Quantitative analysis of results of measurements of magnetoresistance of Bi_2Sr_2CaCu_2O_x single crystals for magnetic fields 0.2-0.8 T, in the temperature interval 78.5-84 K has been made. For temperatures lower than 81 K the dissipation is predominantly determined by the suppression of the long-range superconducting order, whereas closer to T_c the flux-flow contribution becomes significant. We obtain very good agreement between the experiment and the model in which two dissipation mechanisms are additive

Babić, Dinko

Ivkov, Jovica

Leontić, Boran

Repository of Faculty of Science, University of Zagreb

ISSN1330–0008CODENFIZAE4TWO–DIMENSIONAL VORTEX PLASMA IN Bi2Sr2CaCu2Ox SINGLECRYSTALS IN THE VICINITY OF TcDINKO BABIC´, BORAN LEONTIC´ and JOVICA IVKOVaDepartment of Physics, Faculty of Science, University of Zagreb, POB 162, 10001Zagreb, CroatiaaInstitute of Physics, POB 304, 10001 Zagreb, CroatiaDedicated to Professor Mladen Paic´ on the occasion of his 90th birthdayReceived 1 June 1995UDC 538.945PACS 74.72.-Hs, 72.15.Gd, 74.40.+kQuantitative analysis of results of measurements of magnetoresistance of Bi2Sr2-CaCu2Ox single crystals for magnetic fields 0.2–0.8 T, in the temperature interval78.5–84 K has been made. For temperatures lower than 81 K the dissipation ispredominantly determined by the suppression of the long-range superconductingorder, whereas closer to Tc the flux-flow contribution becomes significant. We ob-tain very good agreement between the experiment and the model in which twodissipation mechanisms are additive.1. IntroductionAlthough in the course of the past few years much effort has been devoted tofind an explanation of the magnetoresistance of high-Tc superconductors, a satis-factory solution of this and other related transport phenomena has been elusive.It is clear, however, that the vortex-lattice dynamics in these layered compoundsis governed by rather complicated mechanisms, which depend very much on a par-ticular experimental arrangement. In the presence of a magnetic field higher thanthe critical field Bcr, the anisotropic three-dimensional (3D) vortex lattice becomesFIZIKA A 4 (1995) 3, 511–518 511babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .unstable with respect to the fully two-dimensional (2D) arrangement of vortices[1-4]. The lowering of the effective dimensionality can thus lead to some phenom-ena which are inherent to 2D-superconductivity. In addition, by investigating theproperties of the isolated CuO2 sheets, common to all high-Tc compounds, valu-able information about the nature of high-temperature superconductivity can beprovided.Critical fluctuations are greatly enhanced in systems with lower dimensionality.An important consequence of such fluctuation phenomena is the creation of boundvortex-antivortex pairs, which dissociate above the Kosterlitz-Thouless (KT) tem-perature T2D [5-7]. In such systems, the existence of the vortex plasma in theabsence of any applied field makes the analysis of dissipation phenomena rathercomplicated. Strongly fluctuating vortices greatly influence the phase-coherenceof the superconducting pairs and suppress the long-range superconducting order.Also, the density of free vortices, which contribute to the flux-flow, is determinednot only by the magnetic field, but also by the thermally stimulated dissociationof the bound vortex-antivortex pairs.In our previous article [1], we made a detailed analysis of the magnetoresistancecurves of Bi2Sr2CaCu2Ox (BISCO) single crystals, for the magnetic field appliedparallel to the c-axis, in terms of dissipation due to the suppression of the long–range superconducting order in decoupled double CuO2 planes. From our results,we estimated T2D at about 37 K and Bcr between 0.1 T and 0.2 T. This we did byinterpreting the effective critical temperature of an isolated superconducting sheetas the T2D, while the critical field value was deduced from the obvious discrepancybetween the experimental results and the two-dimensional theory at low-field values.However, good agreement was achieved only at temperatures at least a few degreesbelow the mean-field transition temperature Tc, while closer to Tc the dissipationwas found to be noticeably higher then that predicted by the theory of fluctuationsalone. We made a tentative conclusion that in this temperature region the flux-flowtakes considerable part in the total dissipation, and we based this assumption onthe observed change of the sign of the Hall voltage in the proximity of Tc, which iscommonly attributed to the flux-flow [8,9].Here we report the continuation of our work on the data of Ref. 1. We find verygood agreement with the model based on the assumption that the two dissipationmechanisms are additive, in which the flux-flow contribution is determined mostlyby thermal dissociation of the bound vortex-antivortex