- Email: [email protected]
Influence of the anisotropy in the c-axis resistivity measurements of high-Tc superconductors
Physica C 315 Ž1999. 271–277
Influence of the anisotropy in the c-axis resistivity measurements of high-Tc superconductors J.L. Gonzalez ´ a
a,b,)
, J.S. Espinoza Ortiz a , E. Baggio-Saitovitch
a
Dpto. DME, Brazilian Center for Research in Physics, Rua Dr. XaÕier Sigaud 150, URCA, Rio de Janeiro, CEP: 22290-180 RJ, Brazil b SuperconductiÕity Laboratory, IMRE-Physics Faculty, UniÕersity of HaÕana, 10 400 La HaÕana, Cuba Received 26 November 1998; received in revised form 3 February 1999; accepted 8 February 1999
Abstract The cross-measurement configuration, commonly used in the determination of the c-axis resistivity was the matter of our theoretical analysis. We obtained an analytical expression for the measured resistivity which depends on a ‘effective anisotropy’ factor, identified here as G , and takes into account the non-homogeneous current distribution into the sample. We showed that the best experimental results are obtained for superconductor samples with the largest G . In particular, for samples with G ) 1, the distance between the voltage contacts and the current ones should be a third of the length of the sample in order to obtain the real rc and to avoid the use of other complicated experimental set-up. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 74.25.Fy; 74.72.y h Keywords: Electrical resistivity; Anisotropy; High-Tc superconductors
1. Introduction The high-Tc superconductors have a large anisotropy in their magnetic and transport properties that drastically modify its features in the H–T plane w1x. In these materials, the superconductivity is principally determined by the conduction in the CuO 2 planes, these are Josephson coupled along the c-axis and consequently, the dissipation mechanisms for currents flowing in the plane, and perpendicular to it, are very different w2x. Then, in resistivity measurements the temperature dependence of the resistivity in the ab-plane, r a b , and out of the plane, namely rc , presents different values and markedly different behaviors w2,3x. Despite the superconductivity to be mainly determined by the CuO 2 planes, many important physical properties are influenced by the interlayer coupling w2x, thus, this mechanism and its own essence have been issued experimentally by using different techniques, among them, the c-axis resistivity. Numerous efforts have been made to measure the c-axis resistivity w4–11x, furthermore the two ones Ž r a b and rc . in a same sample and under the same experimental conditions. )
Corresponding author. Centro Brasileiro de Pesquisas Fisicas ŽCBPF., Rua Xavier Sigaud a 150, URCA, Rio de Janeiro, CEP: 22290-180, RJ, Brazil. E-mail: [email protected] 0921-4534r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 1 9 3 - 8
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
272
Fig. 1. Ža. Experimental configuration used in the determination of the c-axis resistivity Ž rc .. Žb. Experimental configuration used in the determination of the in-plane resistivity Ž r a b ..
A difficulty which conspires against the determination of rc Žalso r a b ., arises from the high anisotropy in these HTSC materials. It is expected to be a non-uniform current distribution inside the sample, then in a typical experiment, the measured resistivity Žout of plane. is a mixture of the true rc and r a b . This fact produces overestimated values of this quantity, furthermore, it can provoke resistivity curve deformations which can be erroneously attributed to intrinsic physical effects. For example, it has been claimed to be a possible cause of the ‘resistive shoulder’ which have been observed in the in-plane measurements of BSSCO samples w12x. To overcome this problem and to measure a non-distorted value of rc , some experimental modifications have been made in experiments trying to induce a uniform current distribution into the sample w13,14x. On the other hand, in samples not very thick, more complex experimental designs like multi-terminal techniques have proved to be another option for computing these quantities w15–18x. In this work, we explore the simplest experimental approach for the rc determination, the four probe method with a cross-contact configuration commonly used in experiments Žsee Fig. 1a.. Our starting point is a linear model for the potential distribution into the sample and within this theoretical frame, we provide possible experimental results by analyzing our analytical expressions. We found that the experimental error Ž rc,meas. rrc,true . depends on a parameter called here ‘effective anisotropy’ Ž G ., which take into account the resistivity anisotropy and the sample dimensions. It was concluded that there is a voltage contacts position which permits us to obtain the true resistivity, avoiding the use of complicated multi-terminal techniques. In particular, for samples with G ) 1 the voltage contacts should be located at a third of the length of the sample in order to obtain the real resistivity.
