Competition between superconductivity and localization in two-dimensional ultrathin a-MoGe films

Physica 135B (1985) 113-119 North-Holland, Amsterdam

C O M P E T I T I O N B E T W E E N S U P E R C O N D U C T I V I T Y AND L O C A L I Z A T I O N IN TWO-DIMENSIONAL ULTRATHIN a-MoGe FILMS J.M. G R A Y B E A L A T& T Bell Laboratories, Murray Hill, NJ 07974, USA

The competition between superconductivity and localization in two dimensions has been systematicallystudied in ultrathin homogeneous superconducting films of amorphous Mo-Ge. These films were model 2D systems, with essentially constant equivalent bulk properties down to film thicknesses of -10/k. A strong and systematic reduction in the superconducting transition temperature Tc with increasing 2D disorder (i.e., sheet resistance R•) has been observed. Experimental evidence argues aginst "proximity effects" as the primary cause of this behavior. Furthermore, results of superconductive tunneling indicate that this reduction does not arise from simple pairbreaking effects alone. Superconductivity appears particularly sensitive to disorder in 2D, far more so than the normal state properties. For example, by sheet resistances of R= ~- 2000 II, Tc has been suppressed by 97% from the bulk value, whereas the normal state resistivity has only increased 40%. The results are found to be consistent with recent formulations of the theory, which not only take into account the effects of localization and the related enhancement-ifi the Coulomb interaction, but also the dynamical nature of the screening.

As is now well established, increasing disorder leads to localization and the related enhancements of the C o u l o m b interaction [1-3]. It is therefore not surprising that such an enhancem e n t of the C o u l o m b interaction would inherently c o m p e t e with superconductivity. F u r t h e r m o r e , it is also apparent that the coupling to disorder increases in reduced dimensions. Loosely speaking, it is " h a r d e r " for an electron to go around a defect in 2D than in 3D, where the electron can also go over or under it. T h e r e f o r e , the fact that disorder will c o m p e t e with superconductivity in reduced dimension seems quite reasonable. W h a t perhaps is surprising is that even in the weakly localized regime ( W L R ) , relatively far from the m e t a l - i n s u l a t o r transition, the magnitude of the reduction of superconductivity brought on by increasing disorder in 2D is indeed very large. As we shall show, the effects upon 2D superconductivity are m a n y times larger than for the normal state properties. In order to best examine the complex issues involved, one desires a model system approach, where the relevant p a r a m e t e r s can be varied in a known and controlled fashion. T o be specific, the initial effects of disorder on T c (compared with the Tc0 of the unaffected material) are predicted from perturbation theory to be universally re-

duced as A T J Too ~ (1~EFt') ° - 1

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The relevant measure of disorder is 1/EFT , which is proportional to the resistivity p in 3D and the sheet resistance R n in 2D. Therefore, as shown in fig. 1, in order to unambiguously separate the predicted 2D effects from those occurring due to changes in the bulk (3D) properties, the "equivalent b u l k " properties must be held constant as a function of film thickness. Certainly the maintenance of a constant film p is a reassuring (if not sufficient) condition that this is so. Note that by 2D, we here m e a n that the thermal diffusion length ( h D / k a T ) 1/2 is greater than the film thickness d, even though with respect to electronic p a r a m e t e r s (mean free path l; kF1; etc.) the film remains 3D. H e r e we present the results of a systematic study of the effects of disorder upon 2D superconductivity in ultrathin h o m o g e n e o u s films of aMo79Ge21. The bulk properties, both structural [4] and superconducting [5], have b e e n carefully examined by other workers. We chose to use a h o m o g e n e o u s system over a granular one, as we felt the results would perhaps lead to cleaner interpretation. We chose an a m o r p h o u s system as

0378-4363 / 85 / $03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

114

J.M. Graybeal / Competition between superconductivity and localization in 2D ATc ( I / D - I ~C a\EFT/

