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The Nernst effect in high-Tc cuprate superconductors
Physica C 328 Ž1999. 230–240 www.elsevier.nlrlocaterphysc
The Nernst effect in high-Tc cuprate superconductors K. Yamafuji
a,)
, T. Fujiyoshi b, T. Kiss c , M. Inoue c , T. Sasaki d , N. Kobayashi
d
a
c
Ariake National College of Technology, 150 Higashihagio-machi, Omuta 836-8585, Japan b Faculty of Engineering, Kumamoto UniÕersity, Kumamoto 860-8555, Japan Graduate School of Information Science and Electrical Engineering, Kyushu UniÕersity, Fukuoka 812-8581, Japan d Institute for Material Research, Tohoku UniÕersity, Sendai 980-8577, Japan Received 5 August 1999; received in revised form 27 August 1999; accepted 13 October 1999
Abstract Detailed measurements of both the Nernst coefficient, Ne , and the current vs. voltage characteristics were carried out in a YBCO film by varying systematically the flux density, B, and the temperature, T. The Nernst coefficient was observed over the temperature region of a fairly wide superconducting vs. normal resistive transition under the applied flux density, B. The transport entropy of a fluxoid, Sf , estimated directly from the present measurements showed a small tail decreasing exponentially with increasing temperature, even above the mean-field critical temperature, Tc Ž B ., whereas the existing theory predicted Sf to become zero above Tc Ž B .. Then the expression for Sf above Tc Ž B . was derived. The expression for the resistivity around the temperatures of the resistive transition was also derived, including the resistivity due to the flux flow. With the aid of these expressions, the observed Nernst coefficient was explained quantitatively by the present theory. q 1999 Elsevier Science B.V. All rights reserved. Keywords: High-Tc superconductors; Nernst effect; Resistive transition; Flux pinning; Transport entropy; Thermal fluctuation
1. Introduction In high-Tc cuprate superconductors, the effects of thermal fluctuation on the flux depinning phenomena in the superconducting mixed state under the applied flux density, B, is quite severe especially at higher temperatures due to mainly the extremely short coherence length along c-axis w1x. For example, the so-called glass–liquid transition w2x appears at Tg Ž B . far below the mean-field critical temperature, Tc Ž B .,
and this kind of transition can be interpreted as a limiting case of the thermally depinning transition w3–5x, while Tg Ž B . locates in the very vicinity of Tc Ž B . in the metallic superconductors w6x. Furthermore, the mean-field critical temperature, Tc Ž B ., itself can no more be regarded as the clear transition line between the superconducting mixed state and the normal state, differing from the case of the metallic low temperature superconductors. The Nernst coefficient, Ne , for type-2 superconductors is defined by Ref. w7x as Ne s r Sf rf 0 ,
Ž 1.1 .
)
Corresponding author. Tel.: q81-944-53-8601; fax: q81944-53-1361. E-mail address: [email protected] ŽK. Yamafuji.
where r is the electric resistivity for a very small value of the transport current density, Sf is the
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 5 5 3 - 5
K. Yamafuji et al.r Physica C 328 (1999) 230–240
transport entropy per unit length of a single fluxoid, and f 0 is the flux quantum. As the temperature is increased, the resistivity for a very small current density, r , recovers suddenly at Tg Ž B . from an extremely small value induced by the flux creep to a relatively small value induced by the thermally-assisted flux-flow w2–5x, and approaches gradually the normal resistivity, r N ŽT ., near Tc Ž B .. On the other hand, the transport entropy, Sf , decreases with the increase of temperature and becomes to be below the experimentally detectable level near the uppermost temperature, Tc Ž B .U , of the resistive transition region between the superconducting mixed state and the normal state w7–13x. Thus Ne of this type given by Eq. Ž1.1. can only be observed in the temperature range of Tg Ž B . Q T Q Tc Ž B .U and, hence, may be measurable quantitatively with a high reliability only in high-Tc cuprate superconductors, for which the observable temperature range of Ne is fairly wide. Then many observed data have been presented for Ne in high-Tc cuprate superconductors w7–13x. Unfortunately, however, one can find no theory which is available for the quantitative comparison with these observed data. Only the theoretical expression for Sf presented so far was derived by Maki w14a,14bx based on the Calori–Maki theory w15x. However, his theory is available only for the BCS superconductors and, hence, Sf becomes zero at Tc Ž B ., besides that the quantitative reliability of his expression for Sf has not been confirmed experimentally for high-Tc superconductors even below Tc Ž B .. As for the resistivity, r , Ikeda et al. w16x have presented the theoretical expressions for single crystals of high-Tc cuprate superconductors by starting from the G–L functional Hamiltonian and by taking account of the effects of thermal fluctuation explicitly. However, the effect of the flux depinning near the glass–liquid transition temperature, Tg Ž B ., is not considered explicitly in their theory, and their expression shows noticeable deviation from the observed data at lower temperature for bulk single crystals with, may be, some pinning centers w16x. Furthermore, many physical quantities including the material constants are contained in their expression. For the quantitative comparison of their theory with the observed data, therefore, many kinds of observations are necessary in the sample, in which the
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observation of Ne was made, for estimating the values of these physical quantities. The purpose of the present paper is to provide the theory to be able to explain quantitatively the observed data of Ne . Since at least the resistive measurement in the same sample is also necessary for the analysis and the discussion of the observed results of Ne , we shall try to derive the theory that contains only the parameters evaluable by the set of these two kinds of measurements. For this purpose, a theoretical expression for the transport entropy, Sf , above Tc Ž B . is at first derived. Whereas the present expression for Sf approaches asymptotically the Maki expression w14a,14bx below Tc Ž B ., the semi-quantitative reliability of his expression has been supported by existing observation w7– 12x, though the magnetic-field dependence near Tc Ž B . might be somewhat questionable w13x. Based on our previous work w17x, however, the result of the comparison of the observed data presented by Zeh et al. w7x with our preliminary theory showed a quantitative agreement below Tc Ž B ., where the Maki expression of Sf was used and the reasonable values were assumed for the pinning parameters contained in our theory of resistivity, r , below Tc Ž B .. Next, we derived theoretical expression for the resistivity, r , including that above Tc Ž B ., because the expression presented by Ikeda et al. w16x is not adequate for the present purpose. Besides that all the parameters contained in their expression cannot be estimated only by these two kinds of measurements, their expression is hardly applicable for the resistivity near Tg Ž B .. In fact, the behavior of the flux flow near Tg Ž B . becomes different almost entirely from the usual flux flow in metallic superconductors at T < Tc Ž B . due to severe effect of thermal fluctuation, and becomes as that may be called the thermally-assisted flux-flow w3–5x, as will be discussed in Appendix A in detail. The resistivity in our theory approaches asymptotically to that in the thermallyassisted flux-flow region near Tg Ž B ., where the values of flux-pinning parameters contained in our theory can be evaluated experimentally. Whereas the observed data of Ne presented by Zeh et al. w7x seem to be reliable for the quantitative comparison with the present theory, the values of parameters necessary for the comparison with the present theory are unfortunately not specified for
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their sample. Thus, we carried out detailed measurements of the current vs. voltage characteristics in a YBCO film for the purpose of estimating the values of parameters included in the present theory, which are relating to the flux pinning, the thermally-assisted flux-flow and the resistive transition. It will be shown that the total curve of Ne against temperature observed in the YBCO film is successfully explained by the present theory. 2. Theoretical expression for Ne Let us consider a film of a high-Tc cuprate superconductor at the temperature, T, in the mixed-state under the applied flux density, B, in the direction of the z-axis, which is chosen to be along the direction of the film thickness. When the transport current with the current density, J, is applied to the direction of the y-axis, the fluxoids per unit length per unit area suffer the driving force of the Lorentz force type, FL , given by FL s J = B s Ž FL ,0,0 . , FL s JB. Ž 2.1 . Then, the fluxoids move to the direction of the x-axis with the average velocity, zJ , and the electric field, E J , is induced in the direction of the y-axis as E J s B = zJ s Ž 0, E J ,0 . ; EJ s Jr Ž T , B ; J . , Ž 2.2 . where r is the formally defined resistivity as E JrJ, which is usually dependent on J. When the temperature gradient, = T, is applied in the direction of the x-axis, on the other hand, the fluxoids per unit length per unit area suffer the driving force, FT , given by FT s FT= Tr=T s Ž FT ,0,0 . ; FT s =TBSf Ž T , B . rf 0 , Ž 2.3 . where f 0 is the flux quantum and Sf ŽT, B . is the transport entropy of a single fluxoid per unit length. Then these fluxoids move to the direction of the x-axis with the average velocity, zT , and hence the electric field, ET , is induced in the direction of the y-axis as ET s B = zT s Ž 0, ET ,0 . ; ET s ET Ž T , B ; =T . . Ž 2.4 . When both the measurements of E J and ET are carried out in the same sample under the above-men-
tioned experimental conditions, the directions of the driving force, FL and FT , and those of the induced electric field, E J and ET are respectively the same as each other, as can be seen from Eqs. Ž2.1. and Ž2.3. and also Eqs. Ž2.2. and Ž2.4.. Even for strongly anisotropic samples as in the high-Tc cuprate superconductors, therefore, the behavior of flux motion would be the same as each other for the same values of FL and FT . As can be seen from Eqs. Ž2.1. and Ž2.3., the value of FL is equal to the value of FT when the transport current density, J, takes the value of J s JT given by JT s=TSf Ž T , B . rf 0
Ž 2.5 .