2. The model In the rc experimental determination, the current is injected onto the top surface of the sample and extracted at the bottom. Two aligned contacts recording the voltage are located according to Fig. 1a. In the quasistatic limit the following differential equation must be satisfied: 1 E2 V Ž x , z .
ra b
Ex2
q
1 E2 V Ž x , z .
rc
Ez2
s 0.
Ž 1.
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
273
Here, we impose the natural boundary conditions: EVrE x s 0 at x s 0, L. At the top surface Ž z s D ., the current density Jz Ž x, D . must correspond to the injected current I Žsee Fig. 1a., where I is the transport current applied in the measurement on the current contacts of size a. The expression which satisfies Eq. Ž1. and the first boundary condition is: `
V Ž x , z . s V0 z q
Ý Vn cos Ž np xrL . sinh ž np(rc Ž Dr2 y z . r(ra b L / .
Ž 2.
ns1
Hence, the V0 and Vn coefficients can be found through the Fourier methods using the last boundary condition on the transport current applied. Here we just give the explicit coefficients, leaving detailed development for Appendix A: V0 s I rcrbL and Vn s y2 Irnp b coshŽ np rc Dr2 r a b L.. Finally, the expression for the voltage in the sample in terms of the parameters I, r a b , rc , b and L is:
(
(
`
V Ž x , z . s I rc zrbL y
( (
(
(
ž (
(
2 I rc r a b cos Ž np xrL . sinh np rc Ž Dr2 y z . r r a b L
Ý
ž (
(
nbp cosh np rc Dr2 r a b L
ns1
/
/.
Ž 3.
Establishing that the voltage signal in a measurement is DV s V Ž x 0 , D . y V Ž x 0 ,0. and taking in account Eq. Ž3., we can deduce an expression for the measured resistance as a function of the true resistivities and the parameters of the sample: `
R m s rc DrbL q
( (
ž (
nbp
ns1
Rm s
rc D bL
`
1q
Ý
(
4 rc r a b cos Ž np x 0rL . tanh np rc Dr2 r a b L
Ý
4cos Ž np x 0rL . tanh Ž npGr2 . npG
ns1
(
,
/; Ž 4.
(
where we define the parameter G s rc Dr r a b L. This is our principal result. Considering Eq. Ž4., we observe that R m has an expected term rc DrbL and a ‘complicated’ correction factor, which depends on G and the contact configuration. Note that if G 4 1 the tanhŽ npGr2. can be approximate to 1 Ževen for the first term in the series., and the correction factor takes the closed form: Ý`ns 1cosŽ np x 0rL.rn s yŽ1r2. lnw2Ž1 y cosŽ x 0 prL..x, and as a consequence R m acquire the final form Rm s
rc D bL
1y
4 ln w 2sin Ž x 0 pr2 L .
Gp
.
Ž 5.
Note that this expression just depends on the distance between the current contact and the voltage one, and the parameter G . At this point, it is straightforward to compare our result with a previous work by Logan et al. based on the Montgomery method w19,20x. These results are a particular case of our general expression presented here. They obtained a similar quantity, namely R t for a fixed configuration of current–voltage contacts. In the limit of G ) 1 and taking the voltage contact at the end of the sample as in Ref. w19x Ži.e., x 0 s L. in Eq. Ž5., we recovered the expression obtained by them, Rm s
rc D bL
y
(
4 rc r a b ln 2
pb
.
274
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
'
'
Fig. 2. Dependence of rme as. r rc with the effective anisotropy parameter rc Dr r a b L for a voltage contact located at the end of the sample Ž x s L.. rm is the measure resistivity in the experiment and rc is the expected value of the magnitude. In the insert, we show the curve for G -1 and a fitting following the expression obtained in Ref. w17x.
