arXiv:1605.04273v2 [cond-mat.supr-con] 10 Oct 2016 antcfil n hi plcto o microwave for application spectroscopy their and parallel field in magnetic resonators stripline Pb Superconducting Abstract. 2016 10th, October E-mail: Germany Pfaff Stuttgart, Universit¨at Stuttgart, Institut, Physikalisches 1. Dressel Martin Scheffler Thiemann, Marc Markus and Ebensperger, G Nikolaj esrmns(ihapeapidpwrt oki ierres linear a well in as work depth microwave penetration to parts, par performed resistance, variable, power a surface we in determined Therefore applied resonators stripline Pb ample superconducting understood. fi (with well in resonators measurements be micr such to for of has basis electrodynamics the a measurements), as material suitable a so present resonators. is sensitive in as Pb interest act of superconducting materials also various and of applications spectroscopy technical of variety bsrpiersntr nteeprmna td fohr( other of study field. experimental magnetic variable the a applicability in the in demonstrate materials resonators to magneti stripline resonator furt and We Pb Pb 6.5K a predictions. and in cr BCS 1.6K sample the with between Sn agreement and range good field temperature showing magnetic 140mT, a critical in the Pb determine of also we data, fmgei ed eew n etrsrmnseto hs i in those maximum of reminiscent a features as find such we Here measurements, dependent field. magnetic of natraiewyt lmnt hs vnat even f these ar eliminate are that to ze effects areas way Besides conducting alternative Hysteresis phase. normal superconducting an to the in due Pb. persist is therefore ha the This field magnetic within swept field. the magnetic filaments once remaining resonances of of to factor assign quality form superc we type-I the which this in response, of microwave field low-loss critical a the above fields magnetic ouiieP tiln eoaosi aibemgei fie magnetic variable a in resonators stripline Pb utilize To σ 1 [email protected] and lnrsprodcigmcoaersntr r e eleme key are resonators microwave superconducting Planar σ 2 ftecmlxcnutvt fsprodcigP safunc a as Pb superconducting of conductivity complex the of , T < T c nadig5,D-70569 57, enwaldring ae normicrowave our on Based . σ 1 ofil-oln eshow we ro-field-cooling le antcfil.We field. magnetic allel i tt eerh Here research. state lid chrnepa) At peak). (coherence rbsfrmicrowave for probes xeddtecritical the exceeded s nutrw tl find still we onductor eladimaginary and real s iemgei field magnetic nite fsuperconducting of tcltemperature itical super-)conducting superconductivity indadcan and pinned e emr td a study hermore d(..i ESR in (e.g. ld temperature- n wv stripline owave ed pto up fields c transmission udi the in ound os)on ponse) t na in nts tion Superconducting Pb stripline resonators in parallel magnetic field and their application for microwave spectroscopy2
1. Introduction onators in finite magnetic field. This is not only rel- evant for spectroscopy, but like any exposure of a su- Building a microwave device (or just a simple element perconductor to a magnetic field is also an interesting of certain geometry) from a superconducting material issue on its own. Operating planar microwave devices and obtaining detailed knowledge of its properties is in magnetic field has been studied in the context of useful in several different contexts. Firstly, it can vortex motion, dissipation, and pinning in type-II su- help for the full understanding of the superconducting perconductors, where a pronounced field dependence is material itself. Microwave frequencies, with a found that is unwanted for applications [43–46]. In con- typical range of 1-100GHz, correspond to thermal trast, for type-I superconductors such questions have, energies of 0.05-5K and are in the range of or so far, attracted much less attention. In fact, one mo- below critical temperatures Tc and energy gaps tivation for our work on Pb resonators is that for a of many superconductors. Thus the microwave type-I superconductor, up to the critical field, there response of a superconductor depends on both should be very weak field dependence, which is con- the dissipationless Cooper pairs and the dissipating venient for spectroscopic studies. However, in earlier thermally excited quasiparticles [1, 2]. Accordingly, work it has been found that normal-conducting areas microwave experiments probe the different charge similar to vortices can persist in bulk superconductors carriers, and give access to quantities such as of type I [47] and especially in Pb [48–51]. This flux superconducting penetration depth (related to Cooper trapping effect, as well as an increased critical mag- pair density and superfluid stiffness), energy gap, netic field for resistance restoration and a rather broad surface resistance, and complex optical conductivity resistance transition commonly occur in “hard” super- [3, 4], but they can also