Then the observed value of the electric field induced by Nernst effect, ET s ET Ž =T ., would be equal to the value of E J s E J Ž J . observed by the resistive measurement at J s JT ; i.e., ET Ž =T . s E J Ž JT . .
Ž 2.6 .
Eq. Ž2.6. with Eq. Ž2.5. provides a method of estimation of the transport entropy, Sf , directly form the both measurements of the Nernst effect and the E J vs. J characteristics, even at lower temperatures where the E J vs. J characteristics are strongly nonlinear. With the aid of Eqs. Ž2.2., Ž2.5. and Ž2.6., we get ET Ž T , B ;=T . ' =TNe Ž T , B ;=T . s =Tr Ž T , B ; JT . Sf Ž T , B . rf 0 Ž 2.7 . Eq. Ž2.7. can be the theoretical expression for the electric field induced by the Nernst effect, ET , when the theoretical expressions for r ŽT, B; JT . and Sf ŽT, B . are given explicitly. 2.1. Expression for the transport entropy Only the theoretical expression for the transport entropy of a single fluxoid per unit length, Sf , proposed so far is the one provided by Maki w14a,14bx based on the Calori–Maki theory w15x, which is given by SfM Ž T , B . s
Bf 0 L D Ž T .
bm 0
T
Bc 2 Ž T . y B ,
Ž 2.8a .
K. Yamafuji et al.r Physica C 328 (1999) 230–240
where m 0 is the magnetic permeability of the vacuum, L D ŽT . is a monotonically increasing function of T from L D Ž0. s 0 up to L D ŽTc Ž B .. s 1, Bc2 is the mean-field upper critical flux density, b s 1 q 2 1.16Ž2 k GL y 1. is a numerical constant with k GL denoting the Ginzburg–Landau parameter, and the superscript M represents that the right-hand-side of Eq. Ž2.8a. is the expression presented by Maki w14a,14bx. If we adopt the usual temperature dependence of Bc2 near Tc Ž B . given by Bc 2 Ž T .
U s Bc2
Ž 0. 1 y
ž
T Tc Ž B .
2
/
,
Ž 2.9 .
then, at T Q Tc Ž B ., Eq. Ž2.8a. is reduced to SfM
ŽT , B. s
Bf 0
1
2
Tc Ž B . y T 2
bm 0 Tc Ž 0 . Tc Ž 0 . 2 y Tc Ž B . 2
.
Ž 2.8b .
U Ž0. in Eq. Ž2.9. is not Bc2 Ž0. It is to be noted that Bc2 but an adjustable parameter for describing the behavior of Bc2 ŽT . at T Q Tc Ž B ., and we used an approximation of L D ŽT . ; TrTc Ž0. at T Q Tc Ž B .. As can be seen directly from Eq. Ž2.8b., SfM ŽT, B . becomes zero at T s Tc Ž B .. Even above Tc Ž B . under the applied flux density, B, however, the fluxoids would appear locally due to thermal fluctuation, and hence a small tail of Sf should appear even above Tc Ž B ., at least within the superconducting vs. normal resistive transition region, as has been indicated experimentally w7–12x. Then let us define the fraction of the local superconducting mixed-state regions by QS . Above Tc Ž B ., QS decreases due to thermal agitation. If we consider the general process that the local superconducting mixed-state regions ‘die’ by an external disturbance with the occurrence probability, f, then the simplest equation for describing the behavior of the ‘dead’ fraction of the superconducting mixed-state regions, Q N s 1 y QS , is given by
233
the occurrence of the external disturbance, f, is given by f s t m , and hence Eq. Ž2.10a. is reduced to dQ N rdt m s QS .
Ž 2.10b .
The solution of Eq. Ž2.10b. is given by QS s 1 y Q N s exp Ž yt m . .
Ž 2.11a .
Then the probability density function of the ‘dead’ fraction, P N , which is defined by differential of Q N with respect to t, is given by P N s mt my 1exp Ž yt m . .
Ž 2.11b .
This type of probability density function has been known well as the Weibull distribution function, which has been known to appear widely for fairly simple ‘dying’ stochastic phenomena. One of the typical examples is the thermal depinning process of the pinned fluxoids w3–5x, which will be discussed in Appendix A for the derivation of the thermally-assisted flux-flow resistivity in Section 2.2. Since the ‘dying’ process of the superconducting mixed-state phase due to thermal fluctuation is a single stochastic process, we can put m s 1 and t s TrD M Tc Ž B . in Eq. Ž2.11a., where D M Tc Ž B . is a quantity of the order of the temperature width of the superconducting vs. normal transition. Above Tc Ž B ., therefore, QS , is expected to be given by an exponentially decreasing linear function of TrD M Tc Ž B .. Since Sf above Tc Ž B . is expected to be proportional to QS , the simplest approximation for Sf at the temperatures around the superconducting vs. normal transition may be given by
Sf Ž T , B .
° ¢
SfM Ž T , B . ;
s~
SfM Ž Tp Ž B . , B . exp yg
T FTp Ž B . ,
ž
T y1 Tp Ž B .
/
;
T GTp Ž B . ,
Ž 2.10a .
Ž 2.12a .
When the external disturbance is a kind of the stochastic process composed of m unit stochastic processes independent of each other, each of which occurs with the probability, t, then the probability of
where Tp Ž B . is a slightly lower temperature than Tc Ž B ., and we put as D M Tc Ž B . s Tp Ž B .rg . The value of g can be determined from the request that the differential of Sf with respect to T is also
dQ N rd f s QS .
K. Yamafuji et al.r Physica C 328 (1999) 230–240
234
continuous at T s Tp Ž B .. If we tentatively adopt Eq. Ž2.8b. as the expression for Sf , then g is given by
gs
2Tp Ž B . 2
2
Tc Ž B . y Tp Ž B .
2
.
Ž 2.12b .
The present approximation for Sf is somewhat rough at the temperatures of T ; Tc Ž B ., because the present expression for QS given by Eq. Ž2.11a. may be hold well only at T 4 Tc Ž B ., while the behavior of Sf at T ; Tc Ž B . is not discussed here theoretically in detail. However, the variation of Sf at T ; Tc Ž B . is expected to be fairly smooth, and hence the insufficiency of the present approximation at T ; Tc Ž B . may be removed partly by choosing the value of the adjustable parameter, Tp Ž B ., adequately. 2.2. Expression for resistiÕity In the temperature region of the superconducting vs. normal resistive transition under the applied flux density, B, the electric field, E J , under the applied transport current with the density, J, is induced both in the normal regions and in the superconducting mixed-state regions. Since each of these two kinds of regions is locating alternatively along the direction of the transport current, the total resistivity, r , defined by r s E JrJ is given by
r s r F Ž 1 y Q˜ N . q r N Q˜ N ,
TrDR Tc Ž B ., where DR Tc Ž B . is a quantity of the order of the temperature width of the resistive transition. Then the simplest approximation for Q˜ N s 1 y Q˜S is given by Q˜ N s
½
1;
T GTcU Ž B . ,
1yexp yg w Ž T yTcU Ž B . . r TcU Ž B . x 4 ;
T FTcU Ž B . ,
Ž 2.14 . where TcU Ž B . is a temperature in the vicinity of the uppermost temperature of the resistive transition region, and we put DR Tc Ž B . s TcU Ž B .rg by using g given by Eq. Ž2.12b.. The normal resistivity at T ; Tc Ž B . can usually be approximated by
r N s aN T q b N ,
where a N and b N are almost independent of B. Since the Nernst effect is usually observed by applying a very small =T, the corresponding value of JT given by Eq. Ž2.5. is also quite small. At the temperatures far above the glass–liquid transition temperature, Tg Ž B ., where the electric field induced by the Nernst effect, ET , can be observed, the fluxflow resistivity, r F , for small values of J is approximately given by
r F Ž T , B ; J . s r FF 1 y 1r2exp yf Ž T , B ; J .