In contrast, in the opposite limit Ž G - 1., the same authors obtained for the transversal resistivity: R t s Ž rc DrbL.Ž16expŽyprG .rpG .. To discuss this limit, we prefer to show in Fig. 2 the numerical behavior of rmeas. rrc Žs R t bLrrc D . vs. the parameter G , according to Eq. Ž4., where we took the first 100 000 terms in the series. As it can be observed, when G decreases, rmeas. is greatly distorted from the real resistivity, while for G 4 1, there is no discrepancy between the measured and the expected values. Still in Fig. 2, in the insert, we show the curve for low values of the parameter G following Eq. Ž4. and the curve fitted by the expression of Logan et al. Ž16expŽyprG .rpG .. In summary, we remark that our expression reproduces the same results of Ref. w19x in all the range of G . In Fig. 3, we plot rmeas. rrc vs. the distance between the current contact and the voltage one for three different typical values of the parameter G . These values were taken as representing approximately the YBCO Ž G ; 0.6. w20x, Bi-2212 Ž G ; 10. w14x and Tl-2212 Ž G ; 2.5. w21x systems. As we can observe, for larger G , the more reliable the measurement; this is the case of the BSCCO system where the voltage contact position almost does not modify rmeas., which is always close to the expected value rc . The opposite is the YBCO, which is very sensitive to the contact configuration. In this case, the experimental value can be very distorted from the real one, and experimental modifications Žlike multi-terminal techniques. should be introduced in order to guarantee a uniform current distribution into the sample and a better estimate of rc .
Fig. 3. Dependence of rme as. r rc with the distance between the voltage and the current contacts normalized to the sample length in this direction. Three curves are shown representing the YBCO Ž G f 0.6., Tl-2212 Ž G f 2.5., and BSCCO-2212 Ž G f10. systems, respectively.
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
275
Taking x v as the distance for which the correction factor in Eq. Ž4. is approximately zero Ž rmeas.( rc ., it is interesting to show how x v depends on G ŽFig. 4.. Note that for low values of G , this dependence increases linearly while it reaches a plateau for higher values. The plateau began at Gc f 1 when x vrL f 0.33. This result suggests that for samples with a high anisotropy and low Ž DrL. relation, the voltage contacts should be located at a distance of Lr3 from the current ones in order to guarantee that the measured resistivity will be the true off-plane one Ž rc .. The other configuration in which the contacts are located on the face perpendicular to the c-axis, as in Fig. 1b, was analyzed in Ref. w16x in the single harmonic approximation and for a multi-terminal contact configuration. By using analogous boundary conditions on the current like in the paragraphs above, it is easy to show that the final expression for the voltage in this configuration is: `
V Ž x, z. s
y
Ý
(
ž(
(
4 I rc r a b sin Ž npr2 . sin Ž np xrL . cosh rc np zr r a b L
ž(
(
bnp sinh rc np Dr r a b L
ns1,3,5
/.
/
Ž 6.
This expression is the same as that in Ref. w16x, except for a pr4 factor. Note that our results remove the disagreement between Refs. w16,19x for r a b rc . For G ) 1, we can get tanh G f 1; by taking R m s V Žyx 1 , D . y V Ž x 2 , D .rI and considering the closed form of Ý`ns1Žy1. ny 1 sinŽw2 n y 1xp x 0rL.rŽ2 n y 1. s Ž1r2. lnŽtanwpr4 q p x 0r2 L x., the expression for the measured resistance in the experience becomes:
(
Rm s
(
2 rc r a b bp
ln
tan Ž pr4 q p x 1r2 L . q ln tan Ž pr4 q p x 2r2 L .
4.
Ž 7.
