Testing V3Si for two-band superconductivity
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Testing V Si for two-band superconductivity
M. Zehetmayer and J. Hecher Atominstitut, Vienna University of Technology, 1020 Vienna, Austria
Is V Si a two-band or a single-band superconductor? Everyone who searches the literature forthis question will find conflicting answers, for V Si was claimed to be a perfect example of two-bandand claimed to be a perfect example of single-band superconductivity. In this article we intend toclarify the situation by presenting new experimental facts acquired from the magnetic propertiesof a V Si single crystal. We probe for field dependent two-band effects by analyzing the reversiblemagnetization at different temperatures, and we probe for temperature dependent two-band effectsby analyzing the superfluid density obtained by two different methods at different magnetic fields.All our results are reliably described within the single-band models and thus support the single-bandscenario for V Si but do not completely rule out the presence of a very small second gap.
PACS numbers: 74.25.Ha,74.25.Op,74.70.Ad
INTRODUCTION
Superconductivity in V Si has been studied since1953[1] and was considered to be of conventional s-wave,single-band nature for most of the time. It was only re-cently that, in the wake of MgB , V Si was placed onthe list of potential two-band superconductors to explainunconventional experimental results. In particular, itssuperfluid density was reported [2, 3] not only to de-viate strongly from the single-band BCS behavior butto match a two-band model, whose interband couplingstrength is all but negligible. These conclusions werebacked by infrared spectroscopy results and by calcu-lations of the Fermi surface, which was reported to becrossed by several electronic bands [4]. In contrast, asingle-band description worked well for the field depen-dence of the specific-heat and the thermal conductivity[5]. Furthermore, the field dependence of the magneticpenetration depth and of the vortex core size, determinedby muon-spin rotation, were regarded as single-band be-havior [6], and in [7] the temperature dependence of thespecific heat was reported to be conventional. We con-clude that we face a confusing situation, which we wishedto clarify by further experiments.V Si is a member of the A15 superconductors, whosecrystal structure changes from cubic at room tempera-ture to tetragonal in the superconducting state [8]. Itis a type II superconductor with a Ginzburg-Landau pa-rameter of about 20, a transition temperature of 16 -17 K, and an upper critical field of around 20 T at 0 K[9]; the anisotropy of the superconducting properties ismarginal [10]. The vortex distribution may change froma hexagonal to a cubic lattice as the magnetic field in-creases [11].In this article we will present new experimental resultscapable of probing a possible two-band state of V Si. Inthe next section, the experimental details and the evalu-ation methods will be introduced. We will start with thebasic characterization of the sample, go on with the mea-surements of the magnetic moment, and will then show how the reversible magnetization was obtained and fittedusing the Ginzburg-Landau model. Finally, the direct de-termination of the lower critical field will be explained.In the third section, we will present the results. First, wewill summarize how two-band effects show up in MgB and will then compare this with our findings on V Si.Potential modifications [12] of the field dependencies willbe discussed by analyzing the reversible magnetization,those of the temperature dependencies by analyzing thesuperfluid density and the lower critical field. We will re-port differences between literature data and our resultsand present possible reasons. In the final section, we willsummarize and once again explain why we believe thatV Si is a single-band superconductor.
EXPERIMENT AND EVALUATION
A V Si single crystal was cut into two pieces of equalsizes of about 3 × . × .
