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Intrinsic pinning in cuprate superconductors
-
w-
~
Pereamon
INTRINSIC
PINNING M.
Institute
for Materials
Research,
Applied Superconductivity Vol. 2, No. 314. pp. 305-313, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0964-1807/94’7.00 + 0.00
I
IN CUPRATE TACHIKI
Tohoku
and S.
University,
SUPERCONDUCTORS
TAKAHASHI 2-l-l
Katahira.
Aoba-ku,
Sendai 980, Japan
Abstract-We review our theoretical work on the intrinsic pinning in cuprate superconductors. The superconductors are constructed by stacking of strongly and weakly superconducting layers. The vortices are strongly pinned in the weakly superconducting layers. Since the transport current flows mainly in the strongly superconducting layers, the driving force acting on the vortices is considerably reduced from that in a uniform current system. These effects combine to yield strong vortex pinning and to enhance the critical current. Dependence of the critical current on the direction of applied magnetic field is discussed in connection with the vortex structure in the cuprate superconductors.
1.
INTRODUCTION
It has been shown experimentally that c-axis oriented films of yttrium and bismuth cuprate oxides carry large critical current densities at low temperatures and the current densities stand up to high magnetic fields, especially when the magnetic field is applied parallel to the films [l-14]. The critical current strongly depends on the direction of the magnetic field and shows a sharp peak at the field direction parallel to the layers [l-16]. The above experimental results suggest that the vortices parallel to the layers are strongly prevented from moving perpendicular to the layers. The cuprate superconductors are constructed by stacking of strongly superconducting CuO, layers and weakly superconducting spacing layers. We consider that the layer structure plays an important role for the vortex pinning. In the magnetic field applied parallel to the layers, the cores of vortices thread through the weakly superconducting layers, since the vortex energy is lowest there. The strongly superconducting layers work as potential barriers for the vortices moving perpendicular to the layers [17-241. Along with the weakly superconducting layers working as potential barriers, the inhomogeneous current in the layer structure works to enhance the vortex pinning [25]. In the layered superconductors the transport current flows mainly in the strongly superconducting layers, and the current density in the weakly superconducting layers is much smaller than that in the strongly superconducting layers. In the inhomogeneous current, the vortices are not driven by the average of the transport current density over the region where the vortex current spreads, but by the transport current density flowing just at the vortex center. As a result, the driving force acting on the vortex becomes very weak when the vortex center is pinned in the weakly superconducting layers. These two effects cooperatively work to strongly pin the vortices in the weakly superconducting layers. We call the pinning which originates from the above two effects intrinsicpinning. When the magnetic field is tilted from the layers, the vortices may penetrate stepwise into the sample [17,20]. Each vortex consists of vortex segments parallel and perpendicular to the layers as shown in Fig. 3. The parallel segments are locked strongly in the weakly superconducting layers by the intrinsic pinning. The perpendicular segments (so-called pancake vortices [26]) are pinned by some extrinsic pinning centers such as twin boundaries [27] and columnar defects created by heavy ion irradiation [28-301. If the pinning force due to the extrinsic pinning centers is weaker than the intrinsic pinning force, the critical current density in the tilted magnetic field is determined by the depinning current density of the pancake vortices. In this paper, we review the intrinsic pinning effect in the cuprate superconductors.
2.
INTRINSIC
PINNING
FORCE
Let us first examine the nature of the intrinsic pinning force acting on a single vortex in a cuprate superconductor. In the superconductor, the metallic CuO, layers are strongly superconductive, 305
306
M. TACHIKIand S. TAKAHASHI
-
%(r;
20)
----- lJJo(z)
-2
c-axis
Fig. 1. The spatial variation of the superconducting order parameter along the c-axis. The solid curve indicates the order parameter in the presence of the vortices at positions (a) and (b). The dashed curve indicates the order parameter without the vortices.
while the other layers (spacing layers) are weakly superconductive. Therefore the superconducting order parameter is modulated in the period of the lattice constant along the c axis. Let us use a simple model that the amplitude of the superconducting order parameter Y = 1+4,, exp(icp) takes a spatial variation of the form [17]
with the lattice constant a, along the c-axis. When a single vortex parallel to the x direction in the layers runs through the position (0, zO) in the yz plane, the order parameter will be modified into the form $,(r; zO) = $,(z)tanh[(;y
+ tT>‘]“’
where 5, and tub are the coherence lengths in the direction parallel and perpendicular to the c-axis, respectively. Figure 1 shows the spatial variation of the order parameter along the c-axis when the vortex is located at the positions (a) and (b). If we write the condensation energy density of the uniform superconductor with $i as H~/Src, the condensation energy densities of the superconductors with $,,(z) and *Jr; z,,) may be given by (H,“/~J~)[$~(z)/$,]~ and (H@r)[$,(r; z,,)/$~)]~, respectively. Then, the increase in the condensation energy density by including the vortex is given by (Hz/Sn){($,(r; z&91)2 - [ll/0(z)/e1]2). Integration of the energy density in the yz plane gives the intrinsic pinning potential U&z,). Taking a derivative of the intrinsic pinning potential, the intrinsic pinning force is calculated as &(z,) = -(d/dz,)U,(z,). If we write the maximum value of &(z,,) with respect to z0 as fpM, the maximum pinning force is given by [17]
H,227ca, -Lb q.