Again, the works’ results w19,20x for the case G ) 1 in the same configuration, namely R 1 Žs Ž L r a brbD .Ž16 G expŽypG .rp .. can be obtained from Eq. Ž6. by considering the first term in the series, approximating the sinhŽ rc np Dr r a b L. for large arguments and setting the experimental conditions of the last work; i.e., z s 0, x 1 s yLr2 and x 2 s Lr2 with respect to Fig. 1b. In conclusion, in this work we have demonstrated that the current distribution inside the sample during the experimental determination of rc disrupts the real value of this magnitude. Our analysis permits us to have an estimate of the error in the experimental determination of this magnitude and shows that this effect depends on the effective anisotropic parameter G , having a higher influence in samples which have a smaller G , like YBCO, and less influence in samples with larger G , like single crystals of bismuth, where good estimates of rc can be obtained. For this kind of samples, we can infer that optimum results will be obtained if the voltage contacts are located at a distance of approximately Lr3 with respect to the current ones. Finally, we remark that these methods can be extended to analyze measurements performed on another type of anisotropic samples.
(
(
Fig. 4. Dependence with G of x v r L. Here, x v is the distance between the voltage and current contacts for which rmeas.( rc .
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
276
Acknowledgements This work was supported by the CAPES program. JSEO would like to thank FAPERJ support.
Appendix A Given `
V Ž x , z . s V0 z q
Ý Vn cos Ž np xrL . sinh ž np(rc Ž Dr2 y z . r(ra b L /
ns1
satisfying Eq. Ž1. and boundary conditions. The current density Jz Ž x, z . can be obtained through Jz Ž x, z . s yŽ1rrc .ŽEV Ž x, z .rE z ., i.e., Jz Ž x , z . s y
V0
rc
`
q
ž (
(
Vn np cos Ž np xrL . cosh np rc Ž Dr2 y z . r r a b L
Ý
(r
ns1
c
ra b L
/.
The current density in the z direction at the top surface of the sample is: Jz Ž x , D . s y
V0
rc
`
q
ž (
(
Vn np cos Ž np xrL . cosh np rc Dr2 r a b L
Ý
(r
ns1
c
r ab L
/.
Neglecting the contacts’ size Ž a < L., this means in the limit of ‘punctual contact’ Ž a ™ 0. and supposing the contacts are located in an arbitrary position, for example x s x 1 , the coefficients are given by V0 s
rc
rc
L
L
Id Ž x y x 1 .
I rc
d xs , H J Ž x , z . d x s L H0 L 0 b bL 2(r r L Id Ž x y x . V s cos Ž np xrL . d x , H b np cosh ž np(r Dr2(r L / 0 2(r r I cos Ž np x rL . V s . np b cosh ž np(r Dr2(r L / z
c
ab
1
n
c
c
ab
ab
1
n
c
ab
For our specific case, the current contacts are located at the left corner of the sample, that means x 1 s 0 Žsee Fig. 1a., then: V0 s
I rc bL
and Vn s
(
2 rc r a b I
ž (
(
np b cosh np rc Dr2 r a b L
.
/
References w1x G. Blatter, M.V. Feigelman, B. Geshkenbein, A.I. Larkin, V. Vinokur, Rev. Mod. Phys. 66 Ž1995. 1125. w2x S.L. Cooper, K.E. Gray, in: D.M. Ginsberg ŽEd.., Physical Properties of High Temperature Superconductors IV, World Scientific, Singapore, 1993, p. 61. w3x G. Briceno, ˜ M.F. Crommie, A. Zettl, Phys. Rev. Lett. 66 Ž1991. 2164. w4x L. Forro, V. Ilakovac, J.R. Cooper, C. Ayache, Y. Henrry, Phys. Rev. B 46 Ž1992. 6626. w5x H.M. Duan, W. Kiehl, C. Dong, A.W. Cordes, M.J. Saeed, D.L. Viar, A.M. Hermann, Phys. Rev. B 43 Ž1991. 12925.
