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AC conductivity of a niobium thin film in a swept magnetic field M.I. Tsindlekht1, V.M. Genkin1, Sˇ. Gazi2, and Sˇ. Chromik2 1The Racah Institute of Physics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel and 2The Institute of Electrical Engineering SAS, Du´bravska´ cesta 9, 84104 Bratislava, Slovakia (Dated: September27, 2011) We report theresults of themeasurement theac conductivityof a Nb superconductingthin film in a swept dc magnetic field applied parallel to the surface. Analysis of theexperimental data in a 1 mixedstateshowsthatchangesoftheacconductivityareduetogeneration bytheswept magnetic 1 field unpinned vortices on the film surface. The numberof these vortices is much smaller than the 0 pinned ones and their density decreases with increasing of both theamplitude and frequency of an 2 ac field. p PACSnumbers: 74.25.F-,74.25.Op,74.70.Ad e S 6 It is knownthat the ac response oftype II bulk super- (h0 ≈ 0.1 Oe) decreases the number of unpinned vor- 2 conductors in slow ramped dc fields differs qualitatively tices in the film, and the real component of resistivity from the ac response in constant dc field [1]. Increased decreases with h0. n] ac losses in the mixed state [2] and the second harmonic Theimaginarypartoftheconductivityσ2 isdecreased o generation [3] were observed in a swept dc field. The by the swept dc field and it cannot be ascribed to the c swept dc field induces currents in the sample that can changetheCampbellpenetrationdepth[9]bydcthecur- - change the ac response. DC currents, which exist in su- rent in the film. r p perconductorswithpinninginaconstantdcfield,change Nb thin films, 200 nm thick, were deposited by DC u the Campbell penetration depth [4], but do not increase magnetron sputtering on rotated sapphire substrates at s the ac absorption. . room temperature. The size of substrate with rounded t a The underlining physical mechanism of the influence corners (radius 0.2 mm) is 1.5 by 3 by 15 mm. Actually m the sweptfieldisnotclear. Asweptfieldservesto unpin the sample was a thin walled cylinder and external ap- - thevorticesinthesampleandthelosscomponentofmag- pliedfieldswereparalleltothe cylinderaxis. Monitoring d neticsusceptibilityχ′′ shouldbedependentonthevortex the ac field inside this cylinder permits measurements n density,i.e. thedcfield,whileforHc1 <H0 <Hc2, χ′′ is of the conductivity of the film. RRR of the film was o actually constant [2]. In the surface-switching model [2] ≈ 4, T ≈ 8.5 K. The ac response was measured by the c c [ it was supposed that if H˙0/ωh0 < 1 (h0 and ω are the pick-up coil method. The sample was inserted into one ac amplitude and excitation frequency, respectively) the of balanced pair of coils, and the unbalanced signal was 1 instantaneoustime rateofthe fieldchangessignforfrac- measured by a lock-in amplifier. A ”home-made” mea- v tion of each ac period. During this interval the vortices surement cell of the experimental setup was adapted to 9 1 become pinned and the sample is lossless [5]. During a commercial SQUID magnetometer. A block-diagram 5 the reminder of the period the loss mechanism should of the experimental setup was published elsewhere [10]. 