fpM = -
871
t,
The parameter q is a function of the ratio &/a, and the modulation parameter 6 = $2/+1. The value of rl is shown as a function of &/ac for four values of 6 in Fig. 2. As seen in Fig. 2, the q vs &/a, curve has a peak at &/uc 2 0.3. The peak height increases almost linearly as 6 increases. From the facts mentioned above, we see that the intrinsic pinning effect is most effective for the layered superconductors with a coherence length comparable to the layer spacing and with strongly modulated order parameter. When the coherence length is longer than the layer spacing, q becomes very small and thus the pinning force becomes very weak. We introduce a crossover temperature T* at which the coherence length r, is nearly equal to the layer spacing between the CuCO, layers. Above T*,the vortex core spreads over several layers, and thus the intrinsic pinning force is very much reduced. Below T*,on the other hand, the vortex
Intrinsic
pinning
in cuprate
307
superconductors
0.3
e 0.2
0.1
0
Fig. 2. q as a function
of &/a, The pinning force and the critical current is the modulation parameter.
density
are proportional
to 1. 6
core size becomes smaller than the layer spacing and thus the vortices are locked between the CuO, layers. The drastic change of the critical current density was observed at T* N 60 K in YBa,Cu,O, films [3,11]. In Bi,Sr,CaCu,O,, the BiO and SrO layers are almost semiconductive and the superconducting interaction between the CuO, layers is very weak [23,26]. Therefore the vortex cores are confined in the semiconductive layers up to near the superconducting transition temperature, that is, the crossover temperature is lifted up to near the transition temperature [23,-311.
3.
DRIVING
FORCE
In addition to the mechanism mentioned above, the layer structure of the cuprates has another mechanism to enhance the vortex pinning. In cuprate superconductors, the distribution of the supercurrent is not uniform due to their layer structures, as mentioned in the introduction. In this case, arises a question whether the driving force acting on the vortex in nonuniform current is the same as that in uniform current. To answer this question, we consider a system with a single vortex parallel to the x-axis and an external current in the y-direction. We write the vortex current and the external current j, and jex,, respectively. The currents j, and j,,, induce the magnetic fields b, and b,,,, respectively. Substituting the total current j, + j,,, and the total magnetic field b, + b,,, into the Ginzburg-Landau(GL) free energy, we obtain the electromagnetic interaction energy between the vortex and the external currents as [25,32] F, = 2 sg
[(“Tdr
jv(r).jext(r) + b,(r).b..,(r)]
with A’@;z,,) = mc2/8ne21C/,2(r; zO). Using the Maxwell equation j”(r) = (c/4rc)V x b,(r) and j_,(r) = (c/47r)V x b,Jr) and integrating the first term in the bracket of equation (4), we rewrite equation (4) as Fd =
g
b,,,(r). {V x [A’(r; zO)V x b,(r)] + b,(r)}.
s
The GL equation for the vortex magnetic field in an inhomogeneous c331
system of A(r; z,J is given by
V x [A’@; z,)V x b,(r)] + b,(r) = @,,h(r - r,,)
(6)
308
M. TACHIKI and S. TAKAHASHI
where a,, = hc/2e is the unit flux, 6(r - re) the delta function, and r,, = (0, zO) the position of the vortex center in the yz plane. Substituting equation (6) into equation (5), we have
As seen in equation (7), we notice that the interaction energy is proportional to the magnetic field induced by the external current at the vortex center. Taking a derivative of F, with respect to r,,, we obtain the driving force acting on the vortex per unit length
where the current density j&J
is defined by
itl(ro)= i
FL, x Lt(rO)l.
(9)
The current density j,Jr,) is the transport current density flowing at the vortex center, that is, the current density at the vortex center in the absence of the vortex [25]. Equation (8) indicates that the driving force is determined by the transport current density just at the vortex center, although the vortex current spreads over the penetration depth, which is several thousands A in cuprate superconductors. According to this fact, when the vortices are pinned in the weakly superconducting layers, the driving force acting on them is very weak. Since the critical current is determined by the force balance between the pinning force and the driving force, we expect a significant enhancement of the critical current due to the nonuniform distribution of the transport current according to the mechanism discussed in this section. 4. CRITICAL
CURRENT
Let us consider a cuprate superconductor in a magnetic field parallel to the CuO, layers. The advantage to use the weakly superconducting layers as the pinning centers is their high-density, so that the intrinsic pinning is effective up to high magnetic fields. In a high magnetic field of 10 T, for example, the distance between the vortices is much larger than the layer spacing, and thus all the vortices are pinned in the weakly superconducting layers. In this case, the direct summation of the pinning forces is allowed to obtain the pinning force density per unit volume [34], that is, the pining force density is simply given by the intrinsic pinning force fPMmultiplied by the number of the vortices per unit area (direct summation). In the configuration where the magnetic field is applied parallel to the x-axis and the transport current flows in the y-direction, the critical current density is calculated by balancing the driving force density with the maximum pinning force density as 134,353
f.LB=$f,, 0
(10)
where B is the flux density and j, is the critical current density at the vortex center. The measuring critical current density J, is given by taking the spatial average of the current density. If we introduce a ratio y = J,/j,, J, is expressed as
J, = Y f
0
fpw
(11)
The expression of the critical current density differs by the factor y from the conventional one where the uniform transport current distribution is assumed. For the weak pinning case with a small modulation parameter (6 = $i/~+~ << l), using the relation j,, cc I,&), we obtain the explicit expression of the enhancement factor as y = (1 + 6’/2). This fact indicates that the nonuniform
Intrinsic
pinning
in cuprate
superconductors
309
current distribution enhances the critical current. In equation (1) we assume a sinusoidal variation of the superconducting order parameter. However, when the order parameter steeply changes at the boundary between the strongly superconducting layer and the weakly superconducting layer, the enhancement factor y becomes large [25]. Inserting equation (3) into equation (11) and using the relation H, = @0/(2&r5,&,), the intrinsic critical current density is obtained as [17, 361
(12) where j,, is the depairing current density of the CuO, layers defined by cH,/(3&4,,) which yields j,, E lo9 A/cm2 for the parameter values H, N 1 T and A,, N 1500 A for YBa,Cu,O,. The factor (1 - B/H,,) expresses the decrease of the the condensation energy due to the vortex normal cores. The critical current density J, estimated by using equation (12) is comparable to the depairing current density j,, at low temperatures. Since the magnetic field dependence of J, arises only from the factor (1 - B/H,,), the critical current density is insensitive to a usually applicable magnetic field at low temperatures where H,, is extremely high and thus the factor is nearly unity. This behavior of the critical current density is consistent with the experimental results of YBa,Cu,O, [l, 3, lo] and Bi,Sr,CaCu,O, [2,6]. When the magnetic field is applied perpendicular to the layers (the vortices are parallel to the c-axis) and the external current is parallel to the layers, the vortex driving force is directed parallel to the layers and thus intrinsic pinning has no effect for preventing the vortex motion. In this case introduction of some extrinsic pinning centers such as defects and precipitates is needed to prevent the vortex motion and to bring about a finite critical current density.