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277 w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
R. Jin, H.R. Ott, D.P. Grindatto, Physica C 250 Ž1995. 395. J.H. Cho, M.P. Maley, H. Safar, J.Y. Coulter, L.N. Bulaevskii, Physica C 265 Ž1996. 277. T. Aouaroun, V. Hardy, Ch. Goupil, F. Warmont, G. Villard, Ch. Simon, Supercond. Sci. Technol. 10 Ž1997. 572. N.E. Hussey, K. Nozawa, H. Tagaki, S. Adachi, K. Tanabe, Phys. Rev. B 56 Ž1997. 11423. J.-S. Zhou, J.B. Goodenough, B. Dabrowski, K. Rogacki, Phys. Rev. Lett. 77 Ž1996. 4253. J.R. Kirtley, K.A. Moler, G. Villard, A. Maignan, Phys. Rev. Lett. 81 Ž1998. 2140. A.K. Pradhan, C. Chen, B.M. Wanklyn, Physica C 251 Ž1995. 405. D. Lopez, G. Nieva, F. de la Cruz, Phys. Rev. B 50 Ž1994. 7219. J.H. Cho, M.P. Maley, S. Fleshler, A. Lacerda, L.N. Bulaevskii, Phys. Rev. B 50 Ž1994. 6493. H.C. Montgomery, J. Appl. Phys. 42 Ž1971. 2971. R. Busch, G. Ries, H. Werther, G. Kreiselmeyer, G. Saemann-Ischenko, Phys. Rev. Lett. 69 Ž1992. 522. G.A. Levin, J. Appl. Phys. 81 Ž1997. 714. Y.M. Wan, T.R. Lemberger, S.E. Hebboul, J.C. Garland, Phys. Rev. B 54 Ž1996. 3602. B.F. Logan, S.O. Rice, R.F. Wick, J. Appl. Phys. 42 Ž1971. 2975. S.J. Hagen, T.W. Jing, Z.Z. Wang, J. Horvath, N.P. Ong, Phys. Rev. B 37 Ž1988. 7928. V. Hardy, A. Maignan, C. Martin, F. Warmont, J. Provost, Phys. Rev. B 56 Ž1997. 130.
277
Influence of the anisotropy in the c-axis resistivity measurements of high-Tc superconductors J.L. Gonzalez ´ a
a,b,)
, J.S. Espinoza Ortiz a , E. Baggio-Saitovitch
a
Dpto. DME, Brazilian Center for Research in Physics, Rua Dr. XaÕier Sigaud 150, URCA, Rio de Janeiro, CEP: 22290-180 RJ, Brazil b SuperconductiÕity Laboratory, IMRE-Physics Faculty, UniÕersity of HaÕana, 10 400 La HaÕana, Cuba Received 26 November 1998; received in revised form 3 February 1999; accepted 8 February 1999
Abstract The cross-measurement configuration, commonly used in the determination of the c-axis resistivity was the matter of our theoretical analysis. We obtained an analytical expression for the measured resistivity which depends on a ‘effective anisotropy’ factor, identified here as G , and takes into account the non-homogeneous current distribution into the sample. We showed that the best experimental results are obtained for superconductor samples with the largest G . In particular, for samples with G ) 1, the distance between the voltage contacts and the current ones should be a third of the length of the sample in order to obtain the real rc and to avoid the use of other complicated experimental set-up. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 74.25.Fy; 74.72.y h Keywords: Electrical resistivity; Anisotropy; High-Tc superconductors
1. Introduction The high-Tc superconductors have a large anisotropy in their magnetic and transport properties that drastically modify its features in the H–T plane w1x. In these materials, the superconductivity is principally determined by the conduction in the CuO 2 planes, these are Josephson coupled along the c-axis and consequently, the dissipation mechanisms for currents flowing in the plane, and perpendicular to it, are very different w2x. Then, in resistivity measurements the temperature dependence of the resistivity in the ab-plane, r a b , and out of the plane, namely rc , presents different values and markedly different behaviors w2,3x. Despite the superconductivity to be mainly determined by the CuO 2 planes, many important physical properties are influenced by the interlayer coupling w2x, thus, this mechanism and its own essence have been issued experimentally by using different techniques, among them, the c-axis resistivity. Numerous efforts have been made to measure the c-axis resistivity w4–11x, furthermore the two ones Ž r a b and rc . in a same sample and under the same experimental conditions. )
Corresponding author. Centro Brasileiro de Pesquisas Fisicas ŽCBPF., Rua Xavier Sigaud a 150, URCA, Rio de Janeiro, CEP: 22290-180, RJ, Brazil. E-mail: [email protected] 0921-4534r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 1 9 3 - 8
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
272
Fig. 1. Ža. Experimental configuration used in the determination of the c-axis resistivity Ž rc .. Žb. Experimental configuration used in the determination of the in-plane resistivity Ž r a b ..
