Depinning frequency in a heavily neutron-irradiated MgB2 sample
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Depinning frequency in a heavilyneutron-irradiated MgB sample M. Bonura a , A. Agliolo Gallitto a , M. Li Vigni a , ∗ ,A. Martinelli b , a CNISM and Dipartimento di Scienze Fisiche e Astronomiche, Universit`a diPalermo, via Archirafi 36, 90123 Palermo, Italy b CNR-INFM-LAMIA and Dipartimento di Fisica, Universit`a di Genova, ViaDodecaneso 33, I-16146 Genova, Italy
Abstract
The magnetic-field-induced variations of the microwave surface resistance have beeninvestigated in a heavily neutron-irradiated MgB sample, in which the irradiationhas caused the merging of the two gaps into a single value. The experimental resultshave been analyzed in the framework of the Coffey and Clem model. By fittingthe experimental data, we have determined the field dependence of the depinningfrequency, ω , at different values of the temperature. Although the pinning is notparticularly effective, the value of ω obtained at low temperatures is considerablyhigher than that observed in conventional low-temperature superconductors. Key words:
Depinning frequency, MgB , Microwave surface resistance PACS:
Investigation of fluxon dynamics in type-II superconductors is of great interestfor both fundamental and applicative aspects. From the basic point of view, itgives information on the relative magnitude of elastic and viscous forces, whichrule the motion regime of the fluxon lattice [1,2,3,4,5,6]. From the technologicalpoint of view, it allows determining the magnetic-field-induced energy losses,which have important implication in a large variety of superconductor-baseddevices [7]. ∗ Tel.:+39 0916234208; fax: +39 0916162461; e-mail: livigni@fisica.unipa.it
Preprint submitted to Elsevier 11 November 2018 suitable method to investigate the fluxon dynamics consists in determiningthe magnetic-field-induced variations of the microwave (mw) surface resis-tance, R s [2]. In the absence of static magnetic fields, the variation with thetemperature of the condensed-fluid density determines the temperature depen-dence of R s . On the other hand, the field dependence of R s in superconductorsin the mixed state is determined by the presence of fluxons, which bring alongnormal fluid in their cores, as well as the fluxon motion [1,2,3,4,5,6,8,9,10].Measurements of the high-frequency em response allow to conveniently inves-tigate the vortex dynamics because they probe the vortex response at very lowcurrents, when vortices undergo reversible oscillations and are less sensitive toflux-creep processes.In a fluxon lattice driven by mw currents, the regime of vortex motion is ruledby the relative magnitude of the viscous-drag force, due to the presence ofthe normal cores, and the restoring-pinning force, which hinders the motionof fluxons. A very important parameter of the vortex dynamics is the depin-ning frequency, ω , which separates two regimes of vortex motion. When thefrequency of the driving field, ω , is much larger than ω , the viscous-drag forcedominates the restoring-pinning force; in this case, the vortex resistivity [2]is real and the motion of fluxons is highly dissipative. On the contrary, for ω ≪ ω the vortex resistivity is imaginary and the energy losses are stronglyreduced. Measurements of the depinning frequency have been performed inboth conventional [1,3,11] and high- T c superconductors [3,4,6,12]. For tem-peratures lower enough than T c and applied magnetic fields smaller enoughthan H c , conventional superconductors exhibit depinning frequency of theorder of MHz, while much higher values ( &
10 GHz) have been reported forcuprate high- T c superconductors.Since the first studies on MgB , different authors have highlighted severalanomalies in the field-induced variations of the mw surface impedance, espe-cially at low temperatures and magnetic fields much lower than the uppercritical field [13,14,15,16,17,18]. These studies have established that the stan-dard theories are inadequate to describe the fluxon dynamics in the two-gapMgB , in wide ranges of temperatures and magnetic fields. This is most likelydue to the peculiar properties of the mixed state of MgB , related to thepresence of the two distinct gaps [19,20,21,22,23].Recently, polycrystalline Mg B samples irradiated up to very high neutronfluence have extensively been