5.
ANGULAR
DEPENDENCE
OF
CRITICAL
CURRENT
In an external magnetic field tilted from the layers by 8, the vortex core may penetrate stepwise into the sample as shown in Fig. 3, since the intrinsic pinning by the layers is so strong [17]. In this case, the vortex consists of the segments parallel and perpendicular to the layers as seen in Fig. 3. We consider the angular dependence of the critical current density in the following two cases. When the transport current flows along the y-axis (case I), the driving force acts on both segments: the parallel segments (“the parallel vortices”) are driven in the direction of the z-axis and the perpendicular segments (“the pancake vortices”) are driven in the direction of the x-axis, as indicated by the arrows in Fig. 3. We suppose that the two kinds of the vortex segments move independently. Under this assumption, the critical current density for the parallel vortices, J,,,(B), is given by the critical current density for the external field parallel to the x-axis. The critical current density for the pancake vortices, J,,(B), is given by the critical current density for the field parallel to the z-axis. Then, the angular dependence of the critical current density is expressed in terms of
Fig. 3. Vortex penetrating stepwise into the sample. The bold solid curve indicates the path of the vortex core. The vertical dotted lines indicate some extrinsic pinning centers. When the external current is parallel to the y-axis (case I), the driving forces act on the vortex segments parallel and perpendicular to the layers as indicated by the arrows. When the external current is parallel to the x-axis (case II), the driving force acts only on the vortex segments perpendicular to the layers.
M. TACHIKI and S. TAKAHASHI
310
J,,,(B) and J,,(B). Since the average flux density from the parallel vortices is B, = B cos 8, the critical current density for the parallel vortices is expressed by J,,,(B cos Q). Since the average flux density from the pancake vortices is B, = B sin 0, the critical current density for the pancake vortices is given by J,,(B sin 0). The measuring critical current density in the magnetic field tilted from the layers by 8, J,(B; 0), is determined by the smaller one of either J,,,(B cos 0) or J,,(B sin 0) J,(B;
e) = min{J,,,(B cos e), J,,(B sin e)}. (case I)
(13)
When .J,,,(B cos 0) is larger than J,,(B sin f3),the critical current density is regulated by the critical current density of the pancake vortices: J&B; 0) = J,,(B sin 0). Experimental results show that J,,(B) is usually fitted well by B-“, CIbeing a positive constant, except for B - 0. The value of c( depends on the type of the extrinsic pinning center. When the pinning center is of a planar-type such a twin-boundary, a is - 0.5 and thus the angular dependence of the critical current is given by J&B; f3)cc l/fin(B). This simple formula well reproduces most of experimental data of the angular dependence of the critical current density at low temperatures. However, the experimental values deviate from those of the formula for 8 - 0. Some experimental results show that the exponent o! deviate from 0.5 for all region of 8. This deviation may come from the pinning centers of other kinds such as point defects, precipitates of second phase, and screw dislocations. Because of uncertainty in the pinning centers in a sample, we use the experimental values for J,,,(B) and J,,(B) shown in Fig. 4 and obtain the angular dependence of the critical current density shown in Fig. 5. The solid dots are the experimental data measured by Roas et al. using thin films of YBa,Cu,O, [l]. This kind of the angular dependence of the critical current density was also observed in thin films of YBa,Cu,O, [8-12, 141, in thin films of Bi,Sr,CaCu,O, [2,5,6], in thin films of Nd,_,Ce,CuO, [lo], in superlattices of YBa,Cu,O,/PrBa,Cu,O, [12], and in superlattices of YBa,Cu,O,/(Y, --x, Pr,)Ba,Cu,O, [13]. When the transport current flows along the x-axis (case II), the driving force does not act on the parallel vortices because of the Lorentz-force free configuration, and acts only on the pancake vortices in the y-direction. Accordingly, the critical, current density is determined by the critical current density for the pancake vortices in the effective field B sin 0 parallel to the c-axis J,(B;
e) = J,,(B sin e), (case II)
(14)
which is the same as that for J,,,(B) > J,,(B) in case I. As seen in the above, in the strong intrinsic pinning case both the critical current densities in cases I and II are scaled with the field component perpendicular to the layers. The critical current density J,(B; 0) in the cuprate superconductors with extremely large anisotropy such as Bi,Sr,CaCu,Os [2,23] and YBa,Cu,O,/PrBa,Cu,O, [12] exhibits this scaling behavior almost at all
1.0 0.8 0.6 0.4 0.2
Fig. 4. Field dependence of the critical current densities J,,,(B) and J,,(B) in the external magnetic parallel to the x- and z-axes, respectively. The external current is parallel to the y-axis.
fields
311
Intrinsic pinning in cuprate superconductors P7C-rr
Fig. 5. Angular dependence of the critical current densities. The external magnetic field is rotated in the plane perpendicular to the current direction. 0 is the angle between the external magnetic field and the layers. The solid curves indicate our theoretical values and the solid dots indicate the experimental data obtained by Roas ef al. using YBa,Cu,O,_, thin films Cl].