A difficulty which conspires against the determination of rc Žalso r a b ., arises from the high anisotropy in these HTSC materials. It is expected to be a non-uniform current distribution inside the sample, then in a typical experiment, the measured resistivity Žout of plane. is a mixture of the true rc and r a b . This fact produces overestimated values of this quantity, furthermore, it can provoke resistivity curve deformations which can be erroneously attributed to intrinsic physical effects. For example, it has been claimed to be a possible cause of the ‘resistive shoulder’ which have been observed in the in-plane measurements of BSSCO samples w12x. To overcome this problem and to measure a non-distorted value of rc , some experimental modifications have been made in experiments trying to induce a uniform current distribution into the sample w13,14x. On the other hand, in samples not very thick, more complex experimental designs like multi-terminal techniques have proved to be another option for computing these quantities w15–18x. In this work, we explore the simplest experimental approach for the rc determination, the four probe method with a cross-contact configuration commonly used in experiments Žsee Fig. 1a.. Our starting point is a linear model for the potential distribution into the sample and within this theoretical frame, we provide possible experimental results by analyzing our analytical expressions. We found that the experimental error Ž rc,meas. rrc,true . depends on a parameter called here ‘effective anisotropy’ Ž G ., which take into account the resistivity anisotropy and the sample dimensions. It was concluded that there is a voltage contacts position which permits us to obtain the true resistivity, avoiding the use of complicated multi-terminal techniques. In particular, for samples with G ) 1 the voltage contacts should be located at a third of the length of the sample in order to obtain the real resistivity.
2. The model In the rc experimental determination, the current is injected onto the top surface of the sample and extracted at the bottom. Two aligned contacts recording the voltage are located according to Fig. 1a. In the quasistatic limit the following differential equation must be satisfied: 1 E2 V Ž x , z .
ra b
Ex2
q
1 E2 V Ž x , z .
rc
Ez2
s 0.
Ž 1.
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
273
Here, we impose the natural boundary conditions: EVrE x s 0 at x s 0, L. At the top surface Ž z s D ., the current density Jz Ž x, D . must correspond to the injected current I Žsee Fig. 1a., where I is the transport current applied in the measurement on the current contacts of size a. The expression which satisfies Eq. Ž1. and the first boundary condition is: `
V Ž x , z . s V0 z q
Ý Vn cos Ž np xrL . sinh ž np(rc Ž Dr2 y z . r(ra b L / .
Ž 2.
ns1
Hence, the V0 and Vn coefficients can be found through the Fourier methods using the last boundary condition on the transport current applied. Here we just give the explicit coefficients, leaving detailed development for Appendix A: V0 s I rcrbL and Vn s y2 Irnp b coshŽ np rc Dr2 r a b L.. Finally, the expression for the voltage in the sample in terms of the parameters I, r a b , rc , b and L is:
(
(
`
V Ž x , z . s I rc zrbL y
( (
(
(
ž (
(
2 I rc r a b cos Ž np xrL . sinh np rc Ž Dr2 y z . r r a b L
Ý
ž (
(
nbp cosh np rc Dr2 r a b L
ns1
/
/.
Ž 3.
Establishing that the voltage signal in a measurement is DV s V Ž x 0 , D . y V Ž x 0 ,0. and taking in account Eq. Ž3., we can deduce an expression for the measured resistance as a function of the true resistivities and the parameters of the sample: `
R m s rc DrbL q
( (
ž (
nbp
ns1
Rm s
rc D bL
`
1q
Ý
(
4 rc r a b cos Ž np x 0rL . tanh np rc Dr2 r a b L
Ý
4cos Ž np x 0rL . tanh Ž npGr2 . npG
ns1
(
,
/; Ž 4.