Ž 2.13 .
where r F is the flux-flow resistivity in the superconducting mixed-state region, r N is the normal resistivity, and Q˜ N is the fraction of the normal region in the resistive transition. The contribution of the fluxcreep resistivity can safely neglected compared with the thermally-assisted flux-flow resistivity at these temperatures w3,4x. It is to be noted that Q˜ N may be slightly different from Q N s 1 y QS discussed in Section 2.1, because of a slight difference between the transitions of the resistivity and the bulk quantity such as the magnetization, the latter of which is closely related to the behavior of the transport entropy, Sf w14a,14bx. If we assume for simplicity that the above difference may only reflect to the difference of the temperature width of the transition, then the fraction of the superconducting mixed-state region in the resistive transition, Q˜ N , is given by putting m s 1 and t s
Ž 2.15 .
y1r2exp yf Ž T , B ;0 .
4,
Ž 2.16a .
where r FF is the thermally-assisted flux-flow resistivity and f ŽT, B; J . is defined by
f ŽT ,B; J . s
J q J0
J0T J0
ž
T TgU Ž B .
Õ Ž Dy1 . my 1
y1
/
.
Ž 2.16b . In Eq. Ž2.16b., D is the dimension of the fluxoids, Õ and z s mŽ D y 1. y 1 are the static and dynamic exponents near Tg Ž B ., respectively, and J0 and J0T are the parameters characterizing the flux pinning. The detailed derivation of Eqs. Ž2.16a. and Ž2.16b. are given in Appendix A, together with the concrete definitions of these parameters, all of which can be estimated by the measurements of the E J vs. J characteristics.
K. Yamafuji et al.r Physica C 328 (1999) 230–240
Thus, we get the expression for the resistivity around the superconducting vs. normal resistive transition, which given by Eq. Ž2.13. with Eqs. Ž2.14., Ž2.15., Ž2.16a., and Ž2.16b.. Accordingly, the present expression of the electric field, ET , induced by the Nernst effect is given by Eq. Ž2.7. with Eqs. Ž2.12a. and Ž2.13..
3. Comparison of the present theory with experimental results Typical examples of the present observed results for the electric field, ET , induced by the Nernst effect are shown in Fig. 1. The specimen used for the measurement of the Nernst effect is a YBCO film with 1 mm = 0.1 mm = 0.2 mm. Typical examples of the observed results for the current density, J, vs. the induced electric field, E J , are shown in Fig. 2a and b, where the used specimen was a YBCO film with 0.8 mm = 0.1 mm = 0.2 mm. Whereas the both
235
kinds of measurements are desirable to be carried out in the same specimen, we used two specimens for the convenience of the measurements, where both specimens were fabricated at the same time under the same condition. The transport entropy, Sf , estimated from these observed data with the aid of Eq. Ž2.6. with Eq. Ž2.5. is shown in Fig. 3a by white circles for B s 8 T, where the observed data below 78 K may not be reliable because the observed data of ET at B s 8 T cannot be very reliable as can be seen from Fig. 1. As can be seen directly from Fig. 3b, Sf above Tc Ž B . s 82.5 K at B s 8 T shows an exponentially linear decrease with temperature, as predicted in the present theory described in Section 2.1. The theoretical curve of Sf given by Eqs. Ž2.12a. and Ž2.12b. is also shown by solid curves in Fig. 3a and b for B s 8 T by choosing the parameter, Tp Ž B ., as Tp Ž B . s 79.5 K at B s 8 T. The agreement of the theory with the observed data seems to be satisfactory above Tp Ž B ., while the reason of a small deviation of
Fig. 1. Temperature dependence of Nernst coefficient, Ne s ET r=T, at B s 14, 8 and 4 T.
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K. Yamafuji et al.r Physica C 328 (1999) 230–240
Ž2.13., w1 y Ž rrr N .x s Ž1 y Q˜ N .w1 y Ž r Frr N .x becomes to be proportional to Ž1 y Q˜ N . at T 4 Tc Ž B ., where the value of Ž r Frr N . becomes negligibly small, and Ž1 y Q˜ N . is expected to show an exponentially linear decrease with temperature as can be seen from Eq. Ž2.14.. An example of the confirmation of the theoretical prediction is shown in Fig. 5 at B s 8 T: one really see the exponential decrease of w1 y Ž rrr N .x at T 4 Tc Ž B . s 82.5 K at B s 8 T. In Fig. 6a and b, the theoretical curve of the resistivity, r , given by Eq. Ž2.13. with Eqs. Ž2.14., Ž2.15., Ž2.16a. and Ž2.16b. plotted by solid curves are compared with the present observed data at B s 8 T. The agreements seems to be satisfactory in the both of the linear and the logarithmic scales. Finally, the theoretical expression for the Nernst coefficient, Nc s ETr=T, given by Eq. Ž2.7. is com-
Fig. 2. E J y J characteristics under the various temperatures at B s 4 T Ža. and B s8 T Žb..
the observed data from Maki’s expression is not clear at present because the observed data of Sf below Tp Ž B . themselves are not very reliable as mentioned above. In the investigation of the behavior of resistivity near the superconducting vs. normal resistive transition, we measured the resistivity for a small current density of J s 1.0 = 10 5 Arm2 , and the results are shown in Fig. 4a and b. As can be seen from Eq.
Fig. 3. Transport entropy, Sf , at B s8 T plotted linearly Ža. and logarithmically Žb.. Sf , is estimated from the observed data with the aid of Eq. Ž2.6. with Eq. Ž2.5.. Solid line represents the theoretical curve of Sf given by Eqs. Ž2.12a. and Ž2.12b.. The parameters used are Tc0 s90 K, Tc Ž8 T. s82.5 K, Tp Ž8 T. s 79.5 K, and k s 36.
K. Yamafuji et al.r Physica C 328 (1999) 230–240
Fig. 4. Temperature dependence of resistance at B s 0, 4 and 8 T plotted linearly Ža. and logarithmically Žb.. The current density for measurements is J s1.0=10 5 Arm2 .
pared with the observed data B s 8 T in Fig. 7. The agreement may be said as satisfactory, if one would
Fig. 5. w1yŽ r r r N .x vs. temperature at B s8 T which decreases exponentially with temperature at T 4Tc Ž B .. The normal resistivity is given by r N s 4.7=10y8 T y1.7=10y6 V m.
237
Fig. 6. Theoretical curve of resistivity, r , given by Eq. Ž2.13. with Eqs. Ž2.14. Ž2.15. Ž2.16a. Ž2.16b. plotted linearly Ža. and logarithmically Žb.. Open circles represent the observed data at B s8 T. The parameters used in the calculation are r FF s8.5=10y7 V m, J0T s 3.0=10 8 Arm2 , J0 s1.0=10 7 Arm2 , ms 3.5, and Ž Dy1. Õ s1.4.
consider the fact that Ne is proportional to the product of Sf ŽT, B . and r ŽT, B; J ..
Fig. 7. Comparison between observed data and theoretical expression given by Eq. Ž2.7. for the Nernst coefficient, Ne s ET r =T.
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K. Yamafuji et al.r Physica C 328 (1999) 230–240
4. Summary The Nernst coefficient, Ne , in high-Tc cuprate superconductors defined by Ne s r Sf rf 0 w7x is discussed experimentally as well as theoretically, where r is the electric resistivity for a very small current density, Sf is the transport entropy per unit length of a single fluxoid, and f 0 is the flux quantum. It was found that the transport entropy, Sf , estimated directly from the resistive measurement and the measurement of Ne in a YBCO film showed an exponentially linear decrease with temperature above the mean-field critical temperature, Tc Ž B . as shown in Fig. 3b. The present observed fact could be explained theoretically by simply assuming that, above Tc Ž B ., the superconducting mixed state regions and normal regions disappear andror appear fluctuatingly due to thermal fluctuation and the disappearance probability of the superconducting mixed state region is constant everywhere, and also that Sf is proportional to the fraction of the superconducting mixed state region, QS . Since the superconducting mixed state regions and the normal regions are locating alternatively along the direction of the driving force of the Lorentz force type, the total resistivity, r , is given by r s r F Q˜S q r N Ž1 y Q˜S ., where r F is the resistivity in the superconducting mixed state, r N is the resistivity in the normal state, and Q˜S is slightly different from QS by reflecting the difference between the resistive transition and the transition of the magnetization. However, the resent resistive measurement in the YBCO film indicated that Q˜S also showed an exponentially linear decrease with temperature as can be seen from Fig. 5, which could also be explained by the assumption mentioned above. An expression for the resistivity in the superconducting mixed state, r F , in the temperature region of Tg Ž B . Q T Q Tc Ž B .U was derived based on the theory of the E vs. J characteristics near Tg Ž B . w3–5x, as described in Appendix A. Then the expression for the total resistivity, r , showed a quantitative agreement with the present observed data not only in the linear plot but also in the log–log plot as shown in Fig. 6a and b. Finally, the present theoretical expression for Ne showed a quantitative agreement with the present observed data, as shown in Fig. 7. It is to be noted
that the present theoretical expression for Ne can also be applied for Ne in metallic superconductors, because the glass–liquid transition also appears w6x. However, the width of the observed temperature range of Ne , Tg Ž B . Q T Q Tc Ž B .U , is very narrow in the metallic superconductors, and hence very careful and sensitive measurements may be necessary for obtaining the observed data as a target of the quantitative comparison with the theories for this type of Ne . Appendix A Under the Lorentz-type driving force with the density given by FL s JB, the motional unit of the fluxoids has been known as the flux bundle ŽFB. surrounded by the correlation lengths between pinned fluxoids w18x. One of the important effects of thermal fluctuation on the flux pinning phenomena is the effective decrease of the critical current density of pinned fluxoids due to the active thermal motion of fluxoids inside the pinning potential, which can be formally expressed by w19x JcFB s Jc0FB 1 y ² u 2th :rd p2 Jc0FB
3r2
,
Ž A.1 .
