Field dependent alternating current susceptibility of metalorganically deposited YBa[sub 2]Cu[sub...

Field dependent alternating current susceptibility of metalorganicallydeposited YBa2Cu3O7−� films

D.-X. Chena�

ICREA, Psg. Lluís Companys 23, 08010 Barcelona, Spain and Grup d’Electromagnetisme,Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

E. Pardo and A. SanchezGrup d’Electromagnetisme, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,Barcelona, Spain

M. N. IlievDepartment of Physics and Texas Center for Superconductivity and Advanced Materials,University of Houston, Houston, Texas 77204-5002

S.-S. Wang and Z.-H. HanApplied Superconductivity Research Center, Department of Physics, Tsinghua University, Beijing 100084,China

�Received 5 November 2006; accepted 23 January 2007; published online 5 April 2007�

The field amplitude and frequency dependent complex alternating current susceptibility ��Hm , f� ofYBa2Cu3O7−� films metalorganically deposited on different substrates with different processes hasbeen measured at 77 K and studied in relation with their microstructures and some modeling results.It is shown that ��Hm� for films with well aligned grains and a high Jc is of Bean type with acharacteristic f dependence for a power-law E�J�, so that thermally activated collective flux creepis the dominant dissipation mechanism. The Jc of these films may be well determined by ��Hm , f�measurements. For films with misaligned grains and intermediate values of Jc, ��Hm , f� isanomalous and able to be roughly simulated by a linear-exponential E�J�. This phenomenon shouldbe related to the presence of weak links and Josephson vortices, but to look for its physicalmechanism is still challenging. © 2007 American Institute of Physics. �DOI: 10.1063/1.2713938�

I. INTRODUCTION

Solution techniques have various advantages in obtain-ing high-quality YBa2Cu3O7−� �YBCO� films in the coatedconductor development, such as a precise controllability ofcompositions, a wide flexibility to coating object, a highdeposition rate, and a low cost of the required nonvacuumprocessing equipment.1–6 Among all the solution techniques,metalorganic decomposition using thrifluroacetates is one ofthe most promising methods for fabricating YBCO filmswith critical-current density Jc well above 1010 A/m2.7–10

In the studies of preparation technology of metalorgani-cally deposited YBCO films, the direct four-point direct cur-rent �dc�-voltage technique was traditionally used for testingthe conducting performance.1–6 For measuring Jc, the filmsoften had to be necked to reduce the current-carrying crosssection and proper silver electrodes had to be deposited ontothe fired films before further annealing at 450 °C inoxygen.2–4 Such a complex sample preparation proceduremade this technique practical only for limited cases. On theother hand, some contactless inductive techniques have beendeveloped to conveniently measure Jc and current-voltagecharacteristic of YBCO films, so that a detailed research maybe carried out for every film sample.11–13 In spite of this, the

second type of techniques has been utilized mainly as a toolto determine Jc but not to study the involved dissipationmechanisms, except for cases where only low alternatingcurrent �ac� fields are needed.14

Recently, using a home-made high-field acsusceptometer,15 we have measured the ac susceptibility �

=��− j�� of YBCO films as a function of field amplitude Hm

and frequency f . Comparing ��Hm , f� with modeling results,we have found that the thermally activated collective vortexcreep is the dissipation mechanism for a high-quality singlecrystal film deposited by high rate pulsed laser deposition�PLD� on a SrTiO3 single crystal, the flux flow along thegrain boundaries �GBs� corresponds to the dissipation ofsome coated conductors made by PLD technique, and thethermally activated Josephson-vortex creep may be the re-sponsible factor for the dissipation of some metalorganicallydeposited films.16,17 Since the last case is very interesting, wewill make a systematic study on it in the present article.

The studied samples and their ac susceptibility and re-sistance measurements are described in Sec. II, where rel-evant experimental results for microstructures are also given.Since it is a necessary condition for presenting the features ofthe measured ��Hm , f� and for understanding them, severalmodels of ac susceptibility of hard superconductors are in-troduced in Sec. III. Our discussion is made in Sec. IV bycommenting and interpreting phenomena of great interest.Finally, our conclusions are stated in Sec. V.a�Electronic mail: [email protected]

JOURNAL OF APPLIED PHYSICS 101, 073905 �2007�

0021-8979/2007/101�7�/073905/12/$23.00 © 2007 American Institute of Physics101, 073905-1

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http://dx.doi.org/10.1063/1.2713938

http://dx.doi.org/10.1063/1.2713938

http://dx.doi.org/10.1063/1.2713938

II. SAMPLES AND MEASUREMENTS

A. Samples

Each studied YBCO film was deposited on a polished�001� surface of a LaAlO3 �LAO� single-crystal substrate ofdimensions 5�or 10��10�0.5 mm3 with or without a bufferlayer. Four samples were �A� YBCO-Au-LAO with a 37 nmgold buffer layer, �B� YBCO-BST-LAO with a 60 nmBa0.5Sr0.5TiO3 �BST� buffer layer, �C� YBCO-LAO without abuffer layer, and �D� YBCO-LAOR without a buffer layerprepared by a refined process. The YBCO film thickness was170, 100, 220, and 220 nm for A, B, C, and D, respectively.

The BST buffer layer was metalorganically deposited onthe polished LAO surface. For the precursor solution, thestarting materials were strontium acetate, barium acetate, andtitanium�IV� butoxide. Glacial acetic acid and acetyl acetonewere used as chelating agents and 2-metoxyethanol was usedas solvents. A concentration of 0.4 M was used for deposit-ing onto the LAO substrate by using a photoresist spinner atroom temperature with a spin rate of 5000 rpm for 30 s; theambient humidity was controlled to be below 50%. The as-coated gel film was inserted into a quartz tube furnace,ramped to 200 °C slowly under 4% H2-Ar gas flow, and keptat this temperature for a certain time, and then ramped againto 1000 °C, at which kept for 4 h before cooled to roomtemperature. The Au buffer layer was sputter-deposited onthe polished LAO surface in a Gatan 682 precision ionetching/deposition system at room temperature.

