Magnetoresistance measurements of highly underdoped superconducting La2−xSrxCuO4 films with x = 0.051 and x = 0.048, performed in dc magnetic fields up to 20 T and at temperatures down to 40 mK, reveal a magnetic–field induced transition from weak to strong localization in the normal state. The normal–state conductances per CuO2–plane, measured at different fields in a single specimen, are found to collapse to one curve with the use of a single scaling parameter that is inversely proportional to the localization length. The scaling parameter extrapolates to zero near zero field and possibly at a finite field, suggesting that in the zero–field limit the electronic states may be extended. Typeset using REVTEX 1 The unusual normal–state transport properties of high–Tc superconductors include strongly anisotropic resistivities, a temperature–dependent Hall effect, and anomalous magnetoresistance [1]. The character of the electronic ground state underlying superconductivity is the subject of experiment and speculation, and is expected to be different according to different models, with suggestions that it is insulating [2], or weakly localized in two dimensions (2D) [3], as recently reported in La2−xSrxCuO4 (LSCO) in very high magnetic fields [4]. Extremely high magnetic fields are required to quench superconductivity when the transition temperature is high. We have therefore investigated strongly underdoped, but still superconducting specimens, with values of Tc reduced below 4 K, in magnetic fields up to 20 T, at temperatures down to 40 mK. We show that the field localizes the carriers, leading to variable–range hopping at the highest fields and lowest temperatures, so that the behavior observed in strong fields is not a reliable guide to the nature of the zero–field electronic “ground state” in the absence of superconductivity. We also show that the normal–state conductance per CuO2 plane, measured at different fields in one specimen, may be collapsed to a single curve by adjusting a single scaling parameter. The scaling indicates a gradual transition from weak localization at low fields to strong localization at high fields, similar to the disorder–induced localization observed in conventional 2D and 3D metals and semiconductors [5–8]. However, within our experimental accuracy, the scaling parameter T0, which is inversely proportional to the localization length, extrapolates to zero at fields that are close to zero and possibly finite, suggesting that in the zero–field limit the electronic state underlying superconductivity may be extended. The specimens were c–axis aligned epitaxial films, grown by pulsed laser deposition on SrLaAlO4 substrates [9]. They were patterned by photolithography, and silver pads were evaporated for four–point resistivity measurements. The magnetoresistance (MR) measurements were made in magnetic fields up to 20 T, generated by Bitter magnets, in two different low–temperature setups to check for consistency. One was a He cryostat with dc measurements and temperatures down to 600 mK. The other was a dilution refrigerator in which 2 3 Hz–ac was used, with temperatures down to 40 mK. Most of the data were accumulated by sweeping the field. Several runs were also made by sweeping the temperature, and were found to be consistent with the others. The data from the two setups differred by less than 5%. This difference reflects slightly different field calibrations and small differences in current, and is insignificant for the discussion of this paper. We measured two films with a nominal composition given by x = 0.051. The values of Tc, ρ and MR differred only slightly, so that we present the results for only one of them. It is designated as specimen S1, with a value of Tc of 3.8 K, and ab–plane resistivity, ρab, at 40 K equal to 2.9 mΩcm. The temperature dependence of ρab in zero field is shown in the inset of Fig. 1. We also measured the MR of a third film, S2, with nominal composition x = 0.048, and Tc = 400 mK, which was measured earlier up to 6 T [10]. In all cases the magnetic field was perpendicular to the ab–plane. In Fig. 1 ρab for specimen S1 is plotted against ln T for fields from 7 to 10.6 T, and in Fig. 2 for fields from 7 to 20 T. It is apparent that the field gradually quenches superconductivity and induces a superconductor–insulator transition, similarly to the behavior described previously for specimen S2 [10], except for the higher fields that are necessary to suppress superconductivity in S1. In Ref. [10] we analyzed the nature of this S–I transition, and found that it differs from the Cooper–pair localization predicted by Fisher [11]. The inset to Fig. 2 shows ρab for specimen S1, plotted as ln ρab against T . For the highest fields the data follow straight lines to T = 2 (corresponding to T = 60 mK), consistent with 3D Mott variable–range hopping [12]. The slopes of the straight lines increase with increasing field, pointing to field–induced localization of the