frequency window (“bypass band”), where transmission is at its maximum, while at all other frequencies transmission is strongly suppressed. In practice, such a microwave cavity is realized by a series of inductively coupled narrow copper “strips” (“poles”) that can be arranged in a variety of configurations. For pertinent disclosure, see, e.g, Zhi-Yuan Shen,  High - Temperature Superconducting Microwave Circuits , Artech House, Boston, May 1994, incorporated herein by reference, esp. Chapter Four. The terms “strip” and “pole” have been used interchangeably in the technical realm of HTS-based filters. 
     Shen at page 116 illustrates one such geometry (Shen, FIG. 4.14) and its corresponding bypass band (Shen, FIG. 4.15). More specifically, Shen states that his FIG. 4.14 shows “[a]four-pole superconductive microstrip filter layout. The substrate is LaAlO 3 .” Shen states that his FIG. 4.15 shows “[m]easured transmission response at 77K of a filter fabricated with a postannealed YbaCuO on 425-μm-thick substrate.” The box-like bypass band is roughly comprised of the sum of slightly shifted gaussian-shaped bypass bands from each of the poles. The more poles, the closer the bypass band shape is to an ideal “box” shape. On the other hand, increase in the number of poles is associated with increase in losses, which in turn degrades the bypass band sharpness (e.g., introduces “skirts”). The latter characteristics determine the degree to which signals outside the nominal bypass band are mixed in. 
     The same kind of unwanted mixing is also generated whenever the poles&#39; surface impedance is nonlinear, i.e., dependent on the microwave signal power level. The magnitude of these undesirable admixtures, known as “intermodulation products,” is a key performance measure of a filter. For the common copper-based filters, in which copper is to a high degree a linear material, a popular approach to minimization of intermodulation products is to increase the number of poles. This approach constitutes a tradeoff between the bypass band sharpness on the one hand, and an increase in the device volume and its noise level on the other hand. The latter effects represent disadvantages that are particularly important for military and civilian applications in which size and bypass band sharpness are key. Examples of such applications are those involving antennae arrays in radar, as well those involving compact, sensitive antennae. 
     The discovery in 1986 of the HTS family of materials led to the suggestion of replacing the copper strips in microwave filters with HTS strips. The distinct advantages of HTS-based filters are their significantly lower losses (which are, depending on temperature, by about two to three orders of magnitude) and their relatively small size. Due to these and other advantages (such as affordability, reliability and low maintenance), as well as due to the advent of maintenance-free small-volume closed-cycle coolers, HTS-based filters have become a commercial reality. Notable among the companies involved in this line of business are Superconductor Technologies Inc. (STI) of Santa Barbara, Calif., and Conductus, Inc. of Sunnyvale, Calif. STI recently completed a merger with Conductus, and can be contacted at 460 Ward Drive, Santa Barbara, Calif., 93111. The current deployment of HTS-based filter units in over a thousand commercial wireless base-stations represents the first large scale application of such devices. See pertinent disclosure by B. Willemsen, “HTS Technology for Commercial Wireless Applications,”  Journal of Superconductivity  (in press, 2003), incorporated herein by reference. Moreover, the sharp bandpass and low losses characterizing HTS-based filters are expected to become increasingly influential in the increasingly crowded civilian cellular phones spectrum. HTS-based filters have military implications as well. For instance, the United States Navy is interested in extending the applicability of HTS-based filters to higher power emit antennae, such as may be used for radar. 
     The technological progress in HTS-based filter utilization is hampered, however, by the observed HTS surface impedance nonlinearity, i.e., its dependence on the microwave field power level. See pertinent disclosure by the following papers incorporated herein by reference: P. P. Nguyen, D. E. Oates, G. Dresselhaus, M. S. Dresselhaus and A. C. Anderson,  Phys. Rev. B  51, 6686 (1995); Y. M. Habib, C. J. Lehner, D. E. Oates, L. R. Vale, R. H. Ono, G. Dresselhaus and M. S. Dresselhaus,  Phys, Rev. B  57 13833 (1998). Surface impedance nonlinearity has been observed in low temperature superconductor (abbreviated “LTS”) films such as Niobium Nitrate (NbN). See pertinent disclosure by the following papers, incorporated herein by reference: P. P. Nguyen, D. E. Oates, G. Dresselhaus, M. S. Dresselhaus and A. C. Anderson,  Phys. Rev. B  57, 6686 (1995); Y. M. Habib, C. J. Lehner, D. E. Oates, L. R. Vale, R. H. Ono, G. Dresselhaus and M. S. Dresselhaus,  Phys, Rev. B  57, 13833 (1998). Surface impedance nonlinearity has also been observed in HTS films of YBCO and BSCCO and TBCCO (the Bi 2 Sr 2 CACuO and Tl 2 Ba 2 CaCuO groups). See pertinent disclosure by the following papers, incorporated herein by reference: J. H. Claasen, J. C. Booth, J. A. Beall, L. R. Vale, D. A. Rudman and R. H. Ono,  Supercond. Sci. Technol.  12, 714 (1999); J. C. Booth, L. R. Vale, R. H. Ono and J. H. Claasen,  Supercond. Sci. Technol.  12, 711 (1999); H. Claasen, J. C. Booth, J. A. Beall, D. A. Rudman, L. R. Vale and R. H. Ono,  App. Phys. Lett.  74, 4023 (1999); J. C. Booth, J. A. Beall, D. A. Rudman, L. R. Vale and R. H. Ono,  Journal App. Phys.  86, 1020 (1999). As of yet there is no consensus regarding the origin of surface impedance nonlinearity. See pertinent disclosure by D. E. Oates, M. H. Hein, P. J. Hirst, R. G. Humphreys, G. Koren and E. Polturak, “Nonlinear Microwave Surface Impedance of YBCO Films: Latest Results and Present Understanding,”  Physica C  (Superconductivity), volumes 372–376, part 1, pages 462–468 (1 Aug. 2002; available online 9 Apr. 2002), incorporated herein by reference. 