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Fig. 2. Measured low-temperature (T~ 15 K, above fluctuation conductivity effects) p vs. R[] (lower scale) and film thickness d (upper scale) for the MovgGe21films. or gross inhomogeneities. As will b e c o m e clearer later, this may well be important, as any such film structure that would alter the nature of the electron screening at long wavelengths could influence the suppression of superconductivity in 2D. F u r t h e r m o r e , the films may even be better than indicated above. In fact, the entire rise in the resistance as a function of R D has both the correct behavior and magnitude expected from localization and interaction effects alone. This can be seen in fig. 3 where we show the normalized change in sheet conductance Ag G = - h Ap/e2pR[] as a function of R[]. In the W L R , it is predicted that Ag D should be independent of R D, as is observed all the way to d = 9 A. F u r t h e r m o r e , using values for Ts0 and 7~, obtained from normalstate m e a s u r e m e n t s [AR(H) and AR(T)] upon these same films, one can estimate the value of Ag[] from theory [7] to be 0.5 < - A g a < 0.7. Note that due to their a m o r p h o u s nature, even at d = 12 A the film is several m e a n free paths thick, so boundary scattering effects are relatively small. Therefore, we now turn to the effects of disorder upon superconductivity for these very same ultrathin films. As shown in fig. 4, the suppression in the transition t e m p e r a t u r e T c is strong and initially linear, extrapolating to zero at only R~ = 1 k ~ . Recall our normal state results where we

J.M. Graybeal / Competition between superconductivity and localization in 2D

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found Ap/p of only 5 and 45%, respectively, for the 36/~ (R~ = 470 1)) and 12 A (R~ = 1960 12) films. For the same films we find T~ suppressions of 45 and 97%, respectively, below the bulk value. In addition, note that the size of the To~Too suppressions observed here are far larger than those expected from Kosterlitz-Thouless [8].

115

This reduction of T c with increasing R~ has, of course, been observed before. The works of Naugle et al. [9] (AI, Ga, Bi films), and Strongin et al. [10] (Bi, Pb) certainly displayed this behavior in disordered soft metals long ago, although the results were not definitively interpreted. More recently, works by Raffy et al. [11] (W-Re), and Okuma et al. [12] (Zn) interpret this behavior as we have here. The fact that the T c data is so similar for all these systems implies that the behavior is universal. Early explanations of this reduction of Tc with increasing R D (decreasing d) included "proximity effects", whereby the superconductivity is degraded by the presence of neighboring non-superconducting layers. Certainly the simplest of these is the phenomenological model of Cooper [13], which assumes a degraded normal layer on each side of the film, into which some of the superconducting electrons will "leak". Many more involved models exist; for example, the calculation of Halbritter [14] considers the hybridization of electron states in the conduction band with localized states in neighboring dielectric layers. Unfortunately, the parameters involved are often very difficult to obtain from direct measurements. However, it seems reasonable that the nature of these proximity coupling should be rather sensitive to the nature of the interface(s) and neighboring layer(s). Certainly this is one aspect which can be experimentally examined. In fig. 5 we show just a few examples of the parameters varied for films thicker than - 4 0 A. Unfortunately, the ultrathin films depend upon the precise nature of the multilayer structure for their existence. Here we display samples without protective over- and underlayers and deposited upon crystalline sapphire substrates, and films with differing Ge percentages (and thus, different density of states). Note that the initial slopes are identical to the results of fig. 4. Additionally (not shown here), we fixed the M o - G e film thickness at d = 46/~ and varied the a-S• overlayer thickness from 8 to 100/k. We found that the film p, R[] and T~ were unchanged within better than 5%. Therefore, although we agree that proximity effects surely must be relevant at some level, we feel that the evidence argues against their being