reveal different dynamical conductors of type I that contain many impurities and properties, such as quasiparticle relaxation, critical imperfections [52–54]. However, this has so far rarely fluctuations, or collective excitations [5–10]. been investigated with microwaves in a variable mag- Another field of interest for superconducting mi- netic field, yet a detailed understanding is required as crowave devices are technical applications such as mi- residual normal-conducting areas in Pb could gravely crowave filters, but they also include numerous types influence the electrodynamic properties. of resonators, detectors, amplifiers, and related devices In this work we examine Pb stripline resonators in fundamental research [11–14]. Depending on ap- in varying external magnetic field up to 140mT, be- plication, e.g. in astrophysics or quantum information low and above the critical magnetic field Bc, at tem- science, planar superconducting resonators might be peratures of 1.6K to 6.5 K. With the determina- operated with vastly different power levels, and, corre- tion of penetration depth and surface resistance we spondingly, different microscopic processes govern the are able to access the complex conductivity and find resonator performance and possible strategies for de- striking similarities to measurements in variable tem- vice improvements [15–19]. Microwave spectroscopy in perature. We study possible influences of remaining condensed matter physics is yet another resonator ap- normal-conducting areas in the superconducting phase plication. For many materials of present research, such by first suppressing superconductivity with magnetic as magnetic and heavy-fermion materials, there exist fields above Bc and then regaining superconductivity important low energy scales that can be accessed with by decreasing the magnetic field. In order to demon- microwave spectroscopy [20–25], and a very prominent strate the possibility to use Pb stripline resonators for case here is the vast range of conventional and uncon- microwave spectroscopy applications in varying mag- ventional superconductors, including cuprate, pnictide, netic field, we study a Sn sample with a critical tem- organic, heavy-fermion, and strongly disordered super- perature of Tc ≈ 3.7 K and a critical magnetic field of conductors [26–29]. To this end, a variety of different Bc ≈ 30 mT [39] with the stripline resonator. microwave spectroscopy techniques have been devel- oped [30–35]. 2. Experimental setup and methods Studying conducting materials is also our mo- tivation for the development of superconducting Pb The geometry of our microwave stripline resonators stripline resonators that can act as compact and sen- is shown in Fig. 1, as three-dimensional schematic sitive probes in cryogenic microwave spectroscopy [36– view (a) and in cross section (b,c). They consist of 38]. Here Pb with Tc ≈ 7.2 K and a critical magnetic a 1 µm thick and 45 µm wide center conductor, which field of about Bc ≈ 80 mT is a well suited basis ma- is embedded between two dielectric sapphire plates terial [39], as it can be processed for thin films rather of 127 µm thickness, which in turn are sandwiched easily. For the spectroscopic study of certain mate- between two conductive ground planes above and rials [21, 22, 40–42], application of a magnetic field in below. The Pb center conductor is evaporated onto one the order of tens of mT is helpful, and therefore we of the sapphire plates and has a meander structure. address the microwave properties of Pb stripline res- Superconducting Pb stripline resonators in parallel magnetic field and their application for microwave spectroscopy3
Two gaps in the center conductor separated of −30 dBm. from each other at a certain distance l reflect At resonance frequencies of the stripline resonator the microwave signal to a large extend, allowing the microwave meets the resonance condition resulting the microwave to form a standing wave between in a maximum of microwave transmission. At the gaps. Fundamental frequencies vary with the frequencies deviating from the resonance frequency the separation l of the gaps and in our case range transmission decreases rapidly, leading to a Lorentzian between 1 GHz to 2 GHz with higher harmonics form of the transmission signal around resonance of the fundamental frequency being measurable up frequencies [58]. After correction of the data performed to 20 GHz. This propagation of the microwaves similar to [38], a fit of a Lorentzian with center allows measurements with superconducting stripline frequency ν and full width at half maximum ∆ν is resonators to be particularly sensitive to the state of possible. This enables the calculation of the quality the material near the surface. In order to study an factor Q = ν/∆ν for any given resonance, defined additional superconducting material (here: Sn), one of as the ratio of power