For preparing the YBCO coating solution, trifluoroaceticacid and the acetates of Y, Ba, and Cu were selected with atwo-step process. In the first step, barium acetate was en-tirely dissolved in de-ionized water, and then yttrium acetateand copper acetate were added at room temperature withstrong stirring. The cation ratios of Y, Ba, and Cu in thesolution were 1:2:3. The solution was stirred for 30 min toensure that the salts were entirely dissolved. After that, astoichiometric amount of trifluoroacetic acid was added tothe solution. The second step was different for samples A–Cand sample D. For samples A–C, the solution was simplyevaporated at 50 °C for about 24 h to yield a glassy blueresidue. For sample D, however, the solution was distilledunder decompression at 100 Pa and water bathed at 40 °C toyield a glassy blue residue. After this, the residual gel wasdissolved in sufficient methanol to obtain a solution againand distillation under decompression was repeated to refinethe gel. For all samples, the resultant residual gel was dis-solved in methanol to obtain a final YBCO coating solutionof certain concentration, which was coated by using the pho-toresist spinner at room temperature onto the given substratewith a spin rate of 4000 rpm for certain time. The coatedsample was heat treated in two stages. The first was a py-rolysis stage, in which the gel film was heated slowly to380 °C in humid oxygen to form a uniform fluoride-containing solid film. During the second sintering stage up to775 °C, fluorine was eliminated and the YBCO supercon-ducting film formed, following a protocol described in Refs.4 and 18 with controlled variations of temperature and atmo-sphere.

B. ac susceptibility and resistance

1. ac susceptibility

The perpendicular external ac susceptibility �=��− j��of the film sample was measured at 77 K as a function of thefield amplitude Hm and frequency f . To do this, a high-fieldac susceptometer using Helmholtz coils as a magnetizer wasdesigned and constructed.15 During measurements, thesample was placed at the center of the Helmholtz coilswound with a thick enamel-covered copper wire and sur-rounded by two concentric measuring-compensating coils ofthin enamel-covered copper wire connected in series opposi-tion with different areas and numbers of turns. The entireassembly was immersed in liquid nitrogen. The mutual im-pedance between the Helmholtz coils and the measuring-compensating coils was adjusted �by winding a proper num-ber of turns� to be nearly zero when the sample was absentand the susceptibility of the sample was calculated from thedifference of the mutual impedance between the cases withand without the sample. The ac magnetizing current was pro-vided by a bipolar operational power amplifier driven by theinternal oscillator of a lock-in amplifier, which was also usedfor measuring the magnetizing current �through a resistorconnected in series to the Helmholtz coils� and the electro-motive force induced in the measuring circuit. By carefullymeasuring the dimensions and counting the numbers of turnsof all the coils and by a final calibration with a copper disk asa standard sample, absolute results for � of the film sampleswere obtained in Systeme International units.15,19

The measured �� and �� of samples A, B, C, and D asfunctions of �0Hm and f are plotted in Figs. 1�a�–1�d�, re-spectively. We can see quite different features for differentsamples, whose details will be described and discussed later.

2. Resistance

The samples were checked by four-point resistance mea-surements with 0.5 mA current during cooling; they weresuperconducting below Tc with a metallic type of R�T� curveabove Tc. The reduced resistance curves for three samplesare shown in Fig. 2, from which we see Tc�90, 81, and 91.5K for films A, B, and C, respectively, and there is a smallresidual resistance between Tc and 86 K for film B.

C. Microstructures

1. Raman spectra

The local phases and their crystallographic orientations,as well as the oxygen concentration and occupations, may betested by Raman spectra measurements.20 In the presentwork, Raman spectra of films A, B, and C were measuredunder a microscope with objective with �50 magnification.The laser spot size was 3−4 �m. The sample was laying ona rotation stage, and by rotation of the stage the angle be-tween incident light polarization and the sample edge couldbe rotated, thus making it possible to work in different par-allel and crossed scattering configurations.

Some conclusions may be made from the measurementsas follows. For film A, the oxygen content is close to 7, andthere is a small amount of impurities. However, the peaknear 500 cm−1 corresponding to apex oxygen �O4� vibrations

073905-2 Chen et al. J. Appl. Phys. 101, 073905 �2007�


along the c axis is much stronger in parallel polarizationsthan the 334 cm−1 peak, which is a clear indication that partof the film is not c oriented. From the variations of the500 cm−1 peak intensity with the rotation angle one can con-clude that the projection of the c axis on the film surface isalong the sample edges. Film B is well c oriented with a andb axes parallel to the sample edges. Its oxygen content isabout 6.8, and there is a relatively small amount of BaCuO2

impurity phase as indicated by the structure near 630 cm−1.The most prominent difference of film C from films A and Bis the presence of additional relatively strong structures at230, 264, 295 and 630 cm−1 at the most spots probed. The

630 cm−1 structure can be assigned to BaCuO2. The rest ofthe additional structures likely indicate a large number ofbroken chains. For full chains the vibrations of Cu1 and O1are Raman forbidden as these atoms are at centrosymmetricalsites. If some of O1 atoms are missing or shifted to vacancysites, the vibrations of chain end Cu1 and O1 atoms becomeRaman active and their vibrations give Raman lines near 230and 595 cm−1, respectively. In an alternative explanation,these additional peaks may be related to other minorityphases. If these lines are neglected, the sample is perfectly coriented and the a and b axes are along the sample edges.The oxygen content is close to 7.

Several spectra recorded with both incident and scatter-ing light polarizations parallel to an edge of the sample areplotted in Fig. 3.

FIG. 1. �� and �� measured at 77 K as functions of ac field amplitude Hm

and frequency f . Plots �a�, �b�, �c�, and �d� are for films A, B, C, and D,respectively.

FIG. 2. Reduced resistance as a function of temperature for films A �YBCO-Au-LAO�, B �YBCO-BST-LAO�, and C �YBCO-LAO�.

FIG. 3. Room-temperature Raman spectra for films A, B, and C at severalspots. Both incident and scattering light polarizations are parallel to an edgeof the film.



