Fluxon-induced losses in niobium thin-film cavities

Physica C 339 (2000) 231±236

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Fluxon-induced losses in niobium thin-®lm cavities W. Weingarten * European Organization for Nuclear Research, CERN, CH-1211 Geneva 23, Switzerland Received 2 August 1999; received in revised form 11 February 2000; accepted 12 May 2000

Abstract New data obtained with niobium ®lm cavities allow a re®nement of the physical explanation of RF losses by trapped magnetic ¯ux at small RF ®eld amplitudes. A stationary model explains these data. As the diameter of the ¯uxons decreases, the RF bypasses them, and the RF losses are due to the voltage, to which the normal conducting electrons, located in the ¯uxon core, are subject to. This loss mechanism is similar in origin to that of the so-called BCS losses. Ó 2000 Elsevier Science B.V. All rights reserved. PACS: 74.25 Nf Keywords: Coherence length; Mixed state; Surface impedance; Thin ®lms

1. Introduction Recent data, much more complete (large sample size) [1] than the previously available ones, have made it possible to verify the assertions concerning ¯uxon-induced losses at a low RF ®eld amplitude in the superconducting (sc) thin niobium ®lm cavities, as used for the LEP electron positron collider (Fig. 1). The following assertion was made earlier [2±4]: These losses are essentially similar, with regard to the physical mechanism, to those in bulk niobium cavities [5,6]. 1 They are thought to be caused by the current passing through the normal *

Tel.: +41-22-767-6615; fax: +41-22-767-9465. E-mail address: [email protected] (W. Weingarten). 1 Excellent surveys of the work performed up till about 1974 are presented in these references. The authors quote the relevant original publications, which for conciseness are omitted in this paper.

conducting (nc) cores of the ¯uxons. The additional surface resistance, RH , due to these losses is described by the relation RH ˆ

RN Happ ; Hc2

…1†

where RN is the Ônc surface resistanceÕ and Happ , the ambient magnetic ®eld component applied perpendicular to the ®lm surface. In ®lm cavities, these losses are smaller in magnitude because of two e€ects: the larger upper critical ®eld, Hc2 , of thin niobium ®lms, and pinholes or nc or insulating spots in the ®lm. These are ®lled up for small magnetic ®elds and do not contribute to additional losses up to saturation. From that point, the increasing magnetic ®eld will be trapped in the form of ¯uxons 2 and hence contribute to RH . 2 The reduction of the increase of the surface resistance with the applied ®eld due to pinholes, etc. can be taken into account by a factor smaller than and close to unity: RH ˆ a…RN Happ =Hc2 †.

0921-4534/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 0 3 6 5 - 8

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W. Weingarten / Physica C 339 (2000) 231±236

3. Remarks on the history of the explanation for ¯uxon-induced losses

Fig. 1. Fluxon-induced losses Rfl0 vs. the mean free path l at 1.8 K (data taken from Ref. [1] but plotted di€erently), showing a minimum near l ˆ 30 nm. Error bars are not included for reasons of clarity, but the lines indicate an error band.

2. Impact of the new data on the widespread explanation for ¯uxon-induced losses However, the new data suggest that the assertion, as summarised in Eq. (1), is based on a ÔmisconceptionÕ because of the following two reasons [1]: (i) The value of the critical magnetic ®eld, Hc2 , necessary to explain the new data is too high. (ii) The plot of the ratio RH =Happ (ˆRf10 ) vs. the mean free path, l, displays a minimum, which cannot be explained by Eq. (1) (Fig. 1). Argument (i) in itself however would not be sucient to contest the assertion. Part of the trapped ¯ux was attributed to pinholes [4], etc., as explained earlier. The upper critical ®eld, Hc2 , as derived from Eq. (1), would then be an upper bound, which is a much weaker condition and no longer in contradiction with the new data. In what follows, we shall also see that argument (ii) does not constitute a contradiction to this assertion either, providing the notion that Ônc surface resistanceÕ is interpreted correctly (i.e., as the surface resistance of the nc electrons present in the sc host metal). Nevertheless, the new data allow a re®nement of the physics of ¯uxon-induced losses in niobium ®lm cavities.