stored to power dissipated each the ground planes is replaced by a sample, as depicted cycle. It is a measure of losses in the resonator in Fig. 1(c). and dominantly influenced by the conductors. By replacing one ground plane with a superconducting (a) Pb ground plane or sample of different material that has lower critical Sn sample temperature and lower critical magnetic field than magnetic field sapphire the underlying resonator, the losses in the resonator direction become dominated solely by the losses in the sample, giving experimental access for measurements explicitly on the sample. The determination of the quality factor Q also Pb center allows for the calculation of the surface resistance [38] Pb ground conductor plane coupling gap Rs = πµ0Γν/Q , (1) with Γ a geometrical constant of the resonator [38,59] (b) (c) and µ0 the permeability constant. Additionally the Sn surface reactance [1]
h εr X = −Z0νλ/(2πc) (2) B ε B s h t w r Pb Pb can be calculated with λ the penetration depth into the superconducting material and Z0 ≈ 377Ω the Figure 1. Schematic geometry of a microwave stripline impedance of the vacuum. Calculation of the real resonator in 3D view (a) and in cross section without sample (b) and imaginary parts of the complex conductivityσ ˆ = and with additional Sn sample (c). As indicated, the direction of the magnetic field is parallel to the resonator. σ1 + iσ2 is then possible using 2 2 2 σ1 = −Z02ω(XsRs)/(c(Xs + Rs) ) (3) Depending on the field of application in which superconducting microwave resonators are used, the and 2 2 2 2 2 power that is applied to drive them has to be adjusted σ2 = Z0ω(Xs − Rs)/(c(Xs + Rs) ) , (4) properly. Circuit QED experiments in quantum with c the speed of light. information science, for example, might employ an We studied the stripline resonator experimentally average number of microwave photons in the resonator with a vector network analyzer (VNA) in transmission, that can be much smaller than one [12] whereas the recording the complex transmission parameter S21. drive/readout powers for superconducting resonators 4 We performed the measurements in a He cryostat, can be many orders of magnitude higher for other reaching down to temperatures of about 1.6 K. A static applications [56]. At very small power, the resonator magnetic field was applied parallel to the resonator response might be non-linear, i.e. depend on the as indicated in Fig. 1 ‡. Before each measurement applied microwave power, due to two-level fluctuators with variable magnetic field, zero field cooling (ZFC) [15] whereas at very high powers there can be non- has been performed, meaning the resonator has been linear behavior due to the superconductor [57]. For cooled down below its superconducting transition our experiments, we want the applied power to be temperature in the absence of a magnetic field and the high enough for good signal-to-noise ratio in the magnetic field has been applied subsequently at a fixed transmission measurement but yet well below the temperature. power range where our stripline resonators becomes non-linear, [38], and we eventually used an input power Superconducting Pb stripline resonators in parallel magnetic field and their application for microwave spectroscopy4
3. Results and discussion 120000
1.6 K
2.5 K
100000
For a resonator made entirely out of Pb (center 3.0 K conductor as well as ground planes) the quality factor 5.0 K
6.0 K Q in a varying magnetic field has been determined as 80000
6.5 K shown in Fig 2 for several different temperatures. Q 3.29 GHz
60000
was obtained for the second mode of the resonator at 3.29 GHz with a magnetic field up to 120 mT. At fields
40000 below the critical field Bc(T ), Q only decreases little. Qualityfactor This regime resembles the Meissner state. Following a
20000 drastic decrease, indicating increasing losses close to the normal conductor transition, a plateau emerges 0 leading to two kinks in the course of Q (e.g. in
0 20 40 60 80 100 120 the range of 77-84mT at T = 1.6K). This area Magnetic field [mT] is the intermediate state where superconducting and normal conducting areas coexist [49]. Since a perfectly Figure 2. Quality factor Q for a resonance of the Pb parallel alignment of the resonator to the magnetic resonator with increasing magnetic field up to 120 mT at field is practically impossible, an intermediate phase different temperatures in the range between 1.6 K and 6.5K. Q shows only a small decrease in the Meissner state with is always existent [60] and was found throughout all B < Bc(T ). Close to Bc(T ), Q decreases drastically before our measurements. This has also been found in earlier entering the intermediate state. Q still shows a smaller but publications on Pb microwave resonators [22, 61], but substantial remaining value even at B > Bc(T ), indicating was not discussed there. possible superconductivity in the form of filaments.