where is the essential critical current density of FB without taking account of the thermal fluctuation and ² u 2th : is the square average of the thermally fluctuating displacements of the fluxoids inside the effective pinning potential of FB with the half width of d p . Hereafter, let us denote JcFB as Jc and Jc0FB as Jc0 , for simplicity. For high-Tc cuprate superconductors, the irreversibility temperature, Tirr Ž B ., above which the magnetization becomes reversible, has been observed w20x near the so-called glass–liquid transition at Tg Ž B . w2x, and hence Jc given by Eq. ŽA.1. is expected to become zero near Tg Ž B .. Even when the values of Jc0 have the distribution with a very narrow width, the width of the distribution of the value of Jc near Tg Ž B . becomes to be of the comparable order of magnitude as nearly zero Jc ’s. It has been shown by a computer simulation w4x and by theoretical investigation w5x that the thermally depinning transition at Tp Ž B . occurs when the minimum value of Jc becomes zero. Since the correlation length between depinned fluxoids diverges at Tp Ž B . w4x and the thermally depinning transition is of
K. Yamafuji et al.r Physica C 328 (1999) 230–240
the 2nd order w5x, the so-called glass–liquid transition can be regarded as one of the limiting cases of the thermally depinning transition w3x, because the thermally depinning transition can in principle occur in high-Tc cuprate superconductors however the configuration of the pinned fluxoids is. When the effects of thermal fluctuation on the flux pinning can be disregarded as in the metallic superconductors at T < Tc Ž B ., the electric field induced by the flux flow is given by E0FF s r 0 FF Ž J y Jc0 . .
Ž A.2a .
In high-Tc cuprate superconductors, on the other hand, Jc decreases remarkably from Jc0 and becomes to be nearly zero near Tg Ž B .. Then the flux flow can occur when the value of the applied current density, J, exceeds Jc and the electric field induced by flux flow is given by w3,4x E FF s r FF Ž J y Jc . ,
Ž A.2b .
where the value of the thermally-assisted flux-flow resistivity, r FF , is usually two decades smaller than the usual flow resistivity, r 0FF w3,4x. This kind of flux flow can occur even when the value of J is smaller than the essential critical current density, Jc0 , so far as J is larger than Jc , and hence may be called the thermally-assisted flux-flow w3x. This kind of flux pinning phenomena can also occur in metallic superconductors because Tg Ž B . is also observed w6x, though Tg Ž B . locates in the very vicinity of Tc Ž B . w6x. Since the values of Jc have a wide distribution near Tg Ž B . as mentioned above, the electric field due to thermally-assisted flux-flow can be induced so far as the value of J exceeds the minimum value of Jc denoted by Jcm . Then the theoretical expression for the distribution of the values of Jc is necessary for the calculation of the total electric field, E J , induced by the thermally-assisted flux-flow. This is the essential difference of the E vs. J characteristics in high-Tc cuprate superconductors at T ; Tg Ž B . from those in metallic superconductors at T < Tc Ž B .. The flux flow can be regarded as a kind of stochastic ‘dying’ process of the pinned fluxoids, called the depinning process under the external disturbance for the flux pinning in the form of the
239
Lorentz-type driving force with the density given by FL s JB. Near the superconducting vs. normal transition where the coherence length of the superconducting hole pairs is very large, we have seen in the Section 2 that the ‘dying’ process of the local superconducting region becomes so simple as to be described by a probability density function of the Weibull type which describes a fairly simple stochastic ‘dying’ processes given by Eq. Ž2.10a.. Near the glass–liquid transition temperature, Tg Ž B ., where the correlation lengths of pinned fluxoids diverge w2,4x and, hence, the local detailed depinning process does not affect seriously the flux flow of the flux bundles with the correlated volumes, it is not very curious, even if the depinning process is described by the probability density function of the Weibull-type given by Eq. Ž2.11b.. The probability density function of the Weibull type appears when the concerning stochastic process is composed of the m unit stochastic processes independent of each other. When m unit stochastic processes are dependent of each other, however, the ‘equivalent’ numbers of independent unit process should be smaller than m. Thus the number, m, in Eq. Ž2.10b. needs not be an integer for the depinning process. This anticipation was confirmed by a simulation for the depinning process w3,21x, and was shown that the current density, J, vs. the induced electric field, E J , characteristics near the glass–liquid transition could be described well by using the Weibull-type probability density function where t in Eq. Ž2.11b. is given by w3,4x t s jc y jcm ;
jc s JcrJ0 ,
jcm s Jcm rJ0 ,
Ž A.3 .
with various non-integer of m according as the strength of the flux pinning. In the actual specimens, many pinning centers with various pinning strengths are distributed spatially. For the appearance of the electric field, E J , induced by the flux flow, at least the total motion of some flux-bundle raw must occur along the direction of the Lorentz-type driving force. Then the minimum value of the critical current density of flux-bundle raws, Jc , is denoted by Jcm , and hence E J begins to appear when the applied current density, J, exceeds
K. Yamafuji et al.r Physica C 328 (1999) 230–240
240
Jcm . According to the simulation w4x, the temperature dependence of Jcm above but near Tg Ž B . is given by X
Jcm s
½
J0T w 1y Ž T r Tg Ž B . . x
Õ Ž Dy 1 .
y J0T w Ž T r Tg Ž B . . y1 x
T -Tg Ž B . ,
;
Õ Ž Dy 1 .
;
Ž A.4 .
T )Tg Ž B . ,
References
where J0T s J0T Ž B . can be regarded as a constant with respect T only near Tg Ž B .. From Eqs. Ž2.11b. and ŽA.3., the temperature dependence of the probability density function, P Ž jc . is given by P Ž jc . s m Ž jc y jcm .
my1
exp y Ž jc y jcm .
m
.
Ž A.5 . The area of P Ž jc . below j s JrJ0 is the depinned fraction of the flux-bundle raws, where J0 is a normalization constant as a measure of the distribution width of Jc . Then E J at T ) Tg Ž B . is given by E J s J0
Hj
j
r FF Ž j y jc . P Ž jc . d jc ,
Ž A.6a .
cm
where r FF is the thermally-assisted flux-flow resistivity of the flux-bundle raws. By carrying out a partial integration, Eq. ŽA.6a. is reduced to
r s E JrJ s r FF 1 y
1
j
H exp y Ž j y j j 0 c
cm
.
enough for an approximate description of the behavior of E J for small J at T 4 Tg Ž B ..
m
d jc .