At this stage, a word on the notion of the Ônc surface resistanceÕ, RN , appearing in Eq. (1), will be provided. In his review article of 1974 [5,6], Pierce mentioned two kinds of loss mechanism, which are eventually described by the same formula (Eq. (1)). On the one hand, the losses can be due to the RF current ¯owing through the nc cores of the ¯uxoids. In another review article that appeared at the same time, Hartwig and Passow called them ``stationary'' ¯uxoids [5,6]. On the other hand, the RF current may force the ¯uxoids to move, creating a voltage and hence dissipation (``non-stationary'' ¯uxoids). He stated that the ®rst explanation is mainly valid for type I superconductors with larger cores, and the second for type II superconductors with smaller cores. However, the explanation for the latter case is less obvious. In conclusion, he stated that ``very roughly then, one can take RH  RN Happ =Hc2 as an estimate of loss due to small amounts of trapped ¯ux passing through the surface of both type I and type II superconductors''. In addition, he mentioned a di€erence between these two types: depending on the ¯uxon size or whether they are arranged into bundles or not, the ``RF ®eld can be expected to sag into these vortices, and the RF currents will ¯ow through the normal core rather than detouring it''. Hence, Pierce was aware of the fact that the path of current ¯ow depends on the ¯uxon diameter, and therefore considered Eq. (1) only as an approximation to the particular physical situation of a type II superconductor. In the meantime, also inspired by a review article of Gough [7], the conditions of current ¯ow around or through any nc ÔspotÕ, have been clari®ed [8]: if their diameter is small compared to the nc penetration depth (or alternatively, if the resistance is large), the current will bypass them and vice versa. We see no reason why this result should not be applied for ¯uxons. In conclusion, the Ônc surface resistanceÕ depends on the situation of current ¯ow through or around these ¯uxons. The main di€erence is that, if the current passes through them (small resistance), the surface resistance increases with their

W. Weingarten / Physica C 339 (2000) 231±236

resistivity. If the current detours them (large resistance), the surface resistance increases with their conductivity. After the discovery of the high temperature superconductor, interest in ¯uxon-induced losses rose again. Clem and Co€ey have explained these losses by ¯uxon movement [9]. Within the appropriate limits, their results go over into those obtained previously for conventional type II superconductors. They restrict their analysis to non-stationary ¯uxons. As we will see, however, stationary ¯uxons can entirely account for the losses observed for small RF ®eld amplitudes. This case is not discussed by Clem and Co€ey, and therefore, their model will not be examined further. 4. Re®nement of model We shall now apply the two-¯uid model, concentrate on the main ideas, and content ourselves with approximate calculations. The two-¯uid model can describe the physics in the usual way and hence does not have to be further substantiated (Fig. 2). The inductance L describes the sc electrons, and the resistance R describes the nc ones present in the superconductor. The RF losses are iÿ1 h 2 I 2; …2† P ˆ …1=2†R 1 ‡ … R=xL† x being the angular frequency, and I, the total current ¯owing. First, we assume that no ¯uxons are present. We translate the lumped circuit model of Fig. 1 into the equivalent physical situation of a current

233

passing through a quadratic slab of width w, within its penetration depth (k for an sc metal and d for an nc one, cf. Appendix A). We use the replacements R ! …rk†ÿ1 , L ! Ls ˆ l0 k, and I ! H0 w: r is the conductivity of the nc electrons still present in the sc metal; r…T † ˆ r0 f …T †, where r0 is the conductivity just above the critical temperature Tc ; f(T) is the fraction of nc electrons still 4 present in the sc metal, f …T † ˆ …T =Tc † . H is the magnetic ®eld, and Ls , the surface inductance of the sc electrons. Since rk2 xl0  1, one ®nds for the losses of this square (in W per m2 ) P ˆ …1=2† r0 x2 k3 l20 f …T † H 2 |‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚} w2

…3†

RBCS

with the two-¯uid model representation of the ÔBCS surface resistanceÕ, RBCS ˆ r0 x2 k3 l20 f …T †, which we shall need later. Secondly, we assume that there is a ¯uxon present. In a similar manner, but using a di€erent translation (justi®ed in Appendix A) R ! RN ˆ ÿ1 …r0 d† , we obtain from Eq. (2) the surface resistance of ¯uxon-induced losses, R H ˆ RN