Leaving this state towards higher fields, Q
3.0 becomes comparably small, however it does not vanish ] 3.14 K
3.81 K
2.5
4.48 K
80 completely. This could very well be caused by 5.51 K
5.82 K
2.0 remaining superconductivity in the resonator, which is 6.49 K
7.16 K typical for hard superconductors of type I [52]. Several 1.5 Resistance[
60 similar effects are found, e.g. in tantalum where a 1.0 gradual transition of the resistance at B>Bc happens 0.5 and is associated with superconducting “filaments” [53, 0 20 40 60 80 100 120 140
Magnetic field [mT]
54]. This gradual transition has also been theoretically 40 critical field predicted [62] and found in alloys of type II [63,64]. In upper limit of intermediate phase
dc measurement
our measurements these effects lead to the small values Fit to critical field
Fit to upper limit
20
of Q, often persisting to magnetic fields way above Criticalmagnetic field [mT]
fit parameter (Fit to critical field):
B (T = 0 K) = 81.3 mT
Bc(T ), e.g. in the measurement at 1.6 K up to about c
T = 7.43 K 120 mT, while Bc(T = 1.6 K) ≈ 77 mT, showing that c our Pb specimen contains impurities and imperfections 0 making it a hard superconductor. 0 2 4 6 8 From the data in Fig. 2 the critical magnetic field Temperature [K] of Pb could be determined for each temperature at Figure 3. Bc(T ) and upper limit of the intermediate phase of half the maximum value of Q as well as the upper Pb in dependence of temperature as determined from microwave field limit of the intermediate state. This is shown measurements (squares). A DC measurement of the resistivity in in Fig. 3. Additionally DC-measurements using the magnetic field (using a transmission line with the same geometry as the resonator without the gaps) is shown in the inset. Bc(T ) same geometry as the resonator (without the gaps for derived at the first increase in resistivity is shown in the main microwave reflectance) are shown in the inset. Here panel as empty circles. The empirical equation for Bc(T ) has Bc(T ) has been determined at the first increase in been fitted to the microwave data and the values for Tc and resistivity. The subsequent course of the resistivity at Bc(0) show good agreement with earlier measurements [65–67]. higher fields shows a gradual increase typical for hard superconductors and the existence of superconducting resulting values Bc(0) = 81.3 mT and Tc = 7.43K are filaments at B >Bc(T ) [52–54]. The Bc(T ) data of in good agreement with earlier measurements [65–67]. the microwave measurement has been fitted with the Strong hysteresis could be seen when increasing empirical equation the external magnetic field to values higher than 2 Bc(T )= Bc(0)(1 − (T/Tc) ) , (5) Bc(T ). Q for a resonance at 3.29 GHz with an increasing and afterwards decreasing field is shown as with B (T ) the critical field at a given temperature c filled symbols in Fig 4 at T ≈ 1.6K. Before this and Bc(0) the critical field at zero temperature. The Superconducting Pb stripline resonators in parallel magnetic field and their application for microwave spectroscopy5 measurement zero field cooling (ZFC) was performed, is therefore a well-fitting procedure to circumvent these to allow for full restoration of superconductivity unwanted effects. and maximum Q. During the increase of the field The lines in Fig 4 show a measurement using the same resonator and mode at 3.29 GHz now at 120000 T ≈ 3.0K. The external magnetic field has been increasing magnetic field repeatedly increased to a maximum positive value and 100000 decreasing magnetic field increasing magnetic field afterwards decreased to a maximum negative value. 3.29 GHz The first maximum field exceeded Bc(T ) and thus 80000 T 1.6 K Q again shows hysteresis. The following iterations of this measurement however, lowered the maximum 60000 field each time to a value lower than the previous
iteration (both positive and negative). Since Bc(T ) is
40000 not exceeded during later iterations, Q increases again, Quality factor Quality first iteration reaching almost initial values. This is explained by the last iteration introduction of normal conducting areas of opposite 20000 3.29 GHz polarity into the resonator with the second half of each T 3.0 K iteration, where the magnetic field has negative value 0 -100 -80 -60 -40 -20 0 20 40 60 80 100 [45]. Normal conducting areas of opposing polarity Magnetic field [mT] annihilate leading to a reduction of overall normal conducting area. Figure 4. Filled symbols show the quality factor for a resonance The regaining of the original state with this of the Pb resonator with variable magnetic field up