2. Scanning electromicroscopy

Six scanning electromicroscopy �SEM� photos areshown in Fig. 4, where �a�, �c�, �e�, and �f� were taken at highmagnifications for films A, B, C, and D, respectively, and �b�and �d� were taken at low magnifications for films A and B,respectively. All of them are from the same samples as thoseused for � measurements, except �f�, which is for anothersample of the same batch. We observe that all films have agranular structure with certain porosity �voids are presentedby dark spots� and the YBCO grain �gray� sizes are on theorder of 1 �m. The bright areas are impurities of BaCuO2 or�Y, Ba�CuO2, as confirmed by elemental analysis. There is amosaic pattern parallel to the film edges in �c�, which wasreported in the literature as an indication of the existence ofgrains with the c axis lying in the film surface.4 Such apattern has also been observed in some regions of film A,which is consistent with the Raman spectra described earlier.Comparing �e� with �f� for two films without a buffer layer, itseems that the latter corresponds to a more homogeneousstructure containing less porosity and impurities. Anotherimportant fact is that long cracks are found in �d� for YBCO-BST-LAO but not for the other films, as in �b�.

3. X-ray diffractions

For film B, full width at half maximum �FWHM� valuesof 1.2° and 1.8° were obtained from the out-of-plane � andin-plane � scan, respectively. These values were around 1°for films C and D. All this shows good c and a �or b� align-ments, which is in agreement with the earlier-mentioned Ra-man spectra. However, without enough resolution, the dif-fraction patterns did not show the existence of grains withthe c axis parallel to the film surface for samples A and B.For sample A, �−2� x-ray diffraction measurements showeda clear evidence of the overall c-axis alignment. As plotted in

Fig. 5, however, there is a unique feature in the � scan offilm A, which shows the existence of many grains with theira or b axes making 30°, 45°, or 60° angles with the filmedges, being a phenomenon not found from the Raman spec-tra study. Since this � scan spectrum is consistent with thatof the gold film before depositing the YBCO layer, such agrain structure should be a rough crystallographic copy ofthe gold film. Since the misfit in lattice parameter betweenAu and LAO is as large as 7.6%, the creation of such acomplex structure in the gold layer deposited on single crys-tal LAO is possible.

III. MODELS OF AC SUSCEPTIBILITY

A. � for films obeying the Bean model

The magnetic properties of a hard superconductor werecalculated by Bean from what was later called the critical-

FIG. 4. SEM pictures for films A ��a�, �b��, B ��c� ,�d��, C �e�, and D �f�.

FIG. 5. The �-scan x-ray diffraction pattern of film A.



state model,21 assuming that a lossless macroscopic currentmay be sustained up to a field-independent critical-currentdensity Jc �the Bean model�.22 In order to understand � of thestudied films, we first give the results calculated from thismodel. The original Bean model was used for longitudinalmagnetic properties of infinitely long bodies, where demag-netizing effects were absent. In relation to the perpendicular� of our films, the relevant cases that have been treated arefor thin strips and thin disks, whose � as a function of Hm

has been numerically calculated and tabulated in Refs. 23and 24 based on the analytical formulas derived in Refs.25–28. Considering a thin strip of width w and thickness tand a thin disk of radius r and thickness t, both having acritical-current density Jc, the perpendicular �� /�0 and�� /�0, −�0 being the �� in the complete shielding state, asfunctions of Hm / �Jct� are plotted in Fig. 6. The characteristicquantities �0, the maximum �� /�0, �m� /�0, and the �� /�0 andHm / �Jct� corresponding to �m� are listed in Table I.

The foregoing results may be satisfactorily used for ourrectangular films of width w, length l, and thickness t�w /10 000, of which �0 may be calculated with an accuracyabout 1% by

�0 = 0.44�1 + 0.294� l

w− 1�w

t�1

l

w 2� . �1�

This formula has been obtained by extrapolating the datanumerically calculated for completely shielded prisms, re-ported partially in Ref. 29.

B. � for thermally activated collective vortex creep

In the Bean model, a constant Jc is assumed as a prop-erty of hard superconductors regardless of its origin. In this

case, the local electric field is E=0 when the magnitude ofcurrent density obeys J�Jc, whereas any finite values of Ecan be induced when J=Jc. As a consequence, the magneticproperties deduced from the critical-state model are hyster-etic without a time or frequency dependence. Such a depen-dence will occur when a specific loss mechanism is involved.For example, even if the origin of Jc itself is not considered,a time dependence can occur owing to normal eddy currentssince the normal conductivity n of the bulk should set amaximum E /J, above which one has E=J /n.30 Consideringthe origin of Jc, a model of thermally activated Abrikosovvortex �AV� depinning and creep was given by Kim et al.and Anderson soon after Bean proposed his model.21,31 Forhigh-temperature superconductors being doped insulatorsrather than conventional metals, the coherence length � issmall and pinning centers are mainly provided by point de-fects, e.g., oxygen vacancies, rather than by generic extendedprecipitates or GBs. As a result, the AV pinning is weak anda collective creep can take place.32 From the J-dependentactivation energy of collective depinning

U�J� = U0 lnJc/J , �2�

Brandt has derived from the Arhennius law a power-lawcurrent-voltage dependence33,34

E�J� = sgn�J�EcJ/Jcn, �3�

where n=U0 / �kT� and Jc may be conveniently defined as theJ when the electrical field meats the criterion E=Ec

=10−4 V/m. He has used this model with n�1 to approxi-mately characterize the behavior of collective AV depinningand creep and computed the perpendicular ��Hm , ��, �=2�f , of disks that obey this power-law E�J�. It turns outthat in this case for a finite n, ��Hm� at a fixed � is similar tothat of the Bean model �i.e., for n→ � and it shifts to higherHm with increasing � in such a way that ��Hm , �� dependson Hm /�1/�n−1� only. This last statement is referred to as thescaling law.33,34

The differences of ��Hm� for the model of collective AVdepinning and creep from that for the Bean model may befound in the values of �m� and ��m� �, i.e., the �� correspond-ing to �m� . Normalized to the Bean values for a thin disk, they

FIG. 7. For the perpendicular ac susceptibility of a thin disk of radius r andthickness t obeying the power-law E�J�, the maximum ��, �m� , the �� atwhich �m� occurs, ��m� �, and the field amplitude at which �m� occurs,Hm��m� �, normalized to their Bean values as functions of 1 /n. For the lastquantity, dimensionless frequency �c

2��0�rtc /2=1 is set.

FIG. 6. Perpendicular �� /�0 and �� /�0 as functions of normalized fieldamplitude Hm / �Jct�, Jc and t being the critical-current density and thickness,for a thin disk and a thin strip calculated from the Bean model.

