**6. The combined superconducting theory and experiment of microwave losses**

The electrodynamics theory of E and B fields and losses, Q, in resonant cavities is well developed. Jackson [31], Csősz et al. [55], Kwon et al. [56], and other books [1, 2] have explicit equations for losses due to dielectric, metallic, or superconducting materials. Following Kwon et al., here we present in compact form (block diagram) (**Figure 11A**) a summary of the electrodynamics and losses of superconductors in resonant cavities. We CONNECT theoretical expressions with the experimental block diagram that measures Q (**Figure 11B**). The quantity that articulates both parts is the quality factor, Q, of the loaded cavity. On one hand, it can be arrived at from theory (BLUE OVAL), taking into account dissipative processes that develop in a superconducting sample or a Josephson junction. The SC theory represented by the leftmost blocks of the diagram feed the electromagnetic theory of Q with fundamental SC parameters, such as the coherence length, ξ; the London penetration depth, λ; critical temperatures; energy gap, Δ; values and critical fields, Hc, H1, and H2; their temperature and field behaviors; and so forth. On the other hand, we obtain Q by measurement (RED OVALS) with the equipment shown in **Figure 11B** or any other version of measurement with the perturbation cavity method.

**Figure 11** shows two block diagrams connected by the quantity Q. Block A includes the fundamental theory of superconductivity and t, the main phenomena, effects, and fundamental parameters, and block B shows a commonly used experimental arrangement for measuring microwave losses by means of the technique of cavity perturbation. Here, we include a field modulation stage that can bring much more sensitivity to details of the recordings [35–37]. But, the modulation stage can be bypassed and record directly the microwave absorption (see orange arrows in **Figure 11**). If temperature is slowly varied, then one obtains Q(T) (from PABS(T)). Likewise, if magnetic field is varied (and excite a lot of vortex dynamics and quasiparticle dynamics), then one obtains Q(H) (from PABS(H)).

Experimentally, it is possible to measure the Q of the cavity directly as an output of the amplifier or directly from the detector used. If the output is being modulated, then the measurement is the derivative with respect to varying magnetic field, Ho < H1, or Hc1 < Ho < Hc2, if the superconductor is in the Meissner state or the SC sample is a type II superconductor in the mixed state. Our block diagram includes low-temperature equipment coupled to the resonant cavity, so the sample could be set at any temperature between 4.2 K and 77 K, or 300 K, or any desired temperature. Temperature can also be gradually changed in order to register Q(T). By field modulation, the derivative of Q(T) with respect to magnetic field can also be recorded. Much more structure is captured in this way. SC losses, as measured inside resonant cavities, continue to be vigorous due to the wealth of novel information that continues to be discovered.

Next, we take a SC system, granular, crystalline JJ, with strong or weak links, JJ networks, SQC, and so forth, that is bathed with microwaves coming from a particular direction; several portions of the sample will respond to microwaves differently. Take the JJ represented in **Figure 10**; quasiparticles and electrons and holes in the sandwiched (the "middle") material will execute dissipative accelerated motions due to the Lorentz force they experience. Vortices will also execute

**23**

*Superconductivity and Microwaves*

3 + L4 + …, and 1/Qt = 1/Q1 + ….

(1/Q )T = 22% < (1/Q )T = 77%.

theorem and in consequence into the Q expression:

*DOI: http://dx.doi.org/10.5772/intechopen.86239*

translational viscous-dissipative motions because, again, the Lorentz force (or Magnus force) drives them [1–3, 26]. Pinned vortices will execute damped oscillations, and this will absorb electromagnetic energy. The normal electrons inside the fluxoid cores will move along with the fluxoids themselves; their motion is, again, dissipative. If we, abstractly, enumerate the different loss processes we have mentioned and some others that could operate (as the electrons and the holes produced in the multiple Andreev reflections), then we can write: total loss = Lt = L1 + L2 + L

We notice that several of these terms are of the same form with respect to the expression of the dissipated electromagnetic energy that enters into Poynting's