investigated [24,25,26,27,28]. It has been shownthat irradiation up to exposure levels of 2 × cm − leads to an improvementin the upper critical field and in the field dependence of the critical currentdensity. On further increasing the neutron fluence, all the superconductingproperties, such as T c , H c , J c , are reduced. Furthermore, measurements ofspecific heat, as well as point-contact spectroscopy, have shown that in thesample irradiated at the highest fluence (1 . × cm − ) the two gaps merge2nto a single value [27,28]. Very recently [26], transmission-electron-microscopystudies have shown that neutron irradiation creates nanometric amorphousregions within the MgB crystallites, whose density increases on increasing theneutron fluence. In samples irradiated with neutron fluence ≤ cm − , suchdefects act as additional pinning centers. The field dependence of the criticalcurrent density observed in these samples has been quantitatively justifiedby considering the contribution of two pinning mechanisms, one arising fromgrain boundaries, which is also present in the pristine sample, and the otherarising from the defects induced by irradiation. On the contrary, the resultsof J c ( B ) obtained in the samples irradiated with neutron fluence larger than10 cm − have not fully been justified; in this case, the measured J c valuesare even lower than those expected from the grain-boundary contribution. Athorough understanding of the pinning mechanisms that come into play in theheavily irradiated samples is not yet achieved.In this paper, we report a detailed investigation of the magnetic-field-inducedvariations of the microwave surface resistance of a MgB sample irradiated atthe neutron fluence of 1 . × cm − . Preliminary results obtained in thissample have shown that the mw losses can be justified in the framework ofstandard models for vortex dynamics [18]. Here, we report the results obtainedin a wide range of temperatures (4 . ÷ T c ), from which we determine thetemperature and the magnetic-field dependencies of the depinning frequency.The investigation allowed us to determine also the field dependence of thepinning coefficient and the radius of action of the pinning potential. The field-induced variations of the mw surface resistance has been investigatedin a bulk sample of Mg B irradiated at very high neutron fluence (1 . × cm − ). The procedure for the preparation and irradiation of the sample isreported in detail elsewhere [25,27]. The sample has been prepared by di-rect synthesis from Mg (99.999% purity) and crystalline isotopically enriched B (99.95% purity), with a residual B concentration lower than 0.5%. Theuse of isotopically enriched B makes the penetration depth of the thermalneutrons greater than the sample thickness; this guarantees that the irradia-tion effects are almost homogeneous over the sample. Several superconduct-ing properties of the sample have been reported in Refs. [24,25,26,27,28]. Forsimplicity and easy of comparison, we label the sample as P6, according toRefs. [18,24,25,26,27,28]. Point-contact-spectroscopy [28] and specific-heat [27]measurements have shown that the neutron-irradiation process determined inthis sample a merging of the two gaps into a single value.The sample has a nearly parallelepiped shape with w ≈ . t ≈ . h ≈ . T onsetc ≈ . T c ≈ . J c ≈ × A/cm , it exhibits a monotonicdecrease with the magnetic field, following roughly an exponential law. Theupper critical field is isotropic and its value at T = 5 K is µ H c ≈ ρ n (40 K) = 130 µ Ω cm; how-ever, as suggested by Rowell [29], the real value of the residual normal-stateresistivity can be different because of reduction of the effective current-carryingcross-sectional area of the sample due to the grain boundaries. The rescaledvalue of the residual normal-state resistivity, corrected by the Rowell’s crite-rion, is ρ n ( T c ) ≈ µ Ω cm [25].Although the superconducting transition of the sample is sharp, it results∆ T c /T c ≈ .
03. Since the distribution of T c may affect the temperature de-pendence of the mw surface resistance near T c , we have determined the T c distribution function by measurements of the AC susceptibility at 100 kHz.We have found that the first derivative of the real part of the AC suscepti-bility can be described by a Gaussian distribution function of T c , centered at T c = 8 . ± . σ T c = 0 . ± .