to 100 mT procedure can be useful if ZFC can not be performed after performing ZFC. The magnetic field has first been increased (e.g. if it is unpractical to exceed T ), since a simple to a positive maximum value and afterwards decreased to a c negative maximum value. Q shows hysteresis both for positive magnetic field sweep procedure can allow for complete magnetic fields and for negative magnetic fields as it does not restoration even at very low temperatures. reach initial values after once exceeding Bc(T ). Lines show With increasing temperature the resonance fre- Q of the Pb resonator for variable magnetic field alternating quency ν of each resonance shifts to lower frequencies. between positive and negative maximum values with decreasing maximum value of the swept field for consecutive iterations. This is due to the penetration depth increasing with After showing strong hysteresis in the first iterations, Q rises temperature [69], which leads to a change of the ef- almost to initial values with following iterations, indicating fective geometry of the resonator. The frequency de- annihilation of normal conducting areas. creases following the expression [38] −1/2 Q drops to low values similar to Fig 2 at fields ν = ν0 [1 + λ(T )πµ0/Γ] , (6) higher than B (T ). When decreasing the magnetic c with ν0 the resonance frequency at zero penetration field again, Q starts to rise again at similar B (T ), c depth. λ(T ) is the temperature-dependent penetration however it does not reach its initial values of about 5 depth, which we can assume to change as λ = 1.16 × 10 , but instead only reaches a maximum value 4 −1/2 4 λ0 1 − (T/T ) near T [70]. By fitting (6) to of about 7.97 × 10 at B = 0. Additionally Q still c c a shift of a resonance with rising temperature, ν0 increases substantially below Bc(T ) with decreasing field compared to the only small change during the of the unpenetrated stripline is acquired. ν0 is now initial field increase. Increasing the field to negative used to calculate λ(B) by assuming the penetration depth magnetic field dependent in equation (6) and values and exceeding Bc(T ) shows similar behavior 2 concerning hysteresis although lower in magnitude. solving for λ(B) leading to λ(B)=Γ h(ν0/ν(B)) − 1i. Similar to the superconducting filaments at B > The resulting λ(B) is shown in Fig. 5(b). It Bc(T ), these hysteresis effects are caused by normal increases with rising magnetic field. This increase conducting filaments trapped in surrounding supercon- is comparably small for magnetic fields B
part σ2 in Fig. 5(d). Looking at several higher har-
5000
Sn sample monics of the Pb resonator, frequency-dependentσ ˆ was 1.85 K
in Pb stripline resonator calculated and its real and imaginary parts for differ- 2.30 K ent frequencies are depicted in Fig 5(e) and Fig. 5(f). 2.50 K σ1 shows finite value at magnetic fields lower than 4000 7.63 GHz Bc(T ). Close to and above Bc(T ) it increases rapidly and forms a peak similar to a coherence peak seen in 30
3000 temperature-dependent σ1 measurements [38, 73, 74]. The coherence peak in temperature-dependent mea-
20 Qualityfactor surements however, is always located below Tc, whereas fi t parameters: B (T = 0 K) = 31.3 mT
c
2000
T = 4.04 K the peaks we measured are located above Bc. At even c
10 Criticalmagnetic [mT] field higher fields, σ1 decreases again. Similar to the tem- 1.0 1.5 2.0 2.5 3.0
Temperature [K] perature dependent Bc(T ) from Fig. 3, the peak shifts
to lower fields with increasing temperature and also 0 20 40 60 80 100
decreases in magnitude. Magnetic field [mT] σ2 (Fig. 5(d,f)) shows high values at low magnetic fields and a continuous decrease towards higher fields. Figure 6. Quality factor Q for a Pb stripline resonator with At B > B (T ) the change in value gets more a Sn sample in dependence of external magnetic field. Q shows c an initial sharp decrease at magnetic fields of about 19 mT to pronounced, approaching σ2(B) ≈ 0 at fields well 24 mT, which is due to the sample transitioning into the normal above Bc(T ). The general shape of the course conducting state, enhancing losses in the resonator. Inset: Bc(T ) again resembles the shape of measurements in variable of Sn determined from the initial decrease of Q fitted with equation (5). The values for Bc(0) and Tc are in good agreement temperature well, although σ2(T ) ≈ 0 already at with literature [39, 76]. T ≈ Tc. The non-vanishing values of σ1 and σ2 at fields higher than Bc(T ) are caused by the remaining superconducting filaments as well. Additional kinks in 4. Conclusion the course of σ1 and σ2 are located at magnetic fields