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PHYSICAL REVIEW B
VOLUME 59, NUMBER 14
1 APRIL 1999-II
Scaling of the Hall resistivity in epitaxial HgBa2CaCu2O61d thin films with columnar defects W. N. Kang Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045 and Texas Center for Superconductivity, University of Houston, Houston, Texas 77204
B. W. Kang Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045
Q. Y. Chen* Texas Center for Superconductivity, University of Houston, Houston, Texas 77204
J. Z. Wu, S. H. Yun, and A. Gapud Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045
J. Z. Qu and W. K. Chu Texas Center for Superconductivity, University of Houston, Houston, Texas 77204
D. K. Christen and R. Kerchner Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
C. W. Chu Texas Center for Superconductivity, University of Houston, Houston, Texas 77204 ~Received 1 April 1998; revised manuscript received 19 November 1998! We have studied the Hall scaling relation r xy 5A r bxx for the mixed-state Hall effects of HgBa2CaCu2O61d thin films before and after irradiation by high-energy Xe ions. b increases from 1.060.1 to 2.060.1 as a perpendicular magnetic field increases from 0 to 8 T, and increases from 1.060.1 to 1.560.05 as the angle between the columnar tracks and a 4 T field decreases from 75° to 0°. The observation of b 51 is consistent with a recent theory based on d-wave superconductivity, in which the Hall angle was predicted to show a universal behavior that was independent of the mean free path of charge carriers in the low-field region. @S0163-1829~99!50114-8#
One of the most controversial issues of mixed-state Hall effects in high-T c superconductors ~HTS’s! is the puzzling scaling relation between the Hall resistivity r xy and longitudinal resistivity r xx . 1–17 Recent experiments have investigated the role of strong pinning based on the mixed-state Hall effects for samples containing columnar defects created by heavy-ion irradiation.4–6 Heavy ions amorphize the superconductor and form fine columnar defects of 5–10 nm in diameter along their linear trajectory. These defects provide strong flux-pinning strength within certain ranges of temperature T and magnetic field H.18 Thus, measuring the Hall effect of samples before and after heavy-ion irradiation in a field, either parallel or oblique to the linear tracks, may help find direct correlation between the mixed-state Hall scaling behavior and the atomic disorder. An interesting scaling relation, r xy 5A r bxx , has been observed on Bi2Sr2CaCu2O8 ~Bi-2212! crystals2 and Tl2Ba2CaCu2O8 ~Tl-2212! films for which b was ;2.3 But other similar studies found b 51.5– 2.0 for YBa2Cu3O7 ~YBCO! films1 and crystals,6 much the same as HgBa2CaCu2O61 d ~Hg-1212! films7 and HgBa2Ca3Cu4O101 d crystals.9 Budhani et al.5 observed that the Hall scaling remained unaffected even after irradiation and the anomaly of sign reversal diminished as defect density increased. Samoilov et al.4 measured the Hall conductiv0163-1829/99/59~14!/9031~4!/$15.00
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ity s xy of YBCO crystals and Tl-2212 films before and after irradiation, and suggested that the pinning enhancement did not modify s xy . In our previous work6 on YBCO crystals with columnar defects, however, we found that the pinning strength affected not only the Hall scaling but also s xy . The model proposed by Vinokur, Feigel’man, Geshkenbein, and Larkin ~VFGL! ~Ref. 13! suggested that b should be 2 in the thermally assisted flux-flow ~TAFF! region regardless of the pinning. Their results were in fair agreement with both the weakly pinned systems of Bi-2212 crystals ( b 52.060.1) ~Ref. 2! and the rather strongly pinned systems of heavy-ion irradiated Tl2Ba2Ca2Cu3O10 films ( b 51.8560.1), 5 though not with the recent results on heavyion irradiated YBCO crystals6,8 or Hg-1212 films ( b 51.5– 1.9). 7 Wang, Dong, and Ting ~WDT! ~Ref. 14! explained the scaling behavior and the anomalous sign reversal of the Hall effect by taking into account the backflow current due to pinning. In their model, as the pinning strength increases, b may change from 2 to 1.5, since the vortex-flow velocity ( v L ) dependent damping coefficient G( v L ), which is related to the time average of pinning force by ^ Fp & 5 2G( v L ) vL , is roughly proportional to v 20.5 in the strong L pinning regime.13 In a recent work, Kopnin and Volovik ~KV! ~Ref. 20! showed b (51) to be a universal number R9031
©1999 The American Physical Society
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FIG. 1. r xx vs T in Hg-1212 film before (B f 50 T, open symbols! and after (B f 51 T, solid symbols! heavy-ion irradiation.