05 K. In the following, we will use thisdistribution function to quantitatively discuss the results.The mw surface resistance has been measured using the cavity-perturbationtechnique [2]. A copper cavity, of cylindrical shape with golden-plated walls,is tuned in the TE mode resonating at ω/ π ≈ . µ H ≈ H , is perpendicular to the mw magnetic field, H ω . When the sample isin the mixed state, the induced mw current causes a tilt motion of the vortexlattice [9]; Fig. 1b schematically shows the motion of a flux line.The surface resistance of the sample is given by R s = Γ Q L − Q U ! , where Q L is the quality factor of the cavity loaded with the sample, Q U thatof the empty cavity and Γ the geometry factor of the sample.The quality factor of the cavity has been measured by an hp-8719D NetworkAnalyzer. The surface resistance has been measured as a function of the DCmagnetic field, at fixed temperatures. All the measurements have been per-formed at very low input power; the estimated amplitude of the mw magneticfield in the region in which the sample is located is of the order of 0 . µ T.4 w H w H F L F L (a) (b) th w Fig. 1. (a) Field and current geometry at the sample surface. (b) Schematic repre-sentation of the motion of a flux line. H (T) R s ( H ) / R m a x s T =4.2K Fig. 2. Field-induced variations of R s at T = 4 . R s ( H ) ≡ R s ( H , T ) − R res ,where R res is the residual mw surface resistance at T = 2 . H = 0;∆ R maxs ≡ R n − R res , where R n is the normal-state value of R s at T = T c . The linesare the best-fit curves obtained, as explained in Ref. [18], with µ H c = 1 .
71 T, ω /ω = 0 .
67 and the field dependence of the critical current density reported inRef. [25]. The inset shows a minor hysteresis loop obtained by sweeping H from 0to 0.25 T and back, along with the best-fit curve. The field-induced variations of R s have been investigated at different tem-peratures. For each measurement, the sample was ZFC down to the desiredvalue of temperature; the DC magnetic field was increased up to a certainvalue and, successively, decreased down to zero. Figs. 2, 3 and 4 show thefield-induced variations of R s , at different temperatures. In all the figures,∆ R s ( H ) ≡ R s ( H , T ) − R res , where R res is the residual mw surface resistanceat T = 2 . H = 0; moreover, the data are normalized to the maximumvariation, ∆ R maxs ≡ R n − R res , where R n is the normal-state value of the sur-face resistance at T = T c . In the figures, the continuous lines are the best-fitcurves obtained by the model described in Sec. 4.5 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.00.00.20.40.6 T =5K R s ( H ) / R m a x s H (T) T =6K T =7K Fig. 3. Field-induced variations of R s for the MgB sample, at different values ofthe temperature. ∆ R s ( H ) ≡ R s ( H , T ) − R res ; ∆ R maxs ≡ R n − R res . The lines arethe best-fit curves obtained as explained in Sec. 4. From Fig. 2 one can see that at T = 4 . R s ( H ) curve exhibits amagnetic hysteresis for H lower than ≈ .