     Recent YBCO data provides clear evidence that nonlinearity is observed in high quality films and is enhanced by factors such as magnetic/non-magnetic impurity doping (Zn, Ni) and oxygen underdoping. See pertinent disclosure by the aforementioned D. E. Oates et al.,  Physica C  372–376, 462 (2002). It has been suggested that vortex motion in large-angle grain and twin boundaries, which are ubiquitous in HTS, are the root cause for the nonlinearity phenomenon. See pertinent disclosure by the following papers, incorporated herein by reference: M. Coffey and J. R. Clem,  Phys. Rev. B  48, 342 (1993); M. Benkraouda and J. R. Clem,  Phys. Rev.  53, 5716 (1996); J. McDonald, J. R. Clem and D. E. Oates,  Phys. Rev. B  55, 11823 (1997); J. Halbritter,  Journal of Superconductivity  10 91 (1997). This attribution to vortex motion is motivated by the observation that such defects act as Josephson junctions; since vortex motion in a Josephson junction is nonlinear, it is reasoned, the ensuing surface impedance is nonlinear as well. While this mechanism is certainly at work it has proved to be quantitatively unable to account for the observed nonlinearity at elevated power levels. See pertinent disclosure by the aforementioned D. E. Oates et al.,  Physica C  372–376, 462 (2002). See pertinent disclosure also by H. Xin, D. E. Oates, A. C. Anderson, R. L. Slattery, G. Dresselhaus and M. S. Dresselhaus,  IEEE Trans. Microwave Theory and Techniques,  48, 1221 (2000), incorporated herein by reference. 
     A competing proposition is that the nonlinearity is intrinsic to the highly correlated electron state (the condensate state) that underlies the superconductivity phenomenon. This proposition (“intrinsic nonlinearity”) is consistent with the observed nonlinearities in LTS and HTS that are qualitatively different, and with the observed nonexistent frequency dependence of the nonlinearity effect. See pertinent disclose by the aforementioned D. E. Oates et al.,  Physica C  372–376, 462 (2002). This proposition was further corroborated by calculations that quantitatively explain recent data of high quality, optimally oxygenated HTS sample. See D. Agassi and D. E. Oates, “Nonlinear Surface Reactance of a Superconductor Strip,”  Journal of Superconductivity , Vol. 16 No. 5, pp. 905–961 (October 2003). 
     Encased YBCO (YBa 2 Cu 3 O 7-δ ) strips (typically 300 nm thick, 100μ wide and on the order of several centimeters long) exhibit, at elevated power levels, an amount of nonlinearity sufficiently large to introduce intermodulation products to a degree that is deleterious for radar applications. If HTS emit-filters are to become a reality, then, nonlinear microwave response must be controlled and reduced. It is thus essential that the nonlinear surface impedance of HTS filters be reduced at microwave frequencies, in order to enhance the performance of HTS filters and extend their operational range to power applications such as those involving transmitting antennae and radar. 
     The following United States patent documents, incorporated herein by reference, pertain to superconductivity and superconductor devices, especially such involving high-temperature superconducting materials: Fuke et al. U.S. Pat. No. 6,529,092 B2 issued 4 Mar. 2003; Eden U.S. Pat. No. 6,516,208 B1 issued 4 Feb. 2003; Wikborg et al. U.S. Pat. No. 6,463,308 B1 issued 8 Oct. 2002; Talanov et al. U.S. Pat. No. 6,366,096 B1 issued 2 Apr. 2002; Hershtig U.S. Pat. No. 6,212,404 issued 3 Apr. 2001; Adam U.S. Pat. No. 6,094,588 issued 25 Jul. 2000; Mansour U.S. Pat. No. 6,041,245 issued 21 Mar. 2000; Larson et al. U.S. Patent Application Publication 2002/0130729 A1 published 19 Sep. 2002; Larson et al. U.S. Patent Application Publication 2002/0130716 A1 published 19 Sep. 2002. 
     SUMMARY OF THE INVENTION 
     In view of the foregoing, it is an object of the present invention to provide a methodology for lowering the nonlinear surface impedance of high temperature superconducting (HTS) filters at microwave frequencies. 
     In accordance with many embodiments of the present invention, a superconductor strip (e.g., a high temperature superconductor strip), suitable for situation upon a substrate, comprises superconductor material (e.g., high temperature superconductor material). The inventive superconductor strip has three domains, viz., a pair of lateral domains and a medial domain (intermediate the lateral domains). When the inventive superconductor strip conducts electrical current: the amount of current that is conducted by the inventive superconductor strip is the sum of the current that is conducted by its medial domain and the current that is conducted by its lateral domains; and, more than fifty percent of the current that is conducted by the inventive superconductor strip is conducted by its lateral domains (equivalently expressed, less than fifty percent of the current that is conducted by the inventive superconductor strip is conducted by its medial domain). According to typical inventive embodiments, the superconductor strip is characterized by a strip width, the medial domain is characterized by a medial width, and each lateral domain is characterized by a lateral width. The strip width is the sum of the medial width and the lateral widths. Each lateral width approximately equals the penetration depth of the inventive HTS strip. 
     Generally according to inventive practice, the inventive superconductor strip has associated therewith a strip nonlinearity, the medial domain has associated therewith a medial nonlinearity, and the lateral domains have associated therewith a lateral nonlinearity. The medial nonlinearity is greater than the lateral nonlinearity. The strip nonlinearity is the sum of the medial nonlinearity and the lateral nonlinearity. The strip nonlinearity varies in accordance with the proportions, with respect to the conducting of the current by the strip, of the conducting by the medial domain and the conducting by the lateral domain. According to many inventive embodiments, the medial domain is attributed with at least one element that reduces the conductivity of the medial domain as compared with what the conductivity of the medial domain would be in the absence of such at least one element. The element or elements is or are the following: (i) the presence of plural apertures in the medial domain; and/or, (ii) the presence of plural depressions in the medial domain; and/or, (iii) the presence of insulating oxide material in the medial domain. 
     The present invention also provides a method for reducing the nonlinearity associated with a superconductor strip (which is at least substantially composed of a superconductor material) placed on a substrate. Many embodiments of the inventive method are for reducing the nonlinearity associated with a high temperature superconductor strip (which is at least substantially composed of a high temperature superconductor material). The inventive method comprises: identifying two longitudinal lateral domains and one longitudinal medial domain (located between the lateral domains) into which the superconductor strip is widthwise divided, each lateral domain having a width approximately equal to the penetration depth of the superconductor strip; and, modifying the medial domain so that the conductivity of the medial domain decreases, the conductivity of the lateral domains proportionately increasing relative to the overall conductivity of the superconductor strip, the overall nonlinearity of the superconductor strip commensurately decreasing. Generally according to the inventive method, the modifying of the medial domain results in the conductivity of the lateral domains being greater than one-half of the overall conductivity of the superconductor strip. According to typical inventive embodiments, the modifying of the medial domain includes performing at least one of the following: (i) providing plural apertures in said medial domain; and/or, (ii) providing plural depressions in said medial domain; and/or, (iii) degrading said medial domain. Usually, the degrading of the medial domain includes heating the medial domain so as to deoxygenate the medial domain, the degrading resulting in the medial domain being at least substantially composed of a dielectric oxide material. 