carriers. In the field and temperature range of this experiment the MR was positive for all films, approaching an approximately linear dependence on field at the highest fields. This differs from the results of Ref. [4] on single crystals of LSCO with x = 0.08 and 0.13, where the MR in the limit of high fields was found to be negative. The saturation of ρab below 60 mK is presumably a result of superconducting fluctuations. For lower fields the fluctuations occur at higher temperatures, and the T–dependence of ρab 3 becomes weaker than exponential. It may be seen that for some fields and temperatures the T–dependence is close to ln(1/T ), as observed in Ref. [4] down to about 0.7 K. It is apparent, however, that this is only an intermediate stage in the gradual evolution from variable–range hopping to weakly localized and eventually metallic behavior, so that the logarithmic dependence by itself does not seem to have any special significance. Fig. 3a shows the data for film S1 as a log–log graph of the conductance per single CuO2 plane, G, against temperature, at different fields. We now adopt the scaling procedure used in several previous studies of disorder–induced localization in various 2D and 3D systems [5–8]. We find that shifting the data for the different fields along the lnT–axis allows the collapse of the normal–state data to a single curve, as shown in Fig. 3b. In this procedure we plot the data against ln αT , where α(B) is set equal to one for B = 20 T, and chosen for other fields so as to superimpose the curves, as on Fig. 3b. The deviations on the low–T side result from superconducting fluctuations. The ln G scale is normalized by the constant G00, equal to e /2πh̄, to allow a direct comparison with the results of Ref. [6], where G00 was found to separate the strong and weak localization regimes in metallic disordered 2D films. We find variable–range hopping in the limit G/G00 ≪ 1, changing to a weaker T– dependence as G/G00 approaches one. This behavior resembles the transition from strong to weak localization described in Refs. [5] and [6]. The strong–localization region is characterized by the parameter T0 in the Mott variable– range hopping law, which is inversely proportional to the localization length. Since the conductance depends on temperature only in the combination T0/T , a shift from T to αT is equivalent to a shift from T0 to T0/α, so that T0(B) = T0(20T)/α. We therefore use the factor α(B), determined as the factor in the scaling procedure that superposes the curves on Fig. 3b, to determine also T0(B), even in the regime where the charge carriers are no longer strongly localized and Mott’s law no longer applies. At some lower, possibly inaccessible temperature, Mott’s law can be expected to hold again, with this value of T0. This definition of T0 allows us to plot the scaled conductance as a function of ln T/T0, as on the upper scale of Fig. 3b. 4 In Fig. 4 we plot the normal–state curve constructed in this way for the two films S1 (x = 0.051) and S2 (x = 0.048). In order to cause the two curves to be superimposed to form a single curve, it is necessary to rescale not only the horizontal, but also the vertical axis. This is different from the case of disorder–induced localization in metallic 2D films [5,6], where the conductance approaches a constant value, independent of disorder, in the high–T limit. This difference may be related to the fact that the metal–insulator transition in LSCO is inherently different, apparently driven primarily by band filling [13]. The figure also shows the data from Ref. [4], for an LSCO crystal with x = 0.08, in a pulsed field of 50 T, scaled to join the other curves at the high–T end of their data. It may be seen that for this specimen even the 50 T–field does not induce strong localization, and that the curve departs from the shape of the other two curves as the temperature is lowered. If, as stated in Ref. [4], the field is sufficient to suppress superconducting fluctuations, the curve from their data on Fig. 4 seems to indicate the likelihood of metallic character for their specimen. This conclusion differs from that of Ref. [4], where specimens up to the optimally doped, i.e. for values of x ≤ 0.15, are said to be insulating. There are two reasons for the difference in our conclusions. First, Ando et al. characterize a specimen as “insulating” when the slope of R(T ) is negative, while we use the much more stringent criterion that there must be a finite value for T0 and hence for the localization length. Second, as we show, the localization in the field does not necessarily imply localization in the absence of a field. These differences in interpretation are not likely to be affected by differences in the specimen characteristics, such as that implied by the negative MR of the specimen of Ref. [4], which suggests differences in spin scattering, presumably resulting from differences in structure and composition. Strict adherence to scaling would lea