5 operate. The sample switches back and forth from the Boththeacanddcfieldswereparalleltoeachother. The . dissipativestatetothenon-dissipativestateandresistiv- magnetic susceptibility of the sample at frequencies 293 9 0 ity should be some average value. The difficulty of this and 1465 Hz and h0 from 0.04 to 1.2 Oe was measured 1 model is the lack oflosses in the mixed state in constant in a point-by-point mode while during the measurement 1 dc fields [1, 6–8]. thedcfieldwaskeptconstantandinasweptmodewhen : the dc field was ramped at a given rate. v Interpretation of the experimental results that were i obtained with bulk samples is hampered by the inho- Inzerodcfieldandoflowenoughtemperaturesthesu- X mogeneity of an ac electric field which induces the ac perconducting film completely shields the small external ar current. The amplitude of the ac electric field decreases ac field [11], and the susceptibility of the sample should with the depth. On the other hands, for a thin film we be −1/4π. In orderto testthis, we comparedthe phases can achieve the homogeneity of electric field with accu- ofthesignalsfromthesampleandfromabulksupercon- racy ≈ d/R, where d is the thickness of the film and ductorwherecompleteshieldingisobserved. Itturnsout R is some macroscopical size, for example, the size of a that the phases of these two signals coincide with high substrate. In this paper we reports the results of mea- accuracy. This allowed us to obtain the ac susceptibility surement the conductivity, σ(ω) = σ1(ω)+iσ2(ω), of a for any field and temperature in absolute units. Mea- superconducting Nb thin film in swept dc fields applied surements were performed at two temperatures, 7 and 8 parallelto the surface. We considerthatthis experiment K. Results for both temperatures are similar, and here shows that the additional dissipation in a swept dc field we will discuss only the data for 7 K. is due to the unpinned vortices that were generated at Theacconductivityofthefilmwascalculatedfromex- the surface. These vortices move into the film and pro- perimentaldatafollowingthis simple consideration. The videflux-flowresistance. Thepresenceofasmallacfield thickness of the film was 200 nm and its volume was 2 0.04 ) 4.0x103 1040 ’’( 0.02 /1 a ) -00..0002 11088 O OOe/ees//;ss ;;2 21994336 a 5Hn Hdz z1465 Hz 33..05xx110033 1100230 1 2b 3 H40 (kOe5) -0.04 ( ’ -0.06 3 2.5x10 293 Hz -0.08 1465 Hz 3 0 2 4 6 2.0x10 0 1 2 3 4 5 Magnetic field (kOe) Magnetic field (kOe) FIG. 1: (Color online) Real and imaginary components of magneticsusceptibilityatfrequencies293and1465Hzinthe FIG.2: (Coloronline)Realpartofresistivityρ1asafunction sweep rate 0 and 18 Oe/s. of dc field for frequencies 293 and 1465 Hz, h0 = 0.04 Oe in the swept rate 18 Oe/s, σ0 ≈ 1.4×1026 CGS, ρ0 = 1/σ0 . Insetshows fielddependenciesofρ1 at frequency293 Hzand sweep rate 5 Oe/s (curve a); frequency 1465 Hz and sweep 2.7×10−5 cm3, while the volume of the substrate was rate 25 Oe/s (curve b). Parameter q = H˙0/ωh0 is the same 6.7×10−2 cm3. The ac magnetic moment of the sample for both curves. is defined actually by the magnetic field inside the sample, i. e. by the total ac current in the film that shields the interior. We can thus write χ(ω)h0S = js(ω)dS/c, superconductors [2], there is a large plateau-like region whereLandS aretheperimeterandareaoftheperpen- ′′ ofmagnetic fields where χ is approximatelyconstantin dicular to the field direction cross section of the sample, ′′ value. Here, the measured χ depends on the frequency j (ω) is the average current density in the film, and d is s and in accordance with Eq.