     The present invention further provides a combination including a substrate and at least one conductive strip disposed on the substrate. Each conductive strip includes a superconductive material (e.g., a high temperature superconductive material) and is characterized by two lateral domains and a medial domain therebetween. The lateral domains of a conductive strip at least substantially consist of superconductive material (e.g., a high temperature superconductive) material. Each lateral domain of a conductive strip has a width corresponding to the penetration depth the conductive strip. The lateral domains of a conductive strip conduct more than half of the electricity that is conducted by the conductive strip. In other words, the medial domain of a conductive strip conducts less than half of the electricity that is conducted by the conductive strip, since the total amount of electricity that is conducted by the conductive strip is the amount of electricity that is conducted by the conductive strip&#39;s medial domain plus the amount of electricity that is conducted by the strip&#39;s lateral domains; correspondingly, the total amount of nonlinearity that is associated with conduction by the conductive strip is the amount of nonlinearity that is associated with conduction by the conductive strip&#39;s median domain plus the amount of nonlinearity that is associated with conduction by the conductive strip&#39;s lateral domains. According to typically inventive practice, the conductive strip is a “thin film” variety of conductive strip. Generally preferred inventive embodiments include one or more conductive strips that are at least substantially made of high temperature superconductor material. 
     Nonlinearity results in unwanted intermodulation products that are harmful in contexts of radar and communication applications. The present inventor&#39;s scientific understanding is that nonlinearity is an inherent characteristic of HTS and therefore cannot be eliminated. The present invention features new strip (pole) geometries that represent unique modifications of old strip (pole) geometries, the present invention thus affording the benefits of substantial reductions in nonlinearities. The inventive implementation of novel strip geometries provides an effective strategy for approaching the ideal form of a compact linear sharp bypass band device. The present invention&#39;s strip geometries succeed in lowering or minimizing the nonlinear surface impedance of HTS filters at microwave frequencies, thereby enhancing their performance and extending their operational range to high power emission applications such as transmission antennae and radar. 
     The aforementioned D. Agassi and D. E. Oates,  Journal of Superconductivity , Vol. 16 No. 5, pp. 905–961 discloses that the operative mechanisms for nonlinearity are qualitatively distinct at the two opposite borders or margins (i.e., the HTS strip&#39;s two outside extreme regions), as compared with the middle (i.e., the HTS strip&#39;s single inside region, located between the HTS strip&#39;s two extreme outside regions). The HTS strip&#39;s two bordering/marginal regions are synonymously referred to herein as the “lateral domains” or “near-edge domains” of the HTS strip. The HTS strip&#39;s middle/intermediate region, located between the HTS strip&#39;s two near-edge domains, is synonymously referred to herein as the “medial domain” or “midsection domain.” Each near-edge domain is a region that is delimited by an edge and that covers a longitudinal portion of the HTS strip that is proximate or adjacent to the edge. 
     In the strip&#39;s midsection domain, intrinsic nonlinearity dominates. Since the midsection domain typically covers most of the strip width, good agreement with the nonlinearity data follows. The present invention avails itself of the distinction in mechanisms operating in these two domain types, viz., the HTS strip near-edge sections versus the HTS strip midsection. That is, there are two distinct types of nonlinearities in a strip: (i) the intrinsic nonlinearity that emanates from the strip midsection; and, (ii) the Lorentz-force nonlinearity that emanates from the strip near-edge sections due to the large current density and gradients at such locations. The strip&#39;s midsection type of nonlinearity is “large” (considerably larger) and is frequency-independent. In contrast, the strip&#39;s near-edge type of nonlinearity is “small” (considerably smaller), and decreases in accordance with f −2 , where f is the frequency. The present inventor believes that the intrinsic nonlinearity that is present in the strip&#39;s midsection may also be present, but to an as yet unknown extent, in the strip&#39;s near-edge sections (in addition to the Lorentz-force nonlinearity that is present in the strip&#39;s near-edge sections). 
     The present invention provides HTS strips that, for many applications, have superior performance as compared with conventional HTS strips. As compared with a conventional superconductor strip, the inventive superconductor strip&#39;s overall current conduction is decreased by decreasing the medial domain&#39;s current conduction while not changing the lateral edge domains&#39; current conduction; in this manner, the respective current conductions, and hence the associated nonlinearities, are “re-proportioned” in favor of the lateral edge domains. According to frequent inventive practice, the majority portion (more than half) of the electricity is caused to be conducted via the two lateral (“near-edge”) domains, while the minority portion (less than half) of the electricity is caused to be conducted via the medial (“midsection”) domain. The present invention effects these ratios or proportions of conductibility (of the midsection domain relative to the edge domains in an HTS strip) through new modifications that are broadly characterized herein as being “geometric” or “degradative” in character. The present invention&#39;s “geometric” modes include “discontinuous” and “dimensional” modes of HTS strips. The “discontinuous” geometric modes of HTS strips include “apertural” modes and “depressional” modes of HTS strips. 
     Whether an HTS strip is conventional or inventive, the HTS strip will usually describe the aforenoted three regions (“domains”), viz., two marginal regions (“lateral domains” or “near-edge domains”) and one intermediate region (“medial domain” or “midsection domain”). According to typical “wide-strip” geometric modes of the present invention (such as shown herein in  FIG. 4  and  FIG. 5 ), the dimensions of the inventive HTS strip are about the same as those of a conventional HTS strip; however, in significant contrast to a conventional HTS strip, the conductibility of the midsection domain of the inventive HTS strip is reduced (e.g., minimized) through the formation of discontinuities (e.g., perforations or trenches) in the midsection domain of the inventive HTS strip. According to typical “narrow-strip” geometric modes of the present invention (such as shown herein in  FIG. 3 ), the dimensions of the inventive HTS strip are markedly different from those of a conventional HTS strip; that is, in contrast to a conventional HTS strip, the conductibility of the midsection domain of the inventive HTS strip is reduced (e.g., minimized) through the narrowing of the midsection domain of the inventive HTS strip. According to typical “wide-strip” degradative modes of the present invention (such as shown herein in  FIG. 6 ), the dimensions of the inventive HTS strip are about the same as those of a conventional HTS strip; however, in significant contrast to a conventional HTS strip, the conductibility of the midsection domain of the inventive HTS strip is reduced (e.g., minimized) through the material degradation of the midsection domain of the inventive HTS strip. 