FIG. 2. r xy vs T/T c in Hg-1212 films before (B f 50 T, open symbols! and after (B f 51 T, solid symbols! heavy-ion irradiation.
which is independent of the mean free path of charge carriers at low T and H regions for d-wave superconductors. Meanwhile, Kouznetsov et al.21 found strong evidence for mixed d- and s-wave pairing in YBCO by Josephson tunneling experiments between YBCO crystal and Pb junction. Among the numerous HTS’s, Hg-based cuprates are especially suitable for studying the effect of pinning on the mixed-state Hall effect because, in this series of materials, the pinning strength strongly depends on H. The unperturbed pinning potential (U 0 ) of the Hg-based system was observed to follow U 0 (H)}1/H, 15 which is similar to the case of YBCO,16 though U 0 (H)}1/AH ~Ref. 17! for the Bi-2212 and Tl-2212 systems. We report here a study of the mixedstate Hall effect in highly c-oriented Hg-1212 films before and after irradiation. The pinning strength dependence of the scaling behavior would help provide new insight, we believe, into the flux dynamics in the TAFF region. The fabrication process19 and transport properties7 of Hg1212 films used in this study were described in detail in our previous papers. The ion irradiation was performed at the Superconducting Cyclotron Center at the Michigan State University using 5 GeV Xe ions. Irradiation was done at room temperature along the normal direction of the film surface. The film was irradiated up to a dose of 5.0 31010 ions/cm2, which was chosen so that the columnar defects would be able to maintain a magnetic-flux density of B f 51 T. r xy and r xx were measured simultaneously using a two-channel nanovoltmeter ~HP34420A! based on the standard five-probe dc method. H was applied in parallel or at an angle to the c axis of the Hg-1212 films. r xy was extracted from the antisymmetric part of Hall voltages measured under opposite H. The nominal current densities used in these measurements were about 250 A/cm2. Both r xx and r xy were measured in the ohmic regime, in which there was no dependency on the applied current. The T dependence of r xx near the mid-transition temperature (T c ) before (B f 50 T) and after heavy-ion irradiation (B f 51 T) for H up to 4 T is shown in Fig. 1. T c was determined according to the typical 10–90 % criterion. The enhancement of T c , judged from the r xx vs T plot in various H, is clearly observed. This agrees with previous findings4–6 on
the samples with columnar defects. At H50, however, T c has been enhanced by ;0.8 K after irradiation, contrary to that reported on the irradiated HTS’s,4–6 in which a suppression of T c by 0.3–0.7 K was found for the same extent of irradiation. In Fig. 2, we show the corresponding r xy for H51,2 and 4 T. In order to compare the r xy for samples different in T c , the figure is presented in reduced temperature T/T c rather than in real T. For H,2 T, double sign reversals in r xy were observed upon lowering T both before and after irradiation. Note that after irradiation the peaks at lower T are shifted to high T while the dips at higher T remain largely unchanged compared to the unirradiated data. Furthermore, all r xy values below T c are suppressed by irradiation. These data show that the columnar defects do play an important role as strong pinning centers even at relatively high T. In Fig. 3, we show the scaling between r xy and r xx for B f 51 T with a series of H up to 8 T. As mentioned before, r xy and r xx have been measured simultaneously at the same T; plots of r xy as a function of r xx were thus done regardless
FIG. 3. H dependence of scaling behavior between r xy and r xx in Hg-1212 film with columnar defects. The solid lines are power laws, b 51.0, 1.5, and 2.0, for the sake of clarity.
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SCALING OF THE HALL RESISTIVITY IN . . .
FIG. 4. Tilted angle dependence of scaling behavior between r xy and r xx in Hg-1212 film with columnar defects. The solid lines are power laws, b 51.0 and 1.5.