J.M. Graybeal / Competition between superconductivity and localization in 2D

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the primary origin of the T c suppression. As stated earlier, an integral part of our vertical multilayer structure was a protective overlayer of a-Si, which was subsequently oxidized to form an a-SiO x tunnel barrier. Therefore, using photolithographic techniques to define the tunnel junction area and subsequently depositing a Pb counter-electrode (A--~ 1.3 meV), we were able to examine the tunneling curves of our ultrathin films. In fig. 6 we show an example of the fine S - I - S tunneling curves thus obtained. Clearly this points to the quality of our ultrathin films. It is interesting to note that if the T~ reduction from the bulk value of T c 0 = 7 . 2 5 K (for Mov9Ge21 ) were completely due to simple pairbreaking effects, then the gap structure would have been driven completely gapless for this film. Although there is apparently some gap width observed, we can clearly rule out complete gaplessness. Therefore, our T c reduction is due to more than just simple pairbreaking effects. Let us simply add here that we also observe the expected cusp in the normal-state density of states.

Additionally, we have examined the superconducting properties of these films in the presence of a perpendicular magnetic field. Fig. 7 displays our critical field results for 4 different Mo79Ge21 films with decreasing d (and To) , where here we have defined Tc(H ) to be that temperature where R ( T ) = RN/2. Note that for the thick films (highest To) the behavior can be well described by the standard G L A G behavior. However, with de-

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J.M. Graybeal / Competition between superconductivity and localization in 2D

creasing film thickness, the measured Hc2(T ) behavior progressively deviates from G L A G behavior. Looking at the thinnest film ( 2 9 ~ ) shown, we see that if we fit G L A G to the initial critical field slope at T c, then we appear to substantially overestimate H~2(T ) at lower temperatures. If instead we try to fit G L A G to the low temperature behavior, then we overestimate T~. However, there are subtleties present, as we also observe increasingly strong field-dependent fluctuations with decreasing film thickness. For H = 0, all of the resistive transition widths are very small; ATe{H=0) < 5 m K for the thickest film, and AT~{~=0)<75mK for the thinnest film. In finite field, the transition widths for the thickest films remain manageable, of order 250 m K for H = 75 kOe. However, for the ultrathin films, the widths rapidly balloon to enormous proportions. As noted in the figure, the resistive transition width ( 2 0 - 8 0 % RN) for the 29 ~ film in a 30 kOe filed is about 1.2 K wide. Such an increase in the fluctuations is not unexpected, however. Lee and Shenoy [15] note that with increasing field there will be a crossover in the dimensional behavior for the fluctuations, with a resultant growth of the critical region of order ATc(H)/T ~- (HI/Tcd) 3 1/2, where l is the mean free path. Simply stated, with increasing field the Landau orbit radius will constrain the fluctuation size, and therefore for large enough H the dimensionality will cross over from D to (D - 2) dimensions. Checking the numbers, we indeed find that their predicted ATe(H) gives a critical region of about 1 K for our 29/~ film in a 3 0 k O e field. Furthermore, if we examine the functional behavior of the measured transition widths upon H (shown in fig. 8), we indeed observe that they are proportional to H 1/2, as predicted. This therefore seems to explain the rapid broadening of the transition widths with increasing H and decreasing d. Therefore, in the absence of a known way to precisely determine the Tc in finite field, we must be careful about our conclusions. However, for any definition of T~(H) based upon R(T) equal to a fixed percentage of RN, we find progressive deviation in the critical field behavior for G L A G with increasing R D. These deviations include an increase in the initial critical field slope and an

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increasing amount of negative curvature just below Tc(H---0). We also observe very strong H - d e p e n d e n t fluctuations as mentioned above. We now turn to recent theories of the consequence of localization effects upon superconductivity. Calculations have been done by several workers in both 2D and 3D, including Anderson et al. [16], Maekawa and Fukuyama [17], and Kapitulnik and Kotliar [18]. Many new formulations exist. Although we have not discussed the results here, we have found the localization corrections to the normal state properties to be well described by calculations valid in the weakly localized regime. Therefore, it seemed very reasonable to turn to the results of Fukuyama and co-workers, which are perturbation theory calculations valid in the WLR. Their calculations are first order in A = ( 2 7 r E v r ) -1, which in 2D is equal to RD/81 k ~ . An important parameter is the measure of "dirtiness", too r, by which a-MoGe is classified as a very dirty material (too r ~ 1). They predict several reasons for the suppression of superconductivity by localization. First, there is