-60
-30
0
30
60
B(degree) Fig. 6. Angular dependence of the critical current densities at a high temperature. The solid and dashed curves indicate the critical current densities in case I and II, respectively. The external magnetic field is rotated in the xz-plane. t? is the angle between the external magnetic field and the layers.
312
M.
TACHIKI
and S. TAKAHASHI
temperatures. On the other hand, the critical current density J&B; 0) in YBa,Cu,O, with weak anisotropy shows this scaling behavior only at low temperatures much below the crossover temperature T* [lO-123. In case I, the critical current deviates gradually from the scaling behavior as the temperature increases, since the intrinsic pinning becomes weakened and the parallel vortices take part in the depinning process [lS, 18, 19,22,36,37]. At high temperatures above T*, the critical current density is very much diminished from that expected from the scaling. On the other hand, the critical current in case II follows the scaling behavior at all temperatures, since the driving force does not act on the parallel vortices. Figure 6 shows schematic representation of the angular dependence of the critical current densities above T*. The solid and dashed curves indicate the critical current densities in case I and II, respectively. In contrast to the monotonic angular dependence in case II, the sharp distinct peak appears around 8 = 0 superimposed on the broad peak in case I. The appearance of the sharp peak around 0 = 0 may come from the fact that since the intrinsic pinning is very weak above T* the intrinsic pinning works only around 0 = 0 where the vortex lines are almost parallel to the layers. This type of the peak structure in the angular dependence of the critical current has experimentally been observed in thin films of YBa,Cu,O, [8-12,151, in bulks of oriented-grained YBa,Cu,O, [7], in untwinned single crystals of YBa,Cu,O, [16], in thin films of Bi,Sr,CaCu,O, [38], and in thin films of Nd,_,Ce,CuO, [lo]. Acknowledgements-We would like to thank Dr T. Nishizaki at Tohoku University for providing us with his experimental data prior to publication. This work is supported by a Grant-in-Aid for Scientific Research on Priority Area, “Science of High T, su~rconductivity” given by the Ministry of Education, science and Culture, Japan.
REFERENCES 1. B. Ross, L. Schultz and G. Saemann-Ischenko, Phys. Rev. Left. 64,479 (1990). 2. P. Schmitt, P. Kummeth, L. Schultz and G. Saemann-Ischenko, Phys. Rev. Lett. 67, 267 (1991). 3. S. Awaji, K. Watanabe, N. Kobayashi, H. Yamane and Ii. Hirai, Japan J. Appl. Phys. 31, L1539 (1992). 4. B. D. Biggs, M. N. Kunchur, J. J. Lin, J. Poon, T. R. Askew, R. B. Flippen, M. A. Subramanian, J. Go~lak~shanan and A. W. Sleight, Phys. Rev. B39, 7309 (1989). 5. H. Raffy, S. Labdi, 0. Laborde and P. Monceau, Physica B 16%166, 1423 (1990). 6. H. Yamasaki, K. Endo, S. Kosaka, M. Umeda, S. Yoshida and K. Kajimura, P&s. Rev. Letr. 70,333l (1993). 7. J. W. Ekin, Appi. Phy,r. Lett. 59, 360 (1991). 8. C. Tome-Rosa, G. Jakob, A. Walkenhorst, M. Mauf, M. Schmitt, M. Paulson and H. Adrian, 2. Phys. B83,221(1991). 9. A. Walkenhorst, C. Tome-Rosa, P. Wagner, Th. Kluge, C. Stolzel, G. Adrian, G. Jakob and H. Adrian, Europhys. L&r. 18, 641 (1992). 10. T. Nishizaki, F. Ichikawa, T. Fukami, T. Aomine, T. Terashima and Y. Bando, Phy&a C 204,305 (1993); T. Nishizaki, private communications. 11. R. M. Schalk, H. W. Weber, Z. H. Barder, P. Przyslupsky and J. E. Evetts, Physica C 199, 311 (1992). 12. G. Jakob, M. Schmitt, Th. Kluge, C. Tome-Rosa, P. Wagner, Th. Hahn and H. Adrian, Phys. Rev. B47,12099 (1993). 13. Qu Li, C. Kwon, X. X. Xi, S. Bhattacharya, A. Walkenhorst, T. Venkatesan, S. J. Hagen, W. Jiang and R. L. Greene, Phys. Rev. Z&t. 69, 2713 (1992). 14. 2. X. Gao, E. Osquiguil, M. Maenhoudt, B. Wuyts, S. Libbrecht and Y. Bruynseraede, Phyr. Rev. Left. 71,3210(1993). 15. Y. Iye, T. Trrashima and Y. Bando, Physica C 184,326 (1991). 16. W. K. Kwok, U. Welp, V. M. Vinokur,.S. Fleshier, J. Downey and G. W. Crabtree, Phys. Rev. L&t. 67, 390 (1991). 17. M. Tachiki and S. Takahashi, Solid &ate Commun. 70.291 (1989): 72.1083 (1989). ’ ’ 18. B. I. Iviev and N. B. Kopnin,~J. few. temp. Phys. 77,4i3 (1989). ” 19. J. Friedel, J. Phys. 1, 7757 (1989). 20. D. Feinberg and C. Villard, Phys. Rev. Lett. 65, 919 (1990). 21. A. Barone, A. I. Larkin, and Yu. N. Ovchinnikov, J. Supercond. 3, 155 (1990). 22. S. Chakravarty, B. I. Ivlev and Yu. N. Ovchinnikov, Pfzys Rev. L&t. 64, 3187 (1990). 23. P. H. Kes, J. Aarts, V. M. Vinokur and C. J. van der Beek, Phys. Reo. Len. 64, 1063 (1990). 24. K. E. Gray and D. H. Kim, Physica C 180, 139 (1991). 25. M. Tachiki, S. Takahashi, and K. Sunaga, Phys. Rev. B47, 6095 (1993); M. Tachiki and S. Takahashi, Cryogenics 32, 923 (1992). 26. J. R. Clem, Phys. Rev. B43, 7838 (1991). 21. S. Fleshler, W. K. Kwok, U. Welp, V. M. Vinokur, M. K. Smith, J. Downey and G. W. Crabtree, phys. Rev. B47, 1448 (1993). 28. L. Civale, A. M. Marwich, T. K. Worthington, M. A. Kirk, J. R. Thompson, L. Krusin-Elbaum, Y. Sun, J. R. Clem and F. Holtzberg, Phys. Rev. Left. 67, 648 (1991). 29. W. Gerhauser, G. Ries, H.-W. Neumiiller, W. Schmitt, 0. Eibl, G. Saemann-Ischenko and S. Klaumiinzer, Phys. Rev. Left. 68, 879 (1992). 30. B. Holzapfel, G. Kreiselmeyer, M. Kraus, G. Saemann-Ischenko, S. Bouffard, S. Klaumiinzer and L. Schultz, Phys. Rev. B48, 600 (1993). 31. R. A. Klemm, Layered Superconductors. Oxford Univ. Press (1993). 32. A. A. Abrikosov, Fundamentals of the Theory of Metals, p. 435, North-Holland, Amsterdam (1988).