TABLE I. Characteristic quantities for perpendicular ac susceptibility ��− j�� of thin strip of width w and thickness t and thin disk of radius r andthickness t calculated from the Bean model.

Quantity Strip Disk�0 �w / �4t� 8r / �3�t��m� /�0 0.2365 0.2408−�� /�0��=�m� � 0.3571 0.3807Hm / �Jct��=�m� � 0.785 0.970



are plotted in Fig. 7 as functions of 1 /n. We see that withincreasing 1/n from 0, both �m� and ��m� � increasesmoothly. In order to compare both models in Hm��m� �, acritical frequency has to be defined, since Hm��m� � is fre-quency dependent for the creep model according to theabove-mentioned scaling law. Defining a dimensionless criti-cal frequency �c

2��0�rtc /2=1 for the thin disk of radius rand thickness t, where c=Jc /Ec, the Hm��m� � normalized toits Bean value �denoted by a subscript B� as a function of1/n is plotted in Fig. 7, from which we see that Hm��m� � at�c

2=1 decreases smoothly with increasing 1/n. We shouldnote that the Hm��m� � vs 1/n relation at any value of �c

2 maybe obtained from this curve using the scaling law mentionedearlier. Although the choice of �c

2=1 is arbitrary, it is conve-nient since too low a �c

2 will make Hm��m� ��Hm��m� �B at1 /n=0.2 and too high a �c

2 will lead Hm��m� � to decrease andthen increase with increasing 1/n.

It is worth explaining why the dimensionless frequencyis symboled as �c

2. In the classical work of eddy-current per-meability and impedance calculation, � appeared as an anglein radians in the analytical solution of Maxwell equationsand the Ohm law, and �2 was named as the normalized ordimensionless frequency for it was proportional tofrequency.35,36 In fact, the physical meaning of �2 is simplythe square of the ratio of a characteristic dimension, alongwhich the induced currents penetrate, to the skin depth �s,

�s = ��/2�−1/2, �4�

where � and are the scalar permeability and conductivityof the material and �=2�f is the angular frequency. Thiskind of notations has been conveniently and consistentlyused in our previous studies on problems related to eddycurrents or circulating supercurrents with a certain E�J�relation,19,37–42 including � f

2 used later for the flux flow case.

C. � for flux flow above a finite Jc

Analogous to the Bean model, the power-law E�J� forthe collective AV creep model must have an upper limit setby the normal conductivity of the bulk n. However, as stud-ied in grain-aligned YBCO films, there can be Abrikosov-Josephson vortices �AJVs� along the strong-link GBs andthese AJVs can start to flow at values of E much smallerthan those set by n and even smaller than the Ec used in thepower-law E�J� for the collective AV creep inside thegrains.16

In biaxially textured YBCO coated conductors Jc is lim-ited by networks of low-angle GBs. Different from the AVsin the grains with a core size of �, the vortices in the low-angle GBs are AJVs with a core size along the GB greaterthan �.43 Owing to the smoothing off of pinning potentialsrelated to the presence of more pinning centers over thelarger core dimension along GB and to the increase of theelastic energy relevant to the AJV bending parallel to GB, theAJVs are less pinned by point defects along the GB than AVsare in the grains, but mainly pinned by dislocations alignedin the GB.44 If pinning of dislocations is small and uniformenough, with increasing current, the dissipation will first oc-cur from thermally activated AJV creep and quickly chang-

ing into vortex flow channeling along GBs, with a basicallylinear E�J� curve above Jc. This behavior has been inferredfrom systematic experimental studies that have been carriedout mainly on transport properties of bicrystal films.45–47

For this case, we can assume an approximate dissipationmodel, where

E = 0�J Jc� , �5�

E = �J − sgn�J�Jc�/ f�J � Jc� . �6�

In order to know the features of the ac susceptibility forsuch a type of flux-flow process, longitudinal ��Hm , �� hasbeen calculated for a long cylinder of radius r obeying thisE�J� characteristic.42 The results are shown in Fig. 8, whereHm is normalized to Jcr and � is replaced by a dimensionlessfrequency � f

2��0�r2 f. We see that when � f2�0, the results

coincide with those of the Bean model, and with increasing� f

2, all the three values of �m� , −��m� �, and Hm / �Jcr��=�m� � increase.

D. � for Kim–Anderson type of thermally activatedvortex creep

Although thermally activated vortex depinning and creepwas the primary explanation of the critical state in hardtype-II superconductors proposed by Kim and Anderson,21,31

the E�J� corresponding to this mechanism does not lead to a��Hm� that is similar to the Bean curves in general. As de-rived in Refs. 31, 32, and 34, we have for the Kim-Andersontype of thermally activated vortex creep that

U�J� = U0�1 − J/Jc� , �7�

E�J� = sgn�J�Ec exp�m�J/Jc − 1�� , �8�

where U0 is the pinning barrier for a vortex �or a vortexbundle�,31 Jc is the critical-current density for vortex depin-ning at T=0, and m=U0 / �kT�. It gives a finite E�±0�= ±Ece

−m at J=0, so that the conductivity will cross zerowhen the induced E changes its sign during ac magnetiza-tion. Since the actual conductivity cannot be lower than thenormal conductivity n for the bulk, this model has to be

FIG. 8. The calculated longitudinal �� and �� of a long cylinder of radius robeying the flux-flow E�J�, Eqs. �5� and �6�, with critical-current density Jc

as functions of normalized field amplitude Hm / �Jcr� and dimensionless fre-quency � f

2��0�r2 f. Arrows indicate the direction of increasing frequency.Dashed lines show results of the Bean model.



replaced by E=J /n when Ec exp�m�J /Jc−1�� J /n. Asa result, ��Hm , f� will have complicated and m-dependentfeatures. In order to study this phenomenon qualitatively, wehave calculated the longitudinal ��Hm , f� of a long cylinderof radius r=5 mm with such an E�J� characteristic. As donein Ref. 17, it is arbitrarily assumed that n= �Jc /10� /E�J=Jc /10�. The calculated results for Jc=104 A/m2 and m=2,5, and 10 are plotted in Figs. 9�a�–9�c�, respectively. We cansee that the low-Hm �� increases with decreasing f in allcases and there is a pronounced minimum in the ��Hm�curve corresponding to a two-peak ��Hm� feature if m issmall.