118

J.M. Graybeal / Competition between superconductivity and localization in 2D

the reduction in the density of states due to the Coulomb interaction. Second, there is increased pairbreaking due to retardation effects. However, the real "muscle" in their calculation comes from the enhancement of the screened Coulomb interaction at long wavelengths. This is a consequence of the dynamical nature of the screening brought about by the disorder. Since the electrons are diffusing, that implies that density fluctuations are not instantaneously screened, as the electrons need time to respond. And since diffusion is particularly slow at small q (long wavelengths) due to localization effects, we therefore are left with a large effective coulomb interaction at large distances. We therefore turned to the most recent 2D calculation in the dirty limit (toD~-~ 1) by Ebisawa et al. [19]. Their calculation involve parameters such as wo (obtained by Carter [5] for M o - G e by specific heat measurements), the diffusion constant D (obtained from bulk critical field slopes), the mean free path l (estimated to be - 3 ~ for amorphous Mo-Ge), the effective Coulomb interaction /~* (estimated to be -~0.1) and the inverse screening length K (estimated from free electron results). Of the estimated parameters, the theory is only weakly sensitive to /~* and l; however, it is very sensitive to K. Estimating K to be of order 0.25 ~ from free electron theory we find that the theory overestimates the initial slope of the Tc suppression versus R~ by only about 30%. Shifting K by only a factor of two brings the theory into excellent accord with the experimental results down to film thicknesses of =30 (R~ ~ 500 1)). Therefore we find that the recent theoretical calculations agree well with the data with a very reasonable value for the only "free" parameter, K. With this understanding, it is therefore reasonable to ask who wins as T--* 0, superconductivity or localization? Somewhat nai'vely, since localization effects continue to grow with decreasing temperature as 1 / T to some power, while superconductivity essentially saturates as T--~0, it therefore begs the question ast to whether there could be localization-induced reentrant superconductivity. Going even further, one wonders what impact localization will have upon the quasiparticles. With this in mind, we monitored the resis-

tance of our thinnest superconducting films down to T--30 inK, both with and without a magnetic field. At least down to these temperatures, we did not observe any signs of reentrance, although this is not a conclusive answer to our question. Certainly this remains an interesting possibility to pursue. Returning to the predicted reduction in screening at large distances, we find many interesting points for further consideration. As we alluded to earlier, any actual structure built into the film that would influence this long-wavelength screening (e.g., such as percolation between connecting "islands") could be of consequence to the T~ suppression versus R[]. Furthermore, one could conceivably improve this screening by placing a good conductor nearby. This could by easily done by first evaporating an insulating film, and then a good metal onto the 2D superconductor. The only point requiring care would be that the tunneling resistance through the insulator be large enough such that electrons could not easily tunnel across into the metal, thereby avoiding impurity scattering sites in the superconductor (and raising the effective dimensionality). The results of such a study would indeed be interesting. Certainly the magnitude of the localization effects upon 2D superconductivity appear surprisingly large. One cannot help wondering how relevant such effects are in bulk systems of technological importance. Recall that the relevant measure of disorder in 3D is the resistivity p. Certainly a great amount of experimental results exist in 3D systems. A careful study of such data with an eye to localization effects deserves consideration. Another interesting topic for further study is the importance of disorder to superconductivity in 1D. As discussed earlier, perturbation theory calculations predict that the suppression of T~ is proportional (1/EF'c) D-1 in D dimensions. Therefore, perturbation theory breaks down in 1D. Experimentally, this could be achieved utilizing the present state-of-the-art electron-beam lithographic techniques. Such a system would be an interesting model system for studying the effect of disorder upon some of the 1D organic superconductors.