Ž A.6b . Whereas the last term cannot be integrated analytically, the mean-value approximation of the integral can be used for small value of j, and the result yields Eq. Ž2.16a. with Eq. Ž2.16b.. For describing the general behavior of E J at T ) Tg Ž B ., the value of J0 and J0T appearing in Eqs. ŽA.2a. – Ž3. should be dependent on T and B w21x. However, E J is nearly linear to J for small values of J at T 4 Tg Ž B ., and hence a simple modification of using TgU Ž B . instead of Tg Ž B . without changing the value of J0 and J0T may be
w1x G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 Ž1994. 1125. w2x D.S. Fisher, M.P.A. Fisher, D.A. Huse, Phys. Rev. B 43 Ž1991. 130. w3x K. Yamafuji, T. Kiss, T. Fujiyoshi, in: A. Narlikar ŽEd.., Studies in High Temperature Superconductors, vol. 36, Nova Science Publishers, New York, 1999, in press. w4x K. Yamafuji, T. Kiss, Physica C 290 Ž1997. 9. w5x T. Matsushita, T. Kiss, Physica C 315 Ž1999. 12. w6x B. Ni, T. Wakuda, K. Funaki, E.S. Otabe, T. Matsushita, Adv. Supercond. 8 Ž1996. 631. w7x M. Zeh, H.-C. Kober, F. Ri, R.P. Huebener, A.V. Ustinov, J. Mannhart, R. Gross, A. Gupta, Phys. Rev. Lett. 64 Ž1990. 3195. w8x M. Galffy, A. Freimuth, U. Murek, Phys. Rev. B 41 Ž1990. 11029. w9x S.J. Hagen, C.J. Lobb, R.L. Greene, M.F. Forrester, J. Talvacchio, Phys. Rev. B 42 Ž1990. 6777. w10x H.-C. Ri, R. Gross, F. Gollnik, A. Beck, R.P. Huebener, Phys. Rev. B 50 Ž1994. 3312. w11x R.P. Huebener, Supercond. Sci. Technol. 8 Ž1995. 189. w12x M.R. Cimberle, C. Ferdeghini, D. Marre, Putti, S. Siri, G. Grasso, R. Flukiger, Phys. Rev. B 52 Ž1995. 9727. ¨ w13x T. Sasaki, K. Watanabe, N. Kobayashi, Sci. Rep. RITU A 42 Ž1996. 351. w14ax K. Maki, Prog. Theor. Phys. 41 Ž1969. 902. w14bxK. Maki, J. Low. Temp. Phys. 1 Ž1969. 45. w15x C. Caroli, K. Maki, Phys. Rev. Lett. 21 Ž1967. 591. w16x R. Ikeda, T. Ohmi, T. Tsuneto, J. Phys. Soc. Jpn. 60 Ž1991. 1051. w17x K. Yamafuji, T. Fujiyoshi, T. Kiss, Physica C 311 Ž1999. 253. w18x A.I. Larkin, Yu.N. Ovchinnikov, J. Low Temp. Phys. 34 Ž1978. 409. w19x K. Yamafuji, T. Fujiyoshi, K. Toko, T. Matsuno, K. Kishio, T. Matsushita, Physica C 212 Ž1993. 424. w20x M. Suenaga, D.O. Welch, R. Budhani, Supercond. Sci. Technol. 5 Ž1992. S1. w21x T. Kiss, T. Nakanura, K. Hasegawa, M. Inoue, H. Okamoto, K. Funaki, M. Takeo, K. Yamafuji, I. Irie, in: D. Dew-Huges ŽEd.., Proc. ICEC17, IOP Publishing, London, 1998, p. 427.
The Nernst effect in high-Tc cuprate superconductors K. Yamafuji
a,)
, T. Fujiyoshi b, T. Kiss c , M. Inoue c , T. Sasaki d , N. Kobayashi
d
a
c
Ariake National College of Technology, 150 Higashihagio-machi, Omuta 836-8585, Japan b Faculty of Engineering, Kumamoto UniÕersity, Kumamoto 860-8555, Japan Graduate School of Information Science and Electrical Engineering, Kyushu UniÕersity, Fukuoka 812-8581, Japan d Institute for Material Research, Tohoku UniÕersity, Sendai 980-8577, Japan Received 5 August 1999; received in revised form 27 August 1999; accepted 13 October 1999
Abstract Detailed measurements of both the Nernst coefficient, Ne , and the current vs. voltage characteristics were carried out in a YBCO film by varying systematically the flux density, B, and the temperature, T. The Nernst coefficient was observed over the temperature region of a fairly wide superconducting vs. normal resistive transition under the applied flux density, B. The transport entropy of a fluxoid, Sf , estimated directly from the present measurements showed a small tail decreasing exponentially with increasing temperature, even above the mean-field critical temperature, Tc Ž B ., whereas the existing theory predicted Sf to become zero above Tc Ž B .. Then the expression for Sf above Tc Ž B . was derived. The expression for the resistivity around the temperatures of the resistive transition was also derived, including the resistivity due to the flux flow. With the aid of these expressions, the observed Nernst coefficient was explained quantitatively by the present theory. q 1999 Elsevier Science B.V. All rights reserved. Keywords: High-Tc superconductors; Nernst effect; Resistive transition; Flux pinning; Transport entropy; Thermal fluctuation
1. Introduction In high-Tc cuprate superconductors, the effects of thermal fluctuation on the flux depinning phenomena in the superconducting mixed state under the applied flux density, B, is quite severe especially at higher temperatures due to mainly the extremely short coherence length along c-axis w1x. For example, the so-called glass–liquid transition w2x appears at Tg Ž B . far below the mean-field critical temperature, Tc Ž B .,
and this kind of transition can be interpreted as a limiting case of the thermally depinning transition w3–5x, while Tg Ž B . locates in the very vicinity of Tc Ž B . in the metallic superconductors w6x. Furthermore, the mean-field critical temperature, Tc Ž B ., itself can no more be regarded as the clear transition line between the superconducting mixed state and the normal state, differing from the case of the metallic low temperature superconductors. The Nernst coefficient, Ne , for type-2 superconductors is defined by Ref. w7x as Ne s r Sf rf 0 ,
Ž 1.1 .
)
Corresponding author. Tel.: q81-944-53-8601; fax: q81944-53-1361. E-mail address: [email protected] ŽK. Yamafuji.
where r is the electric resistivity for a very small value of the transport current density, Sf is the
0921-4534r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 5 5 3 - 5
K. Yamafuji et al.r Physica C 328 (1999) 230–240
transport entropy per unit length of a single fluxoid, and f 0 is the flux quantum. As the temperature is increased, the resistivity for a very small current density, r , recovers suddenly at Tg Ž B . from an extremely small value induced by the flux creep to a relatively small value induced by the thermally-assisted flux-flow w2–5x, and approaches gradually the normal resistivity, r N ŽT ., near Tc Ž B .. On the other hand, the transport entropy, Sf , decreases with the increase of temperature and becomes to be below the experimentally detectable level near the uppermost temperature, Tc Ž B .U , of the resistive transition region between the superconducting mixed state and the normal state w7–13x. Thus Ne of this type given by Eq. Ž1.1. can only be observed in the temperature range of Tg Ž B . Q T Q Tc Ž B .U and, hence, may be measurable quantitatively with a high reliability only in high-Tc cuprate superconductors, for which the observable temperature range of Ne is fairly wide. Then many observed data have been presented for Ne in high-Tc cuprate superconductors w7–13x. Unfortunately, however, one can find no theory which is available for the quantitative comparison with these observed data. Only the theoretical expression for Sf presented so far was derived by Maki w14a,14bx based on the Calori–Maki theory w15x. However, his theory is available only for the BCS superconductors and, hence, Sf becomes zero at Tc Ž B ., besides that the quantitative reliability of his expression for Sf has not been confirmed experimentally for high-Tc superconductors even below Tc Ž B .. As for the resistivity, r , Ikeda et al. w16x have presented the theoretical expressions for single crystals of high-Tc cuprate superconductors by starting from the G–L functional Hamiltonian and by taking account of the effects of thermal fluctuation explicitly. However, the effect of the flux depinning near the glass–liquid transition temperature, Tg Ž B ., is not considered explicitly in their theory, and their expression shows noticeable deviation from the observed data at lower temperature for bulk single crystals with, may be, some pinning centers w16x. Furthermore, many physical quantities including the material constants are contained in their expression. For the quantitative comparison of their theory with the observed data, therefore, many kinds of observations are necessary in the sample, in which the
231