     Thus, preferred inventive modes of HTS strips include the following: (1) an inventive dimensional geometric mode of a narrow strip line, wherein the HTS strip&#39;s width is at least about five penetration lengths, but less than about fifty penetration lengths; (2) an inventive discontinuous geometric mode of a wide strip line, wherein the width of the wide strip is two to three orders of magnitudes of the penetration depth, and wherein large apertures (e.g., holes, perforations, openings, voids, etc.) in the midsection domain (the apertures typically covering more than half of the area of the strip midsection domain) impede the current flow along the strip in its midsection domain; (3) an inventive discontinuous geometric mode involving wide strip lines, wherein the width of the wide strip is two to three orders of magnitude of the penetration depth, and wherein an array of wide depressions (e.g., trenches, grooves, furrows, channels, etc.) in the midsection domain impede the current flow along the strip in its midsection domain; (4) an inventive degradative mode involving wide strip lines, wherein material degradation (of the HTS material) of the midsection domain impedes the current flow along the strip in its midsection domain. Inventive practice is not limited to selection of just one inventive mode, as many inventive embodiments will combine elements of two or more inventive modes (e.g., midsection narrowness and/or midsection apertures and/or midsection depressions and/or midsection degradation). 
     The present invention&#39;s unique foundational principle is its achievement of the following functional objective: The reduction (e.g., minimization) of the contribution of the HTS strip&#39;s middle longitudinal region to the overall current conduction by the HTS strip. The present invention&#39;s geometric and material configurations result in the forcing of “most” (more than about half) of the flow of the microwave-induced current to occur in the two extreme longitudinal regions. The various modes of inventive practice, which achieve the inventive functional objective, are classified herein as “narrow” strips (wherein the midsection domain is concomitantly “narrow”) or as “wide” strips (wherein the midsection domain is concomitantly “wide” and is discontinuous, e.g., perforated, trenched or degraded). In the light of the present disclosure, the ordinarily skilled artisan will be capable of practicing the present invention&#39;s HTS strip forms at a micron length scale, following standard fabrication practices such as are known in SQUID design. 
     The present invention&#39;s novel high temperature superconductor (HTS) strip forms, suitable for use as poles of HTS microwave filters, succeed in minimizing the nonlinear surface impedance of such HTS microwave filters, thereby extending the applicability range of such HTS microwave filters to high power emission applications. The present invention lends itself to diverse applications (including but not limited to those involving HTS microwave filter apparatus) wherein HTS material is utilized in electronic apparatus. A main advantage of the present invention&#39;s new HTS filter designs are their suitability for both receiving antennae and emitting antennae, due to a substantial reduction in their nonlinear responses. At the same time, the inventive HTS strip forms retain the advantages traditionally associated with HTS filters. The basic inventive principle underlying the inventive HTS strip forms allows ample “head-space” for inventive performance optimization. Non-HTS-based filters, such as copper-based filters that are commonly used for emit antennae applications, have their place in the world of electronics; nevertheless, HTS filters have greater potential than have traditional non-HTS filters. For a variety of applications, the present invention&#39;s HTS-based filters better realize this potential than do known HTS-based filters. 
     Other objects, advantages and features of this invention will become apparent from the following detailed description of the invention when considered in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order that the present invention may be clearly understood, it will now be described, by way of example, with reference to the accompanying drawings, wherein like numbers indicate the same or similar components, and wherein: 
         FIG. 1  is a diagrammatic plan view of a segment of a conventional “wide” HTS strip. 
         FIG. 2  is a graphical representation, coupled with a diagrammatic transverse cross-sectional elevation view of a conventional “wide” HTS strip, illustrating the current density profile j X (Y) in accordance with the width w of the HTS strip. The HTS strip&#39;s width w is very much greater than the HTS strip&#39;s thickness d. 
         FIG. 3  is a diagrammatic plan view of a segment of an inventive “narrow” HTS strip. 
         FIG. 4  is a diagrammatic plan view of a segment of an inventive “wide” HTS strip characterized by apertures located in the midsection domain of the HTS strip. 
         FIG. 5  is a diagrammatic plan view of a segment of an inventive “wide” HTS strip characterized by depressions located in the midsection domain of the HTS strip. 
         FIG. 6  is a diagrammatic plan view of a segment of an inventive “wide” HTS strip characterized by material degradation in the midsection domain of the HTS strip. 
         FIG. 7  is a diagrammatic plan view of a segment of an inventive “wide” HTS strip characterized by a combination or “conglomeration” of inventive features (including apertures, depressions and material degradation) existing in the midsection domain of the HTS strip. 
         FIG. 8  is a graph comparing calculation (using equation (1), hereinbelow) and data for the lowest nonlinear correction to the penetration depth. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to  FIG. 1 , conventional HTS strip  10  is a “wide” strip that includes a midsection domain  12  and two near-edge domains  14 . Each near-edge domain  14  is bounded on one side by a longitudinal edge  16  and on the other side by a longitudinal delineation  18 . Midsection domain  12  is bounded on both sides by the two delineations  18 . Each delineation  18  separates midsection domain  12  from a near-edge domain  16 . The two edges  16  and the two delineations  18  are all approximately parallel. Near-edge domains  14  are each represented in  FIG. 1  by a hatched area. The dotted area represents midsection domain  12 . Conventional strip  10  is situated upon a substrate  30  that, according to common practice, is dielectric. The combination of substrate  30  and at least one conventional strip  10  is includable in a high temperature superconducting device such as an HTS filter. 