J.M. Graybeal / Competition between superconductivity and localization in 2D

In conclusion, we have presented the results of a systematic study of the consequences of increasing disorder upon superconductivity in two dimensions. For these purposes we have synthesized a model 2D system, with essentially constant equivalent bulk properties down to film thicknesses of ---10/~. We find a strong, initially linear suppression of Tc with increasing R[]. The experimental evidence argues against proximity or simple pairbreaking effects as being the origins of this reduction. We also observe progressively apparent deviations in the critical field behavior with increasing RD, combined with the observation of strong field-dependent fluctuations. We find that recent formulations of the localization theory are in essential agreement with our Tc results. And finally, we do not observe any sign of localization-induced reentrance within the temperature regions that we were able to explore.

Acknowledgments This work was primarily done at Stanford University under NSF-MRL support as part of the author's doctoral research. The author wishes to acknowledge the essential contributions and guidance of Professor M.R. Beasley in the course of this work. Additionally, we wish to thank R.L. Greene, who collaborated in the measurements below 1 K. We also gladly acknowledge helpful and stimulating discussions with H. Fukuyama, S. Maekawa and E. Abrahams. References [1] E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673.

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[2] B.L. Altshuler, A.G. Aronov and P.A. Lee, Phys. Rev. Lett. 44 (1980) 1288. [3] For a good review, see R.C. Dynes and P.A. Lee, Science 223 (1984) 355. [4] J.B. Kortright, Ph.D. Dissertation, Stanford University (1984), unpublished. [5] W.L. Carter, Ph.D. Dissertation, Stanford University (1983), unpublished. [6] J.M. Graybeal and M.R. Beasley, Phys. Rev. B29 (1984) 4167, and to be published. J.M. Graybeal, M.R. Beasley and R.L. Greene, Proc. of the Intern. Conf. on Low Temp. Phys. LT-17, Karlsruhe, FRG, U. Eckern et al., eds., Elsevier Sci. Publ., Amsterdam (1984), and to be published. M.R. Beasley, S.J. Bending and J.M. Graybeal, Proc. of the Intern. Conf. of Localiz., Interactions and Transport Phenom. of Impure Metals, Braunschweig, FRG (1984). J.M. Graybeal, Ph.D. Dissertation, Stanford University (1985), unpublished. [7] S. Hikami, A.I. Larkin and Y. Nagaoka, Prog. Theor. Phys. 63 (1980) 707. H. Fukuyama, J. Phys. Soc. Jpn. 50 (1982) 1105. [8] M.R. Beasley, J.E. Mooij and T.P. Orlando, Phys. Rev. Lett. 42 (1979) 1165. [9] D.G. Naugle and R.E. Glover, Phys. Lett. 28a (1969) 611. D.G. Naugle, R.E. Glover, III, and W. Moormann, Physica 55 (1971) 250. [10] M. Strongin, R.S. Thompson, O.F. Kammerer and J.E. Crow, Phys. Rev. B1 (1970) 1078. [11] H. Raffy, R.B. Laibowitz, P. Chaudhari and S. Maekawa, Phys. Rev. B28 (1983) 6607. [12] S. Okuma, F. Komori, Y. Ootuka and S. Kobayashi, J. Phys. Soc. Jpn. 52 (1983) 2639. [13] L.N. Cooper, Phys. Rev. Lett. 6 (1961) 689. [14] J. Haibritter, Solid State Commun. 18 (1976) 1447. [15] P.A. Lee and S.R. Shenoy, Phys. Rev. Lett. 28 (1972) 1025. [16] P.W. Anderson, K.A. Muttalib and T.V. Ramakrishnan, Phys. Rev. B28 (1983) 117. [17] S. Maekawa and H. Fukuyama, J. Phys. Soc. Jpn. 51 (1981) 1380. [18] A. Kapitulnik and G. Kotliar, Phys. Rev. Lett. 54 (1985) 473. [19] H. Ebisawa, H. Fukuyama and S. Maekawa, to appear in J. Phys. Soc. Jpn.