55 mm . Both samples, namedVA and VB, became superconducting below 16.7 K witha transition width of 0.2 K and had a residual resistanceratio of 33.Using a SQUID magnetometer, we measured the mag-netic moment of sample VB at temperatures from 2 to16 K in 1 K steps as a function of applied magnetic fieldfrom 0 to 7 T. Sample VA was analyzed with a non-commercial rotating sample magnetometer, where thesample is glued on the rim of a circular plate, which ro-tates at a frequency of 15 Hz. During one rotation thesample passes four pick-up coils, where it induces elec-trical voltages proportional to its magnetic moment. Fordetails about the instrument and how the magnetic mo-ment is determined, the reader is referred to [13]. Themain advantage of the rotating sample magnetometer isits fitting into a cryostat with a 15 T magnet. Accord-ingly, magnetization loops up to 15 T were recorded attemperatures of 5.2, 7, 9, 11, 13, and 15 K.The resulting magnetization loops, either measured inthe SQUID or in the rotating sample magnetometer, re-vealed reversible behavior over most of the field range. Ir-reversible effects, caused by flux-pinning, emerged merelyat low fields and became dominant near 0 T. For instance,the critical current density at 5.2 K, evaluated using themethods presented in [14], decreased with field from some10 Am − at 0.1 T to 5 × Am − at 1 T and to a negli-gible value at and above 2 T. Accordingly, more than 85per cent of the loop fell into the reversible regime. Theirreversible effects became slightly larger at lower temper-atures but considerably smaller at higher temperatures.The critical currents, which are proportional to the hys-teresis width of the magnetization loops, were found to besomewhat larger in the SQUID than in the rotating sam-ple magnetometer, yet the reversible parts agreed well.As already mentioned, irreversibility appeared merelyat low fields, but even there, knowing the magnetiza-tion as a function of increasing, M ( H +a ), and decreas-ing applied field, M ( H − a ), allowed us to determine thereversible part via M r ( H a ) ≃ M ( H +a ) + M ( H − a )],where H +a and H − a refer to the same applied field H a .This procedure gives reliable results as long as the hys-teresis width is not much larger than the correspondingreversible signal. To get M r ( B ) from M r ( H a ), we calcu-lated the magnetic induction via B = µ ( H − DM r + M r ),with µ = 4 π × − NA − , H the applied field correctedby the field induced by the macroscopic currents [14], and D the numerically calculated demagnetization factor ofthe sample in the Meissner state.The next step was to compare the reversible magne-tization, acquired from experiment, with theory. Thetheoretical magnetization curves of a single-band super-conductor were taken from approximate equations ofthe Ginzburg-Landau theory, provided by Brandt [15].According to this paper [15] the approximation errorsshould be less than two per cent, for the Ginzburg-Landau parameter of V Si is large. The Ginzburg-Landau curves depend on just two parameters, namelythe upper critical field, at which the magnetization be-comes zero, and the Ginzburg-Landau parameter, whichdetermines the shape of the curve. The well-knownGinzburg-Landau relations [16] allowed us to calculatefurther quantities, such as the lower and the thermody-namic critical field, the coherence length, the magneticpenetration depth, and the superfluid density.The lower critical field was additionally determined bymeasuring the field at which vortices start to penetratethe sample. After having minimized the stray fields in theSQUID cryostat, we cooled the sample below its transi-tion temperature in zero field. Then we enhanced theapplied field stepwise but interrupted each step by mea-suring the corresponding remanent magnetic moment inzero field. The remanent moment should vanish at lowfields and start to increase above the lower critical field,where vortices are created and pinned in the sample. magnetic induction (T) -80-400 m a gn e ti za ti on ( m T ) FIG. 1. The reversible magnetization of MgB as a functionof the magnetic induction. The open circles show the exper-imental data for the applied field oriented parallel ( H k c )or perpendicular ( H k ab ) to the uniaxial sample axis at 10,20, and 30 K (the reduced temperatures are 0.26, 0.51, and0.77). The solid lines are fits of the single-band Ginzburg-Landau model to the experimental data over the whole fieldrange, while the broken lines are such fits either to the low orto the high field part of the experimental data, reflecting thetwo-band nature of MgB . For details see [17]. RESULTS AND DISCUSSION