The higher the impurities and/or defects, the higher the pinning centers, the higher the microwave losses due to both fluxoid dissipative dynamics, (1/Q ) impure. The higher the normal conductivity, n, the lower the normal resistivity, and the larger the local mean free path that the normal charge carriers experience, and

the lower the normal conductivity, the higher the normal resistivity and the higher the losses (1/Q ). The higher the applied static field in the Hc1 < Ho < Hc2 range, the larger the number of fluxoids present or forming at the edge of the SC, Φ = nΦo, the more elements executing dissipative dynamics and more overall microwave loss. Varying temperature: given a superconductor, either HTCS or LTSC, the lower the temperature with respect to its critical temperature (Tc), let us say 22%, the more electrons form Cooper pairs and they will move around without dissipation. The electrons that remain normal are less and less as T departs from Tc; these electrons will produce less losses, (1/Q )T, than the same material at, let us say, 77% of Tc. Hence, for identical materials, all other parameters fixed, we would expect

From the theory, the integrals of electromagnetic energy are related to the complex conductivity, = n − iSC, and with the Q of the cavity. These integrals are a deduction of the electrodynamics of the superconductor based in Maxwell equations and the London equations. The superconducting parameters, ξ, λ, Δ, κ, S, , are involved in these equations, and hence they determine also the microwave absorption. From the experimental side, we have represented by blocks the main constituents of the generation of microwaves, the resonant cavity, and the detection system. The quantity measured is the Q of the cavity. Contrasting the theoretical formulas with the measured absorption and knowing the type of SC sample and/or

As a manner of example, we take the case of magnetic field dependent microwave

Most recent results: Very recently the absorption of microwaves by granular, fine powders of MgB2 and K3C60 type II superconductors was measured and found to

losses in superconducting niobium microstrip resonators reported by Kwon et al. [56]. They find that quasiparticle generation is the dominant loss mechanism for parallel magnetic fields. For perpendicular fields, the dominant loss mechanism is vortex motion or switches from quasiparticle generation to vortex motion, depending on the cooling procedures. They calculate the expected resonance frequency and the quality factor as a function of the magnetic field by modeling the complex resistivity. Key parameters characterizing microwave loss are estimated from comparisons of the observed and expected resonator properties. Based on these key parameters, they find that a niobium resonator whose thickness is similar to its penetration depth is the best choice for X-band electron spin resonance applications. They also detect partial release of the Meissner current at the vortex penetration field, suggesting that the interaction between vortices and the Meissner current near the edges is essential

if it is Josephson junction, an assignment of the losses can be made.

to understand the magnetic field dependence of the resonator properties.

in consequence, the less charge scattering and less energy loss. Conversely,

#### *Superconductivity and Microwaves DOI: http://dx.doi.org/10.5772/intechopen.86239*

*On the Properties of Novel Superconductors*

**microwave losses**

perturbation cavity method.

tion that continues to be discovered.

loss of electromagnetic energy inside the cavity and due to the SC sample inside it, and these losses are quantified by 1/Qs − 1/Qo. Now, how does the measurement of Q connects with the whole of the superconductivity theory and Maxwell equations?

The electrodynamics theory of E and B fields and losses, Q, in resonant cavities is well developed. Jackson [31], Csősz et al. [55], Kwon et al. [56], and other books [1, 2] have explicit equations for losses due to dielectric, metallic, or superconducting materials. Following Kwon et al., here we present in compact form (block diagram) (**Figure 11A**) a summary of the electrodynamics and losses of superconductors in resonant cavities. We CONNECT theoretical expressions with the experimental block diagram that measures Q (**Figure 11B**). The quantity that articulates both parts is the quality factor, Q, of the loaded cavity. On one hand, it can be arrived at from theory (BLUE OVAL), taking into account dissipative processes that develop in a superconducting sample or a Josephson junction. The SC theory represented by the leftmost blocks of the diagram feed the electromagnetic theory of Q with fundamental SC parameters, such as the coherence length, ξ; the London penetration depth, λ; critical temperatures; energy gap, Δ; values and critical fields, Hc, H1, and H2; their temperature and field behaviors; and so forth. On the other hand, we obtain Q by measurement (RED OVALS) with the equipment shown in **Figure 11B** or any other version of measurement with the