of T. Since r xx below B52 T is negative in a certain T range, the plot uses the absolute value u r xy u . b in r xy b 5A r xx was extracted from the slope of the solid lines shown in Figs. 3 and 4. We obtained b by the linear fitting of r xy vs r xx plots from the lower T or r xx to the higher T or r xx regimes. The ranges of data adopted for linear regression, spanning roughly two decades of r xx data, are believed to well represent the TAFF region in which most previous studies and discussions on the HTS’s were conducted. Here, in any event, H dependences of the scaling, which are different from those reported on the Bi-2212 ~Ref. 2! and Tl-2212 systems,3 are clearly demonstrated, with b changing from 1.560.05 to 2.060.1 as H increases from 4 T to 8 T. This is consistent with those observed at relatively low H on the unirradiated YBCO crystals6 and Hg-1212 films.7 Below 4 T, however, b changes from 1.060.1 to 1.560.05 as H increases. The similar trend of the Hall scaling was also shown in Fig. 4 for various tilted angles ( u 50°, 45°, 60°, and 75°! between the heavy-ion tracks and H54 T. That is, b changes from 1.060.1 to 1.560.05 as u decreases from 75° to 0°. Since the c-axis component of field H' is given by H' 5H cos u, H' naturally would decrease with an increasing u. The observations that b ;1.060.1 at u 50° for 1 T ~Fig. 3! and 75° for 4 T ~Fig. 4!, which give identical H' (51 T), are one main focus in the present paper, which we will elaborate later. Figures 3 and 4 clearly show the dependence of Hall scaling on the pinning strength. Figure 5 gives another direct evidence, in which we plot the Hall scaling of Hg-1212 film before and after irradiation for 2 T and 4 T. Note that after irradiation, b changed from 1.560.05 to 1.3 60.1 at 2 T and from 1.860.05 to 1.560.05 at 4 T. VFGL ~Ref. 13! proposed a universal scaling law r xy } a r bxx , where a 5 h tan Q and h is the usual viscous coefficient. It has been contended that, since a is a slowly varying function while r xx is an exponential function of T, b should be 2 regardless of any vortex state in the presence of quenched disorder and thermal noises in the TAFF region. In addition, they also suggested that b could deviate from the expected value of 2 because of the T dependence of a near T c , which, they believed, was the cause for the reported b
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FIG. 5. Scaling behavior between r xy and r xx in Hg-1212 film before and after irradiation for H52 and 4 T.
51.7 on YBCO films.1 Note that, in Fig. 3, for H>6 T, there is a substantial deviation from linearity. This could be due in part to the T dependence of a, which is a strong function of T at higher H.7 On the other hand, WDT ~Ref. 14! recently developed a theory of flux motion taking both pinninginduced backflow and thermal fluctuations into account in the force balance equation. At relatively higher T, their main result is summarized by
r xy 5
b 0 r 2xx ¯ /H c2 # 22G ~ n L !@ 12H ¯ /H c2 # % , $h@ H F 0B
~1!
where b 0 5 m m H c2 with m m being the mobility of the charge ¯ is a field carrier and H c2 being the upper critical field. H averaged over the vortex core. Equation ~1! predicts a negative dip in r xy if H is low enough and the pinning is relatively strong, under which a double sign reversal can be expected. As H increases, the originally negative dips in r xy ¯ /H c2 increases with an increasing H. become positive since H In a relatively strong pinning system, such as Hg-1212, the depth of the dip is greater than that of a weak pinning system such as Bi-2212 ~Ref. 2! or Tl-2212.3 The data in Fig. 2 are in good agreement with this picture. According to Eq. ~1!, there are two distinctive regimes for the Hall scaling behavior. In the weak pinning regime, ¯ /H c2 at relatively high H, Eq. ~1! becomes r xy G( v L )! h H 2 ;A r xx , giving the same scaling exponent b 52 as that of VFGL.13 But in the strong pinning regime, where G( v L ) ¯ /H c2 at low H, b is no longer 2. Since G( v L ); v 20.5 @hH L for the strong pinning case,10 the scaling relation becomes r xy ;A r 1.5 xx . Naturally, for the intermediate case, b can hence vary from 1.5 to 2.0. In fact, the result of b 51.560.05 for H54 T corresponds to the theoretical estimate of 1.5 for the strong pinning scenario, while that of b 52.060.1 for H 58 T corresponds to the weak pinning scenario. This is consistent with our previous observation on the YBCO crystals as well as the Hg-1212 films without columnar defects. However, it is not easy to understand the data for b 51.0– 1.5 observed at low H shown in Figs. 3 and 4 within
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the proposed models by VFGL ~Refs. 10–13! and WDT.14 Using a first-order approximation of perturbation theory over disorder potentials, Vinokur and co-workers10–13 have obtained G( v L ); v 20.5 for the strong pinning case. But for a L weakly disordered system, precise calculation of G( v L ) is rather difficult and remains unresolved at this time. Within the context of the WDT model, our observation of b 51.0 suggests that G( v L ) is proportional to v 21 L in a certain range of pinning strength for the strongly disordered systems such as Hg-1212 and YBCO samples of columnar defects. In our earlier work6 on the irradiated YBCO crystals, however, since the Hall signal in the TAFF region at low H was comparable to the extrinsic noise, we could not observe this scaling with b 51.0– 1.5. Very recently, KV ~Ref. 20! calculated s xy for d-wave superconductors. In the low T and H regimes, the Hall angles have a universal value independent of the mean free paths of electrons and holes. Their main conclusion for the relation between r xy and r xx is given by
r xy 5 ~ p /221 !