(1) the frequency dispersion the film thickness. We neglected the influence of demag- ∝1/ω of the conductivity practically disappears, Fig. 2. netizing fields because the demagnetizing factor for our In the surface superconducting state, before the transi- geometricalarrangementis0.036. FromMaxwell’sequa- −→ −→ tion into the normal state, the observed χ does not de- tion, curlE = iωB/c we can obtain the average electric pend on frequency, Fig. 1, and the dispersion ρ1 ∝ 1/ω field in the film as E = iωS(1+4πχ)h0/cL. And then is restored. for average conductivity, σ(ω)=js(ω)/E, we have In [2] it was noted that χ′′ is a function of the single σ(ω)=σ1(ω)+iσ2(ω)=−σ0 iχ(ω)ω0 , (1) pmaeraasmureetderreqsi=stiHv˙i0ty/ωohf0th.eTfihlme ifnosrettwtoosFetisg.of2frsehqouwesnctihees [1+4χ(ω)]ω and sweep rates but with the same value q. There is where σ0 =c2L/ω0Sd≈1.4×1026 CGS,for our sample, a difference approximately six times between these two and ω0/2π =1 Hz. curves in the plateau region. The field dependencies of the ac susceptibility were DC magnetization data provide us with the value of measuredin point-by-pointandin sweptmodes. For the the critical current of the film. The measured dc mag- employedfrequencies,acamplitudes,andsweeprates,we neticmomentofthesampleisproportionalthedifference neglect the variation of the dc field during an ac period. between the magnetic field inside the substrate H and i Fig. 1 shows the magnetic susceptibility at frequencies theapplieddcfieldH0,∆H ≡Hi−H0 =4πM/V.Fig.3 293 and 1465 Hz, sweep rates 0 and 18 Oe/s, and ac presents ∆H as a function of H0 after zero field cool- amplitude 0.04 Oe. ing. Here M is the magnetic moment of the sample in In point-by-point data we do not see any observable emu and V is the sample volume. It is seen that near difference of χ for 293 and 1465 Hz. Application of 4.6 kOe ∆H and the observed magnetization disappear. Eq.(1) shows that the frequency dispersion conductivity Wecanconsiderthismagneticfieldasthesecondcritical is∝1/ω. Thisiscorrectonlyforlargedcfieldswherein- fieldHc2 orasthe irreversibilityfieldwhereonlypinning complete shielding is observed. For small dc fields, more disappears. In the latter case we should expect the in- ′′ precisemeasurementsarerequiredbecauseinthesefields creasingof χ due to increasingthe number ofunpinned weobservecompleteshielding,andinthedenominatorof vortices when the dc field exceeds 4.6 kOe. However, Eq.(1) we get the small value 1+4πχ≈0. In the swept χ(ω) does not have any peculiarities for point-by-point ′′ dcfieldsχ differsfromzeroifH0 >Hc1 and,asforbulk data near this field, and H0 = 4.6 kOe is probably the 3 2/ 0 1/ 0 20 Oe)0.5 10-2 (a) 10-1 (b) H (0.0 10 T = 7 K -0.5 10-2 e) H0 (kOe) -3 O 4.2 4.4 4.6 4.8 5.0 10 ( H 0 1 0-3 -4 10 -10 Hc2 10-4 5 Oe/s, 0.04 Oe 5 Oe/s, 0.12 Oe 23 Oe/s, 0.04 Oe 23 Oe/s, 0.12 Oe -5 -5 0 1 2 3 4 5 10 10 0 2 4 6 0 2 4 6 Magnetic field (kOe) Magnetic field (kOe) Magnetic field (kOe) FIG.3: (Coloronline)Magneticfieldjump,∆H,fielddepen- FIG.5: (Coloronline)Theconductivityatthefrequency293 dence. Inset: ∆H near H0 =4.6 kOe. See text. Hz,sweeprates5and23Oe/s,andamplitudes0.04and0.12 Oeasafunctionofdcfield. Both components,σ1 andσ2 are increasing as theamplitude of an ac field increases. -2 10 2 Oe/s 0 -3 5 Oe/s in the ac field with amplitudes 0.04 and 0.12 Oe. Both /210 1243 OOee//ss components of σ are increasing with h0. These results -4 showthatforthesweptfieldmodedataweactuallyhave 10 to deal with nonlinear conductivity, while the point-by- -5 10 point mode data (not shown here) did not reveal any nonlinear behavior in the mixed state. For the point-by- 010-2 point mode there is a complete shielding in the mixed /1 -3 stateandthenonlinearresponseinthesurfacesupercon- 10 ducting state, as for bulk superconductors [6, 8]. The 10-4 ac response in the parallel to the surface magnetic fields 10-5 larger than Hc2 is usually