observation of Ne was made, for estimating the values of these physical quantities. The purpose of the present paper is to provide the theory to be able to explain quantitatively the observed data of Ne . Since at least the resistive measurement in the same sample is also necessary for the analysis and the discussion of the observed results of Ne , we shall try to derive the theory that contains only the parameters evaluable by the set of these two kinds of measurements. For this purpose, a theoretical expression for the transport entropy, Sf , above Tc Ž B . is at first derived. Whereas the present expression for Sf approaches asymptotically the Maki expression w14a,14bx below Tc Ž B ., the semi-quantitative reliability of his expression has been supported by existing observation w7– 12x, though the magnetic-field dependence near Tc Ž B . might be somewhat questionable w13x. Based on our previous work w17x, however, the result of the comparison of the observed data presented by Zeh et al. w7x with our preliminary theory showed a quantitative agreement below Tc Ž B ., where the Maki expression of Sf was used and the reasonable values were assumed for the pinning parameters contained in our theory of resistivity, r , below Tc Ž B .. Next, we derived theoretical expression for the resistivity, r , including that above Tc Ž B ., because the expression presented by Ikeda et al. w16x is not adequate for the present purpose. Besides that all the parameters contained in their expression cannot be estimated only by these two kinds of measurements, their expression is hardly applicable for the resistivity near Tg Ž B .. In fact, the behavior of the flux flow near Tg Ž B . becomes different almost entirely from the usual flux flow in metallic superconductors at T < Tc Ž B . due to severe effect of thermal fluctuation, and becomes as that may be called the thermally-assisted flux-flow w3–5x, as will be discussed in Appendix A in detail. The resistivity in our theory approaches asymptotically to that in the thermallyassisted flux-flow region near Tg Ž B ., where the values of flux-pinning parameters contained in our theory can be evaluated experimentally. Whereas the observed data of Ne presented by Zeh et al. w7x seem to be reliable for the quantitative comparison with the present theory, the values of parameters necessary for the comparison with the present theory are unfortunately not specified for
232
K. Yamafuji et al.r Physica C 328 (1999) 230–240
their sample. Thus, we carried out detailed measurements of the current vs. voltage characteristics in a YBCO film for the purpose of estimating the values of parameters included in the present theory, which are relating to the flux pinning, the thermally-assisted flux-flow and the resistive transition. It will be shown that the total curve of Ne against temperature observed in the YBCO film is successfully explained by the present theory. 2. Theoretical expression for Ne Let us consider a film of a high-Tc cuprate superconductor at the temperature, T, in the mixed-state under the applied flux density, B, in the direction of the z-axis, which is chosen to be along the direction of the film thickness. When the transport current with the current density, J, is applied to the direction of the y-axis, the fluxoids per unit length per unit area suffer the driving force of the Lorentz force type, FL , given by FL s J = B s Ž FL ,0,0 . , FL s JB. Ž 2.1 . Then, the fluxoids move to the direction of the x-axis with the average velocity, zJ , and the electric field, E J , is induced in the direction of the y-axis as E J s B = zJ s Ž 0, E J ,0 . ; EJ s Jr Ž T , B ; J . , Ž 2.2 . where r is the formally defined resistivity as E JrJ, which is usually dependent on J. When the temperature gradient, = T, is applied in the direction of the x-axis, on the other hand, the fluxoids per unit length per unit area suffer the driving force, FT , given by FT s FT= Tr=T s Ž FT ,0,0 . ; FT s =TBSf Ž T , B . rf 0 , Ž 2.3 . where f 0 is the flux quantum and Sf ŽT, B . is the transport entropy of a single fluxoid per unit length. Then these fluxoids move to the direction of the x-axis with the average velocity, zT , and hence the electric field, ET , is induced in the direction of the y-axis as ET s B = zT s Ž 0, ET ,0 . ; ET s ET Ž T , B ; =T . . Ž 2.4 . When both the measurements of E J and ET are carried out in the same sample under the above-men-
tioned experimental conditions, the directions of the driving force, FL and FT , and those of the induced electric field, E J and ET are respectively the same as each other, as can be seen from Eqs. Ž2.1. and Ž2.3. and also Eqs. Ž2.2. and Ž2.4.. Even for strongly anisotropic samples as in the high-Tc cuprate superconductors, therefore, the behavior of flux motion would be the same as each other for the same values of FL and FT . As can be seen from Eqs. Ž2.1. and Ž2.3., the value of FL is equal to the value of FT when the transport current density, J, takes the value of J s JT given by JT s=TSf Ž T , B . rf 0
Ž 2.5 .
Then the observed value of the electric field induced by Nernst effect, ET s ET Ž =T ., would be equal to the value of E J s E J Ž J . observed by the resistive measurement at J s JT ; i.e., ET Ž =T . s E J Ž JT . .
Ž 2.6 .
Eq. Ž2.6. with Eq. Ž2.5. provides a method of estimation of the transport entropy, Sf , directly form the both measurements of the Nernst effect and the E J vs. J characteristics, even at lower temperatures where the E J vs. J characteristics are strongly nonlinear. With the aid of Eqs. Ž2.2., Ž2.5. and Ž2.6., we get ET Ž T , B ;=T . ' =TNe Ž T , B ;=T . s =Tr Ž T , B ; JT . Sf Ž T , B . rf 0 Ž 2.7 . Eq. Ž2.7. can be the theoretical expression for the electric field induced by the Nernst effect, ET , when the theoretical expressions for r ŽT, B; JT . and Sf ŽT, B . are given explicitly. 2.1. Expression for the transport entropy Only the theoretical expression for the transport entropy of a single fluxoid per unit length, Sf , proposed so far is the one provided by Maki w14a,14bx based on the Calori–Maki theory w15x, which is given by SfM Ž T , B . s
Bf 0 L D Ž T .
bm 0
T
Bc 2 Ž T . y B ,
Ž 2.8a .
K. Yamafuji et al.r Physica C 328 (1999) 230–240
where m 0 is the magnetic permeability of the vacuum, L D ŽT . is a monotonically increasing function of T from L D Ž0. s 0 up to L D ŽTc Ž B .. s 1, Bc2 is the mean-field upper critical flux density, b s 1 q 2 1.16Ž2 k GL y 1. is a numerical constant with k GL denoting the Ginzburg–Landau parameter, and the superscript M represents that the right-hand-side of Eq. Ž2.8a. is the expression presented by Maki w14a,14bx. If we adopt the usual temperature dependence of Bc2 near Tc Ž B . given by Bc 2 Ž T .
U s Bc2
Ž 0. 1 y
ž
T Tc Ž B .
2
/
,
Ž 2.9 .
then, at T Q Tc Ž B ., Eq. Ž2.8a. is reduced to SfM
ŽT , B. s
Bf 0
1
2
Tc Ž B . y T 2
bm 0 Tc Ž 0 . Tc Ž 0 . 2 y Tc Ž B . 2
.
Ž 2.8b .
U Ž0. in Eq. Ž2.9. is not Bc2 Ž0. It is to be noted that Bc2 but an adjustable parameter for describing the behavior of Bc2 ŽT . at T Q Tc Ž B ., and we used an approximation of L D ŽT . ; TrTc Ž0. at T Q Tc Ž B .. As can be seen directly from Eq. Ž2.8b., SfM ŽT, B . becomes zero at T s Tc Ž B .. Even above Tc Ž B . under the applied flux density, B, however, the fluxoids would appear locally due to thermal fluctuation, and hence a small tail of Sf should appear even above Tc Ž B ., at least within the superconducting vs. normal resistive transition region, as has been indicated experimentally w7–12x. Then let us define the fraction of the local superconducting mixed-state regions by QS . Above Tc Ž B ., QS decreases due to thermal agitation. If we consider the general process that the local superconducting mixed-state regions ‘die’ by an external disturbance with the occurrence probability, f, then the simplest equation for describing the behavior of the ‘dead’ fraction of the superconducting mixed-state regions, Q N s 1 y QS , is given by
233
the occurrence of the external disturbance, f, is given by f s t m , and hence Eq. Ž2.10a. is reduced to dQ N rdt m s QS .
Ž 2.10b .
The solution of Eq. Ž2.10b. is given by QS s 1 y Q N s exp Ž yt m . .
Ž 2.11a .
Then the probability density function of the ‘dead’ fraction, P N , which is defined by differential of Q N with respect to t, is given by P N s mt my 1exp Ž yt m . .
Ž 2.11b .
This type of probability density function has been known well as the Weibull distribution function, which has been known to appear widely for fairly simple ‘dying’ stochastic phenomena. One of the typical examples is the thermal depinning process of the pinned fluxoids w3–5x, which will be discussed in Appendix A for the derivation of the thermally-assisted flux-flow resistivity in Section 2.2. Since the ‘dying’ process of the superconducting mixed-state phase due to thermal fluctuation is a single stochastic process, we can put m s 1 and t s TrD M Tc Ž B . in Eq. Ž2.11a., where D M Tc Ž B . is a quantity of the order of the temperature width of the superconducting vs. normal transition. Above Tc Ž B ., therefore, QS , is expected to be given by an exponentially decreasing linear function of TrD M Tc Ž B .. Since Sf above Tc Ž B . is expected to be proportional to QS , the simplest approximation for Sf at the temperatures around the superconducting vs. normal transition may be given by
Sf Ž T , B .
° ¢
SfM Ž T , B . ;
s~
SfM Ž Tp Ž B . , B . exp yg
T FTp Ž B . ,
ž
T y1 Tp Ž B .
/
;
T GTp Ž B . ,
Ž 2.10a .
Ž 2.12a .
When the external disturbance is a kind of the stochastic process composed of m unit stochastic processes independent of each other, each of which occurs with the probability, t, then the probability of
where Tp Ž B . is a slightly lower temperature than Tc Ž B ., and we put as D M Tc Ž B . s Tp Ž B .rg . The value of g can be determined from the request that the differential of Sf with respect to T is also
dQ N rd f s QS .
K. Yamafuji et al.r Physica C 328 (1999) 230–240
234
continuous at T s Tp Ž B .. If we tentatively adopt Eq. Ž2.8b. as the expression for Sf , then g is given by
gs
2Tp Ž B . 2
2
Tc Ž B . y Tp Ž B .
2
.
Ž 2.12b .
