InOx thin-films superconductors

Physica 135B (1985) 120-123 North-Holland, Amsterdam

S C A L I N G BEHAVIOR OF THE M A G N E T O C O N D U C T A N C E OF H I G H L Y D I S O R D E R E D In/InO x THIN-FILM SUPERCONDUCTORS

A.F. H E B A R D and M.A. P A A L A N E N A T& T Bell Laboratories, Murray Hill, New Jersey 07974, USA

Magnetoconductance data on composite amorphous In/InO x thin (100/~)-film superconductors have been analyzed near critical disorder where the mean-field transition temperature To0 is rapidly suppressed with increasing normal-state sheet resistance. For a factor of 2.5 decrease in T~o, from 2.49 K to 0.97 K, the magnetoconductance data are observed to be scale-invariant with respect to the parameter ratios H / T , the perpendicular magnetic field divided by the temperature, and T/T~, the temperature Tnormalized to To0. This behavior is consistent with the functional forms of the Aslamazov-Larkin and Maki-Larkin contributions to the magnetoconductance providing the electron diffusivity is constant and the inelastic scattering rate is linear in temperature. These restrictive conditions are found to be valid for independent experimental determinations. Deviations in the scaling behavior are observed when weak localization contributions dominate at high temperature and when data for films with even lower To0 are included in the analysis.

Only recently has it been recognized that magnetoconductancc (MC) data can provide detailed and useful information about electronic transport processes in disordered thin-film superconductors [1]. This recognition was essentially stimulated by the work of Larkin [2] who pointed out how the inclusion of impurity vertex corrections to the Maki [3] diagram could lead to a direct determination of the inelastic electron lifetime ~'i from MC measurements. (Physically, the Maki diagram describes the reduced lifetime of quasiparticle states in the presence of superconducting fluctuations.) In addition to this Maki-Larkin contribution to the MC there are a number of other quantum corrections including the effect of the magnetic field H on the conductivity of the fluctuation-induced superconducting pairs [4-6] (based on the Aslamazov-Larkin diagram contribution [7]), a weak localization correction to the normal state properties [8], and Coulomb interaction contributions [9] which can become important at high fields. Accordingly, the analysis of MC data can become considerably more complicated than its acquisition, particularly when temperature and field regimes of two or more different contributions overlap. In this paper we show that when MC data is

plotted with respect to well-chosen scaling variables an insightful and useful assessment of the consistency between theory and experiment can often be obtained. As shown below, this scaling analysis is particularly appropriate for the comparison of MC data of a sequence of films with different transition temperatures. To illustrate these ideas, we present MC data for five I n / I n O x thin-film superconductors near critical disorder where the mean-field transition temperatures Tc0 are rapidly degraded with increasing disorder as measured by the normal-state sheet resistance Ro. The resistive transitions at H = 0 corresponding to the films of this study are shown in fig. 1. The rapid decrease in To0 near critical disorder where R n ~ - h / e 2 = 4114gl/D has been reported previously [10]. The resistive transitions for the fully superconducting films (films (a)-(c)) were analyzed using the Aslamazov-Larkin theory to determine T~0 for each film [11, 12]. These values are indicated by vertical arrows in fig. 1. Although a Tc0 could not be assigned to films (d) and (e), it will become readily apparent from the following discussion that the MC data for all of the films is qualitatively similar. This fact alone suggests that there is a more pronounced effect of disorder on the vortex fluctuations, which domi-

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If there is a sufficient amount of disorder in the films, then one might expect that, at low enough temperatures, the dominant inelastic processes arise from electron-electron interactions which give a scattering rate ~'(~ in two dimensions proportional to the product R , T [13]. If this is the case, and if Rn and D are not strong functions of disorder, then H/Hioc H / T and we see from eq. (1) that the MC should scale with respect to the variables t = T/T~o and H~ T. Interestingly, recent theoretical treatments [14, 15] which extend the range in H and T over which a M a k i - L a r k i n analysis is applicable and which also include weak localization corrections to the Aslamazov-Larkin contribution [15] can similarly be written as a function of the same scaling variables. These ideas are tested by replotting the MC isotherms for film (a), shown in fig. 2 as a lagarithmic plot with respect to H, on a similar logarithmic plot (open circles) in fig. 3 but with an H~ T abscissa. Note that the negative of the MC is plotted on the ordinate. For comparison, the MC isotherms of film (b) are also plotted in fig. 3 as open triangles. Inspection of fig. 3 thus reveals that the scaled magnetoconductances of these two films are self-similar with respect to H~ T and are interleaved with an essentially monotonic increase in t over the range 1.1 ~ t ~ 6.1 (cf. fig. 3

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where the magnetic length determined by the characteristic field H i = hc/4eD,6 is roughly equal 1/2 to the inelastic diffusion length (D~'i) for electrons with inelastic scattering time ~i and diffusivity D. Because the function fl(T/Teo ) diverges logarithmically at T~0, it is clear that for normalized temperatures t = T/Tco sufficiently close to unity, the M a k i - L a r k i n contribution to the MC can be large. The function f ( H / H i ) is written in terms of digamma functions and has the same form as the weak localization correction to the normal state conductance. This weak localization contribution is expected to dominate at higher temperatures. It also includes a functional dependence on the spin-orbit rate, a dependence which is not present in eq. (1) because of the spin-singlet nature of the Cooper pairs.