     Strip  10  is characterized by a width w and a length l.  FIG. 1  is a partial view, as length l may be envisioned to extend well beyond the longitudinal ends of strip  10  as depicted. Midsection domain  12  is characterized by a width w M  that is smaller than the overall width w of strip  10 . Each near-edge domain  14  is characterized by a width w N  that is smaller than the width w M  of midsection domain  12 . Thus, w=w M +2w N . HTS strip  10  has a flat (planar) upper (top) surface  20  and two longitudinal side (top-to-bottom) surfaces  22 . Each edge  16  is a junction of upper surface  20  and a side surface  22 . Accordingly, midsection domain  12  has an upper surface  20   M  having width w M . Each near-edge domain  14  has an upper surface  20   N  having width w N .  FIG. 1  is diagrammatically idealized insofar as the edges  16  of a conventional superconductor strip  10  are shown to be perfectly straight, but in fact are rough or uneven. 
     Reference is now made to  FIG. 2 , which correlatively illustrates the profile of microwave current-density j X (Y) together with the widthwise cross-section (well distanced from the longitudinal ends of HTS strip  10 ) of a conventional wide, long HTS strip  10 . As illustrated in  FIG. 2 , the microwave current-density profile is strongly peaked in the two near-edge domains  14 , but is much lower and constant in the mid-section domain  12  as well as the associated magnetic field wrapped around strip  10 . See pertinent disclosure at pages 125–131 of T. Van Duzer and C. W. Turner,  Principles of Superconductive Devices and Circuits , Elsevier, N.Y., 1981, said book incorporated herein by reference. The two demarcation  18  lines correspond to the current-density delineations illustrated in  FIG. 2 . The current-density peaks at the near-edge domains  14  extend over a penetration depth λ, which typically is orders of magnitude smaller than the width w of strip  10 . That is, each near-edge domain  14  of strip  10  has a width w N  that approximately equals the penetration depth λ of strip  10 . Thus, w≐w M +2 λ, since λ≐w N . An approximative equivalence in width w N  of both near-edge domains  14  is inferred from the symmetrical current-density profile shown  FIG. 2 . 
     The magnetic field H wraps around strip  10  such that, at the midsection domain  12  and the two near-edge domains  14 , the magnetic field H is both parallel and perpendicular to the longitudinal top-to-bottom surfaces  22  of strip  10 . These current and field distributions clearly indicate the existence of two physically distinct domain genres, viz., (i) the midsection domain  12  and (ii) the pair of near-edge domains  14 . Midsection domain  12  is characterized by an almost constant, small current-density j X (y), and is further characterized by a magnetic field H that is aligned approximately parallel to the surface  20  of strip  10 . The two near-edge domains  14  are where the current density j X (y) and its spatial gradient are high, while the magnetic field H is aligned approximately perpendicular to the surface  20  of strip  10 . In a wide strip  10 —i.e., wherein w&gt;&gt;λ—the area of midsection domain  12  is significantly larger than the area in total of the two near-edge domains  14 . 
     At the midsection domain  12  of strip  10  the configuration of current density j X (y) together with magnetic field H conforms with an infinite, perfectly flat superconductor in the presence of a transversal field. See pertinent disclosure in Section 39 of A. Abrikosov, L. Gorkov and I. Dzyaloshinski,  Methods of Quantum Field Theory in Statistical Physics , Dover Publications, Inc., New York, 1963, said book incorporated herein by reference. As noted hereinabove, the condensate state in midsection domain  12  yields a “large” nonlinear response. On the other hand, at the near-edge domains  14  the physics is dominated by the large current-density values and gradients. In this case, the paired-electrons current is particularly affected by the Lorentz force, which is largest at the edges, since the Lorentz force is proportional to the (vector) product of the current and magnetic field, whereas the magnetic field is proportional to the current (curl) derivative by virtue of the second London equation. See pertinent disclosure in Section 39 of the aforementioned book Abrikosov et al.,  Methods of Quantum Field Theory in Statistical Physics , Dover Publications, Inc., New York, 1963. The ensuing response is manifestly nonlinear. See the aforementioned paper included herein as Appendix A, coauthored by the present inventor and D. E. Oates and as yet unpublished by the  Journal of Superconductivity.    
     The frequency dependencies and magnitudes of these two nonlinearity contributions are qualitatively and quantitatively distinct. For a wide strip  10 , the intrinsic contribution associated with midsection domain  12  is “large” and practically independent of frequency f. See the aforementioned paper included herein as Appendix A, viz., Y. D. Agassi et al., in press for publication in 2003 by the  Journal of Superconductivity . In contradistinction, the Lorentz force contribution associated with the near-edge domains  14  is “small” and proportional to 1/f. Based on this analysis, the present invention seeks to minimize the intrinsic “large” contribution of midsection domain  12  to the aggregate nonlinearity, thereby maximizing the Lorenz force-related “small” contribution of the two near-edge domains  14  to the aggregate nonlinearity. The present invention uniquely accomplishes this minimization of the cumulative nonlinearity by providing various embodiments of inventive HTS strip  100 , which represent new HTS strip forms in terms of geometry and/or materiality. 
     With reference to  FIG. 3  through  FIG. 7 , the present invention takes advantage of the scientific principles discussed hereinabove in connection with  FIG. 2 . The inventive HTS strips  100  shown in  FIG. 3  through  FIG. 7  each parallel the conventional strip  10  shown in  FIG. 1  insofar as being characterized by “regionalization.” Each inventive strip  100  embodiment includes a midsection domain  112  and two near-edge domains  114 . Each near-edge domain  114  is bounded on one side by a longitudinal edge  116  and on the other side by a longitudinal delineation  118 . Midsection domain  112  is bounded on both sides by the two delineations  118 . Midsection domain  112  is bounded on both sides by the two delineations  118 . Each delineation  118  separates midsection domain  112  from a near-edge domain  116 . The two edges  116  and the two delineations  118  are all approximately parallel. Inventive HTS strip  100  has an upper surface  120  and two longitudinal side (top-to-bottom) surfaces  122 . According to most inventive embodiments, upper surface  120  essentially describes an imaginary geometric plane, albeit inventive practice is variable in terms of continuity or discontinuity of upper surface  120 . Similarly as conventional strip  10  is depicted in  FIG. 1 , inventive strip  100  is situated upon a substrate  30 ; in inventive practice, substrate  30  will typically be dielectric, but need not be, depending on the embodiment. The combination of substrate  30  and at least one strip  10  is includable in a high temperature superconducting device such as an HTS filter.  FIG. 3  through  FIG. 7  are diagrammatically idealized insofar as the edges  116  of an inventive superconductor strip  100  are shown to be perfectly straight, but in fact are rough or uneven. 