Intrinsic pinning in cuprate superconductors
313
33. By taking variation of the free energy equation (1) of Ref. [25] with respect to the vector potential, the second GL equation is derived as
1
2%; z,)v x a(r) = - A(r) - 2 Vcp(r)
where A(r) is the vector potential and cp(r)is the phase of the order parameter. A vortex located at r0 is described by introducing the phase singularity expressed as [32] V x Vcp(r)= 2x&r - rO).
(A2)
Taking rotation of equation (Al) with respect to r and using equation (A2), we derive equation (6). 34. A. M. Campbell and J. E. Evetts, Adv. Phys. 21, 199 (1972). 35. H. Ullmaier, Irreversible Properties of Type II Superconductors, Vol. 76, Springer Tracts in Modern Physics (Edited by G. Hohber). Springer, Berlin (1975). 36. G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin and V. M. Vinokur, Rev. Mod. Phys. To be published. 37. S. Doniach, in High Temperature Superconductivity (Edited by K. S. Bedell et a[.), p. 406. Addison-Wesley, Reading, Mass. (1990). 38. Y. Iye, A. Fukushima, T. Tamegai, T. Terashima and Y. Bando, Physica C 18%189, 297 (1991).
w-
~
Pereamon
INTRINSIC
PINNING M.
Institute
for Materials
Research,
Applied Superconductivity Vol. 2, No. 314. pp. 305-313, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0964-1807/94’7.00 + 0.00
I
IN CUPRATE TACHIKI
Tohoku
and S.
University,
SUPERCONDUCTORS
TAKAHASHI 2-l-l
Katahira.
Aoba-ku,
Sendai 980, Japan
Abstract-We review our theoretical work on the intrinsic pinning in cuprate superconductors. The superconductors are constructed by stacking of strongly and weakly superconducting layers. The vortices are strongly pinned in the weakly superconducting layers. Since the transport current flows mainly in the strongly superconducting layers, the driving force acting on the vortices is considerably reduced from that in a uniform current system. These effects combine to yield strong vortex pinning and to enhance the critical current. Dependence of the critical current on the direction of applied magnetic field is discussed in connection with the vortex structure in the cuprate superconductors.
1.
INTRODUCTION
It has been shown experimentally that c-axis oriented films of yttrium and bismuth cuprate oxides carry large critical current densities at low temperatures and the current densities stand up to high magnetic fields, especially when the magnetic field is applied parallel to the films [l-14]. The critical current strongly depends on the direction of the magnetic field and shows a sharp peak at the field direction parallel to the layers [l-16]. The above experimental results suggest that the vortices parallel to the layers are strongly prevented from moving perpendicular to the layers. The cuprate superconductors are constructed by stacking of strongly superconducting CuO, layers and weakly superconducting spacing layers. We consider that the layer structure plays an important role for the vortex pinning. In the magnetic field applied parallel to the layers, the cores of vortices thread through the weakly superconducting layers, since the vortex energy is lowest there. The strongly superconducting layers work as potential barriers for the vortices moving perpendicular to the layers [17-241. Along with the weakly superconducting layers working as potential barriers, the inhomogeneous current in the layer structure works to enhance the vortex pinning [25]. In the layered superconductors the transport current flows mainly in the strongly superconducting layers, and the current density in the weakly superconducting layers is much smaller than that in the strongly superconducting layers. In the inhomogeneous current, the vortices are not driven by the average of the transport current density over the region where the vortex current spreads, but by the transport current density flowing just at the vortex center. As a result, the driving force acting on the vortex becomes very weak when the vortex center is pinned in the weakly superconducting layers. These two effects cooperatively work to strongly pin the vortices in the weakly superconducting layers. We call the pinning which originates from the above two effects intrinsicpinning. When the magnetic field is tilted from the layers, the vortices may penetrate stepwise into the sample [17,20]. Each vortex consists of vortex segments parallel and perpendicular to the layers as shown in Fig. 3. The parallel segments are locked strongly in the weakly superconducting layers by the intrinsic pinning. The perpendicular segments (so-called pancake vortices [26]) are pinned by some extrinsic pinning centers such as twin boundaries [27] and columnar defects created by heavy ion irradiation [28-301. If the pinning force due to the extrinsic pinning centers is weaker than the intrinsic pinning force, the critical current density in the tilted magnetic field is determined by the depinning current density of the pancake vortices. In this paper, we review the intrinsic pinning effect in the cuprate superconductors.
