Journal of Superconductivity, Vol. 4, No. 5, 1991

The Bardeen-Stephen Theory John R. Clem ~

Received 5July1991

The experimental situation that prompted the theory is reviewed, the essentials of the BardeenStephen theory are sketched, and several extensions are mentioned, particularly those that are needed to understand the high-temperature copper-oxide superconductors. KEY WORDS: Bardeen-Stephen theory; vortex dynamics.

The Bardeen Stephen theory [1] had a significant impact on the field of superconductivity when it was presented in 1965. The basic ideas presented in that paper have stood the test of time and are now permanently part of the lore of type-II superconductivity. According to the Science Citation Index, the paper to date has received over 250 citations. The 1986 discovery of superconductivity in the copper oxides [4] generated an enormous amount of renewed interest in superconductivity, and the Bardeen-Stephen paper's citation history reflects this phenomenon. My purpose in this paper is to sketch the experimental situation that required a theory, briefly explain the essentials of the Bardeen-Stephen theory, and touch upon some of the extensions of the theory, particularly those that are required for the understanding of the high-temperature copper-oxide superconductors. When a small magnetic field is applied to a superconductor, screening currents spontaneously arise to screen the magnetic field from the interior of the sample. These currents flow within the penetration depth ,~ of the surface and decay exponentially with distance from the surface. For a sufficiently large applied field, however, the surface currents break down, and larger amounts of magnetic flux are admitted into the material. The spatial distribution of this magnetic flux is not uniform, as in nonmagnetic normal metals, but is quite inhomogeneous. The parameter that determines

the scale of the resulting field inhomogeneity is the Ginzburg-Landau parameter [5] ~; = ,~/~, where ~ is the coherence distance. According to the Ginzburg Landau theory, as developed by Abrikosov [6], when tr < 1/x/2, the superconductor is called a type-I superconductor, the wall energy between the superconducting and normal phases is positive, and the entering magnetic flux enters in the form of large normal domains. The domain size adjusts so as to minimize the overall Gibbs free energy; the chief competing terms are the wall energy, which favors large normal domain sizes, and the magnetic energy, which favors a state of uniform magnetic field. On the other hand, when tc > 1/.,/2, the superconductor is called a typeII superconductor, the wall energy between the superconducting and normal phases is negative, and magnetic flux enters in domains of the smallest possible size, subject to the constraint of fluxoid quantization [7]. The elementary flux quantum in a superconductor is Oo=hc/2e=2.07 x 10 - 7 G c m 2. The quantized entities that carry this magnetic flux into a type-II superconductor go by many names: vortices, flux lines, fluxons, and fluxoids. The center of a vortex is where the local magnetic field reaches its peak value. The magnetic field distribution is maintained by a local vortex-like supercurrent distribution, which reaches its peak value at a distance of about from the vortex axis. The region within about ~ from the vortex axis is referred to as the vortex core. The superconducting order parameter [5-6] is strongly suppressed in the vicinity of the core and is zero on the vortex axis. The Bardeen-Stephen theory relates to the question of how these vortices move under the influence

~Ames Laboratory-USDOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 5001 I.

337 0896-1107/91/ 1000-0337506.50/0 9 199l PlenumPublishingCorporation

338

Clem

of an applied current. At the time of the development of this theory, a number of observers had studied this problem, but the most extensive work had been done by Kim, Hempstead, and Strnad [8 10]. These authors had studied flux flow in long flat sheets of type-II superconductors with the external magnetic field H applied perpendicular to the broad surface. This produced an array of vortices of density n = B / ~b0, where B is the macroscopic magnetic induction in the sample. A transport current of density J r was passed through the sample along the long dimension. Kim et al. found characteristics of the voltage versus current or electric field E versus transport current density J r as follows: (a) E was zero (or nearly zero) for values of J r up to a certain critical current density Jc, which depended upon both the applied magnetic field H and the absolute temperature T and (b) for J r > Jc, the plots of E versus J r were linear (or nearly so) with slope Pl, which the authors called the flow resistivity. At low temperatures, pf followed the empirical rule

pf/p, = H/Hc2(O)