1 RN …k=d†f …T † RBCS

1‡ |‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}

Happ : Hc2 …T †

…4†

R1

We have found in Eq. (4) a relation between the BCS surface resistance RBCS and that of the ¯uxoninduced losses RH . The corresponding data are shown in Figs. 1 and 3. For small values of RN , compared to RBCS …d=k†=f …T †, the current passes the nc ¯uxons, the losses increase with the resistivity, and Eq. (4) transforms into Eq. (1). For large values of RN , the current avoids the nc ¯uxons, and the losses increase with the conductivity. Hence Eq. (4) represents a generalisation of Eq. (1) for the current splitting into a component passing through the nc ¯uxons and another component avoiding them. 5. Comparison with experiment

Fig. 2. Two-¯uid model representation of the current ¯ow through a superconductor.

The measured surface resistance hRH i ( ˆ G/Q with the geometry factor G and the unloaded

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W. Weingarten / Physica C 339 (2000) 231±236

Fig. 3. The measured ÔBCS surface resistanceÕ vs. the mean free path l at 4.2 K (from Ref. [1]). The lines indicate an error band.

®gure of merit Q due to the magnetic losses) is an average value over the surface exposed to di€erent directions of the ambient magnetic ®eld vector. A factor b < 1 takes account of the di€erence between this average value and hypothetical surface resistance RH of the cavity as if it were entirely exposed to a perpendicular magnetic ®eld vector: hRH i ˆ bRH . It is RH we are interested in because it describes the ®lm properties. Hence, from Eq. (4), hRH i ˆ bRH ˆ bR1

Happ : Hc2 …T †

…5†

Our aim now is to compare Hc2 determined from Eq. (4) or Eq. (5) with the measured values of Hc2 . The ratio hRH i=Happ is taken from the new data, 3 as well as b…ˆ 0:69† and RBCS . The calculation is based on the data represented in between the lines on Figs. 1 and 3. The result is shown in between the lines of Fig. 4. We shall now compare these values of Hc2 with the measured ones. There exists a large amount of data on the sc quantities of bulk niobium. A representative sample [9] of measurements on the 3 According to Ref. [1], at a temperature of 4.2 K, which is the reference temperature for the calculation to follow, this ratio is larger by a factor of 2.7 with regard to that at 1.7 K.

Fig. 4. The upper critical ®eld Hc2 of bulk niobium, as measured by C.C. Koch et al., of niobium ®lms according to the references marked in Table 1, and calculated from Eq. (4). The lines are derived from the error bands shown in Figs. 1 and 3. The arrows indicate how the data of Ref. [11] for very thin ®lms are shifted, when extrapolated for thicker ®lms, following Eqs. (2), (2a) and (3) of Ref. [11].

upper critical ®eld Hc2 is also shown in Fig. 4. Data on Hc2 of sputtered niobium ®lms are much scarcer. We have collected the data (Table 1), which are also indicated in Fig. 4.

6. Concluding remarks The calculated values of Hc2 are very close to the measured ones for niobium ®lms in the regime of small mean free path l, and for bulk niobium in the regime of large l. This can be understood according to Ref. [10], as for small l (compared to n0 ), the upper critical ®eld Hc2 of the thin ®lm is determined by the very small grain size produced by sputtering (and not by additional impurities). This result has been independently corroborated elsewhere [4]. For larger l, the upper critical ®eld of the ®lm should approach that of the bulk, which is the case. In conclusion, the new data are not in contradiction with the assertion leading to Eq. (1), but they require a generalisation for small ¯uxon diameters, as given in Eq. (4). The lower sensitivity

W. Weingarten / Physica C 339 (2000) 231±236

235

Table 1 Upper critical ®elds of sputtered niobium ®lms Film thickness (lm)

Hc2 , 4.2 K (kg)