     Because inventive strip  100  and conventional strip  10  are analogous in some respects, the same reference characters are used herein for strip dimensions. Inventive strip  100  is characterized by a width w and a length l.  FIG. 3  through  FIG. 7  are partial views, as in each said figure length l may be envisioned to extend well beyond the longitudinal ends of inventive strip  100  as shown. Midsection domain  12  is characterized by a width w M  (which is smaller than the overall width w of inventive strip  100 ), and each near-edge domain  114  is characterized by a width w N  (which is smaller than the overall width w of inventive strip  100 ); therefore, w=w M +2w N . Each edge  116  is a junction of upper surface  120  and a side surface  122 . Accordingly, midsection domain  112  has an upper surface  120   M  having width w M . Each near-edge domain  114  has an upper surface  120   N  having width w N . Similarly as shown for conventional strip  10  in  FIG. 1 , the near-edge domains  114  of inventive strip  100  are each represented by a hatched area, and the midsection domain  112  of inventive  100  is represented in  FIG. 3  through  FIG. 7  by either of two kinds of dotted areas. 
     With particular reference to  FIG. 4 ,  FIG. 5  and  FIG. 7 , inventive strip  100  comprises superconductive material (e.g., high temperature superconductor material) and is divisible into two lateral domains  114  and a medial domain  112  therebetween. Inventive strip  100  is capable of conducting an amount of current that at least substantially consists of an amount of current conducted by lateral domains  114  and an amount of current conducted by medial domain  112 . Medial domain  112  is discontinuous so that medial domain  112 , and hence inventive strip  100 , comprises a lesser amount of superconductive material than if medial domain  112  were continuous. Inventive strip  100  is thereby characterized by a greater proportion of conduction by the lateral domains  114  than if the medial domain  112  were continuous. In inventive practice in general, it may be preferable, depending on the inventive embodiment, to provide for more than half of the current being conducted by lateral domains  114 ; however, this condition is not necessary to inventive practice and is not present in some inventive embodiments. 
     As illustrated in  FIG. 4 ,  FIG. 5  and  FIG. 7 , the discontinuity of medial domain  112  is manifested as at least one of the following properties: (i) a plurality of apertures in the inventive strip&#39;s medial domain, such as shown in  FIG. 4 ; and/or, (ii) a plurality of depressions in the inventive strip&#39;s medial domain, such as shown in  FIG. 5 ; and/or, (iii) a plurality of insulative locations in the inventive strip&#39;s medial domain, such as shown in  FIG. 7 . A combination of these properties is represented in  FIG. 7 . Inventive strip  100  is characterized by nonlinearity that is a function of the amount of current conducted by inventive strip  100 . Lateral domains  114  are characterized by “weak” nonlinearity that is less magnitudinous that which characterizes medial domain  112 . Medial domain  112  is characterized by “strong” nonlinearity that is more magnitudinous than that which characterizes lateral domains  114 . Strip  100  is thereby characterized by a greater proportion of the weak linearity than if medial domain  112  were continuous. 
     Like conventional strip  10  shown in  FIG. 1 , the inventive strips  100  shown in  FIG. 4 ,  FIG. 5 ,  FIG. 6  and  FIG. 7  are “wide” HTS strips, in which the widths w N  of near-edge domains  114  are considerably smaller than the width w M  of midsection domain  112 . The width w of a “wide” HTS strip, whether a conventional strip  10  or an inventive strip  100 , is such that the ratio of width w to penetration depth λ is at least about two hundred; that is, w/λ≧200. In contrast, inventive strip  100  shown in  FIG. 3  is an inventively “narrow” HTS strip. As shown in  FIG. 3 , each width w N  of a near-edge domain  114  is larger than the width w M  of midsection domain  112 . Inventively “narrow” HTS strip  100  replaces traditional strip “wideness” with inventive strip “narrowness.” 
     An inventively “narrow” strip  100  is such that the ratio of width w to penetration depth λ falls within the range between about five and about fifty; that is, 5≦w/λ≦50. In inventively “narrow” strips  100 , the “large” nonlinear contribution by midsection domain  112  is drastically reduced, leaving a drastically increased “small” nonlinear contribution by the two near-edge domains  114 . The present invention establishes this preferred w/λ value range of approximately 5 to 50 because it covers a desired “middle ground” between w/λ values that are too low and w/λ values that are too high. If w/λ is too small (i.e., below 5, e.g., in the 2–3 range) or too large (i.e., above 50, e.g., in the 70–80 range), the near-edge domains&#39; current density peaks will merge, to a considerable extent, into an approximately uniform charge distribution, thus essentially leaving a situation akin to that of a conventional “wide” strip  10  (which is characterized by a uniform current distribution). 
     The uneven topographies of the inventive strips  100  shown in  FIG. 4  and  FIG. 5  significantly differ from the even topography of the conventional strip  10  shown in  FIG. 1 . An inventive HTS strip  100  having a “wide” width dimension (i.e., wherein w/λ≧200) may be easier to fabricate than would an inventive HTS strip  100  having a “narrow” width dimension (i.e., wherein 5≦w/λ≦50). The midsection domain  112  shown in  FIG. 4  is provided with large holes  150 . The midsection domain  112  shown in  FIG. 5  is provided with trenches  160 . According to either of these inventively “discontinuous” configurations, the electromagnetic current will primarily flow along the near-edges  114  of inventive strip  100 . Similarly, the electromagnetic current will primarily flow along the near-edges  114  of “degradative” inventive strip  100  shown in  FIG. 6  and “conglomerative” inventive strip  100  shown in  FIG. 7 . 
     Various approaches can be taken to making inventive strips  100 . In the light of the instant disclosure, the ordinarily skilled artisan will be capable of making inventive strips  100 . The fabrication of inventive geometries such as shown in  FIG. 3  through  FIG. 5  can be achieved using standard techniques such as ion-milling patterning that is routinely practiced in SQUID design. According to the ion-milling patterning technique that is known in SQUID-related applications, holes of various geometries (“antidots”) and narrow line shapes are introduced for different physical reasons. See pertinent disclosure by: S. J. Kim, Yu I. Latyshev, T. Yamashita and S. Kishida,  Physica C  362, 150 (2001); A. S. Katz, S. I. Wood and R. C. Dynes,  Jour. of App. Phys.  87, 2978 (2000); D. Koelle et al.,  Rev. Mod. Phys.  71, 631 (1999); G. Benz, S. Wunsch, T. A. Scherer, M. Neuhaus and W. Jutzi,  Physica C  356, 122 (2001). Other pertinent references are disclosed in the aforementioned SQUID review article D. Koelle et al.,  Rev. Mod. Phys.  71, 631 (1999). 