(1)

where p, is the normal-state resistivity and HeffT) is the upper critical field at temperature T. Further details of these and related experiments, as well as a discussion of related theoretical ideas, are reviewed in [11]. The basic principles are as follows. The transport current density JT produces a Lorentz force per unit length fL = Jrr on each vortex. The direction of this force is perpendicular to both the current direction and the vortex direction. Pinning forces, arising from metallurgical defects in the sample, oppose the Lorentz force up to a maximum value offp = Jcdpo/C per unit length of vortex. ForfL >fp, the net driving force per unit length of vortex is fL--fp. Because of various damping mechanisms, characterized by a viscous drag coefficient I/ and a viscous drag force per unit length r/vL, the combined effect of all forces results in steady vortex motion with speed vL given by the force-balance equation 7Jt)LmfL--fp. Since the motion of an array of vortices of flux density B with velocity vL generates a macroscopic electric field [12] E = B x VL/C, the force balance equation can be reexpressed as the equation E = P / ( J - J c ) , where the flow resistivity is related to the viscous drag coefficient 77 via

p/= B~o/71c 2

(2)

The Bardeen-Stephen theory [1] explained the results of Kim et al. [8-11] using a local model of the vortex in a relatively pure superconductor near

T= 0 K. The order parameter (or gap parameter) was taken to be zero inside a radius a, assumed to be very nearly equal to the coherence distance 4. The authors calculated the total rate of energy dissipation D per unit length for a vortex moving with velocity vL. They then obtained the viscous drag coefficient from the relation

D=qv 2

(3)

Using a few reasonable approximations, they showed that their result reduced to Kim et al.'s empirical rule

[8-10] 7J = ~)OItc2( O )O'n/ C2

(4)

which is obtained from Eqs. (1) and (2) under the conditions that B = H and a n = l / p n . Bardeen and Stephen [1] also showed that the Hall angle expected in the mixed state is the same as that in the normal state for a magnetic field equal to that in the vortex core. Other phenomenological theories of flux flow were proposed by van Vijfeijken [13] and by Nozieres and Vinen [14]. These models also assume a simple structure of the vortex core, which is treated as being fully normal and where dissipation takes place via quasiparticle scattering by the lattice. Although all three theories are fairly successful in their predictions for the flux-flow resistivity, they generally do not account for the experimentally observed Hall effect data, nor do they explain the thermomagnetic transport phenomena. Detailed calculations of the flow resistance of type-II superconductors have been carried out starting from the microscopic theory of superconductivity. The review of Gor'kov and Kopnin [15] summarizes the work done in this area and illustrates the regions where analytic expressions are available. Other mechanisms of dissipation have been examined, including thermal effects [ 16,17] and those describable via timedependent Ginzburg-Landau theory [18-21]. Some of the enormous amount of research on copper-oxide superconductors has been spent on attempts to understand dissipative processes in these materials. A topic that has received a considerable amount of attention is flux creep [22,23], which for a number of reasons is more prominent [24] than in the conventional superconductors [25]. The effects of the extreme anisotropy in oxides also have received attention. Such effects can be calculated either from anisotropic London theory [26] using an extension of the Bardeen-Stephen model [1], or, when the vortices are parallel to the CuO2 layers and move parallel to the

The Bardeen-Stephen Theory layers, f r o m a n e x t e n s i o n [27] o f the L a w r e n c e D o n i a c h t h e o r y [28]. I n the l a t t e r c a l c u l a t i o n [27] the c a l c u l a t i o n o f the viscous d r a g o n a v o r t e x is a n a l o g ous to t h a t for a v o r t e x in a large J o s e p h s o n j u n c t i o n [29]. In s u m m a r y , the B a r d e e n - S t e p h e n t h e o r y [1] has h a d , a n d c o n t i n u e s to have, a n i m p o r t a n t influence o n o u r t h e o r e t i c a l u n d e r s t a n d i n g o f the d i s s i p a t i o n a s s o c i a t e d with flux m o t i o n in t y p e - I I superconductors.