pairs. Together with ourprevious results, we are now able to quantitatively describe the magnetoresistancecurves in the temperature range from 78.5 K to 84 K (which is just below the pointR(H = 0) = 0), and for applied fields 0.2–0.8 T.2. Results and discussionThe experimental setup has been described in detail in the previous paper [1].Briefly, the samples were produced by the KCl-method and the magnetoresistancewas measured by a low-frequency (33 Hz) AC-method in a standard four-contactconfiguration, with the magnetic field applied parallel to the c-axis of the crystal.The critical temperature Tc was determined by linear extrapolation of the steep512 FIZIKA A 4 (1995) 3, 511–518babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .part of the transition curve in zero magnetic field to R = 0, and its value wasfound to be about 84.5 K. Special care was taken to determine the inclusions ofthe Bi2Sr2Ca2Cu3Ox phase. Measurements of the magnetoresistance in the fullyresistive state above 90 K showed that this phase is slightly present. But for thegiven temperatures (T ≤84 K) and the magnetic field values (H < 1 T) its influenceon the magnetoresistance can be considered negligible. However, if one wants tohave very accurate extrapolated normal-state resistivity below Tc, which is essentialin the flux-flow calculations, the normal-state resistivity has to be extrapolated fromthe linear part of the resistance curve below 105 K, where the higher-temperaturephase mentioned above is in the superconducting state and does not contribute tothe normal resistivity. This is the procedure we adopted in our present analysis.We note that our measurements (I = 24 µA) correspond to an ohmic dissipation,since we have found that in the above mentioned part of the H − T phase diagramV − I curves are linear at least up to 1 mA.In our previous analysis, we did not succeed to describe quantitatively the dis-sipation above 81 K within the fluctuation model alone, the observed resistanceclose to Tc being higher than that predicted by the theory. We explained the ap-parent deviation of the phase-coherence-breaking-field Hϕ from the exponential lawHϕ=H0 exp (−T/T0) and its rapid fall above 81 K as the manifestation of an in-creased flux-flow contribution to the total dissipation [1]. The simplest model ofthe dual dissipation, i.e., due to the suppression of the long–range superconduct-ing order and the vortex motion, is that these two contributions should be added.Thus, the total dissipation can be expressed as R = Rc+Rf , where Rc is the resis-tance due to the critical fluctuations above T2D in the decoupled superconductingsheets and Rf is the flux-flow resistance. The flux-flow resistance can be written asRf = 2πξ2(n++n−)Rn, where ξ is the coherence length, Rn the normal-state resis-tance, and nf = n+ + n− is the total density of free vortices (n+) and antivortices(n−).When we extrapolate the dissipation due to the critical fluctuations in the regionof the mixed dissipation (close to Tc) and subtract it from the total dissipation,we observe a remarkable thing: the additional dissipation depends mostly on thetemperature, and very weakly on the magnetic field (Fig. 1). At a given temperaturethe field dependence of the additional dissipation for the fields 0.2–0.8 T is almostwithin the experimental error (because the total dissipation is still predominantlydetermined by the fluctuations of the superconducting order-parameter). So, in thefirst approximation, we neglect the field dependence of Rf .This rather surprising observation indicates that the density of the free vorticesis determined by some scaling that is strongly dependent on the specific proper-ties of the thermally created 2D-vortex-plasma in the decoupled superconductingsheets. The density of the free vortices in the conventional 3D-superconductors isdetermined simply by the ratio B/Φ0, where B is the magnetic field and Φ0 isthe magnetic flux quantum, which leads to the relation Rf = (B/Hc2)Rn. Thus,in this simple model, the magnetic–field dependence of the flux-flow resistance islinear and its temperature dependence is given by the temperature dependence ofthe upper critical field Hc2. It seems that the situation in the high-Tc compoundsFIZIKA A 4 (1995) 3, 511–518 513babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .is much more complicated.Fig. 1. Normalized flux-flow contribution Rf (H,T )/R(0, 90 K) to the total dis-sipation, for applied magnetic fields 0.2–0.8 T, extracted by subtracting thefluctuation resistance obtained in our previous work from the total resistanceR(H,T )/R(0, 90 K). Solid curve is the mean–temperature dependence of the flux-flow resistance, neglecting the small field dependence. Temperature dependence canbe described by the law Rf (T ) ≈ Rn[exp(−U˜/T )]ν(t), where Rn is the normal-stateresistance, ν(t) = 