In this section we will analyze whether superconductiv-ity in V Si is more reliably described by a single or a two-band model. If V Si is a two-band superconductor, weexpect some properties to deviate from the single-bandBCS behavior [12]. Possible effects on the field depen-dence will be discussed via the reversible magnetizationand possible effects on the temperature dependence viathe superfluid density. The results will be compared withthe behavior of MgB , a well-known two-band material.We will see that the single-band models provide a fairdescription of our results in V Si.We start by summarizing some two-band effects ofMgB [18]. This material consists of the near-isotropic π -band and the anisotropic σ -band, having similar elec-tronic densities of states. Due to interband coupling,the gaps are expected to close at the same field. Yetthe superconducting properties of the π -band are heavilysuppressed above a particular field, which is commonlycalled the upper critical field of the π -band and whosevalue is about a third of that of the σ -band. Accordingly,the field dependence of several superconducting proper-ties deviates significantly from the single-band behavior.This is illustrated in figure 1, where the reversible mag- magnetic induction (T) -50-250 m a gn e ti za ti on ( m T ) FIG. 2. The magnetization of V Si as a function of magneticinduction. The open circles show the reversible data at 15,13, 11, 9, 7, and 5.2 K (the reduced temperatures are 0.90,0.78, 0.66, 0.54, 0.42, and 0.31) and the solid lines the corre-sponding Ginzburg-Landau model fits. The solid lines of theinsets show the irreversible data. netization of an MgB single crystal, indicated by opencircles, is shown as a function of the magnetic induction.The solid lines present the single-band Ginzburg-Landaufits. Barring the 30 K results, the agreement between the-ory and experiment is poor and the differences are notmerely of quantitative but also of qualitative nature. Incontrast, the curves can be nicely fitted by two single-band Ginzburg-Landau curves, as shown by the brokenlines in the diagrams. One is adjusted to the low andthe other to the high field region of the experiment, thusreflecting the two bands with their different upper crit-ical fields [17]. Applying the methods to different fieldorientations reveals the different anisotropies of the twobands. Similar effects were observed in NbSe [19].We are now prepared to shift our focus to V Si. Fig-ure 2 shows the magnetization as a function of the mag-netic induction at temperatures of about 15, 13, 11, 9, 7,and 5 K. The open circles indicate the reversible magneti-zation acquired from the rotating sample magnetometermeasurements of a V Si single crystal and the solid lines the single-band Ginzburg-Landau behavior adjusted tothe experimental data. In the insets, the solid lines in-dicate the irreversible magnetization, though only at lowfields, where a significant hysteresis shows up. At nottoo low temperatures, say about 9 - 15 K, we considerthe agreement between experiment and single-band the-ory very good. As the temperature is reduced, the differ-ences between theory and experiment get larger, becom-ing apparent at low fields, for we adjusted the fits mainlyto the high field regions, where the experiments revealthe reversible data directly.Next, we will analyze the quality of the fits in more de-tail. First, we determined the areas under the reversiblemagnetization curves, which are proportional to the con-densation energies. The ratio of the condensation energyobtained from the Ginzburg-Landau fit to that from theexperimental curves is considered a sensible measure forthe fit quality. In MgB , the differences in the conden-sation energies of the high-field fit and the experimentaldata were some 18 per cent at 30 K and 30 per cent at10 and 20 K. In V Si, we found much smaller deviations,namely 3 - 8 per cent at 9 - 15 K and 10 - 12 per cent at 7and 5.2 K. So, even the low-temperature fits of V Si agreewith the experimental data better than any single-bandfit of MgB .There are several possible reasons for the larger de-viations between theory and experiment in V Si at lowtemperatures. To begin with, as shown in figure 2, thelow-field irreversible magnetization becomes large at lowtemperatures, thus enhancing possible errors in the cor-responding reversible data. The insets of figure 2, how-ever, illustrate that the irreversible magnetization doesnot become much larger than the reversible data evenat low temperatures and small fields, and hence the er-rors in calculating the reversible magnetization are notserious. On the other hand, we are faced with the short-comings of the Ginzburg-Landau theory. As this theoryis derived from BCS theory in the vicinity of the transi-tion temperature, the potential errors grow when we goto lower temperatures. Yet the theory has been success-fully applied to evaluating and describing experimentaldata at high and low temperatures, as a function of tem-perature and as a function of field in a large number ofpublications. In particular, using adjustable parametersinstead of the microscopic BCS ones apparently extendsthe applicability to much lower temperatures (e.g. [20]).Watanabe et al. [21] calculated the field dependence ofthe reversible magnetization of an s-wave system usingthe Eilenberger equations, which hold at arbitrary tem-peratures. We verified that the Ginzburg-Landau modelwith a Ginzburg-Landau parameter of 49 reproduces theEilenberger curve of [21] close to the transition temper-ature (see also [22]). Reducing the temperature makesthe Eilenberger curves slightly steeper at low fields andslightly flatter at high fields. This is basically what wefind for the experimental curves in figure 2, namely re-ducing the temperature makes the experimental curvesslightly steeper at low fields and slightly flatter at highfields compared