IV. DISCUSSION

A. Granular superconductor

For YBCO bicrystal films with the c axis perpendicularto the film surface, the Jc across the GB decreases exponen-tially with the a�b�-axis misorientation angle �,

Jc�� = Jc�0�exp�− �/�0� , �9�

where �0�4° at 77 K, if ��4°, whereas the intergranular Jc

is relatively insensitive to � when ��4°.48 As described inSec. II C, for films B, C, and D, the c axes of most grains areperpendicular to the surface and their a�b�-axis misalignmentangles are less than 2°. Therefore, from the Jc point of view,these films should behave as a multiconnected single crystaland their dissipation mechanism should be dominated by thethermally activated collective flux creep with a power-lawtype of E�J�. ��Hm� for a power-law E�J� is similar to that ofthe Bean model shown in Fig. 6 but with a frequency-dependent and higher �m� /�0 according to Table I and Fig. 7.Compared with these features, we notice in Fig. 1 that onlysample D may be basically regarded as a single crystal,whereas the �m� /�0 of sample C is too low and remarkableanomalies occur in ��Hm , f� at low Hm for samples A, B,and C. This implies that at least films A, B, and C are granu-lar in nature.

As described in Sec. III C, there can be AJVs along theGBs when � is not too large and the flux flow of less pinnedAJVs will cause �m� to increase with increasing f . This be-havior is not seen in Fig. 1 for any sample, which suggeststhat either typical AJVs are not present in these samples orthey are tightly pinned to GBs if in case they exist. Weshould mention that the flux-flow type of ��Hm , f� for acoated conductor reported in Ref. 16 is not a general phe-nomenon for all coated conductors, since a dominance offlux-flow dissipation mechanism requires an intermediate�-scan FWHM value �equal to 6.3° in Ref. 16� and somecontinuous vortex-flow paths across the entire film.45

When ��20°, the intergranular Jc will be more thanthree orders of magnitude reduced according to Eq. �9�. Thisyields that film A is composed mainly by weak-linked grainswith many GBs having a misorientation �=15° or 30°. It isinteresting that although the x-ray diffraction pattern con-firmed that the a or b axes of a lot of grains were not parallelto the film edges, no such evidence was found by Ramanspectra for a dozen of spots. An explanation of this may bethat the sizes of misaligned grains are appreciably less than1 �m and they are very uniformly distributed in the film, sothat the spectra for each spot of size 3−4 �m are dominatedby bigger grains whose a and b axes are parallel to the filmedges.

In the above, we have considered neighboring grains tobe well connected. However, the existence of porosity, non-superconducting impurities, and mosaic structures �shown inSEM pictures or implied by Raman spectra but not displayedby x-ray results owing to a low resolution�, perpendicular towhich supercurrent can hardly flow, will create many badconnections between neighboring grains. Therefore, the stud-ied films should be granular superconductors with bothstrong-link and weak-link features in general.

B. Josephson vortices and critical state

The anomalous ��Hm , f� shown in Figs. 1�a� and 1�b�was previously reported and briefly analyzed in Ref. 17. Ex-tending the analysis made in Ref. 17, we will see that thisphenomenon brings up an interesting topic to study further.

FIG. 9. The calculated longitudinal �� and �� of a long cylinder of radiusr=5 mm obeying the Kim–Anderson type of exponential E�J�, Eq. �8�, con-necting to a low E linear E�J� with critical-current density Jc=104 A/m2 asfunctions of normalized field amplitude and frequency. Three cases �a�, �b�,and �c� correspond to m=U0 / �kT�=2, 5, and 10, respectively. The relevantcurrent-voltage characteristic is shown in �b� and �c�. Arrows indicate thedirection of increasing frequency.



Attributing this phenomenon to the existence of Joseph-son vortices �JVs� and their thermally activated creep, wemay start our arguments as follows. Since the dc Josephsonequation is a sinusoidal function of the gauge-invariant phasedifference, there is a strong tendency to form JVs in aJosephson-junction �JJ� array �or network�. Such an arraywill be in a JV state after field cooling. However, studies onmagnetic properties of uniform resistively shunted JJ arrayshave shown that a change in the applied field can induce astate with the most effective static shielding and an averagedarray-circulating critical current.49,50 When the critical cur-rent of each JJ is high enough, the shielding currents flowingthrough each JJ can be very close to their maximum valueand a practically pure critical state with only array-circulating current forms. With decreasing the maximum cur-rent of each JJ, such a pure critical state evolves to a JVstate, where the intergranular currents mainly circulatewithin each JV, although a small array-circulating currentcan be present, owing to JV pinning.51

As calculated in Refs. 49 and 50 for resistively shuntedJJ arrays, the field induced state becomes a JV state when�v /�0 is less than about 0.2, where �v is the flux in a voidproduced by a neighboring maximum JJ current and �0 isthe flux quantum. With very strong demagnetizing effects,the films may be modeled by a planar JJ array of cell size a0

equal to their typical grain size and maximum JJ current Im

=Jc, Ba0t, where

Jc, B = 1.03Hm��m� �/t , �10�

being the Jc in Table I calculated for a thin disk using theBean model.28 Jc, B may thus be estimated to be 109 and 4�109 A/m2 for films A and B, respectively, so that �v /�0

=�0a0Im /�0=�0Jc, Ba02t /�0�0.1 and 0.2 for films A and B,

respectively. In this case, both films can be in a JV state.Furthermore, if JVs in films A and B have a pinning barrierU0�2kT and 5kT, respectively, then their anomalous��Hm , f� can be attributed to a Kim–Anderson type of ther-mally activated JV creep. The last statement has been madeaccording to the obvious similarities between the experimen-tal results of Figs. 1�a� and 1�b� and the calculations in Figs.9�a� and 9�b�.

C. Difficulties with JV creep mechanism

The discussion presented in the previous subsection isonly a first approach to the problem. Actually, a similar ar-gument has already been presented in Ref. 17 to justify theJV creep mechanism of the anomalous ��Hm , f�. However, amore detailed quantitative analysis brings about serious dif-ficulties.

First, if the intergranular Jc arises from a modeling pla-nar JJ array with Im=0.17 mA �corresponding to �v /�0

=0.1� for film A, the averaged critical current per JJ, Ic,should be about Im /10=0.017 mA as calculated byRzchowski et al.52 Thus, Im calculated from Jc, B

=109 A/m2 should eventually lead to a Jc=108 A/m2 owingto JV depinning.