1/(t+ xtω), t = [(T/T2D)− 1] and T2D = 37 K.Doniach and Huberman made a careful analysis of the statistical properties ofthe vortex plasma in two-dimensional systems which exhibit the Kosterlitz-Thoulessbehaviour [6]. Assuming that for fields higher than Bcr superconducting sheets inhighly anisotropic high-Tc compounds, such as BISCO, are indeed 2D-systems witha Kosterlitz-Thouless transition, this approach could be applied to explain theirproperties. In the Doniach and Huberman model, the density of the free vorticesand antivortices is determined by n± = ξ−2 exp(−βU±), and β = 1/kBT and U± isthe characteristic energy of creation of a free vortex (+) or antivortex (−). In theabsence of the magnetic field U+ = U− = U0, where U0 is the sum of two parts.One is the chemical potential µ associated with the vortex core (µ = dξ2H2c /8,where d is an effective sheet thickness, ξ the in-plane coherence length, and Hcthe thermodynamic critical field), and the other is the screened vortex-antivortexinteraction US . With no applied field present, the density of the free vortices isdetermined by [6]:nf = 2ξ−2exp[−β(µ+ US)]. (1)The exponential function has the following physical explanation: the term associ-ated with µ describes the probability of a thermal creation of a vortex of any sign,and the term associated with US represents the probability of a thermally stimu-514 FIZIKA A 4 (1995) 3, 511–518babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .lated vortex-antivortex pair dissociation. Arguing that the vortex dielectric functionin the vicinity of T2D behaves as ǫ(t) = ǫ(0)(1+xt1/2), where t = [(T/T2D)−1] andx is a numerical constant, Doniach and Huberman found that nf can be writtenas:nf = 2ξ−2[exp(−βU˜)]ν(t). (2)Here ν(t) = 1/(t + xt1/2) and U˜ = µ + C, where C is a constant which dependson US and measures the strength of the binding energy. It has to be emphasizedthat µ is temperature dependent, since Hc and ξ are temperature dependent, andits dependence in the Ginzburg-Landau theory is approximately (1 + τ)(1 − τ2),where τ = T/Tc. If we take that for BISCO ξ(0) ≈ 2 × 10−9 m, d < 1.5 × 10−9m (smaller than the distance between the superconducting sheets) and Hc(0) ≈Hc2(0)ξ(0)/λ(0) (where λ(0) ≈ 3 × 10−7 m is the in-plane magnetic penetrationdepth and Hc2(0) ≈ 40 T), we obtain µ(0) ≈ 20–30 K, which is rather small,and close to Tc it becomes even smaller. A small value of µ indicates that U˜ inits major part represents the binding energy of the vortex–antivortex pairs. Also,one should expect a slight temperature dependence of the probability exp(−βµ)in the temperature region of our investigation. However, the strongly temperaturedependent flux–flow resistance (Fig. 1) is mostly due to the thermal dissociation ofvortex-antivortex pairs.The presence of the applied field introduces the paramagnetic splitting termUH , so that U± = U0∓UH . The result is a net imbalance in the density of vorticesparallel (+) and antiparallel (−) to the field. The splitting term UH is basicallyequal to mH, where m is the paramagnetic moment of the vortex (there is also anadditional term in UH , which describes the “charging” effects and it is proportionalto (n+−n−); see Ref. 6 for details). If we look at the energy scale, it is obvious thatif the splitting term UH is much smaller than the characteristic field-independentenergy U˜ , the influence of an applied magnetic field on the (anti)vortex densityshould be very weak indeed. The field-induced dissociation of the vortex-antivortexpairs is then low by comparison with the thermal dissociation, so that the influenceof the magnetic field is mostly in making the (anti)vortices two-dimensional.Since the observed Rf is really very slightly field dependent and strongly tem-perature dependent, we tried to represent all data for various fields ranging from0.2 T to 0.8 T, with a single temperature-dependent curve of the form given in Eq.(2). We introduce an experimental exponent ω which should better describe thetemperature dependence of W = νU˜ by replacing xt1/2 with xtω (it is clear thatthe temperature dependence is much stronger than the predicted t1/2-dependence).Thus, as the fitting parameters we use U˜ , x and ω, for Rn we use the extrapolatedlinear resistance in the temperature region 100–104 K, and for T2D we use the pre-viously estimated value of 37 K [1]. We obtained the best results with the followingvalues of the parameters: U˜ = 2800± 300 K, ω = 7± 1, x = 1.4± 0.2.Having this curve, we added the extrapolated resistance Rc to the computedRf and obtained a predicted curve for the total dissipation. As the result of thisprocedure, we obtained a quantitative description of the magnetoresistance curvesFIZIKA A 4 (1995) 3, 511–518 515babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .from 78.5 K to 84 K, i.e., in the region where there is no dissipation