coinciding with the intermediate phase (e.g. at 77 mT In this work we were able to access the electrodynamic to 84mT at 1.6 K). properties of superconducting Pb microwave stripline resonators in variable magnetic field up to 140 mT Both σ1 as well as σ2 show frequency-dependent behavior (see Fig. 5(e,f)), as they decrease with an and in a temperature range between 1.6K and 6.5 K. increase in frequency. This is in good agreement with With the calculation of the penetration depth into the earlier work as both surface resistance and surface superconducting Pb and the surface resistance, the reactance increase with frequency [38, 73, 75]. calculation of the complex conductivity was possible. By replacing one ground plane of the Pb stripline This was done for the first time for a type I resonator with a Sn sample (made of Sn foil) we are superconductor with microwave stripline resonators able to access the superconducting properties of Sn. in variable, parallel magnetic field and revealed As the electromagnetic losses in the resonator become similarities to temperature-dependent measurements, greatly influenced by the sample, the overall value of e.g. features similar in shape to a coherence peak. Q is strongly affected, showing drastically lower values Existence of remaining superconducting filaments than an all-Pb resonator (maximum of about 5 × 103, at magnetic fields above the critical magnetic field of opposed to all-Pb with about 1.2 × 105). Q also shows Pb has been found. The quality factor of the resonator, as well as the complex conductivity showed non- Superconducting Pb stripline resonators in parallel magnetic field and their application for microwave spectroscopy7
5
0.04
(a)
(b)
600
4
0.03 1.6 K
3.0 K
3
] 5.0 K
0.02
0 20 40 60 80 400 6.0 K
1.6 K
2 [m [nm] » 4.92 GHz s 3.0 K R
5.0 K
1 6.0 K
200
» 4.92 G Hz
0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
2.5
12
(d) (c)
1.6 K 1.6 K
2.0 10 3.0 K 3.0 K
5.0 K 5.0 K ] ]
8
-1 -1 6.0 K 6.0 K
1.5 m m
» 4.92 GHz » 4.92 GHz -1 -1
6
1.0 4 [ [
2
0.5
0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
4
(f)
(e)
3.30 GHz
15
3
4.92 GHz
3.30 GHz ] ]
9.74 GHz
-1 4.92 GHz -1
T » 1.6 K m m
9.74 GHz 10
-1 2 -1
T » 1.6 K [ [
5
1
0 0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140
Magnetic field [mT] Magnetic field [mT]
Figure 5. (a) Surface resistance Rs and (b) penetration depth λ for a Pb resonator at different temperatures plotted against the magnetic field. At B > Bc(T ) only superconducting filaments remain in the resonator increasing Rs and λ heavily. Both quantities also increase with temperature. (c) real part of the complex conductivity σ1 and (d) imaginary part σ2. Both show similar shape to temperature-dependent measurements, but do not vanish immediately at B > Bc(T ) indicating superconducting filaments as well. (e) and (f) show σ1(B) and σ2(B), respectively, for different resonator frequencies. vanishing values at magnetic fields above the critical spectroscopy studies in magnetic field. Alternatively, magnetic field and only vanished well above the critical we showed the possibility to regain the original magnetic field. Combined with the gradual resistance superconducting state by repeatedly sweeping the increase only at B >Bc(T ) this indicates that our magnetic field to positive and negative value below Pb specimen contained impurities making it a hard Bc, enabling restoration of Q even at very low superconductor. temperatures. Determination of the critical magnetic field and By modifying the resonator to be able to study the critical temperature of Pb was possible, which have a Sn sample, we could demonstrate the possibility been found to be in good agreement with previous to examine superconducting samples with stripline results. After exceeding the critical magnetic field resonators in a variable magnetic field. The critical once, hysteresis in the quality factor has been found magnetic field and the critical temperature of Sn could on return to lower magnetic fields, as impurities in be determined and fit the BCS predictions well. the resonator act as seeds for normal conducting areas. These have been found to persist even in the Acknowledgments superconducting phase, enhancing resistivity in the resonator, making it advisable to always perform zero- We thank Gabriele Untereiner for the fabrication of the field-cooling before using a Pb resonator in microwave resonators. Financial support by DFG and by Carl- Superconducting Pb stripline resonators in parallel magnetic field and their application for microwave spectroscopy8
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