Second, it is known that the coupling energy of a JJ ofmaximum current Im is

UJ = Im�0/2� , �11�

and that the pinning barrier to a JV in a uniform planar JJarray is typically on the order of52,53

U0 � 0.2UJ. �12�

Assuming U0=2kT=2�10−21 J �from k=1.38�10−23 J /K and T=77 K� for film A, we obtain using Eqs.�11� and �12� that Im is on the order of 0.03 mA, which isonly a fifth part of Im=0.17 mA calculated from Jc, B

=109 A/m2.From both estimates, we conclude that if a thermally

activated JV creep were simply the origin of the observed��Hm , f� for film A, Im for each JJ would be 0.03 mA, lead-ing to an averaged critical current Ic=0.003 mA per JJ andfinally to a negligible intergranular critical-current densityJc�Jc, B /50. A similar conclusion may be obtained by ana-lyzing the case of film B.

D. Two coupled grain-boundary networks

To resolve this quantitative discrepancy, we propose asuperconducting structure of two GB networks coupled toeach other. In this structure, JVs are formed in a weak-linkGB �JJ� network with a very small averaged Im1 for each JJand an even smaller averaged JV depinning current per JJ,Ic1� Im1 /10; the induced film circulating currents flow inanother strong-link GB network where Im2 for each GB is atleast one order of magnitude greater than Im1, so that thecritical current is practically provided by Im2 alone; the cur-rents in both networks are electromagnetically coupled toeach other, so that the dissipation in the first network occur-ring at very small averaged JJ current I1 owing to thermallyactivated JV creep is proportionally related to the much big-ger averaged GB current I2 flowing in the second network.As a result, U0 in Eq. �7� determined by Im1 in the firstnetwork may be small enough but Jc in Eqs. �7� and �8�,limited by Im2 in the second network, can be so large to get afinal Jc as large as the observed Jc, B.

The coexistence of these weak and strong links is naturalin film A; since a majority of GBs in film A has �=30° or45° and the probability for the a�b� axis aligned grains to benearest neighbors is low, there may be a fine-mesh weak-linknetwork and a coarse-mesh strong-link network. Both net-works are extended to the entire film and completelycoupled, so that all the induced circulating currents flowingin the strong-link network are coupled to the irreversible JVmovements in the weak-link network, which makes ��Hm , f�of film A be quite well simulated by the calculations shownin Fig. 9�c�. Being basically well grain aligned, the existenceof weak links in film B may be justified by the existence ofnonsuperconducting phases and many grains with their c axisparallel to the film edges as well as by the relatively lowoxygen content, which yields worse superconducting perfor-mance as indicated by the gradual resistance decrease withcooling. Since all these factors that cause weak links aredistributed throughout the film, the coupling between theweak and strong-link networks is complete. As a result, its��Hm , f� may also be rather well simulated by results in Fig.9�b�. For films C and D, however, this coupling should be



localized to near the film edges, so that only a small portionof the induced currents may drive JV movements, and their��Hm , f� agrees with Fig. 9 only in some features at low Hm.

Although the discrepancy mentioned earlier may be inprinciple resolved by the assumption of such a structure, itsactual existence has to be justified by further modeling, forwhich the coupling between both networks must be carriedout based on the dc and ac Josephson effects through thegauge invariant phase difference.

Up to now, we have used the JV creep of the Kim–Anderson type combined with a normal conductivity n toexplain qualitatively the anomalous ��Hm , f� at low Hm andlow f . Difficulties are encountered for when consideringquantitatively a single JV network, so that a coupled two-network model is proposed. New difficulties will be met inthe theoretical justification of such a model. In fact, there isanother fundamental difficulty in the foregoing explanationas follows. Equations �7� and �8� for the Kim–Anderson typeof flux creep is only the high-E approximation of the generalsolution to the problem of a simple barrier system Andersonassumed. When E is low, its solution yields the thermallyassisted flux flow with a conductivity much higher than then used for our calculation in Fig. 9 �Ref. 54� so that thecalculated anomalies will no longer occur. Hence, we wouldconclude that although the anomalous ��Hm , f� is doubtlessformally related to a superconducting type of E�J� functionwith a certain nonzero E /J at low J, it is still challenging tofind its physical mechanism.

E. Over-low �0� or �m� /�0

�

Comparing the maximum −�� at the highest f , �−��m,with the �0 calculated from Eq. �1� for complete shielding,we find that �−��m /�0=0.72, 0.52, 0.94, and 0.98 for filmsA, B, C, and D, respectively. It is reasonable for �−��m /�0

to be somewhat less than 1, considering the unavoidablestructural imperfection near the film edges and a finite Lon-don penetration depth. Thus, the behavior of film D may beregarded as normal. The behavior of film A with �−��m /�0

=0.72 also can be regarded as reasonable, since 1120 Hz istoo low a frequency to induce currents to shield the filmcompletely. In fact, the measured �� /�0 curve at 1120 Hz isquantitatively similar to the modeling �� curve in Fig. 9�c�for 1000 Hz. �−��m /�0=0.94 for film C could be attributedto the existence of more defects near the film edges thanthose for film D with the same substrate but refined prepara-tion. Thus, the only remarkable anomalous behavior discov-ered here will be for film B, whose �−��m /�0=0.52 is toosmall.

We should comment that although �−��m /�0 of film Bmay be raised by further increasing f , its high-f limit wouldnot be greater than 0.6. This can be justified by comparingthe trend of the frequency dependence shown in Fig. 1�b�with the modeling one shown in Fig. 9�b�, where the low-field −�� has reached 90% of �0�=1� at f =1000 Hz. In thefollowing, we name the high f limit of �−��m as �0

�. Al-though its �0

� is significantly reduced from �0, ��Hm , f� offilm B normalized to �0

� can be quantitatively in agreementwith our JV creep model. On the other hand, the situation of

��Hm , f� of film C shown in Fig. 1�c� is different; its �0�

value is normal, but its �m� /�0��0.14 is much less than the

Bean value 0.24 and with increasing Hm to Hm��m� �, both ��and �� increase more steadily than the Bean case shown inFig. 6.

In the following, we will look for the reasons for suchover-low values of both �0

� and �m� /�0�.