(
where we define the parameter G s rc Dr r a b L. This is our principal result. Considering Eq. Ž4., we observe that R m has an expected term rc DrbL and a ‘complicated’ correction factor, which depends on G and the contact configuration. Note that if G 4 1 the tanhŽ npGr2. can be approximate to 1 Ževen for the first term in the series., and the correction factor takes the closed form: Ý`ns 1cosŽ np x 0rL.rn s yŽ1r2. lnw2Ž1 y cosŽ x 0 prL..x, and as a consequence R m acquire the final form Rm s
rc D bL
1y
4 ln w 2sin Ž x 0 pr2 L .
Gp
.
Ž 5.
Note that this expression just depends on the distance between the current contact and the voltage one, and the parameter G . At this point, it is straightforward to compare our result with a previous work by Logan et al. based on the Montgomery method w19,20x. These results are a particular case of our general expression presented here. They obtained a similar quantity, namely R t for a fixed configuration of current–voltage contacts. In the limit of G ) 1 and taking the voltage contact at the end of the sample as in Ref. w19x Ži.e., x 0 s L. in Eq. Ž5., we recovered the expression obtained by them, Rm s
rc D bL
y
(
4 rc r a b ln 2
pb
.
274
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
'
'
Fig. 2. Dependence of rme as. r rc with the effective anisotropy parameter rc Dr r a b L for a voltage contact located at the end of the sample Ž x s L.. rm is the measure resistivity in the experiment and rc is the expected value of the magnitude. In the insert, we show the curve for G -1 and a fitting following the expression obtained in Ref. w17x.
In contrast, in the opposite limit Ž G - 1., the same authors obtained for the transversal resistivity: R t s Ž rc DrbL.Ž16expŽyprG .rpG .. To discuss this limit, we prefer to show in Fig. 2 the numerical behavior of rmeas. rrc Žs R t bLrrc D . vs. the parameter G , according to Eq. Ž4., where we took the first 100 000 terms in the series. As it can be observed, when G decreases, rmeas. is greatly distorted from the real resistivity, while for G 4 1, there is no discrepancy between the measured and the expected values. Still in Fig. 2, in the insert, we show the curve for low values of the parameter G following Eq. Ž4. and the curve fitted by the expression of Logan et al. Ž16expŽyprG .rpG .. In summary, we remark that our expression reproduces the same results of Ref. w19x in all the range of G . In Fig. 3, we plot rmeas. rrc vs. the distance between the current contact and the voltage one for three different typical values of the parameter G . These values were taken as representing approximately the YBCO Ž G ; 0.6. w20x, Bi-2212 Ž G ; 10. w14x and Tl-2212 Ž G ; 2.5. w21x systems. As we can observe, for larger G , the more reliable the measurement; this is the case of the BSCCO system where the voltage contact position almost does not modify rmeas., which is always close to the expected value rc . The opposite is the YBCO, which is very sensitive to the contact configuration. In this case, the experimental value can be very distorted from the real one, and experimental modifications Žlike multi-terminal techniques. should be introduced in order to guarantee a uniform current distribution into the sample and a better estimate of rc .
Fig. 3. Dependence of rme as. r rc with the distance between the voltage and the current contacts normalized to the sample length in this direction. Three curves are shown representing the YBCO Ž G f 0.6., Tl-2212 Ž G f 2.5., and BSCCO-2212 Ž G f10. systems, respectively.
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275
Taking x v as the distance for which the correction factor in Eq. Ž4. is approximately zero Ž rmeas.( rc ., it is interesting to show how x v depends on G ŽFig. 4.. Note that for low values of G , this dependence increases linearly while it reaches a plateau for higher values. The plateau began at Gc f 1 when x vrL f 0.33. This result suggests that for samples with a high anisotropy and low Ž DrL. relation, the voltage contacts should be located at a distance of Lr3 from the current ones in order to guarantee that the measured resistivity will be the true off-plane one Ž rc .. The other configuration in which the contacts are located on the face perpendicular to the c-axis, as in Fig. 1b, was analyzed in Ref. w16x in the single harmonic approximation and for a multi-terminal contact configuration. By using analogous boundary conditions on the current like in the paragraphs above, it is easy to show that the final expression for the voltage in this configuration is: `
V Ž x, z. s
y
Ý
(
ž(
(
4 I rc r a b sin Ž npr2 . sin Ž np xrL . cosh rc np zr r a b L
ž(
(
bnp sinh rc np Dr r a b L
ns1,3,5
/.