18 T. The hysteresis is ascribableto the different magnetic induction at increasing and decreasing DC fields, dueto the critical state of the vortex lattice [30,31,32]. These results have beenreported and discussed in Ref. [18]. In order to fit the data, we have determinedthe B profile inside the sample due to the critical state and calculated a properaveraged value of R s ( B ) over the whole sample. The lines in the figure are thebest-fit curves.For T ≥ R s ( H ) curves are reversible, indicating that at these tem-peratures the critical-state effects of the fluxon lattice are negligible. However,we would like to remark that, for samples of millimetric size, the sensitivity ofour experimental apparatus allows detecting hysteresis in the R s ( H ) curvesfor J c & A / cm . 6 .00.20.40.60.81.0 T =7.5K T =8K T =8.5K R s ( H ) / R m a x s H (T) Fig. 4. Field-induced variations of R s for the MgB sample, at different values ofthe temperature. ∆ R s ( H ) ≡ R s ( H , T ) − R res ; ∆ R maxs ≡ R n − R res . The lines arethe best-fit curves obtained as explained in Sec. 4. Microwave losses induced by static magnetic fields have been investigated byseveral authors [1,2,3,4,5,6,7,8,9,32,33]. At low temperatures and for appliedmagnetic fields lower enough than the upper critical field, the main contribu-tion arises from the fluxon motion; however, it has been pointed out that anoticeable contribution can arise from the presence of normal fluid, especiallyat temperatures near T c and for magnetic fields of the same order of H c ( T ).In the London local limit, the surface resistance is proportional to the imagi-nary part of the complex penetration depth, e λ , of the em field: R s = − µ ω Im[ e λ ( ω, B, T )] . (1)The complex penetration depth has been calculated in different approxima-tions [8,9]. Coffey and Clem (CC) have elaborated a comprehensive theoryfor the electromagnetic response of superconductors in the mixed state, in the7ramework of the two-fluid model of superconductivity [8]. The CC theory hasbeen developed under the assumption that the induction field, B , is uniformin the sample; so, it is valid for H > H c whenever the fluxon distributioncan be considered uniform within the AC penetration depth.In the linear approximation, H ω ≪ H , e λ ( ω, B, T ) expected from the CCmodel is given by e λ ( ω, B, T ) = vuut λ ( B, T ) + ( i/ e δ v ( ω, B, T )1 − iλ ( B, T ) /δ nf ( ω, B, T ) , (2)with λ ( B, T ) = λ q [1 − w ( T )][1 − B/B c ( T )] , (3) δ nf ( ω, B, T ) = δ ( ω ) q − [1 − w ( T )][1 − B/B c ( T )] , (4)where λ is the London penetration depth at T = 0, δ is the normal-fluidskin depth at T = T c , w ( T ) is the fraction of normal electrons at H = 0; inthe Gorter and Casimir two-fluid model w ( T ) = ( T /T c ) . e δ v is the effective complex skin depth arising from the vortex motion; it de-pends on the relative magnitude of the viscous and restoring-pinning forces. e δ v can be written in terms of two characteristic lengths, δ f and λ c , arising fromthe contributions of the viscous and the restoring-pinning forces, respectively:1 e δ v = i λ c + 1 δ f , (5)where λ c = Bφ µ k p , (6) δ f = 2 Bφ µ ωη , (7)with k p the restoring-force coefficient, η the viscous-drag coefficient and φ thequantum of flux.The effectiveness of the two terms in Eq. (5) depends on the ratio ω = k p /η ,which defines the depinning frequency [1]. In terms of ω , Eq. (5) becomes1 e δ v = 1 δ f (cid:18) i ω ω (cid:19) . (8)When ω ≪ ω , the fluxon motion is ruled by the restoring-pinning force. Onthe contrary, for ω ≫ ω , the fluxon motion takes place around the minimumof the pinning-potential well and, consequently, the restoring-pinning force isnearly ineffective; so, the contribution of the viscous-drag force predominates8nd the induced em current makes fluxons move in the flux-flow regime. Inthe latter case, enhanced field-induced energy losses are expected.The theory above discussed is strictly valid when B is uniform inside thesample. When fluxons are in the critical state, the assumption of uniform B isno longer valid and the CC theory does not correctly describe the field-inducedvariations of R s . Recently, we have investigated the field-induced variationsof the mw surface resistance in superconductors in the critical state and haveaccounted for the magnetic hysteresis in the R s ( H ) curves [32,33]. The detailsof the procedure we have followed to account for the experimental results ofFig. 2, where the critical-state effects are important, are reported in Refs. [18].Since R s ( H ) in the investigated sample does not show hysteresis in a widerange of temperatures, we do not discuss here on this procedure.The expected value of the normalized surface resistance depends on several pa-rameters, such as the ratio λ /δ , the temperature dependence of the normal-fluid density w ( T ), H c ( T ), the depinning frequency and its field dependence.However, λ /δ and the temperature dependence of the normal-fluid densitydetermine the value of R s ( T ) at H = 0. In Ref. [18] we have shown that the R s ( T ) curve at H = 0 can be quite well justified assuming valid the Gorterand Casimir two-fluid model, with λ /δ values ranging from 0.04 to 0.15,provided that distribution of T c in the sample is taken into account. The largeuncertainty of λ /δ is due to the fact that the T c distribution broadens the R s ( T ) curve, hiding the λ /δ effects.At fixed temperature, the expected field-induced variations of R s depend on ω ( B ) and H c ( T ). The temperature dependence of H c has been reported inRefs. [18,25]; it can be described by the law H c ( T ) = H c [1 − ( T /T c ) α ] , (9)with µ H c = (2 . ± .