F. Division effect on �0�

Noticing the fact that there exist long cracks in film B,we might attribute the �0 reduction to the physical divisionof the film. Consider arbitrarily a long superconductingsquare �w�w=6�6 mm2� bar of YBCO single crystal,whose c axis is along the bar length. In this case, the longi-tudinal �0 is ideally equal to 1 in a complete shielding state.However, a London penetration depth �L�0.3 �m at 77 K�Ref. 55� causes a thin surface layer of thickness �L to benormal, so that the actual �0, �0

�, becomes 4�L /w�0.02%smaller. If we divide this bar into 16 subsquare bars of iden-tical cross section, then �0

� for the assembly of subbars willbe 16�L /w�0.08% smaller. As a result, the division doesnot reduce �0 practically. However, we will show below that�0 may be more reduced if such a division is made for asquare film rather than a square bar.

As stated in Refs. 27 and 28, for the perpendicular mag-netic properties of thin superconductors, �L has to be re-placed by the effective penetration depth56,57

�eff = 2�L2/t � 1.8 �m, �13�

for t=0.1 �m. Thus, making the same calculation as for thebar, the reduction of �0 for an integral film of t=0.1 �m andthe assembly of 16 subfilms should be 4�eff /w�0.12% and16�eff /w�0.48%, respectively. It seems that cracks may re-duce �0 slightly. However, this simple analogy between along bar and a thin film is incorrect, since there are essentialdifferences between a bar whose �0 is w independent and afilm whose �0 is a function of w and t owing to demagnetiz-ing effects.

Because of demagnetizing effects, when the film size wis reduced to the subfilm size w /4, �0 will be reduced to�0 /4 according to Eq. �1�. This means that �0 may be sig-nificantly reduced by the existence of cracks if the interac-tions among the divided subfilms are weak. From the studyof demagnetizing factors of cylinders or disks, we know thatthe averaged demagnetizing field is in the same direction ofthe applied field if the material susceptibility is negative.58,59

This makes �0 for a thin disk be much greater than �0=1 foran infinitely long cylinder. Actually, the overall direction ofdemagnetizing fields outside of the disk on the midplane isalso along the applied field direction, so that the magneto-static interaction among the earlier divided subfilms will in-crease again the �0 that has been reduced by the division.Moreover, the �0 reduced by cracks may be completely re-covered by magnetostatic interactions if the gap width isinfinitesimal between all the neighboring subfilms and ifthere is no limit for the current density along the subfilmedges. This last statement is important and it may be justifiedas follows.



When a body made of material with a constant suscep-tibility is placed in a uniform applied magnetic field, mag-netic poles or currents appear only on the surface.60 Sincethere is a unique solution of the distribution of pole densityor current density to the magnetostatic problem for either acontinuous body or the same body but containing grains withinfinitesimally thin insulating boundaries, �0 must be thesame for both cases when the material has a susceptibility of−1 in the complete shielding state. Physically, the completeshielding in our case is a result of circulating supercurrents,so that the same current density distribution after the divisiontogether with the current continuity requires a pair of exactlyequal but opposite currents of infinite density being presenton both sides of each boundary.

In fact, this is the actual mechanism of a complete mag-netostatic coupling among subfilms. A reduction of �0 by theexistence of cracks can occur only when the gap widthand/or �eff is finite, so that the magnetostatic coupling amongsubfilms becomes incomplete. A simplified treatment for ourcase could be carried out by considering completely shieldedsubfilms with an effective gap width about 5 �m �a sum ofthe real gap width and 2�eff� in between. The calculation maybe carried out using a two-dimensional technique recentlydeveloped by Brandt.61 However, we have made the follow-ing experiment to check such an effect directly. The samplewas a square YBCO film of w=10 mm and t=0.32 �m with��Hm , f� curves similar to those of film D. We have intro-duced, with a sharp knifepoint, scratches on its surface; eachscratch was a groove of width about 0.02 mm. The ��Hm�measured at 2430 Hz and 77 K for the original film and aftereach scratch operation is plotted in Fig. 10, from which wesee that �0

� is reduced by the division and it reaches about 1%and 24% when the film is divided into 4 and 16 subfilms,respectively. The latter case shows a remarkable effect of thedivision with incomplete coupling.

G. Division effect on �m� /�0�

We see from Fig. 10 that the film division has a moreimportant effect on the shape of ��Hm� and ��Hm� curves;they are of Bean type with �m� /�0

�=0.25 before the divisionbut have a more gradual increase of �� accompanied by a flat�� peak or two �� peaks with a lower �m� /�0

�. This is a con-sequence of the magnetostatic coupling among subfilmswhen J is limited by Jc. A similar variation of the perpen-dicular ��Hm� of an assembly of three parallel horizontalrectangular bars with their gap width has been calculatedbased on the Bean model.62–64

It is interesting that in Fig. 10 the results for two cases,with two subfilms and with a scratch of length w /2, arealmost identical. Both have the same �0

� as that of the origi-nal film but a reduced �m� /�0

�=0.205. This fact suggests thatthe existence of few separate long scratches would not influ-ence �0

� much but its effect on the reduction of �m� /�0� can be

significant. As can be realized from the curves in Fig. 10 forthe 16 subfilms, a significant reduction in �0

� caused by divi-sions will be accompanied by very remarkable changes in theshape of ��Hm� curves. Thus, the significant reduction in �0

�

of film B should not be caused by the existence of cracks, butinstead it should imply an appreciable portion of the filmarea to be occupied by nonsuperconducting material. Sincefor this film, the low-current resistance becomes zero when Tis reduced to 81 K, and a zero resistance only requires onepercolation path for supercurrent to flow, the presence oflarge normal conducting area at 77 K is expectable.

The effects on the shape of ��Hm� should be similar ifthe scratch of length w /2 is replaced by a poorly supercon-ducting island of similar length. Thus, the over-low �m� /�0

� offilm C can also be attributed to the existence of extendeddefective regions with low Jc, due to structural nonunifor-mity related to the richness of impurities as BaCuO2 andbroken chains indicated by Raman spectra.