     The above-noted ion milling technique can be used to manufacture, from scratch, “narrow” geometrically inventive strips  100  (such as shown in  FIG. 3 ) as well as “wide” geometrically inventive strips  100  (such as shown in  FIG. 4  and  FIG. 5 ). According to inventive methods of fabricating inventive strips  100  in a single operation, HTS material is deposited as a thin film upon substrate  30 . In so doing, an “etching” technique or a “stenciling” technique can be inventively effectuated. According to an “etching” technique, the inventive strip  100  geometries are etched in the initially deposited HTS material situated on substrate  30 . According to a “stenciling” technique, substrate  10  is initially covered with a sheet or mask so that when HTS material is deposited on substrate  30 , the desired inventive strip  100  geometries are reproduced in the HTS film beneath the sheet or mask. 
     Alternatively, inventive “wide” strips  100  can be made via modification of a conventional “wide” strip, such as strip  10 , that has previously been disposed on a substrate  30  such as shown in  FIG. 1 . In practicing “wide” strips  100  in accordance with the present invention, the conventional strips  10  already produced can be inventively modified so as to become one of the following: an aperturally discontinuous inventive embodiment  100  such as shown in  FIG. 4 ; or, a depressionally discontinuous inventive embodiment  100  such as shown in  FIG. 5 ; or, a degradative inventive embodiment  100  such as shown in  FIG. 6 ; or, some combination thereof, such as shown in  FIG. 7 . Production of an apertural inventive embodiment  100  can involve, for example, the perforating of the HTS film in midsection  12  with an array of holes  150 , thereby converting the midsection  12  HTS film into a midsection  112  apertured HTS film such as shown in  FIG. 4 . Production of a depressional inventive embodiment  100  can involve, for example, the impressing of the HTS film in midsection  12  with an array of trenches  160 , thereby converting the midsection  12  HTS film into a midsection  112  trenched HTS film such as shown in shown in  FIG. 5 . Production of a degradative inventive embodiment  100  can involve, for example, the degrading of the HTS film in midsection  12  by locally heating the HTS film in midsection  12  through application of a strong focused electron beam that will deoxygenate the HTS material, thereby converting the midsection  12  HTS film into a midsection  112  insulating oxide film such as shown in  FIG. 6 . Production of a combinative embodiment such as shown in  FIG. 7  can involve a combination of these techniques. 
     Midsection  12  shown in  FIG. 1 , midsection  112  shown in  FIG. 3 , and the areas of midsection  112  other than the inventive apertures  150  shown in  FIG. 4  and the inventive depressions  160  shown in  FIG. 5 , are each demarcated by a coarser, less concentrated dot pattern, indicating a “non-degraded” HTS material. In contrast, midsection domain  112   DEG  shown in  FIG. 6  is demarcated by a finer, more concentrated dot pattern, indicating an inventively “degraded” material, which is an insulating oxide material rather than an HTS material. The inventive tape  100  embodiment shown in  FIG. 7  features apertures  150 , depressions  160  and degraded midsection domain sections  112   a   DEG  and  112   b   DEG . Degraded tape sections  100   a   DEG  and  100   b   DEG  (which include degraded midsection domain sections  112   a   DEG  and  112   b   DEG , respectively) alternate longitudinally with non-degraded tape sections  100   a  and  100   b  (which include non-degraded midsection domain sections  112   a  and  112   b , respectively). Although these inventive features are shown in  FIG. 7  in the context of an inventive “wide” strip  100  embodiment, they can be inventively practiced in association with inventive strips  100  of varying widths w, including inventively “narrow” widths w, or widths w that are wider than inventive “narrow” widths w but narrower than “wide” widths w, or widths w that are narrower than inventive “narrow” widths w. 
     The nonlinearity associated with the midsection domain of a conventional strip (such as conventional strip  10  that shown in  FIG. 1 ) is “large” and is practically frequency-independent; by comparison, other sources for nonlinearity, such as vortex in-out motion, are documented to be much smaller. See pertinent disclosure by the aforementioned D. E. Oates et al.,  Physica C  372–376, 462 (2002). Accordingly, the observed large nonlinearity of a conventional strip derives from the large width w of the conventional strip, this large width w typically falling in the range between about 100μ and about 150μ. Penetration depth λ is typically two to three orders of magnitude smaller than this large width w. See the aforementioned paper, included herein as Appendix A, coauthored by the present inventor and D. E. Oates and in press for publication in 2003 by the  Journal of Superconductivity.    
     According to the present invention&#39;s novel concept, it follows that nonlinearity reduction (e.g., minimization) is expected to reside in an HTS strip configuration that reduces (e.g., minimizes) the proportional midsection domain contribution. The present invention uniquely accomplishes reduction/minimization of the contribution by the midsection domain to the overall conduction along the strip—and hence uniquely accomplishes reduction/minimization of the nonlinearity of the strip—by forcing the microwave current to disproportionately flow along the two near-edge domains, where nonlinearity is considerably smaller. The inventive HTS strip  100  forms shown in  FIG. 3  through  FIG. 7  are examples of inventive embodiments that achieve this inventive goal. 
     The term “strip,” as used herein, refers to any conductor (e.g., superconductor) strip that is characterized by a uniform width. In terms of its geometric outline, an inventive strip can be rectilinear, curvilinear or some combination thereof, so long as it at least substantially, essentially or generally has a uniform width.  FIG. 3  through  FIG. 7  show segments of inventive superconductor strips that are straight but that are readily envisioned as curved. Nor is it necessary that a “strip,” as used herein, be characterized by a single imaginary axis of symmetry between its two edges that describe a uniform width. “I”-shaped, “T”-shaped, “L”-shaped, “U-shaped and various other outline geometries of a conductor strip are possible in inventive practice. 