in zero field,and for the fields from 0.2 T (just above Bcr) [1] to 0.8 T (Fig. 2).Fig. 2. Comparison of the experimental results and computed curves of the normal-ized magnetoresistance R(H,T )/R(0, 90 K) for various magnetic fields. The bottomfitted curve corresponds to H=0.2 T, and each subsequent curve corresponds tothe field increased by 0.1 T, up to 0.8 T (top curve). Dashed line is the flux-flowcontribution to the total dissipation.Fig. 3. Temperature dependence of the energy W (T ) = U˜ν(T ), which is mostly thebinding energy of the vortex-antivortex pair. W (T ) rapidly increases with decreas-ing temperature.516 FIZIKA A 4 (1995) 3, 511–518babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .Apart from the exact values of the fitting parameters ω and x, which describe thetemperature dependence of Rf , and which are just phenomenological parameters,in our present model, the potential W should be very sensitive to the temperature(Fig. 3) in order that the theory fits well to experimental results, (Fig. 2); in thesmall temperature region investigated, the potential W changes by a factor of two.As mentioned before, this is mostly due to the strong temperature dependence ofthe vortex-antivortex dissociation.The large value of U˜ and the strong temperature dependence ofW set the energyscale which makes the small contribution of the magnetic field to the dissociationof vortex–antivortex pairs more understandable (the magnetic correction to thedensity of the free vortices is accounted for by introducing the factor [cosh(βUH)]ν)in the expression for nf in Eq. (2) [6]. One should apply a relatively strong magneticfield to make its influence significant.3. ConclusionWe have made a quantitative analysis of the magnetoresistance of single crys-tals of BISCO superconductors within the model of mixed dissipation due to thesuppression of the long-range superconducting order and vortex motion, for mag-netic fields 0.2–0.8 T (just above the critical field Bcr which separates 2D from3D behaviour) and for temperatures 78.5–84 K. Above 81 K, the model of thefluctuations of the superconducting order-parameter has to be modified by theflux-flow contribution to the dissipation. The flux-flow contribution to the totaldissipation is explained in terms of the Doniach-Huberman statistical model of2D-superconductors which exhibit the Kosterlitz-Thouless transition. Our analysisshows that free vortices which contribute to the flux-flow are mostly produced bythermal dissociation of the bound vortex–antivortex pairs. The free-vortex densityis weakly dependent on the magnetic field, due to the high value of the binding en-ergy in comparison with the paramagnetic dissociation energy set by the magneticfield. This indicates that thermal creation and dynamics of the vortex plasma is avery important mechanism within the superconducting sheets in high–temperaturesuperconductors with a high degree of anisotropy. A full description of the influenceof the vortex plasma on the properties of these materials has to be elucidated byfurther extensive research, mostly by extending the measurements to lower tem-peratures.AcknowledgementWe greatfully acknowledge financial aid of the Ministary of Science of Croatia.References1) D. Babic´, J. Ivkov, N. Heinrich and B. Leontic´, Phys. Rev. B 49 (1994) 15965;2) D. S. Fisher, M. P. A. Fisher and D. A. Huse, Phys. Rev. B 43 (1991) 130;FIZIKA A 4 (1995) 3, 511–518 517babic´, leontic´ and ivkov: two-dimensional vortex plasma . . .3) L. I. Glazman and A. E. Koshelev, Phys. Rev. B 43 (1991) 2835;4) H. Raffy, S. Labdi, O. Laborde and P. Monceau, Physica C 184 (1991) 159;5) J. M. Kosterlitz and D. Thouless, J. Phys. C 5 (1972) L126;6) S. Doniach and B. A. Huberman, Phys. Rev. Lett. 42 (1979) 1169;7) M. R. Beasley, J. E. Mooij and T. P. Orlando, Phys. Rev. Lett. 42 (1979) 1165;8) S. J. Hagen, C. J. Lobb, R. L. Greene and M. Eddy, Phys. Rev. B 43 (1991) 6246;9) S. N. Artamenko, I. G. Gorlova and Yu. I. Latyshev, Phys. Lett. A 138 (1989) 428.DVODIMENZIONALNA PLAZMA MAGNETSKIH VRTLOGA UMONOKRISTALIMA Bi2Sr2CaCu20x U BLIZINI TcIzvrsˇili smo kvantitativnu analizu magnetootpora monokristala Bi2Sr2CaCu2Ox,za magnetska polja 0.2–0.8 T i u temperaturnom podrucˇju 78.5–84 K. Na tempe-raturama ispod 81 K disipacija je uglavnom posljedica potiskivanja dugodosezˇnogsupravodljivog uredenja, dok blizˇe Tc doprinos kretanja vrtloga postaje znacˇajan.Ustanovljeno je vrlo dobro slaganje eksperimentalnih rezultata i modela u kojemsu ta dva disipacijska mehanizma aditivna.518 FIZIKA A 4 (1995) 3, 511–518

Two-dimensional vortex plasma in Bi_2Sr_2CaCu_2O_x single crystals in the vicinity of T_c

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Two-dimensional vortex plasma in Bi_2Sr_2CaCu_2O_x single crystals in the vicinity of T_c

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