2.
INTRINSIC
PINNING
FORCE
Let us first examine the nature of the intrinsic pinning force acting on a single vortex in a cuprate superconductor. In the superconductor, the metallic CuO, layers are strongly superconductive, 305
306
M. TACHIKIand S. TAKAHASHI
-
%(r;
20)
----- lJJo(z)
-2
c-axis
Fig. 1. The spatial variation of the superconducting order parameter along the c-axis. The solid curve indicates the order parameter in the presence of the vortices at positions (a) and (b). The dashed curve indicates the order parameter without the vortices.
while the other layers (spacing layers) are weakly superconductive. Therefore the superconducting order parameter is modulated in the period of the lattice constant along the c axis. Let us use a simple model that the amplitude of the superconducting order parameter Y = 1+4,, exp(icp) takes a spatial variation of the form [17]
with the lattice constant a, along the c-axis. When a single vortex parallel to the x direction in the layers runs through the position (0, zO) in the yz plane, the order parameter will be modified into the form $,(r; zO) = $,(z)tanh[(;y
+ tT>‘]“’
where 5, and tub are the coherence lengths in the direction parallel and perpendicular to the c-axis, respectively. Figure 1 shows the spatial variation of the order parameter along the c-axis when the vortex is located at the positions (a) and (b). If we write the condensation energy density of the uniform superconductor with $i as H~/Src, the condensation energy densities of the superconductors with $,,(z) and *Jr; z,,) may be given by (H,“/~J~)[$~(z)/$,]~ and (H@r)[$,(r; z,,)/$~)]~, respectively. Then, the increase in the condensation energy density by including the vortex is given by (Hz/Sn){($,(r; z&91)2 - [ll/0(z)/e1]2). Integration of the energy density in the yz plane gives the intrinsic pinning potential U&z,). Taking a derivative of the intrinsic pinning potential, the intrinsic pinning force is calculated as &(z,) = -(d/dz,)U,(z,). If we write the maximum value of &(z,,) with respect to z0 as fpM, the maximum pinning force is given by [17]
H,227ca, -Lb q.
fpM = -
871
t,
The parameter q is a function of the ratio &/a, and the modulation parameter 6 = $2/+1. The value of rl is shown as a function of &/ac for four values of 6 in Fig. 2. As seen in Fig. 2, the q vs &/a, curve has a peak at &/uc 2 0.3. The peak height increases almost linearly as 6 increases. From the facts mentioned above, we see that the intrinsic pinning effect is most effective for the layered superconductors with a coherence length comparable to the layer spacing and with strongly modulated order parameter. When the coherence length is longer than the layer spacing, q becomes very small and thus the pinning force becomes very weak. We introduce a crossover temperature T* at which the coherence length r, is nearly equal to the layer spacing between the CuCO, layers. Above T*,the vortex core spreads over several layers, and thus the intrinsic pinning force is very much reduced. Below T*,on the other hand, the vortex
Intrinsic
pinning
in cuprate
307
superconductors
0.3
e 0.2
0.1
0
Fig. 2. q as a function
of &/a, The pinning force and the critical current is the modulation parameter.
density
are proportional
to 1. 6
core size becomes smaller than the layer spacing and thus the vortices are locked between the CuO, layers. The drastic change of the critical current density was observed at T* N 60 K in YBa,Cu,O, films [3,11]. In Bi,Sr,CaCu,O,, the BiO and SrO layers are almost semiconductive and the superconducting interaction between the CuO, layers is very weak [23,26]. Therefore the vortex cores are confined in the semiconductive layers up to near the superconducting transition temperature, that is, the crossover temperature is lifted up to near the transition temperature [23,-311.
3.
DRIVING
FORCE
In addition to the mechanism mentioned above, the layer structure of the cuprates has another mechanism to enhance the vortex pinning. In cuprate superconductors, the distribution of the supercurrent is not uniform due to their layer structures, as mentioned in the introduction. In this case, arises a question whether the driving force acting on the vortex in nonuniform current is the same as that in uniform current. To answer this question, we consider a system with a single vortex parallel to the x-axis and an external current in the y-direction. We write the vortex current and the external current j, and jex,, respectively. The currents j, and j,,, induce the magnetic fields b, and b,,,, respectively. Substituting the total current j, + j,,, and the total magnetic field b, + b,,, into the Ginzburg-Landau(GL) free energy, we obtain the electromagnetic interaction energy between the vortex and the external currents as [25,32] F, = 2 sg
[(“Tdr
jv(r).jext(r) + b,(r).b..,(r)]
with A’@;z,,) = mc2/8ne21C/,2(r; zO). Using the Maxwell equation j”(r) = (c/4rc)V x b,(r) and j_,(r) = (c/47r)V x b,Jr) and integrating the first term in the bracket of equation (4), we rewrite equation (4) as Fd =
g
b,,,(r). {V x [A’(r; zO)V x b,(r)] + b,(r)}.
s
The GL equation for the vortex magnetic field in an inhomogeneous c331
system of A(r; z,J is given by
V x [A’@; z,)V x b,(r)] + b,(r) = @,,h(r - r,,)
(6)
308
M. TACHIKI and S. TAKAHASHI
where a,, = hc/2e is the unit flux, 6(r - re) the delta function, and r,, = (0, zO) the position of the vortex center in the yz plane. Substituting equation (6) into equation (5), we have
As seen in equation (7), we notice that the interaction energy is proportional to the magnetic field induced by the external current at the vortex center. Taking a derivative of F, with respect to r,,, we obtain the driving force acting on the vortex per unit length
where the current density j&J
is defined by
itl(ro)= i
FL, x Lt(rO)l.