ascribed to the surface super- 0 1 2 3 4 5 conducting states. In Fig. 1 it is seen that over 5.7 kOe Magnetic field (kOe) χ=0 and we could consider this field as Hc3. The ratio Hc3/Hc2 ≈ 1.24 which is considerably smaller than the FIG. 4: (Color online) Magnetic field dependencies of con- theoreticalvalueof1.69[12]. ForH0 >Hc2 bothcompo- ductivity of the film at 293 Hz, h0 = 0.04 Oe at the sweep nents σ1 and σ2 are approximately the same. The effect rate as a parameter, σ0 ≈1.4×1026 CGS. of sweeping does not so distinctive as in the mixed state and in H0 > 5.3 kOe σ1 actually does not depend on the sweeprate. Inthese fields σ1 is approximatelylarger by two orders than in the normal state. The resistivity second critical field. Absence of anomaly of χ(ω) near as a function of the sweep rate exhibits approximately a Hc2 at low amplitudes of excitation in the bulk samples power dependence with exponent 1.3 for H0 = 4.5 kOe. with strong pinning was observed in [1, 6–8]. Fig. 6 shows ρ1 = Re(1/σ) and σ2 at frequency 293 Hz Theconductivityofthefilmatfrequency293Hz,h0 = and h0 = 0.04 Oe for several dc fields as a parameter. 0.04 Oe as a function of dc field for some values of the It is seen that there are approximately two regions of dc sweep rate is shown in Fig. 4. While the loss component fields with different characters of the swept field effect. σ1exhibitsplateau-likebehavior,thereactivecomponent These experimental results can be understood using the σ2 is strongly affected by the dc field. Both components followingmodel. Dissipationinthe mixedstateisdue to of the conductivity decrease with increasing the sweep unpinned vortices that were generated at the surface by rate. Theaccuracyofexperimentaldatadoesnotpermit a swept dc field. Vortex generation rate is proportional us to calculate σ2 for a sweep rate of 2 Oe/s in all dc to the sweep rate and doesn’t depend on the equilib- fields because in this case the measured χ(ω) is weakly rium vortex density. The ac field decreases the number different from χ(ω) measured in point-by-point mode. ofunpinnedvorticesinthefilmanditismoreefficientat The conductivity of the film depends on the ac ampli- higherfrequenciesandacamplitudes. Unpinnedvortices tude, Fig. 5. Here we presentσ at a frequency of 293Hz provide losses through the flux-flow mechanism and we 4 trationdepth,respectively. Ourestimationgivesλ ≈61 L nm at 7 K (for λ (0) ≈ 46 nm [14]). It is evident that L /1200-3 34..55 kkOOee (a) λHCzw≫hiλchLisotvheerrywliasregeEiqn.(c2o)mgpivaersisoσn2w≈it1h0t2h7eCeGxpSeraitm2e9n3- 10-4 455...913 kkkOOOeee talvalue≈1023 CGS.Thedecreasingofσ2 withincreas- -5 5.5 kOe ing sweep rate possibly can be ascribed to the change 10 5.6 kOe of λ due to the dependence of the Labusch parameter C on the dc current in the film [4] for nonparabolic pin- 0 ning well. But σ2 depends also on h0 (Fig. 5). It shows /1 4 that the mechanism of the influence the swept field on 10 the conductivity cannotbe reducedtothe simple change 103 (b) of the Labusch parameter by the dc current and has to include some interaction between pinned and unpinned 2 10 1 10 vortices. Sweep rate (Oe/s) In a large dc field, H0 > 5 kOe, both real and imagi- narycomponentsofthe conductivityaredecreasingwith FIG. 6: (Color online) The real part of resistivity ρ1 and increasing dc field for all sweep rates. The character of imaginary part of conductivity σ2 as a function of a sweep the ac response differs from the response in the mixed rate and dc field as a parameter at frequency 293 Hz and state. The conductivity weakly depends on sweep rate