RRR

Mean free path l …nm† ˆ 2:7 RRR

Reference

5 3.7 0.027 0.0245 3 3 1.6±1.8

20 21 48.5 50.5 15 26 28

15 6.7 3 2 13 9 12

41 18 8.1 5.4 35 24 32

[2] [11] [11] [11] [12] [12] [13]

of niobium ®lm cavities ± compared to bulk niobium cavities ± if exposed to a static ambient magnetic ®eld is due to a combination of a larger upper critical ®eld and the fact that the RF current bypasses the ¯uxons. The minimum of hRH i=Happ evoked in Ref. [1] for l ˆ n0 is explained as a re¯ection of the minimum in RBCS according to Eq. (4). The losses can be explained by stationary ¯uxoids. No movement of the ¯uxoids is necessary to explain the experimental data. Appendix A We consider a square of a superconductor as shown in Fig. 5 with a current ¯owing partly through and partly past an nc spot (¯uxon). We want to calculate the surface resistance and the surface inductance of this slab of width w, which represents approximately the region of interest of one-quarter of a ¯uxon (Fig. 5) and the sc metal in the vicinity. The current ¯ow and the RF losses in this slab may be approximated similar to those of Fig. 2. The RF losses for an individual ¯uxon are given by Pf ˆ …1=2†RI22 , where I2 is the current through the ¯uxon (which is purely resistive) and R, its resistivity. This current I2 is determined by the 2 total current I, I2 ˆ I=f1 ‡ ‰ R=… xL†Š g, where L represents the inductance of the sc electrons in the vicinity of the ¯uxon. Hence, the losses per ¯uxon Pf ˆ …1=2†R

1 1 ‡ ‰ R=… xL†Š

2

I 2:

…A:1†

Fig. 5. Schematic current path around a ¯uxon: the current path is shown in plan view. The square of width w shows a quarter of the perturbed region of current due to the presence of the ¯uxon of diameter 2n. The current penetrates the paper plane perpendicularly a distance d into the ¯uxon, yet the current penetrates the sc metal in the vicinity of the ¯uxon to a distance of k.

We shall now calculate the characteristic values for the di€erent elements of the lumped circuit model. The resistance of the ¯uxon is determined by the conductivity r0 and the nc penetration ÿ1 depth d, 4 R  RN ˆ …r0 d† . In this ®rst-order approximation, the current ¯ows perpendicular to a square column of half-width instead of a cylinder of radius n. The inductance of the quadratic slab of width w is determined by the de®nition of the voltage V ˆ ÿixLI, which is equal to V ˆ Ey w. The Maxwell equation curl E ˆ ÿB_ de®nes the electric ®eld component Ey ˆ ixkl0 Hx , with the

4 d should be smaller than the ®lm thickness, which is the case for the data under study.

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W. Weingarten / Physica C 339 (2000) 231±236

Table 2 Physical quantities Quantity

Relation

Numerical value for Nb

Residual resistivity ratio Coherence length Superconducting penetration depth Residual conductivity Fraction of nc electrons Normal conducting penetration depth just above Tc

RRR ˆ l…nm†=2:7 ÿ1 nÿ1 ˆ np ‡ l ÿ1 0  k ˆ k0 n0 =n r0 ˆ RRR rj300 K f …T †p ˆ …T =Tc †4 d ˆ 2=…xr0 l0 †

n=a n0 ˆ 33 nm k0 ˆ 32 nm rj300 K ˆ 7:6  106 …X m†ÿ1 Tc ˆ 9:5 K n=a

magnetic surface ®eld Hx . From the total current I ˆ Hx w, we obtain L ˆ l0 k, independent of its width w. The RF losses per square metre in the slab are Pf 1 1 1 ˆ H2 2 2 r0 d 1 ‡ ‰1=…r0 dxl0 k†Š2 w |‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚} Rf

ÿ1

from which we derive Rf ˆ …r0 d† f1 ‡ ‰1= 2 ÿ1 …r0 dxl0 k†Š g , representing the surface resistance due to one individual ¯uxon. The average surface resistance RH due to the presence of ¯uxons is derived from the losses of one individual ¯uxon summed over the number N of ¯uxons per square metre, Pˆ

N 1X Rf H 2 A ˆ …1=2†NRf H 2 A: 2 iˆ1

As the ¯ux is (nearly) completely trapped upon cool down, the applied ambient ¯ux per square metre Happ is redistributed in the form of ¯uxons of section A and magnetic ®eld Hc2 , N ˆ Happ = …Hc2 A†. Hence, the total losses per square metre ÿ
P ˆ …1=2†Rf Happ =Hc2 H 2 ; and the surface resistance RH ˆ Rf …Happ =Hc2 ), which is Eq. (4). Finally, we present in detail the physical quantities (Table 2) used in the preceding calculations.

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