(9)
The current density j,Jr,) is the transport current density flowing at the vortex center, that is, the current density at the vortex center in the absence of the vortex [25]. Equation (8) indicates that the driving force is determined by the transport current density just at the vortex center, although the vortex current spreads over the penetration depth, which is several thousands A in cuprate superconductors. According to this fact, when the vortices are pinned in the weakly superconducting layers, the driving force acting on them is very weak. Since the critical current is determined by the force balance between the pinning force and the driving force, we expect a significant enhancement of the critical current due to the nonuniform distribution of the transport current according to the mechanism discussed in this section. 4. CRITICAL
CURRENT
Let us consider a cuprate superconductor in a magnetic field parallel to the CuO, layers. The advantage to use the weakly superconducting layers as the pinning centers is their high-density, so that the intrinsic pinning is effective up to high magnetic fields. In a high magnetic field of 10 T, for example, the distance between the vortices is much larger than the layer spacing, and thus all the vortices are pinned in the weakly superconducting layers. In this case, the direct summation of the pinning forces is allowed to obtain the pinning force density per unit volume [34], that is, the pining force density is simply given by the intrinsic pinning force fPMmultiplied by the number of the vortices per unit area (direct summation). In the configuration where the magnetic field is applied parallel to the x-axis and the transport current flows in the y-direction, the critical current density is calculated by balancing the driving force density with the maximum pinning force density as 134,353
f.LB=$f,, 0
(10)
where B is the flux density and j, is the critical current density at the vortex center. The measuring critical current density J, is given by taking the spatial average of the current density. If we introduce a ratio y = J,/j,, J, is expressed as
J, = Y f
0
fpw
(11)
The expression of the critical current density differs by the factor y from the conventional one where the uniform transport current distribution is assumed. For the weak pinning case with a small modulation parameter (6 = $i/~+~ << l), using the relation j,, cc I,&), we obtain the explicit expression of the enhancement factor as y = (1 + 6’/2). This fact indicates that the nonuniform
Intrinsic
pinning
in cuprate
superconductors
309
current distribution enhances the critical current. In equation (1) we assume a sinusoidal variation of the superconducting order parameter. However, when the order parameter steeply changes at the boundary between the strongly superconducting layer and the weakly superconducting layer, the enhancement factor y becomes large [25]. Inserting equation (3) into equation (11) and using the relation H, = @0/(2&r5,&,), the intrinsic critical current density is obtained as [17, 361
(12) where j,, is the depairing current density of the CuO, layers defined by cH,/(3&4,,) which yields j,, E lo9 A/cm2 for the parameter values H, N 1 T and A,, N 1500 A for YBa,Cu,O,. The factor (1 - B/H,,) expresses the decrease of the the condensation energy due to the vortex normal cores. The critical current density J, estimated by using equation (12) is comparable to the depairing current density j,, at low temperatures. Since the magnetic field dependence of J, arises only from the factor (1 - B/H,,), the critical current density is insensitive to a usually applicable magnetic field at low temperatures where H,, is extremely high and thus the factor is nearly unity. This behavior of the critical current density is consistent with the experimental results of YBa,Cu,O, [l, 3, lo] and Bi,Sr,CaCu,O, [2,6]. When the magnetic field is applied perpendicular to the layers (the vortices are parallel to the c-axis) and the external current is parallel to the layers, the vortex driving force is directed parallel to the layers and thus intrinsic pinning has no effect for preventing the vortex motion. In this case introduction of some extrinsic pinning centers such as defects and precipitates is needed to prevent the vortex motion and to bring about a finite critical current density.
5.
ANGULAR
DEPENDENCE
OF
CRITICAL
CURRENT
In an external magnetic field tilted from the layers by 8, the vortex core may penetrate stepwise into the sample as shown in Fig. 3, since the intrinsic pinning by the layers is so strong [17]. In this case, the vortex consists of the segments parallel and perpendicular to the layers as seen in Fig. 3. We consider the angular dependence of the critical current density in the following two cases. When the transport current flows along the y-axis (case I), the driving force acts on both segments: the parallel segments (“the parallel vortices”) are driven in the direction of the z-axis and the perpendicular segments (“the pancake vortices”) are driven in the direction of the x-axis, as indicated by the arrows in Fig. 3. We suppose that the two kinds of the vortex segments move independently. Under this assumption, the critical current density for the parallel vortices, J,,,(B), is given by the critical current density for the external field parallel to the x-axis. The critical current density for the pancake vortices, J,,(B), is given by the critical current density for the field parallel to the z-axis. Then, the angular dependence of the critical current density is expressed in terms of
Fig. 3. Vortex penetrating stepwise into the sample. The bold solid curve indicates the path of the vortex core. The vertical dotted lines indicate some extrinsic pinning centers. When the external current is parallel to the y-axis (case I), the driving forces act on the vortex segments parallel and perpendicular to the layers as indicated by the arrows. When the external current is parallel to the x-axis (case II), the driving force acts only on the vortex segments perpendicular to the layers.