h0 =0.04 Oe, σ0 ≈1.4×1026 CGS, ρ0=1/σ0. and exhibits approximatelythe reciprocalfrequency dispersion. We did not find any peculiarities in ρ1 near Hc2 (=4.6 kOe). This presents some difficulties. The thickness of the film is larger than the London penetra- ′′ observe approximately constant χ as a function of the tiondepth. AboveHc2 thesuperconductivitycouldexist dc field. The flux-flow dissipation should be frequency only in the thin surface layer while the interior of the independent. In Fig. 2 we showed ρ1 as a function of film is in the normal state. The skin depth of Nb in the the dc field for frequencies 293 and 1465 Hz and sweep normal state is very large in comparison with the film rate 18 Oe/s. It is seen that with an accuracy ±20% thickness and the interior of the film is opaque to the ac eρx1admopelse,noint dthepeefinedldon3.f5rekqOueenχcy′′(w29h3il)e/χχ′′′′(1∝4615/)ω≈. F4o.8r field. We should expect some peculiarities near Hc2 be- while ρ1(293)/ρ1(1465)≈ 1.2. Such weak frequency dis- ficalmusewbheilloewovHecr2Hthc2e aocnlryesbpyontsheeisthfoinrmseudpberycothneduwchtionlge persioncouldbeascribedtothedecreasingofthenumber layers. In the point-by-point mode we observe actually ofTunhpeinrenseidstivvoirtyticoefstbhyetfihlemacρfi1e≈ld.10−23 ÷ 10−24 CGS cboymsuprlefateceshlaiyeeldrisnbgelaotwHo0r≈oveHrcH2ct2h[a1t1]coaunlddtbheeopbrosveridveedd for 5 Oe/s, h0 = 0.04 Oe, and 293 Hz is smaller by sev- picture (χ = −1/4π) will not depend on the opaqueness eral orders magnitude than the normal state resistivity of the interior. In the swept dc field the shielding is not ρ of bulk Nb (≈ 10−18 ÷10−19 CGS). In accordance n complete and observed ac response should depend also with the Bardeen-Stephen formula for flux-flow resistiv- on the interior properties. Below Hc2 the whole film is ity ρf = ρnnφ0/Hc2, where n is the vortex density and in superconducting state, above Hc2 there is only a thin φ0 is the flux quantum, we could conclude that the den- surface superconducting sheath. sityofunpinnedvorticesisverysmallincomparisonwith In conclusion, we have presented the results of mea- equilibrium density neq ≈ H0/φ0. From the present ex- suring the conductivity of a superconducting Nb film in perimental data it is not clear that this is due to the swept dc fields. A model of the influence of the swept decreasing generation rate of unpinned vortices at the dc field on the conductivity of a superconducting film in surface or the increasing of the relaxation rate into the the mixed state was proposed. We found that the swept pinned state. dcfieldgeneratedunpinnedvorticesatthesurface,while Experimentshowsthatthereactivecomponentofcon- the weak ac field decreases the number of these vortices. ductivityσ2 decreaseswithincreasingsweeprate,Fig.5. Unpinned vortices move into the film and decrease both Thenumberofunpinnedvorticesinthefilmisconsidered the real and imaginary components of the conductivity. smallincomparisontothenumberofpinnedvorticesand Authors are deeply thankful to J.R. Clem, I. Felner we can write [13] and G.I. Leviev for valuable discussions. We thank J.R. σ2 =c2/(4πω(λ2L+λ2C), (2) Clem who kindly drew our attention to paper [7]. This work was supported by the Klatchky foundation for su- here λ and λ are the London and the Campbell pene- perconductivity. L C [1] M.Strongin,D.G.Schweitzer,A.Paskin,andP.P.Craig, [2] E. Maxwell, W.P. Robbins, Phys.Lett. 19, 629 (1966). Phys.Rev.136 A926 (1964). 5 [3] S.A.Campbell,J.B.Ketterson,andG.W.Crabtree,Rev. [10] G.I. Leviev, V.M. 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