H. Jc calculation from Hm„�m� … vs f

Our initial aim for studying YBCO superconductingfilms using an ac susceptibility technique was to determinetheir Jc. However, the knowledge that may be gained from��Hm , f� measurements goes beyond the Jc determination,being extended to a wide range of interesting phenomena tobe understood. For example, we have seen how �m� may re-main constant, increase, or decrease with increasing fre-quency, corresponding to different dissipation mechanisms; astructure of coupled weak-link and strong-link networks mayyield a thermally activated JV creep mechanism, whichcould be further questioned, though; the actual �0, �0

�, ofsome films may be significantly smaller than that for a com-pletely shielded film, and this indicates either the existenceof some cracks which divide the film into subfilms with anincomplete magnetostatic coupling or the existence of largenonsuperconducting areas; in some cases, �m� /�0

� is remark-ably less than its Bean value although the �0

� value is normal,and this indicates the existence of some extended regionswith significantly lower values of Jc. Although not directly

FIG. 10. �� and �� measured at 77 K as functions of ac field amplitude Hm

at 2430 Hz for a YBCO film before and after scratching. The scratchedconfigurations are shown by the squares with divisions.



leading to the Jc determination, knowledge of this kind couldbe important for material researchers to improve the technol-ogy of film preparation.

In the earlier analysis, we have frequently used Eq. �10�to estimate Jc. This Bean equation for a thin disk may beroughly used for different cases because it is practically in-dependent of r when r / t�103, and the surface shape effectcannot be larger than 20%, as can be realized by a compari-son between thin disk and thin strip given in Table I. How-ever, if the film is divided into subfilms, the Jc of subfilmswill be underestimated by Eq. �10�, as can be seen from Fig.10, where Hm��m� � for the cases of two or four subfilms isabout 20% reduced compared with the original one with thesame Jc. This underestimated Jc is accompanied by a signifi-cantly reduced �m� /�0

�. If �0� is significantly less than the the-

oretical �0 owing to the existence of extended nonsupercon-ducting regions, Jc calculated using Eq. �10� may be that forthe superconducting region but not the average of the entirefilm. Therefore, two necessary conditions to make a reliabledetermination of Jc of the entire film from ��Hm , f� mea-surements are that the measured �0

� be near the �0 given byEq. �1� and that the observed �m� /�0

� be greater than the Bean�m� /�0

� listed in Table I. Moreover, the dissipation should bedominated by thermally activated collective creep or fluxflow. The Kim–Anderson type of flux creep will bring aboutdifficulties, since it must be cooperated with another un-known dissipation mechanism which leads to complexities inthe modeling ��Hm , f�. Actually, most YBCO films with ahigh Jc obey a power-law E�J� relation, which may be deter-mined by Hm��m� � measured at a set of values of f followingRef. 65 by

J = kJ�n�Jc, B, �14�

E = �0�Hm��m� �wl/��2��w + l�� , �15�

where Jc, B is calculated using Eq. �10�, w, l, and t are thewidth, length, and thickness of the film, and

kJ�n� = 1 + 0.086�1 − exp�− 40/n�� . �16�

After having E�J�, Jc is obtained by considering a crite-rion for Ec. The procedure of E�J� determination of film D,which meets all the necessary conditions mentioned earlier,from ��Hm , f� measurements is explained as follows.

We first replot the high Hm portion of Fig. 1�d� into Fig.11�a�, so that the curves around Hm��m� � can be seen with abetter resolution. We next decide an averaged �� correspond-ing to �m� as marked by a short horizontal line in Fig. 11�a�.From the crossover of this line with the �� curves for fourvalues of f , we obtain �0Hm=7.65, 8.10, 8.56, and 9.14 mTfor f =30, 90, 270, and 810 Hz, respectively. Since ��Hm , f�obeys the scaling law stated in Sec. III B, we can get anaveraged n from each pair of frequency �f1 , f2� using

n21 = 1 +log f2 − log f1

log Hm��m� , f2� − log Hm��m� , f1�. �17�

This results in n=20, then the log–log linear E�J� func-tion can be calculated using Eqs. �10� and �14�–�16�, as plot-ted in Fig. 11�b�. Jc=2.6�1010 A/m2 is finally obtained bythe crossover point of fitting straight line and E=Ec.

A final check can be made from the measured �0�

=19300, �m� =5430, and −��m� �=7860, from which �m� /�0�

=0.28 and −��m� � /�0�=0.41 are obtained. Both are close to

but somewhat greater than the theoretical values 0.27 and0.39, obtained from Table I and Fig. 7, which indicate asmall error of this technique.

V. CONCLUSION

The field amplitude and frequency dependent ac suscep-tibility ��Hm , f� of metalorganically deposited YBa2Cu3O7−�

films has been measured at 77 K and studied in comparisonwith their microstructures and several theoretical models.Some conclusions may be stated as follows.

Thin YBCO films metalorganically deposited on aproper single crystal substrate usually consist of well tex-tured grains with their c axis perpendicular to the film sur-face and a�b� axes parallel to each other. Thus, their ��Hm�may be similar to that calculated from the Bean model andtheir ��Hm , f� may be well modeled by a power-law E�J�,consistent with the dissipation mechanism of thermally acti-vated collective Abrikosov vortex creep, taking place instrong-linked grains.

If the orientation of the a�b� axes has a large distributionand/or if there are enough extended defects �nonsupercon-ducting impurities and voids� in the film, then ��Hm� may beremarkably anomalous with a strong frequency dependence.The anomalous ��Hm , f� may be quantitatively deducedfrom a linear-exponential E�J� function. Being probably aresult of the existence of weak links or Josephson vortices,the physical mechanism of this anomaly is still unclear.

FIG. 11. �a� �� and �� measured at 77 K as functions of ac field amplitudeHm and frequency f for film D. �b� The E�J� and Jc determination from themeasured ��Hm , f� for film D. Arrows in �a� indicate the direction of in-creasing frequency.



The existence of weak links may remarkably reduce Jc

or degrade the superconducting performance at low electricfields even if Jc is high.

The Jc of the entire film may be reliably determined by��Hm , f� measurements when the low-Hm �� approaches−�0 of a completely shielded film and �m� /�0 is greater thanthe Bean value 0.24.

ACKNOWLEDGMENTS

The work is partially supported by Spanish Ministerio deEducación y Ciencia Project No. FIS2004–02792 and Cata-lan Projects No. 2005SGR00731 and CeRMAE.

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Field dependent alternating current susceptibility of metalorganically deposited YBa[sub 2]Cu[sub...

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