We have investigated the field-induced superconductivity-destroying quantum transition in amorphous indium oxide films at low temperatures down to 30 mK. It has been found that, on the high-field side of the transition, the magnetoresistance reaches a maximum and the phase can be insulating as well as metallic. With further increasing magnetic field the film resistance drops and approaches in the high-field limit the resistance value at transition point so that at high fields the metallic phase occurs for both cases. We give a qualitative account of this behavior in terms of field-induced destruction of localized electron pairs.The theoretical description of the zero-field and fieldinduced quantum superconductor-insulator transitions (SIT) in a 2D superconductor is based on a concept of electron pairs which are delocalized on the superconducting side and localized on the insulating side of transition [1][2][3]. According to Refs. [1][2][3], the temperature dependence of the film resistance near the field-induced SIT is controlled by deviation δ = B − B c from the critical field B c and the most specific among perceptible features of SIT is fan-like set of resistance-vs-temperature curves R δ (T ). Such a set is expected to collapse onto a single curve as a function of scaling variable δ/T 1/y , where y is the critical index, see review [4]. Many of the SIT studies were performed on amorphous In 2 O x (x < 3) films whose conductivity is caused by oxygen deficiency compared to fully stoichiometric insulating compound In 2 O 3 : by changing the oxygen content one can cover the range from a superconductor to an insulator and thus realize the zero-field SIT. On the insulating side of this SIT, observation was reported of the activation behavior of the resistance R ∝ exp(T 0 /T ) p with p = 1 (Arrhenius law) and activation energy T 0 tending to zero as the phase boundary is approached [5]. It was found later that switching a magnetic field results in decreasing the resistance and weakening its temperature dependence from the Arrhenius law to the Mott law with exponent p = 1/4 [6]. This was explained in Ref.[6] by magnetic-fieldcaused suppression of the binding energy ∆ of localized electron pairs which manifests as a gap at the Fermi level.Field-induced SIT is realized on the superconducting side of zero-field SIT. It is indicated by fan-like structure of experimental curves R δ (T ) such that, in accordance with the scaling analysis, the expected collapse is indeed the case [7]. Above the field-induced SIT, the existence of two insulating phases was postulated based on results of Hall measurements [8]; however, temperature dependences of the resistance of these phases were not studied. Reversal of a zero-bias peak in the differential resistance at the critical field B c was observed and attributed to the granular structure of films [9].

We have studied the magnetic-field-tuned superconductor-insulator quantum transition (SIT) in amorphous In-O films with different oxygen content and, hence, different electron density. While for states of the film near the zero-field SIT the two-dimensional scaling behaviour is confirmed, for deeper states in the superconducting phase the SIT scenario changes: in addition to the scaling function that describes the conductivity of fluctuation-induced Cooper pairs, there emerges a temperature-dependent contribution to the film resistance. This contribution can originate from the conductivity of normal electrons.

We study the response of a thin superconducting amorphous InO film with variable oxygen content to a parallel magnetic field. A field-induced superconductor-insulator transition (SIT) is observed that is very similar to the one in normal magnetic fields. As the boson-vortex duality, which is the key-stone of the theory of the field-induced SIT, is obviously absent in the parallel configuration, we have to draw conclusion about the theory insufficiency.PACS numbers: 05.70 Fh, 74.20 Mn, 74.25 Dw A good deal of work was performed on the investigation of the magnetic-field-induced superconductor-insulator transition (SIT) on superconducting amorphous films of InO [1][2][3], MoGe [4], and MoSi [5] with thickness comparable to the superconducting coherence length. It was found that with increasing field B the resistance R of all studied films rises abruptly at a magnetic field B c and then passes through a maximum followed by a drop in high magnetic fields [2,3]. The film state just above B c was identified as insulating although at lowest temperatures about 30 mK the maximum resistance does not exceed 100 kΩ and the temperature dependences of 1/R correspond to the activation energies which do not exceed by far the lowest temperatures. Near B c at sufficiently low temperatures, the resistance R(T, B) was found to be a function of single scaling variable u = (B−B c )/T y with exponent y ≈ 0.8. The above experimental findings are regarded to confirm the theory of the quantum SIT [6] in two-dimensional (2D) superconducting films subjected to a normal magnetic field. This theory exploits the concept of the hypothetical system of charged bosons in a random potential and is based on the boson-vortex symmetry of the model Hamiltonian. In the vicinity of the SIT point (T = 0, B = B c ) the film resistance R(T, B) is expected to be a universal function of the single scaling variable which is defined as the ratio of the correlation length ξ ∝ (B − B c ) −ν and the dephasing length L φ ∝ T −1/z , where ν and z are the critical indices. The form of the scaling variable implies that the value 1/zν has to be identified with exponent y. The concept of the localization of electron pairs, or bosons [6], has been supported recently by the work of Ref. [7]. There, it is shown that for a 2D superconducting film with strong disorder the region of fluctuation superconductivity, where the electron pairs occur, should extend down to zero temperature. In this region the unpaired electrons are supposed to be localized whereas the bosons can be either localized or delocalized, dependent on the value of magnetic field.Other experimental results were interpreted within the model of electron pair localization: (i) a crossing of Hall isotherms R xy (B) was observed at a field B c0 > B c and attributed to a transition between the Bose-insulator and a Fermi-insulator that consists of localized single electrons, i.e., pairing is presumed to be destroyed at B c0 [1]; (ii) the resistance drop in high fields was explained in terms of the electron pair...

Resistive superconducting zero-field transition in amorphous In-O films in states from the vicinity of the insulator-superconductor transition is analyzed in terms of two characteristic temperatures: the upper one, Tc0, where the finite amplitude of the order parameter is established and the lower one, Tc, where the phase ordering takes place. It follows from the magnetoresistance measurements that the resistance in between, Tc < T < Tc0, cannot be ascribed to dissipation by thermally dissociated vortex pairs. So, it is not Kosterlitz-Thouless-Berezinskii transition that happens at Tc.

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