M. TACHIKI and S. TAKAHASHI
310
J,,,(B) and J,,(B). Since the average flux density from the parallel vortices is B, = B cos 8, the critical current density for the parallel vortices is expressed by J,,,(B cos Q). Since the average flux density from the pancake vortices is B, = B sin 0, the critical current density for the pancake vortices is given by J,,(B sin 0). The measuring critical current density in the magnetic field tilted from the layers by 8, J,(B; 0), is determined by the smaller one of either J,,,(B cos 0) or J,,(B sin 0) J,(B;
e) = min{J,,,(B cos e), J,,(B sin e)}. (case I)
(13)
When .J,,,(B cos 0) is larger than J,,(B sin f3),the critical current density is regulated by the critical current density of the pancake vortices: J&B; 0) = J,,(B sin 0). Experimental results show that J,,(B) is usually fitted well by B-“, CIbeing a positive constant, except for B - 0. The value of c( depends on the type of the extrinsic pinning center. When the pinning center is of a planar-type such a twin-boundary, a is - 0.5 and thus the angular dependence of the critical current is given by J&B; f3)cc l/fin(B). This simple formula well reproduces most of experimental data of the angular dependence of the critical current density at low temperatures. However, the experimental values deviate from those of the formula for 8 - 0. Some experimental results show that the exponent o! deviate from 0.5 for all region of 8. This deviation may come from the pinning centers of other kinds such as point defects, precipitates of second phase, and screw dislocations. Because of uncertainty in the pinning centers in a sample, we use the experimental values for J,,,(B) and J,,(B) shown in Fig. 4 and obtain the angular dependence of the critical current density shown in Fig. 5. The solid dots are the experimental data measured by Roas et al. using thin films of YBa,Cu,O, [l]. This kind of the angular dependence of the critical current density was also observed in thin films of YBa,Cu,O, [8-12, 141, in thin films of Bi,Sr,CaCu,O, [2,5,6], in thin films of Nd,_,Ce,CuO, [lo], in superlattices of YBa,Cu,O,/PrBa,Cu,O, [12], and in superlattices of YBa,Cu,O,/(Y, --x, Pr,)Ba,Cu,O, [13]. When the transport current flows along the x-axis (case II), the driving force does not act on the parallel vortices because of the Lorentz-force free configuration, and acts only on the pancake vortices in the y-direction. Accordingly, the critical, current density is determined by the critical current density for the pancake vortices in the effective field B sin 0 parallel to the c-axis J,(B;
e) = J,,(B sin e), (case II)
(14)
which is the same as that for J,,,(B) > J,,(B) in case I. As seen in the above, in the strong intrinsic pinning case both the critical current densities in cases I and II are scaled with the field component perpendicular to the layers. The critical current density J,(B; 0) in the cuprate superconductors with extremely large anisotropy such as Bi,Sr,CaCu,Os [2,23] and YBa,Cu,O,/PrBa,Cu,O, [12] exhibits this scaling behavior almost at all
1.0 0.8 0.6 0.4 0.2
Fig. 4. Field dependence of the critical current densities J,,,(B) and J,,(B) in the external magnetic parallel to the x- and z-axes, respectively. The external current is parallel to the y-axis.
fields
311
Intrinsic pinning in cuprate superconductors P7C-rr
Fig. 5. Angular dependence of the critical current densities. The external magnetic field is rotated in the plane perpendicular to the current direction. 0 is the angle between the external magnetic field and the layers. The solid curves indicate our theoretical values and the solid dots indicate the experimental data obtained by Roas ef al. using YBa,Cu,O,_, thin films Cl].
-60
-30
0
30
60
B(degree) Fig. 6. Angular dependence of the critical current densities at a high temperature. The solid and dashed curves indicate the critical current densities in case I and II, respectively. The external magnetic field is rotated in the xz-plane. t? is the angle between the external magnetic field and the layers.
312
M.
TACHIKI
and S. TAKAHASHI
temperatures. On the other hand, the critical current density J&B; 0) in YBa,Cu,O, with weak anisotropy shows this scaling behavior only at low temperatures much below the crossover temperature T* [lO-123. In case I, the critical current deviates gradually from the scaling behavior as the temperature increases, since the intrinsic pinning becomes weakened and the parallel vortices take part in the depinning process [lS, 18, 19,22,36,37]. At high temperatures above T*, the critical current density is very much diminished from that expected from the scaling. On the other hand, the critical current in case II follows the scaling behavior at all temperatures, since the driving force does not act on the parallel vortices. Figure 6 shows schematic representation of the angular dependence of the critical current densities above T*. The solid and dashed curves indicate the critical current densities in case I and II, respectively. In contrast to the monotonic angular dependence in case II, the sharp distinct peak appears around 8 = 0 superimposed on the broad peak in case I. The appearance of the sharp peak around 0 = 0 may come from the fact that since the intrinsic pinning is very weak above T* the intrinsic pinning works only around 0 = 0 where the vortex lines are almost parallel to the layers. This type of the peak structure in the angular dependence of the critical current has experimentally been observed in thin films of YBa,Cu,O, [8-12,151, in bulks of oriented-grained YBa,Cu,O, [7], in untwinned single crystals of YBa,Cu,O, [16], in thin films of Bi,Sr,CaCu,O, [38], and in thin films of Nd,_,Ce,CuO, [lo]. Acknowledgements-We would like to thank Dr T. Nishizaki at Tohoku University for providing us with his experimental data prior to publication. This work is supported by a Grant-in-Aid for Scientific Research on Priority Area, “Science of High T, su~rconductivity” given by the Ministry of Education, science and Culture, Japan.
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Intrinsic pinning in cuprate superconductors
313
33. By taking variation of the free energy equation (1) of Ref. [25] with respect to the vector potential, the second GL equation is derived as
1
2%; z,)v x a(r) = - A(r) - 2 Vcp(r)
where A(r) is the vector potential and cp(r)is the phase of the order parameter. A vortex located at r0 is described by introducing the phase singularity expressed as [32] V x Vcp(r)= 2x&r - rO).
(A2)
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