The present approximation for Sf is somewhat rough at the temperatures of T ; Tc Ž B ., because the present expression for QS given by Eq. Ž2.11a. may be hold well only at T 4 Tc Ž B ., while the behavior of Sf at T ; Tc Ž B . is not discussed here theoretically in detail. However, the variation of Sf at T ; Tc Ž B . is expected to be fairly smooth, and hence the insufficiency of the present approximation at T ; Tc Ž B . may be removed partly by choosing the value of the adjustable parameter, Tp Ž B ., adequately. 2.2. Expression for resistiÕity In the temperature region of the superconducting vs. normal resistive transition under the applied flux density, B, the electric field, E J , under the applied transport current with the density, J, is induced both in the normal regions and in the superconducting mixed-state regions. Since each of these two kinds of regions is locating alternatively along the direction of the transport current, the total resistivity, r , defined by r s E JrJ is given by
r s r F Ž 1 y Q˜ N . q r N Q˜ N ,
TrDR Tc Ž B ., where DR Tc Ž B . is a quantity of the order of the temperature width of the resistive transition. Then the simplest approximation for Q˜ N s 1 y Q˜S is given by Q˜ N s
½
1;
T GTcU Ž B . ,
1yexp yg w Ž T yTcU Ž B . . r TcU Ž B . x 4 ;
T FTcU Ž B . ,
Ž 2.14 . where TcU Ž B . is a temperature in the vicinity of the uppermost temperature of the resistive transition region, and we put DR Tc Ž B . s TcU Ž B .rg by using g given by Eq. Ž2.12b.. The normal resistivity at T ; Tc Ž B . can usually be approximated by
r N s aN T q b N ,
where a N and b N are almost independent of B. Since the Nernst effect is usually observed by applying a very small =T, the corresponding value of JT given by Eq. Ž2.5. is also quite small. At the temperatures far above the glass–liquid transition temperature, Tg Ž B ., where the electric field induced by the Nernst effect, ET , can be observed, the fluxflow resistivity, r F , for small values of J is approximately given by
r F Ž T , B ; J . s r FF 1 y 1r2exp yf Ž T , B ; J .
Ž 2.13 .
where r F is the flux-flow resistivity in the superconducting mixed-state region, r N is the normal resistivity, and Q˜ N is the fraction of the normal region in the resistive transition. The contribution of the fluxcreep resistivity can safely neglected compared with the thermally-assisted flux-flow resistivity at these temperatures w3,4x. It is to be noted that Q˜ N may be slightly different from Q N s 1 y QS discussed in Section 2.1, because of a slight difference between the transitions of the resistivity and the bulk quantity such as the magnetization, the latter of which is closely related to the behavior of the transport entropy, Sf w14a,14bx. If we assume for simplicity that the above difference may only reflect to the difference of the temperature width of the transition, then the fraction of the superconducting mixed-state region in the resistive transition, Q˜ N , is given by putting m s 1 and t s
Ž 2.15 .
y1r2exp yf Ž T , B ;0 .
4,
Ž 2.16a .
where r FF is the thermally-assisted flux-flow resistivity and f ŽT, B; J . is defined by
f ŽT ,B; J . s
J q J0
J0T J0
ž
T TgU Ž B .
Õ Ž Dy1 . my 1
y1
/
.
Ž 2.16b . In Eq. Ž2.16b., D is the dimension of the fluxoids, Õ and z s mŽ D y 1. y 1 are the static and dynamic exponents near Tg Ž B ., respectively, and J0 and J0T are the parameters characterizing the flux pinning. The detailed derivation of Eqs. Ž2.16a. and Ž2.16b. are given in Appendix A, together with the concrete definitions of these parameters, all of which can be estimated by the measurements of the E J vs. J characteristics.
K. Yamafuji et al.r Physica C 328 (1999) 230–240
Thus, we get the expression for the resistivity around the superconducting vs. normal resistive transition, which given by Eq. Ž2.13. with Eqs. Ž2.14., Ž2.15., Ž2.16a., and Ž2.16b.. Accordingly, the present expression of the electric field, ET , induced by the Nernst effect is given by Eq. Ž2.7. with Eqs. Ž2.12a. and Ž2.13..
3. Comparison of the present theory with experimental results Typical examples of the present observed results for the electric field, ET , induced by the Nernst effect are shown in Fig. 1. The specimen used for the measurement of the Nernst effect is a YBCO film with 1 mm = 0.1 mm = 0.2 mm. Typical examples of the observed results for the current density, J, vs. the induced electric field, E J , are shown in Fig. 2a and b, where the used specimen was a YBCO film with 0.8 mm = 0.1 mm = 0.2 mm. Whereas the both
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kinds of measurements are desirable to be carried out in the same specimen, we used two specimens for the convenience of the measurements, where both specimens were fabricated at the same time under the same condition. The transport entropy, Sf , estimated from these observed data with the aid of Eq. Ž2.6. with Eq. Ž2.5. is shown in Fig. 3a by white circles for B s 8 T, where the observed data below 78 K may not be reliable because the observed data of ET at B s 8 T cannot be very reliable as can be seen from Fig. 1. As can be seen directly from Fig. 3b, Sf above Tc Ž B . s 82.5 K at B s 8 T shows an exponentially linear decrease with temperature, as predicted in the present theory described in Section 2.1. The theoretical curve of Sf given by Eqs. Ž2.12a. and Ž2.12b. is also shown by solid curves in Fig. 3a and b for B s 8 T by choosing the parameter, Tp Ž B ., as Tp Ž B . s 79.5 K at B s 8 T. The agreement of the theory with the observed data seems to be satisfactory above Tp Ž B ., while the reason of a small deviation of
Fig. 1. Temperature dependence of Nernst coefficient, Ne s ET r=T, at B s 14, 8 and 4 T.
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Ž2.13., w1 y Ž rrr N .x s Ž1 y Q˜ N .w1 y Ž r Frr N .x becomes to be proportional to Ž1 y Q˜ N . at T 4 Tc Ž B ., where the value of Ž r Frr N . becomes negligibly small, and Ž1 y Q˜ N . is expected to show an exponentially linear decrease with temperature as can be seen from Eq. Ž2.14.. An example of the confirmation of the theoretical prediction is shown in Fig. 5 at B s 8 T: one really see the exponential decrease of w1 y Ž rrr N .x at T 4 Tc Ž B . s 82.5 K at B s 8 T. In Fig. 6a and b, the theoretical curve of the resistivity, r , given by Eq. Ž2.13. with Eqs. Ž2.14., Ž2.15., Ž2.16a. and Ž2.16b. plotted by solid curves are compared with the present observed data at B s 8 T. The agreements seems to be satisfactory in the both of the linear and the logarithmic scales. Finally, the theoretical expression for the Nernst coefficient, Nc s ETr=T, given by Eq. Ž2.7. is com-
Fig. 2. E J y J characteristics under the various temperatures at B s 4 T Ža. and B s8 T Žb..
the observed data from Maki’s expression is not clear at present because the observed data of Sf below Tp Ž B . themselves are not very reliable as mentioned above. In the investigation of the behavior of resistivity near the superconducting vs. normal resistive transition, we measured the resistivity for a small current density of J s 1.0 = 10 5 Arm2 , and the results are shown in Fig. 4a and b. As can be seen from Eq.
Fig. 3. Transport entropy, Sf , at B s8 T plotted linearly Ža. and logarithmically Žb.. Sf , is estimated from the observed data with the aid of Eq. Ž2.6. with Eq. Ž2.5.. Solid line represents the theoretical curve of Sf given by Eqs. Ž2.12a. and Ž2.12b.. The parameters used are Tc0 s90 K, Tc Ž8 T. s82.5 K, Tp Ž8 T. s 79.5 K, and k s 36.
K. Yamafuji et al.r Physica C 328 (1999) 230–240
Fig. 4. Temperature dependence of resistance at B s 0, 4 and 8 T plotted linearly Ža. and logarithmically Žb.. The current density for measurements is J s1.0=10 5 Arm2 .
pared with the observed data B s 8 T in Fig. 7. The agreement may be said as satisfactory, if one would
Fig. 5. w1yŽ r r r N .x vs. temperature at B s8 T which decreases exponentially with temperature at T 4Tc Ž B .. The normal resistivity is given by r N s 4.7=10y8 T y1.7=10y6 V m.
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Fig. 6. Theoretical curve of resistivity, r , given by Eq. Ž2.13. with Eqs. Ž2.14. Ž2.15. Ž2.16a. Ž2.16b. plotted linearly Ža. and logarithmically Žb.. Open circles represent the observed data at B s8 T. The parameters used in the calculation are r FF s8.5=10y7 V m, J0T s 3.0=10 8 Arm2 , J0 s1.0=10 7 Arm2 , ms 3.5, and Ž Dy1. Õ s1.4.
consider the fact that Ne is proportional to the product of Sf ŽT, B . and r ŽT, B; J ..
Fig. 7. Comparison between observed data and theoretical expression given by Eq. Ž2.7. for the Nernst coefficient, Ne s ET r =T.