**Figure 11** shows two block diagrams connected by the quantity Q. Block A includes the fundamental theory of superconductivity and t, the main phenomena, effects, and fundamental parameters, and block B shows a commonly used experimental arrangement for measuring microwave losses by means of the technique of cavity perturbation. Here, we include a field modulation stage that can bring much more sensitivity to details of the recordings [35–37]. But, the modulation stage can be bypassed and record directly the microwave absorption (see orange arrows in **Figure 11**). If temperature is slowly varied, then one obtains Q(T) (from PABS(T)). Likewise, if magnetic field is varied (and excite a lot of vortex dynamics and quasi-

Experimentally, it is possible to measure the Q of the cavity directly as an output

of the amplifier or directly from the detector used. If the output is being modulated, then the measurement is the derivative with respect to varying magnetic field, Ho < H1, or Hc1 < Ho < Hc2, if the superconductor is in the Meissner state or the SC sample is a type II superconductor in the mixed state. Our block diagram includes low-temperature equipment coupled to the resonant cavity, so the sample could be set at any temperature between 4.2 K and 77 K, or 300 K, or any desired temperature. Temperature can also be gradually changed in order to register Q(T). By field modulation, the derivative of Q(T) with respect to magnetic field can also be recorded. Much more structure is captured in this way. SC losses, as measured inside resonant cavities, continue to be vigorous due to the wealth of novel informa-

Next, we take a SC system, granular, crystalline JJ, with strong or weak links, JJ networks, SQC, and so forth, that is bathed with microwaves coming from a particular direction; several portions of the sample will respond to microwaves differently. Take the JJ represented in **Figure 10**; quasiparticles and electrons and holes in the sandwiched (the "middle") material will execute dissipative accelerated motions due to the Lorentz force they experience. Vortices will also execute

particle dynamics), then one obtains Q(H) (from PABS(H)).

**6. The combined superconducting theory and experiment of** 

**22**

translational viscous-dissipative motions because, again, the Lorentz force (or Magnus force) drives them [1–3, 26]. Pinned vortices will execute damped oscillations, and this will absorb electromagnetic energy. The normal electrons inside the fluxoid cores will move along with the fluxoids themselves; their motion is, again, dissipative. If we, abstractly, enumerate the different loss processes we have mentioned and some others that could operate (as the electrons and the holes produced in the multiple Andreev reflections), then we can write: total loss = Lt = L1 + L2 + L 3 + L4 + …, and 1/Qt = 1/Q1 + ….

We notice that several of these terms are of the same form with respect to the expression of the dissipated electromagnetic energy that enters into Poynting's theorem and in consequence into the Q expression:

The higher the impurities and/or defects, the higher the pinning centers, the higher the microwave losses due to both fluxoid dissipative dynamics, (1/Q ) impure. The higher the normal conductivity, n, the lower the normal resistivity, and the larger the local mean free path that the normal charge carriers experience, and in consequence, the less charge scattering and less energy loss. Conversely, the lower the normal conductivity, the higher the normal resistivity and the higher the losses (1/Q ). The higher the applied static field in the Hc1 < Ho < Hc2 range, the larger the number of fluxoids present or forming at the edge of the SC, Φ = nΦo, the more elements executing dissipative dynamics and more overall microwave loss.

Varying temperature: given a superconductor, either HTCS or LTSC, the lower the temperature with respect to its critical temperature (Tc), let us say 22%, the more electrons form Cooper pairs and they will move around without dissipation. The electrons that remain normal are less and less as T departs from Tc; these electrons will produce less losses, (1/Q )T, than the same material at, let us say, 77% of Tc. Hence, for identical materials, all other parameters fixed, we would expect (1/Q )T = 22% < (1/Q )T = 77%.

From the theory, the integrals of electromagnetic energy are related to the complex conductivity, = n − iSC, and with the Q of the cavity. These integrals are a deduction of the electrodynamics of the superconductor based in Maxwell equations and the London equations. The superconducting parameters, ξ, λ, Δ, κ, S, , are involved in these equations, and hence they determine also the microwave absorption. From the experimental side, we have represented by blocks the main constituents of the generation of microwaves, the resonant cavity, and the detection system. The quantity measured is the Q of the cavity. Contrasting the theoretical formulas with the measured absorption and knowing the type of SC sample and/or if it is Josephson junction, an assignment of the losses can be made.