2) T, α = 1 . ± . T c = (8 . ± .
2) K.By using this relation for H c ( T ), it is possible to determine the field de-pendence of the depinning frequency by fitting the experimental isothermal R s ( H ) curves.For T ≥ R s ( H ) curves do not show hysteresis, suggesting that theeffects of the critical state are negligible. Since on increasing the temperaturethe effects of the distribution of T c become more and more important, to fit theexperimental data we have averaged the expected R s ( H ) curves [calculatedby Eqs. (1-4)] over the T c distribution; have used Eq. (9) for H c ( T ); havetaken ω as parameter dependent on H . Moreover, we have used the followingapproximate expression for the magnetization: M = − H p + H p H c − H p ( H − H p ) (10)9nd consequently B = µ H p H c − H p ! ( H − H p ) , (11)where H p is the first-penetration field. H p can be directly deduced from the experimental curves, measuring the ap-plied magnetic field at which R s starts to increase; its temperature dependencehas been reported in Ref. [18]. The best-fit curves of R s ( H ) are reported inFigs. 2, 3 and 4; the field dependence of ω /ω , by which the best fit has beenobtained, is reported in Fig. 5, at different temperatures. We remark that wehave investigated the field-induced variations of R s down to T = 2 . T < . T = 4 . ω/ π ≃ . ω /ω we obtain that the depinning frequency at low fields is ω / π ≈ T ≤ ω is independent of B in a wide field range; on increasing thetemperature, this field range shrinks. Moreover, for applied fields larger thana threshold value, dependent on T , the fluxon lattice moves in the flux-flowregime ( ω ≪ ω ); on increasing the temperature, this threshold field decreases;eventually, at T > ω is expectedto be independent of B . Collective pinning is realized at higher fields whenthe vortex concentration is high and there are many vortices per pinning site;in this regime, ω gets lower values and depends on B [2]. Our experimental T=4.2K T=6K T=7.5K T=5K T=7K T=8K / H - H p (T) Fig. 5. Magnetic field dependence of the depinning frequency, obtained by fittingthe experimental R s ( H ) curves, at different temperatures. .0 0.2 0.4 0.6 0.8 1.0050100150200 k p ( N / m ) H - H p (T) T =4.2K T =6K T =7.5K T =5K T =7K T =8K Fig. 6. Deduced field dependence of the pinning coefficient k p , at different temper-atures. results indicate that for T . T c / k p and η , but only the ratio ω = k p /η . However,the investigated sample has shown properties that can be quite well accountedfor by conventional models; so, one can deduce k p by supposing valid theBardeen-Stephen relation [34] η ( T ) = φ µ H c ( T ) ρ n , (12)where ρ n is the normal-state resistivity at T = T c .Both ρ n ( T c ) and H c ( T ) of the investigated sample have been already deter-mined: ρ n ( T c ) ≈ µ Ω cm [25] and H c ( T ) is given by Eq. (9). The deducedfield dependence of k p is shown in Fig. 6, at different values of the temperature.An upper limit of the pinning constant, k maxp , can be obtained by equating theenergy density per unit length of the vortex core, B c ξ π/ µ , to the elasticstored energy density per unit length of the vortex core, k maxp ξ / k maxp = πB c µ , (13)where B c is the thermodynamic critical field.At low temperatures, from B c and B c we estimate that the thermodynamiccritical field is of the order of 100 mT; so, from Eq. (13) we estimate k maxp to be of the order of 10 N/m . The values of k p we have obtained fromthe experimental data are two orders of magnitude lower than k maxp deducedfrom Eq. (13); so, we infer that pinning is not particularly effective in the11nvestigated sample. This finding is consistent with the low J c value responsiblefor the weak hysteretic behavior of R s reported in Fig. 2.From simultaneous measurements of the pinning constant and critical currentdensity it is possible to estimate the average radius of action, r p , of the pinningpotential, U ( r ). The pinning constant, k p = d U/dr , is related to the criticalcurrent density; since J c = φ − dU/dr | r p , one obtains J c = k p r p /φ [2]. By usingthe value of J c (5 K) reported in Ref. [25] and the value we obtained for k p , itresults r p ≈ ≈ q φ /B ≈