ACKNOWLEDGMENTS I t h a n k M. W . Coffey for assistance in p r e p a r i n g the m a n u s c r i p t . A m e s L a b o r a t o r y is o p e r a t e d for the U.S. D e p a r t m e n t o f E n e r g y b y I o w a State U n i v e r s i t y u n d e r C o n t r a c t N o . W-7405-Eng-82. This w o r k was s u p p o r t e d b y the d i r e c t o r for E n e r g y R e s e a r c h , Office o f Basic E n e r g y Sciences.

REFERENCES 1. 2. 3. 4. 5.

J. Bardeen and M. J. Stephen, Phys. Rev. A 140, 1197 (1965). J. Bardeen, Phys. Rev. Lett. 13, 747 (1964). M. J. Stephen and J. Bardeen, Phys. Rev. Lett. 14, 112 (1965). J. G. Bednorz and K. A. Mfiller, Z. Phys. 64, 189 (1986). V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). A complete English translation appears in Men of Physics: L. D. Landau, Vol. 1, D. ter Haar, ed. (Pergamon Press, New York, 1965), pp. 138-167.

339 6. A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957) [Soy. Phys.-JETP 5, 1174 (1957)]. 7. B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961); R. Doll and M. N/ibauer, Phys. Rev. Lett. 7, 51 (1961). 8. Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Rev. Mod. Phys. 36, 43 (1964). 9. Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Rev. Lett. 12, 145 (1964). 10. Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Rev. A 139, 1163 (1965). I1. Y. B. Kim and M. J. Stephen, in Superconductivity, Vol. 2, R. D. Parks, ed. (Marcel Dekker, New York, 1969), p. 1107. 12. B. D. Josephson, Phys. Lett. 16, 242 (1965). 13. A. G. van Vijfeijken, Philips Res. Rep. Suppl. 8, 1 (1968). 14. P. Nozieres and W. F. Vinen, Philos. Mag. 14, 667 (1966). 15. L. P. Gor'kov and N. B. Kopnin, Usp. Fiz. Nauk 116, 413 (1975) [Soy. Phys.-Usp. 18, 496 (1975)]. 16. J. R. Clem, Phys. Rev. Lett. 20, 735 (1968). 17. R. A. Gasparyan and V. Z. Kresin, Zh. Eksp. Teor. Fiz. 69, 377 (1975) [Soy. Phys.-JETP 42, 192 (1975)]. !8. M. Tinkham, Phys. Rev. Lett. 13, 804 (1964). 19. A. Schmid, Phys. Kondens. Mater. 5, 302 (1966). 20. C. Caroli and K. Maki, Phys. Rev. 159, 306, 316 (1967); 164, 591 (1967); 169, 381 (1968). 21. C. R. Hu and R. S. Thompson, Phys. Rev. B6, 110 (1972). 22. P. W. Anderson, Phys. Rev. Lett. 9, 309 (1962). 23. P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39 (1964). 24. Y. Yeshurun and A. P. Malozemoff, Phys. Rev. Lett. 60, 2202 (1989). 25. M. R. Beasley, R. Labusch, and W. W. Webb, Phys. Rev. 181, 682 (1969). 26. Z. Hao and J. R. Clem, IEEE Trans. Magn. 27, 1086 (1991). 27. J. R. Clem and M. W. Coffey, Phys. Rev. B 42, 6209 (1990). 28. W. Lawrence and S. Doniach, in Proceedings of the Twelfth International Conference on Low Temperature Physics, E. Kanda, ed. (Academic Press of Japan, Kyoto, 1971), p. 361. 29. P. Lebwohl and M. J. Stephen, Phys. Rev. 163, 376 (1967).