As a manner of example, we take the case of magnetic field dependent microwave losses in superconducting niobium microstrip resonators reported by Kwon et al. [56]. They find that quasiparticle generation is the dominant loss mechanism for parallel magnetic fields. For perpendicular fields, the dominant loss mechanism is vortex motion or switches from quasiparticle generation to vortex motion, depending on the cooling procedures. They calculate the expected resonance frequency and the quality factor as a function of the magnetic field by modeling the complex resistivity. Key parameters characterizing microwave loss are estimated from comparisons of the observed and expected resonator properties. Based on these key parameters, they find that a niobium resonator whose thickness is similar to its penetration depth is the best choice for X-band electron spin resonance applications. They also detect partial release of the Meissner current at the vortex penetration field, suggesting that the interaction between vortices and the Meissner current near the edges is essential to understand the magnetic field dependence of the resonator properties.

Most recent results: Very recently the absorption of microwaves by granular, fine powders of MgB2 and K3C60 type II superconductors was measured and found to

be giant, much larger than in the normal state, for magnetic fields as small as a few % of the upper critical field, a quite unexpected result [55]. The authors had the care to experimentally have the grains separated in large distances, so no weak links could be formed and no microwave absorption at the Josephson junctions could take place since no JJ were present.

Yet the microwave absorption is very large, and it represent the case in which σ₁ ≫ σ₂ [46]. We need to remember that σ₁ is due to the normal conductivity and σ₂ is the superconducting conductivity.

The effect is predicted by the theory of vortex motion in type II superconductors; however, its direct observation had been elusive due to skin-depth limitations; conventional microwave absorption studies employ larger samples where the microwave magnetic field exclusion significantly lowers the absorption. The authors show that the enhancement is observable in grains smaller than the penetration depth. A quantitative analysis on K3C60 in the framework of the Coffey-Clem (CC) theory (see the connection theory experiment in **Figure 11**) explains well the temperature dependence of the microwave absorption and also allows to determine the vortex pinning force constant P.

Dissipation and radiation of microwaves of the HTSCs have been measured since their discovery. Bednorz and Muller (the very same discoverers) and Blazey and many others [34–38] reported the modulated microwave absorption of YBaCuO and of several other HTSC families. The basic equipment used in these studies is pretty much like the one shown in **Figure 11**. The electron paramagnetic resonance equipments are well suited to carry on these Q and ∆f determinations provided that a zero magnetic field unit that could cross the zero value of the magnetic field is incorporated to a EPR spectrometer, in addition to a low-temperature (helium and/or liquid nitrogen) system. The cuprates, with their 3d9 unpaired electron, were expected to elicit EPR signals, and in consequence, some microwave absorption by this means was also expected. And certainly, these signals and absorptions have been detected and reported many times [34–38]. But in addition, the cuprates show an important non-resonance absorption at zero magnetic field or near-zero magnetic field [34–37]. Now it is clear that the Cu unpaired, bounded electron is not responsible of such microwave absorption. Bednorz and Muller and other authors have identified as the main absorbers, in the cuprates, the fluxoids, their dissipative dynamics and the loss processes in the weak links.

A strong microwave absorption at low magnetic fields is observed for a variety of metallic cuprates below their critical superconducting transition temperatures (T); using conventional electron paramagnetic resonance (EPR), instrumentation has been recorded [35]. The low-field microwave signal was investigated in the following high-temperature superconductors, YBaCuO (T = 95 K), GdBaCu (T = 95 K), etcetera. It has been investigated the usefulness of the technique as a practical nonintrusive method for screening potentially superconducting samples, with particular emphasis on sensitivity, the effect of microwave power, and the inherent problems of studying signals at small magnetic fields [32, 35].

### **7. Conclusions**

The fundamental property of superconductors to absorb microwave energy in spite of the expulsion of static magnetic fields (Meissner effect) has been explored for various SC, and its use in superconducting microwave technology has been indicated. The experimental measurements of the absorption of microwaves in the form of ESR-like measurements were described, and emphasis was put on the quality factor as one central quantity to measure, and we stressed its great theoretical

**25**

**Author details**

Rafael Zamorano Ulloa

Polytechnic Institute, Mexico City, Mexico

provided the original work is properly cited.