with the Ginzburg-Landau fit. Accord-ingly, the qualitative deviations between our experimentsand the fits are just as expected when we assume that theEilenberger model describes experiment at all tempera-tures. The deviations from the Ginzburg-Landau model,assessed via the condensation energies, agree even quan-titatively, i.e., we found some 12 per cent for the exper-imental data and 14 per cent for the Eilenberger curvesat a reduced temperature of about 0.3. Granted, theEilenberger calculations and our experimental data referto systems with different Ginzburg-Landau parameters( κ ), but both systems belong to the high- κ regime andhence should behave similarly. To conclude, the changesof the Eilenberger curves are not substantial when thetemperature is lowered, which affirms the usefulness ofour approach. Still, we do not expect an exact descrip-tion at the lowest temperatures of figure 2.In contrast to V Si, presented in figure 2, the exper-imental data of MgB , presented in figure 1, also qual-itatively disagree with the single-band models. In par-ticular, we found the high-field fit to lead to a muchsmaller lower critical field, i.e. the magnetization valueat B = 0 T, than obtained from experiment via extrapo-lation of the low-field data, while in V Si the two curvesresult in almost the same lower critical fields.Summarizing, we consider the agreement betweensingle-band theory and experiment in V Si as good ascan be expected in view of the uncertainties within boththe theoretical and the experimental data. Accordingly,we conclude that the reversible magnetization of V Sisupports the single-band scenario.Figure 3 presents the upper critical field and theGinzburg-Landau parameter of V Si. The open circlesindicate results from the rotating sample and the fullcircles those from the SQUID magnetometry. The up-per critical field has been evaluated by adjusting theGinzburg-Landau model merely to the high-field regimeof the reversible magnetization and may thus slightly de-viate from the results obtained from full range fits. Thesolid line presents the clean-limit single-band BCS behav-ior [23], is in excellent agreement with our experimentaldata, and leads to 22.7 T at 0 K, which matches literaturedata well [9]. In contrast to many two-band materials, noclear upward curvature near the transition temperatureappears [24–26]. The Ginzburg-Landau parameter, takenfrom fits over the whole field range and presented in theright panel, decreases quite linearly from about 24 at 0 Kto 19 at the transition temperature, a behavior that isalso close to the single-band BCS prediction [23]. Calcu-lating further properties employing the Ginzburg-Landaurelations [16] resulted in about 90 nm for the magneticpenetration depth, 4 nm for the coherence length, 0.6 Tfor the thermodynamic critical field, and 0.07 T for thelower critical field at 0 K. The critical lengths are close temperature (K) upp e r c r iti ca l f i e l d ( T ) temperature (K) G i n z bu r g L a nd a u p a r a m e t e r FIG. 3. The upper critical field, presented in the left panel,and the Ginzburg-Landau parameter, presented in the rightpanel, of V Si as a function of temperature. The open circlesshow experimental data acquired from the rotating samplemagnetometer and the full circles experimental data from theSQUID measurements. The upper critical field follows theclean-limit BCS behavior, depicted by the solid line, leadingto 22.7 T at 0 K. The Ginzburg-Landau parameter decreasesroughly linearly from about 24 to 19 as the temperature in-creases from 0 K to the transition temperature. to literature data [6, 27].We proceed by analyzing the temperature dependenteffects by means of the superfluid density. Having eval-uated the upper critical field, B c2 , and the Ginzburg-Landau parameter, κ , via the above fit procedure, weacquired the magnetic penetration depth, λ , by usingthe Ginzburg-Landau relations, and the superfluid den-sity, ρ s , via ρ s (T) = [ λ (0 K) / λ (T)] , with λ = κξ , ξ = Φ / (2 πB c2 ), and Φ ≃ . × − Vs. To as-sess the penetration depth at 0 K, we used the SQUIDmeasurements, where the temperature could be reducedto 2 K, but the magnetic fields were limited to 7 T. Wetherefore took the inter- and extrapolated upper criti-cal fields from the rotating sample magnetometer resultsfor fitting the SQUID data, so that only the Ginzburg-Landau parameter remained to be determined. To jus-tify the use of these data, we verified that the reversiblecurves from SQUID and rotating sample magnetometeragree in the overlapping field and temperature range.The left panel of figure 4 presents the superfluid den-sity of V Si, indicated by open circles, as a function ofreduced temperature. The solid line shows the expectedbehavior of a single-band BCS superconductor, which isclose to our experimental data. Figure 4 shows also thesuperfluid density of MgB , indicated by full symbols, asan example for two-band superconductivity. In compari-son with V Si, the MgB curve decreases much faster atlow temperatures and then becomes almost linear as thetemperature increases. To explain this behavior, we needto consider that a two-band superconductor has two dis- reduced temperature (T/T c ) s up e rf l u i d d e n s it y temperature (K) r e du ce d l o