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caption). This scaling behavior for these two films thus serves as a consistency check on the assumptions used in the above discussion to obtain such scaling. Independently determined experimental confirmation of these assumptions are: (1) The measured room-temperature sheet resistance R n is nominally unchanged with R n = 2 2 4 8 [ 1 / [ ] for film (a) and 3394 I I / D for film (b). (2) A constant value D = 0.18 cm 2 s -~ was obtained when a pairbreaking model was used in earlier work to explain critical field data on a similar set of films [10]. (3) An approximately linear temperature dependence of ri-1 for films (a)-(c) has been found from a least squares analysis using theory [15] which extends the range in H and T of the original Larkin result. Interestingly, the magnitude of z ( ~ was found to be within a factor of four of the maximum rate k BToo/h consistent with the existence of Cooper pairs with binding energy approximately equal to kBTco [15]. With a further decrease in Tc0 the excellent scaling behavior exhibited in fig. 3 for films (a) and (b) begins to rapidly deteriorate. For example, the MC isotherms of the most resistive film (e) shown in fig. 4 reveal a distinctly different

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H/T (Oe/K) Fig. 4. MC isotherms for film (e) plotted versus H/T. The monotonically increasing temperature assignments are 0.650 (top), 0.890, 1.210, 2.070, 2.720, 3.680, 5.020, 6.887 and 9.800 K (bottom).

behavior. This deviation is especially prominent at low T and high H where the MC appears to rapidly saturate because of the pronounced decrease in the strength of the superconducting fluctuations. An interesting comparison of films (c) and (d) is shown in the MC isotherms of fig. 5. The self-similarity of these isotherms with respect to the H~ T variable is qualitatively the same as that shown in fig. 3. As Tc0 and hence t cannot be i(]0

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A.F. Hebard and M,A. Paalanen / Magnetoconductance in thin-film superconductors determined for film (d), which does not have a zero resistance transition, it is difficult to assess the goodness of scaling with respect to t. By referring to the numerical tags in fig. 5 we can, however, arbitrarily assign T~0 = 0.15 K to film (d) which gives t = 1.3 for curve (3) with T = 0.20 K, a reasonable interpolation b e t w e e n t = 1.206 (curve (2)) and t = 1.531 (curve (4)) of film (c). Curve (6) with t = 3.0, however, does not interleave with curve (5) ( t = 1.837) and curve (7) (t = 2.353), hence indicating the beginning of a b r e a k d o w n of the scaling d e p e n d e n c e on t. We note from fig. 1 that this arbitrary assignment of To0 = 0.15 K to film (d) is in a region where the resistance is still decreasing, albeit slowly, due to the marginal domination of superconducting fluctuations over localization effects. In conclusion, it is no surprise that the scaling behavior predicted by the functional forms of the M a k i - T h o m p s o n and A s l a m a z o v - L a r k i n theories of the M C begin to b r e a k down near critical disorder where superconductivity is rapidly quenched. Clearly the T and H regions over which the paraconductivity description remains tenable are becoming so severely restricted that weak and, most probably, strong localization effects begin to dominate the entire M C behavior. These latter contributions are not expected to scale with either H~ T or t. The results presented here, however, serve as a convincing confirmation that, at least for films with To0 as low as 1 K, the functional dependences e m b e d d e d in the M a k i -

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T h o m p s o n and A s l a m a z o v - L a r k i n theories are good descriptions of the M C behavior and that inelastic e l e c t r o n - e l e c t r o n scattering processes play an important role in the destruction of superconductivity. References [1] G. Bergmann, Phys. Rev. B29 (1984) 6114, and references therein. [2] A.I. Larkin, Pis'ma Zh. Eksp. Teor. Fiz. 31 (1980) 239 [JETP Lett. 31 (1980) 219]. [3] K. Maki, Prog. Theor. Phys. 40 (1968) 193. [4] K. Usadel, Z. Phys. 227 (1969) 268. [5] E. Abrahams, R. Prange and M. Stephen, Physica 55 (1971) 230. [6] M. Redi, Phys. Rev. B16 (1977) 2027. [7] L.G. Aslamazov and A.I. Larkin, Phys. Lett. 26A (1968) 238, [8] S. Hikami, A.I. Larkin and Y. Nagaoka, Prog. Theor. Phys. 63 (1980) 707. [9] B.L. Altschuler, A.G. Aronov, A.I. Larkin and D. Khmelnitskii, Zh. Eksp. Theor. Fiz. 81 (1981) 768 [Soy. Phys. JETP 54 (1981) 411]. [10] A.F. Hebard and M.A, Paalanen, Phys. Rev. B30 (1984) 4063. [11] A.T. Fiory, A.F. Hebard and W.I. Glaberson, Phys. Rev. B28 (1983) 5075. [12] M.A. Paalanen and A.F. Hebard, Appl. Phys. Lett. 45 (1984) 794. [13] E. Abrahams, P.W. Anderson, P.A. Lee and T.V. Ramakrishnan, Phys. Rev. B24 (1981) 6783. [14] J.M.B. Lopes dos Santos and E. Abrahams, Phys. Rev. B31 (1985) 172, [15] W. Brenig, M.A. Paalanen, A.F. Hebard and P. Wolfle, to be published.