\*Address all correspondence to: davozam@yahoo.com

Physics Department, Superior School of Physics and Mathematics, National

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Superconductivity and Microwaves*

devices themselves.

*DOI: http://dx.doi.org/10.5772/intechopen.86239*

understanding. Q(T) and Q(H) magnetically modulated (or not) from microwave absorption were described, and few examples, spanning about 70 years of this field of research, were given. The knowledge gained on microwave losses can inform the design of new superconducting devices, SQC, JJs, SC qubits, microwave detectors, and/or radiators, operating in heavy microwave environments and/or as microwave

### *Superconductivity and Microwaves DOI: http://dx.doi.org/10.5772/intechopen.86239*

*On the Properties of Novel Superconductors*

take place since no JJ were present.

is the superconducting conductivity.

nitrogen) system. The cuprates, with their 3d9

pinning force constant P.

processes in the weak links.

**7. Conclusions**

be giant, much larger than in the normal state, for magnetic fields as small as a few % of the upper critical field, a quite unexpected result [55]. The authors had the care to experimentally have the grains separated in large distances, so no weak links could be formed and no microwave absorption at the Josephson junctions could

Yet the microwave absorption is very large, and it represent the case in which σ₁ ≫ σ₂ [46]. We need to remember that σ₁ is due to the normal conductivity and σ₂

The effect is predicted by the theory of vortex motion in type II superconductors; however, its direct observation had been elusive due to skin-depth limitations; conventional microwave absorption studies employ larger samples where the microwave magnetic field exclusion significantly lowers the absorption. The authors show that the enhancement is observable in grains smaller than the penetration depth. A quantitative analysis on K3C60 in the framework of the Coffey-Clem (CC) theory (see the connection theory experiment in **Figure 11**) explains well the temperature dependence of the microwave absorption and also allows to determine the vortex

Dissipation and radiation of microwaves of the HTSCs have been measured since

unpaired electron, were expected to

their discovery. Bednorz and Muller (the very same discoverers) and Blazey and many others [34–38] reported the modulated microwave absorption of YBaCuO and of several other HTSC families. The basic equipment used in these studies is pretty much like the one shown in **Figure 11**. The electron paramagnetic resonance equipments are well suited to carry on these Q and ∆f determinations provided that a zero magnetic field unit that could cross the zero value of the magnetic field is incorporated to a EPR spectrometer, in addition to a low-temperature (helium and/or liquid

elicit EPR signals, and in consequence, some microwave absorption by this means was also expected. And certainly, these signals and absorptions have been detected and reported many times [34–38]. But in addition, the cuprates show an important non-resonance absorption at zero magnetic field or near-zero magnetic field [34–37]. Now it is clear that the Cu unpaired, bounded electron is not responsible of such microwave absorption. Bednorz and Muller and other authors have identified as the main absorbers, in the cuprates, the fluxoids, their dissipative dynamics and the loss

A strong microwave absorption at low magnetic fields is observed for a variety of metallic cuprates below their critical superconducting transition temperatures (T); using conventional electron paramagnetic resonance (EPR), instrumentation has been recorded [35]. The low-field microwave signal was investigated in the following high-temperature superconductors, YBaCuO (T = 95 K), GdBaCu (T = 95 K), etcetera. It has been investigated the usefulness of the technique as a practical nonintrusive method for screening potentially superconducting samples, with particular emphasis on sensitivity, the effect of microwave power, and the

The fundamental property of superconductors to absorb microwave energy in spite of the expulsion of static magnetic fields (Meissner effect) has been explored for various SC, and its use in superconducting microwave technology has been indicated. The experimental measurements of the absorption of microwaves in the form of ESR-like measurements were described, and emphasis was put on the quality factor as one central quantity to measure, and we stressed its great theoretical

inherent problems of studying signals at small magnetic fields [32, 35].

**24**

understanding. Q(T) and Q(H) magnetically modulated (or not) from microwave absorption were described, and few examples, spanning about 70 years of this field of research, were given. The knowledge gained on microwave losses can inform the design of new superconducting devices, SQC, JJs, SC qubits, microwave detectors, and/or radiators, operating in heavy microwave environments and/or as microwave devices themselves.