~ n e 2n h ! r , ~ n e 1n h ! xx
~2!
where n e and n h are the electronlike and holelike chargecarrier density, respectively. In TAFF region, since n e and n h are slowly varying functions while r xx is an exponential function of T, Eq. ~2! leads to a universal scaling with b 51 independent of the mean free path and H. The observations of b 51 in Figs. 3 and Fig. 4 are in good agreement with this model. In fact, we have observed, though not shown in Fig. 3, b 5160.15 for a 0.5 T. These results are consistent with our recent work in which b 5160.15 below 0.3 T has been observed in Tl-2212 thin films.22 It is indeed very interesting to observe that b would vary from 1 to 2 by a small change in H or u. Also interesting is that certain theories were able to explain the results at certain H limits,
*Electronic address:
[email protected] J. Luo et al., Phys. Rev. Lett. 68, 690 ~1992!. 2 A. V. Samoilov, Phys. Rev. Lett. 71, 617 ~1993!. 3 A. V. Samoilov, Z. G. Ivanov, and L.-G. Johansson, Phys. Rev. B 49, 3667 ~1994!. 4 A. V. Samoilov et al., Phys. Rev. Lett. 74, 2351 ~1995!. 5 R. C. Budhani, S. H. Liou, and Z. X. Cai, Phys. Rev. Lett. 71, 621 ~1993!. 6 W. N. Kang et al., Phys. Rev. Lett. 76, 2993 ~1996!. 7 W. N. Kang et al., Phys. Rev. B 55, 621 ~1997!. 8 A. Casaca et al., Phys. Rev. B 56, 5677 ~1997!. 9 J. Lohle et al., Physica C 266, 104 ~1996!. 10 V. M. Vinokur et al., Phys. Rev. Lett. 71, 1242 ~1993!. 11 V. M. Vinokur et al., Phys. Rev. Lett. 65, 259 ~1990!. 12 M. V. Feigel’man and V. M. Vinokur, Phys. Rev. B 41, 8986 ~1990!. 1
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although these theories were based upon different origins. However, as a plausible explanation, one can consider the existence of dimensional crossover from a three-dimensional ~3D! to a two-dimensional ~2D! vortex-state as H increases, i.e., a 3D system in the low H limit would give b 51, while a 2D system in the high H would have b 52, and, during dimensional crossover, b can change from 1 to 2. As is well known, a 3D to 2D crossover has been observed on the YBCO films23 and Bi-2212 single crystals.24 Nevertheless, further studies are required to link such similarities with the Hg-based systems. In summary, the Hall effect in Hg-1212 films has been studied before and after irradiation by high-energy Xe ions. After irradiation, the scaling behavior showed a strong dependence on H and u. b increases from 1.060.1 to 2.0 60.1 as H increases from 1 to 8 T. For oblique H, we found that b increased from 1.060.1 to 1.560.05 as u decreased from 75° to 0° for H54 T. Based on the VFGL model,10 our high-H results showed that, as a good approximation, the pinning was negligible, but an explicit pinning strength dependence on H should be considered for a strong pinning case. On the other hand, the KV model20 was in good agreement with our low-H results in the strong pinning regime. With the WDT model,14 our results for the intermediate H can be properly interpreted, but more theoretical details on the pinning strength dependence of the scaling behavior are necessary. We believe that these referred theories10–14,20 can be extended to cover a wider range of magnetic field. This work was supported in part by the State of Texas at the University of Houston. Work at the University of Kansas was supported by the U.S. Air Force through USAFOSR Grant No. F49620-96-1-0358, and by the National Science Foundation through Grant No. DMR-9632279 and NSF EPSCoR fund. We thank Rachel Schmidt for useful discussions.
V. M. Vinokur et al., Zh. Eksp. Teor. Phys. 100, 1104 ~1991! @Sov. Phys. JETP 73, 610 ~1991!#. 14 Z. D. Wang, J. Dong, and C. S. Ting, Phys. Rev. Lett. 72, 3875 ~1994!. 15 B. W. Kang, W. N. Kang, S. H. Yun, and J. Z. Wu, Phys. Rev. B 56, 7862 ~1997!. 16 T. Matsuura and Itozaki, Appl. Phys. Lett. 59, 1236 ~1991!. 17 J. T. Kucera et al., Phys. Rev. B 46, 11 004 ~1992!. 18 K. E. Gray et al., Phys. Rev. B 54, 3622 ~1996!. 19 S. H. Yun and J. Z. Wu, Appl. Phys. Lett. 68, 862 ~1996!. 20 N. B. Kopnin and G. E. Volovik, Phys. Rev. Lett. 79, 1377 ~1997!. 21 K. A. Kouznetsov et al., Phys. Rev. Lett. 79, 3050 ~1997!. 22 W. N. Kang, C. W. Chu, D. H. Kim, and J. U. Lee ~unpublished!. 23 Z. X. Gao et al., Phys. Rev. Lett. 71, 3210 ~1993!. 24 Y. Yamaguchi, H. Tomono, F. Iga, and Y. Nishihara, Physica C 273, 261 ~1997!. 13