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4. Summary The Nernst coefficient, Ne , in high-Tc cuprate superconductors defined by Ne s r Sf rf 0 w7x is discussed experimentally as well as theoretically, where r is the electric resistivity for a very small current density, Sf is the transport entropy per unit length of a single fluxoid, and f 0 is the flux quantum. It was found that the transport entropy, Sf , estimated directly from the resistive measurement and the measurement of Ne in a YBCO film showed an exponentially linear decrease with temperature above the mean-field critical temperature, Tc Ž B . as shown in Fig. 3b. The present observed fact could be explained theoretically by simply assuming that, above Tc Ž B ., the superconducting mixed state regions and normal regions disappear andror appear fluctuatingly due to thermal fluctuation and the disappearance probability of the superconducting mixed state region is constant everywhere, and also that Sf is proportional to the fraction of the superconducting mixed state region, QS . Since the superconducting mixed state regions and the normal regions are locating alternatively along the direction of the driving force of the Lorentz force type, the total resistivity, r , is given by r s r F Q˜S q r N Ž1 y Q˜S ., where r F is the resistivity in the superconducting mixed state, r N is the resistivity in the normal state, and Q˜S is slightly different from QS by reflecting the difference between the resistive transition and the transition of the magnetization. However, the resent resistive measurement in the YBCO film indicated that Q˜S also showed an exponentially linear decrease with temperature as can be seen from Fig. 5, which could also be explained by the assumption mentioned above. An expression for the resistivity in the superconducting mixed state, r F , in the temperature region of Tg Ž B . Q T Q Tc Ž B .U was derived based on the theory of the E vs. J characteristics near Tg Ž B . w3–5x, as described in Appendix A. Then the expression for the total resistivity, r , showed a quantitative agreement with the present observed data not only in the linear plot but also in the log–log plot as shown in Fig. 6a and b. Finally, the present theoretical expression for Ne showed a quantitative agreement with the present observed data, as shown in Fig. 7. It is to be noted
that the present theoretical expression for Ne can also be applied for Ne in metallic superconductors, because the glass–liquid transition also appears w6x. However, the width of the observed temperature range of Ne , Tg Ž B . Q T Q Tc Ž B .U , is very narrow in the metallic superconductors, and hence very careful and sensitive measurements may be necessary for obtaining the observed data as a target of the quantitative comparison with the theories for this type of Ne . Appendix A Under the Lorentz-type driving force with the density given by FL s JB, the motional unit of the fluxoids has been known as the flux bundle ŽFB. surrounded by the correlation lengths between pinned fluxoids w18x. One of the important effects of thermal fluctuation on the flux pinning phenomena is the effective decrease of the critical current density of pinned fluxoids due to the active thermal motion of fluxoids inside the pinning potential, which can be formally expressed by w19x JcFB s Jc0FB 1 y ² u 2th :rd p2 Jc0FB
3r2
,
Ž A.1 .
where is the essential critical current density of FB without taking account of the thermal fluctuation and ² u 2th : is the square average of the thermally fluctuating displacements of the fluxoids inside the effective pinning potential of FB with the half width of d p . Hereafter, let us denote JcFB as Jc and Jc0FB as Jc0 , for simplicity. For high-Tc cuprate superconductors, the irreversibility temperature, Tirr Ž B ., above which the magnetization becomes reversible, has been observed w20x near the so-called glass–liquid transition at Tg Ž B . w2x, and hence Jc given by Eq. ŽA.1. is expected to become zero near Tg Ž B .. Even when the values of Jc0 have the distribution with a very narrow width, the width of the distribution of the value of Jc near Tg Ž B . becomes to be of the comparable order of magnitude as nearly zero Jc ’s. It has been shown by a computer simulation w4x and by theoretical investigation w5x that the thermally depinning transition at Tp Ž B . occurs when the minimum value of Jc becomes zero. Since the correlation length between depinned fluxoids diverges at Tp Ž B . w4x and the thermally depinning transition is of
K. Yamafuji et al.r Physica C 328 (1999) 230–240
the 2nd order w5x, the so-called glass–liquid transition can be regarded as one of the limiting cases of the thermally depinning transition w3x, because the thermally depinning transition can in principle occur in high-Tc cuprate superconductors however the configuration of the pinned fluxoids is. When the effects of thermal fluctuation on the flux pinning can be disregarded as in the metallic superconductors at T < Tc Ž B ., the electric field induced by the flux flow is given by E0FF s r 0 FF Ž J y Jc0 . .
Ž A.2a .
In high-Tc cuprate superconductors, on the other hand, Jc decreases remarkably from Jc0 and becomes to be nearly zero near Tg Ž B .. Then the flux flow can occur when the value of the applied current density, J, exceeds Jc and the electric field induced by flux flow is given by w3,4x E FF s r FF Ž J y Jc . ,
Ž A.2b .
where the value of the thermally-assisted flux-flow resistivity, r FF , is usually two decades smaller than the usual flow resistivity, r 0FF w3,4x. This kind of flux flow can occur even when the value of J is smaller than the essential critical current density, Jc0 , so far as J is larger than Jc , and hence may be called the thermally-assisted flux-flow w3x. This kind of flux pinning phenomena can also occur in metallic superconductors because Tg Ž B . is also observed w6x, though Tg Ž B . locates in the very vicinity of Tc Ž B . w6x. Since the values of Jc have a wide distribution near Tg Ž B . as mentioned above, the electric field due to thermally-assisted flux-flow can be induced so far as the value of J exceeds the minimum value of Jc denoted by Jcm . Then the theoretical expression for the distribution of the values of Jc is necessary for the calculation of the total electric field, E J , induced by the thermally-assisted flux-flow. This is the essential difference of the E vs. J characteristics in high-Tc cuprate superconductors at T ; Tg Ž B . from those in metallic superconductors at T < Tc Ž B .. The flux flow can be regarded as a kind of stochastic ‘dying’ process of the pinned fluxoids, called the depinning process under the external disturbance for the flux pinning in the form of the
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Lorentz-type driving force with the density given by FL s JB. Near the superconducting vs. normal transition where the coherence length of the superconducting hole pairs is very large, we have seen in the Section 2 that the ‘dying’ process of the local superconducting region becomes so simple as to be described by a probability density function of the Weibull type which describes a fairly simple stochastic ‘dying’ processes given by Eq. Ž2.10a.. Near the glass–liquid transition temperature, Tg Ž B ., where the correlation lengths of pinned fluxoids diverge w2,4x and, hence, the local detailed depinning process does not affect seriously the flux flow of the flux bundles with the correlated volumes, it is not very curious, even if the depinning process is described by the probability density function of the Weibull-type given by Eq. Ž2.11b.. The probability density function of the Weibull type appears when the concerning stochastic process is composed of the m unit stochastic processes independent of each other. When m unit stochastic processes are dependent of each other, however, the ‘equivalent’ numbers of independent unit process should be smaller than m. Thus the number, m, in Eq. Ž2.10b. needs not be an integer for the depinning process. This anticipation was confirmed by a simulation for the depinning process w3,21x, and was shown that the current density, J, vs. the induced electric field, E J , characteristics near the glass–liquid transition could be described well by using the Weibull-type probability density function where t in Eq. Ž2.11b. is given by w3,4x t s jc y jcm ;
jc s JcrJ0 ,
jcm s Jcm rJ0 ,
Ž A.3 .
with various non-integer of m according as the strength of the flux pinning. In the actual specimens, many pinning centers with various pinning strengths are distributed spatially. For the appearance of the electric field, E J , induced by the flux flow, at least the total motion of some flux-bundle raw must occur along the direction of the Lorentz-type driving force. Then the minimum value of the critical current density of flux-bundle raws, Jc , is denoted by Jcm , and hence E J begins to appear when the applied current density, J, exceeds
K. Yamafuji et al.r Physica C 328 (1999) 230–240
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Jcm . According to the simulation w4x, the temperature dependence of Jcm above but near Tg Ž B . is given by X
Jcm s
½
J0T w 1y Ž T r Tg Ž B . . x
Õ Ž Dy 1 .
y J0T w Ž T r Tg Ž B . . y1 x
T -Tg Ž B . ,
;
Õ Ž Dy 1 .
;
Ž A.4 .
T )Tg Ž B . ,
References
where J0T s J0T Ž B . can be regarded as a constant with respect T only near Tg Ž B .. From Eqs. Ž2.11b. and ŽA.3., the temperature dependence of the probability density function, P Ž jc . is given by P Ž jc . s m Ž jc y jcm .
my1
exp y Ž jc y jcm .
m
.
Ž A.5 . The area of P Ž jc . below j s JrJ0 is the depinned fraction of the flux-bundle raws, where J0 is a normalization constant as a measure of the distribution width of Jc . Then E J at T ) Tg Ž B . is given by E J s J0
Hj
j
r FF Ž j y jc . P Ž jc . d jc ,
Ž A.6a .
cm
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m
d jc .
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