45 nm; so, the deduced value of r p (5 K) is much smallerthan the distance between vortices. This finding confirms that, in our sample,for T ≤ bulk samples the pinning mecha-nism is ruled by grain boundaries [24,25,26,36,37]. Recently, it has been shownthat neutron irradiation introduces defects in the form of amorphous regionsof mean diameter ∼ ≤ × cm − ) these defects give a furthercontribution to the pinning, leading to an improvement of the critical currentdensity, with respect to the values expected from grain-boundary pinning. Thesame effect has not been observed in samples irradiated with higher fluences;in this case, the measured J c values are even lower than those expected byproperly rescaling the grain-boundary contribution. Most likely, this is dueto the different coherence length of the different samples: in samples exposedto neutron fluence ≤ × cm − the defect dimension matches with thecoherence length; in samples exposed to higher fluences the coherence lengthis larger than the defect size and, consequently, the defects are not effectivefor the fluxon pinning. The results we have obtained in the heavily irradiatedsample confirm this conclusion; indeed, we have obtained small values of thepinning coefficient. So, despite the high concentration of defects, they do notcontribute positively to the pinning.The depinning frequency in the investigated sample is considerably higherthan the values reported in the literature for conventional superconductors;for example, the depinning frequency in bulk niobium, which has comparablevalues of H c , is less than 10 Hz [2]. Since the values we have obtained for thepinning constant k p are considerably lower than the upper limit k maxp , the highvalue of ω cannot surely be ascribed to strong pinning effects; so, we inferthat it is ascribable to a low value of the viscosity coefficient. On the otherhand, in this sample the residual normal-state resistivity is ρ ( T c ) = 90 µ Ω cm;this high value of ρ ( T c ) has been ascribed to a reduced value of the electron12ean free path due to the presence of the defects induced by the neutronirradiation [25]. We suggest that it is just the high value of the normal-stateresistivity responsible for the small viscosity coefficient that, in turn, gives riseto the high value of the depinning frequency. We have measured the magnetic-field-induced variations of the mw surfaceresistance in a heavily neutron-irradiated Mg B sample, in which the twogaps merged into a single value. The field dependence of R s , at different valuesof the temperature, have been discussed in the framework of the Coffey andClem model, with the temperature dependence of the normal-fluid densityexpected from the Gorter and Casimir two-fluid model.By fitting the experimental data, we have determined the magnetic-field de-pendence of the depinning frequency at different temperatures. We have foundthat, for T . T c / ω does not depend on the mag-netic field, indicating that individual pinning is realized in the whole fieldrange investigated; on increasing the temperature, the range of H in whichindividual pinning occurs shrinks and ω is field dependent above a certainthreshold value of the applied field, depending on T . By supposing valid theBardeen-Stephen relation for the viscosity coefficient, we have deduced thefield dependence of the pinning coefficient at different temperatures; more-over, considering the value of the critical current density, we have deduced theradius of action of the pinning potential. Although the neutron-irradiationprocess created a high density of defects, our results show that the pinning isnot particularly effective, consistently with the relatively low value of the crit-ical current density reported for this sample; this finding is most likely due tothe fact that the coherence length is larger than the mean size of the defects.Nevertheless, the deduced value of the depinning frequency is considerablyhigher than that reported for conventional SC, as bulk Nb. We suggest thatthis high value of ω is due to the high value of the normal-state resistivityin the investigated sample, which is due to the reduced value of the electronmean free path because of the presence of the defects. Acknowledgements
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