     As conventionally understood, high temperature superconductor (HTS) materials are materials that exhibit superconductivity at temperatures over 24K (i.e., superconducting transition temperature T c &gt;24K), whereas low temperature superconductor (LTS) materials are materials that exhibit superconductivity at temperatures no higher than 24 K (i.e., superconducting transition temperature T c ≦24K). Currently, there are HTS materials having a T c  temperature as high as about 133K, and higher T c  temperatures are expected in the future. Although HTS superconductivity is emphasized herein, the present invention admits of practice using any and all superconductor materials, including HTS materials, LTS materials, and superconductor materials having T c  temperatures therebetween. Notable in the intermediate T c  temperature category (neither HTS nor LTS) is magnesium diboride (MgB2), which has a a T c  temperature of about 39.5K. 
     With reference to  FIG. 8 , the present inventor discloses additional verification that intrinsic nonlinearity predominates in a superconductor strip, in his unpublished paper (coauthored with D. E. Oates) entitled “Non Linear Surface Reactance of YBCO in the Presence of a Strong Microwave Radiation Field,” submitted on or about March 2003 to  Physica C  for publication, and substantively presented at the Seventh International Conference on Materials and Mechanisms of Superconductivity and High Temperature Superconductors, May 25–30, 2003, Rio de Janeiro, Brazil. At this conference in Brazil the present inventor presented a poster, incorporated herein by reference, relating to his unpublished paper. 
     The data illustrated in  FIG. 8  compares favorably in trend and in magnitude with the theoretical model that is proposed and calculated as described hereinbelow. The data relates to YBCO film that was grown on an MgO substrate via pulsed laser deposition. The basics of the experimental procedure are disclosed by the aforementioned D. E. Oates et al.,  Physica C  372–376, 462 (2002). The YBCO film (having an undulating strip configuration of approximate width 150 μm, approximate thickness 350 nm and approximate length 2 cm) was placed in a microwave stripline resonator cavity, and the experiments were performed at frequency f=2.2 GHz. The nonlinear surface impedance was extracted through measurement of the shift and broadening of the cavity resonance frequency or the intermodulation products power. 
     Still referring to  FIG. 8 , the nonlinear surface reactance of a d-wave superconductor is calculated through effectuation of a microscopic approach that is based on a perturbative expansion in the field strength, and is compared to the optimally-doped YBCO film data. The data is analyzed in terms of the two fluid model expressions wherein the reactance in particular is proportional to the penetration depth. The reactance is fitted in terms of an expansion in even powers of the microwave current density j. Therefore, the lowest nonlinear contribution to the reactance is λ 2 j 2 . The objects of the calculation are the extracted coefficients λ 2 . The nonlinear response of the condensate state to a transversal electromagnetic field is carried out in the framework of a microscopic approach. Starting from the perturbative expansion of the green function in the electromagnetic field, the λ 2  coefficient is extracted from the third order term. This differs from a previous analysis, which was based on the Yip-Saul approximation. The calculation is facilitated by consideration only of the static long wavelength limit, consistent with the observed weak frequency dependence. This approach is advantageous in that the result does not invoke free parameters, thus providing a stringent test of the basic premise. The order parameter d-wave symmetry is an essential part of this analysis. The derived expression for λ 2  in cgs units is 
                 λ   2     ⁡     (   T   )       =       -       8   ⁢     e   4     ⁢       λ   0   5     ⁡     (   T   )       ⁢     d   2     ⁢   a   ⁢           ⁢     μ   2           m   ab     ⁢     c   6     ⁢     ℏ   2     ⁢   β   ⁢           ⁢     a   c     ⁢       Δ   0   3     ⁡     (   T   )             *       ∑   n     ⁢       ∫   0     2   ⁢   π       ⁢       ⅆ       θcos   4     ⁡     (   θ   )         ⁢       cos   2     ⁡     (     2   ⁢   θ     )       ⁢           cos   2     ⁡     (     2   ⁢   θ     )       -     4   ⁢       (       ℏω   n     /       Δ   0     ⁡     (   T   )         )     2             (         cos   2     ⁡     (     2   ⁢   θ     )       +       (     ℏ   ⁢           ⁢       ω   n     /       Δ   0     ⁡     (   T   )           )     2       )       7   /   2                         (   1   )             
 
     In equation (1), above: T is the temperature; e is the electron charge; λ 0 (T) is the (linear) penetration depth; d is the strip thickness; α is a dimensionless approximation-related parameter α 10; μ is the Fermi energy; m ab  is the effective mass in the ab plane; β=(k B T) −1  where k B  is the Boltzmann constant; α c  is the lattice constant in the c-axis direction; the d-wave gap is Δ(T,θ)=Δ 0 (T)cos(2θ) where θ is the azimuth cylindrical-coordinates angle; ω n =(2n+1)π/(βh) is the Matsubara frequency where the summation in the above equation over n covers all integers n. 
     FIG.  8 ′ illustrates a comparison of the calculated results and the experimental results. The parameters employed are α=10, Δ 0 (T=0)/(k B T)=3, μ/Δ 0 (T=0)=25, α c =1.1 mm, λ 0 (T)=λ 0 /√{square root over (1−(T/T C ) 2 )} with λ 0 =200 nm[7] and Δ 0 (T)=Δ 0 √{square root over ((1−(T/T C ) 2 ))} [12] with Δ 0 =0.024 eV. With the exception of a, these parameters are standard with no attempted adjustments. The results compare well with the data both qualitatively and absolutely. The upturn at low temperatures (approximately as 1/T) has been predicted and is characteristic of a d-wave order parameter symmetry. At higher temperatures, thermal excitations overshadow the effects of the d-wave nodes, resulting in a dramatic decrease of the difference between the results for an s-wave and d-wave order parameter symmetry. Based on this comparison between calculation and experimental data, the proposition is strengthened that the observed nonlinearity existing in a superconductor strip is predominantly an intrinsic effect. The good agreement of the calculation with the data suggests that the observed nonlinearity is intrinsic. Additional substantiation of this proposition can be sought in an analogous manner involving explanation of the observed dependence of λ 2  on factors such as doping. 
     Other embodiments of the present invention will be apparent to those skilled in the art from a consideration of this specification or practice of the present invention disclosed herein. Various omissions, modifications and changes to the principles described herein may be made by one skilled in the art without departing from the true scope and spirit of the present invention, which is indicated by the following claims.