w e r c r it ca l f i e l d magnetic field (mT) [r e m a n e n t m a gn e ti c m o m e n t ] / FIG. 4. The superfluid density as a function of reduced tem-perature (left panel) and the reduced lower critical field asa function of temperature (right panel). In the left panel,the open circles show our experimental result on V Si, ob-tained from the reversible magnetization, which is in goodagreement with the single-band BCS behavior, indicated bythe solid line. The broken and the dot-dashed lines show thetwo-band-like superfluid density of V Si reported in Refs. [2]and [3] schematically; the full diamonds illustrate the two-band behavior of MgB [17]. In the right panel, the opencircles show the reduced lower critical field of our V Si singlecrystal, evaluated from the reversible magnetization, which isall but identical to the superfluid density of the left panel.The open diamonds and the full squares are obtained fromdirect measurements of the first vortex-penetration field asindicated by the arrows in the inset. The symbols in the insetpresent the square root of the remanent magnetic moment asa function of applied field at 7 K; the solid line is a linear fitto the high-field data. tinct energy gaps. The smaller gap reduces the excitationenergy on the corresponding part of the Fermi surface andhence makes the superfluid density decrease more rapidlyat small temperatures. This may lead to a near-linear be-havior at intermediate temperatures, as found in severaltwo-band materials, such as MgB , NbSe , and the iron-based superconductors [28–30]. We conclude that alsothe superfluid density of V Si supports the single-bandscenario.We now come back to the two-band scenario of V Siproposed in Refs. [2] and [3], based on measurements ofthe superfluid density. The broken line in figure 4 showsthe result presented in [2], which was obtained by mea-suring the microwave surface impedance of a single crys-tal. There is no doubt that this curve is totally differentfrom our result, that is, it decreases more rapidly at lowtemperatures and has a sharp kink at about 0.6 T c . An-alyzing their data with a two-band model, the authorsfound distinctly different energy gaps, though similar in-traband coupling strengths for the two bands and, as in-dicated by the sharp kink at intermediate temperatures,almost negligible interband coupling. The superfluid den- sity published by Kogan et al. [3], shown schematically infigure 4 by the dot-dashed line, was measured by a tun-nel diode resonator technique and is quite different fromthat of [2], yet it also reflects a two-band scenario withsimilar intraband and near-negligible interband couplingstrengths.What are the possible reasons for the differences be-tween those literature [2, 3] and our data? Two points areobvious. First, while we evaluated the penetration depthfrom fits to the reversible magnetization over the wholefield range, save for very small fields, where the experi-mental data are not available or not reliable, the authorsof Refs. [2] and [3] evaluated their data solely at very lowfields. Thus, a second band, but one with a very smallupper critical field, would resolve the contradictions. Thesecond point is that while our method probes the bulk,the methods of Refs. [2] and [3] probe the sample surface.Thus, surface irregularities or inhomogeneities that maychange the properties on the surface would also resolvethe contradictions. Such a statement, however, remainsa speculation, for we know nothing about the surfaces ofthe samples used in the studies. Diener et al. [31] facedthat problem in MgCNi . Acquiring data with the samemethod as used in [3] resulted in a penetration depthbehavior similar to the broken curves of figure 4, whileacquiring the results from measurements of the lower crit-ical field resulted in a BCS-like behavior. The authorssuggested that those differences might be caused by inho-mogeneities at the sample surface and hence consideredthe BCS-like behavior correct.We also determined the superfluid density at very lowmagnetic fields, namely by measuring the lower criticalfield directly. But measuring the lower critical field di-rectly is anything but trivial. To begin with, we canmerely determine the field at which flux lines start to pen-etrate into the sample. This is accomplished by recordingthe remanent magnetic moment of the sample in zero fieldas a function of the maximum applied field, as describedat the end of Sec. . Unfortunately, this first penetra-tion field is usually not simply connected with the lowercritical field via a single-valued demagnetization factor.First, sample edges give rise to very high stray fields,which may surpass the lower critical field and hence en-force the creation of vortices at very low applied fields,but we do not know well how the local induction is re-lated to the applied field in such a configuration. Second,surface irregularities would affect the creation of vortices,while clean surfaces may induce additional barriers.So, how can we acquire useful results from that pro-cedure? To begin with, we are mainly interested in therelative temperature dependence of the lower critical fieldand not so much in its absolute value. Second, we alignedthe longest length of our sample, the size of which is3 × . × .