/
Ž 6.
This expression is the same as that in Ref. w16x, except for a pr4 factor. Note that our results remove the disagreement between Refs. w16,19x for r a b rc . For G ) 1, we can get tanh G f 1; by taking R m s V Žyx 1 , D . y V Ž x 2 , D .rI and considering the closed form of Ý`ns1Žy1. ny 1 sinŽw2 n y 1xp x 0rL.rŽ2 n y 1. s Ž1r2. lnŽtanwpr4 q p x 0r2 L x., the expression for the measured resistance in the experience becomes:
(
Rm s
(
2 rc r a b bp
ln
tan Ž pr4 q p x 1r2 L . q ln tan Ž pr4 q p x 2r2 L .
4.
Ž 7.
Again, the works’ results w19,20x for the case G ) 1 in the same configuration, namely R 1 Žs Ž L r a brbD .Ž16 G expŽypG .rp .. can be obtained from Eq. Ž6. by considering the first term in the series, approximating the sinhŽ rc np Dr r a b L. for large arguments and setting the experimental conditions of the last work; i.e., z s 0, x 1 s yLr2 and x 2 s Lr2 with respect to Fig. 1b. In conclusion, in this work we have demonstrated that the current distribution inside the sample during the experimental determination of rc disrupts the real value of this magnitude. Our analysis permits us to have an estimate of the error in the experimental determination of this magnitude and shows that this effect depends on the effective anisotropic parameter G , having a higher influence in samples which have a smaller G , like YBCO, and less influence in samples with larger G , like single crystals of bismuth, where good estimates of rc can be obtained. For this kind of samples, we can infer that optimum results will be obtained if the voltage contacts are located at a distance of approximately Lr3 with respect to the current ones. Finally, we remark that these methods can be extended to analyze measurements performed on another type of anisotropic samples.
(
(
Fig. 4. Dependence with G of x v r L. Here, x v is the distance between the voltage and current contacts for which rmeas.( rc .
J.L. Gonzalez ´ et al.r Physica C 315 (1999) 271–277
276
Acknowledgements This work was supported by the CAPES program. JSEO would like to thank FAPERJ support.
Appendix A Given `
V Ž x , z . s V0 z q
Ý Vn cos Ž np xrL . sinh ž np(rc Ž Dr2 y z . r(ra b L /
ns1
satisfying Eq. Ž1. and boundary conditions. The current density Jz Ž x, z . can be obtained through Jz Ž x, z . s yŽ1rrc .ŽEV Ž x, z .rE z ., i.e., Jz Ž x , z . s y
V0
rc
`
q
ž (
(
Vn np cos Ž np xrL . cosh np rc Ž Dr2 y z . r r a b L
Ý
(r
ns1
c
ra b L
/.
The current density in the z direction at the top surface of the sample is: Jz Ž x , D . s y
V0
rc
`
q
ž (
(
Vn np cos Ž np xrL . cosh np rc Dr2 r a b L
Ý
(r
ns1
c
r ab L
/.
Neglecting the contacts’ size Ž a < L., this means in the limit of ‘punctual contact’ Ž a ™ 0. and supposing the contacts are located in an arbitrary position, for example x s x 1 , the coefficients are given by V0 s
rc
rc
L
L
Id Ž x y x 1 .
I rc
d xs , H J Ž x , z . d x s L H0 L 0 b bL 2(r r L Id Ž x y x . V s cos Ž np xrL . d x , H b np cosh ž np(r Dr2(r L / 0 2(r r I cos Ž np x rL . V s . np b cosh ž np(r Dr2(r L / z
c
ab
1
n
c
c
ab
ab
1
n
c
ab
For our specific case, the current contacts are located at the left corner of the sample, that means x 1 s 0 Žsee Fig. 1a., then: V0 s
I rc bL
and Vn s
(
2 rc r a b I
ž (
(
np b cosh np rc Dr2 r a b L
.
/
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