55 mm , parallel to the applied field, so thatthe geometry effects were as small as possible and as-sumed those geometry effects to be temperature inde-pendent. As expected, when we increased the appliedfield, the remanent magnetic moment remained zero atthe beginning, then started to rise at a slope that be-came gradually steeper, indicating that first vortices hadbeen formed and pinned in the sample, and eventuallyincreased quadratically with field (see inset of figure 4).Assuming a constant current density parallel to the sam-ple borders and ignoring the stray fields revealed thisquadratic behavior also in calculations (cf. with [32]).We thus consider the extrapolated onset of the quadraticbehavior (cf. with inset of figure 4) as a reliable assess-ment of the lowest field where vortices parallel to theapplied field were in the sample.We determined both the smallest field at which we ob-served a slope in the remanent magnetic moment and thefield obtained from extrapolating the quadratic part tozero (see inset of figure 4). The temperature dependenceof both was found in good agreement with the super-fluid density acquired from the Ginzburg-Landau fits, asshown in the right panel of figure 4. Note that Ginzburg-Landau theory predicts the same temperature depen-dence for the superfluid density and the lower criticalfield when the changes in the logarithm of the Ginzburg-Landau parameter can be ignored, as is the case in oursample. To assess the absolute lower critical field val-ues, we multiplied the penetration fields with the factor(1 − D ) − ≃ D ≃ .
13 is the averaged de-magnetization factor. In comparison to the lower criticalfields from the Ginzburg-Landau fits, leading to 68 mTat 0 K, this led to lower values by some 10 per cent,namely 60 mT at 0 K, from evaluating the onset fields,and to larger values by 40 per cent, namely 95 mT at0 K, from evaluating the fields where the quadratic be-havior started. We consider these differences to be reli-able. Finally, we conclude that also at very low magneticfields, the temperature dependence shows no indicationof a two-band behavior.Having presented clear support for the single-band be-havior, we may ask if our results rule out the two-bandscenario completely. This is not the case, for our ex-periments would not detect a second band if its contri-bution to the measured quantities were marginal or ifits superconducting properties were all but identical tothe first band. These scenarios, however, would be dif-ferent from those proposed in literature [2, 3]. Whichproperties would a hypothetical second band have in oursample? Above, we have discussed the fit quality of thereversible magnetization data in terms of the ratio ofthe condensation energies from the fit and from exper-iment. In MgB the differences, which are some 30 percent at low temperatures, are ascribed to the two-bandscenario. Accordingly, 70 per cent of the condensationenergy are induced by the σ -band, which is the bandprobed by the high-field fits [17], and 30 per cent by thesecond band, the π -band, and interband coupling effects,in rough agreement with [33]. In V Si, the deviations are much smaller and we have shown that they can qual-itatively and quantitatively be explained by the imper-fections of the Ginzburg-Landau model. Nevertheless,we analyzed the results assuming the deviations to becaused by a hypothetical second band. For example, at9 K the fit error was 5 per cent and hence the second bandwould contribute to the condensation energy ( E c ) by less than 5 per cent, for these 5 per cent include the inter-band contributions. Employing BCS theory, where E c =0 . N ∆ , and assuming both bands to have the same den-sity of states ( N ) would result in a gap (∆) ratio largerthan 4.4 : 1 (1 : p / Si [2, 3] came along with a negligible inter-band coupling strength and the high purity of our singlecrystals makes strong interband impurity-scattering un-likely.
SUMMARY AND CONCLUSION
Let us sum up what we have learned from testing V Sifor two-band superconductivity. We have shown that thefield dependence of the reversible magnetization matchesthe single-band Ginzburg-Landau theory reliably well.The differences between the experimental data and thetheoretical fits grow as the temperature is reduced butremain small and can qualitatively and quantitatively beexplained by the imperfections of the Ginzburg-Landaumodel. We have also found that the temperature depen-dence of the superfluid density, determined with differentmethods at different magnetic fields, follows single-bandBCS behavior. Accordingly, all our results support asingle-band scenario for V Si.
ACKNOWLEDGMENTS
This work was supported by the Austrian Science Fundunder Contract No. 21194 and 23996.
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