Abstract
Competing phenomena in low dimensional systems can generate exotic electronic phases, either through symmetry breaking or a nontrivial topology. In twodimensional (2D) systems, the interplay between superfluidity, disorder and repulsive interactions is especially fruitful in this respect although both the exact nature of the phases and the microscopic processes at play are still open questions. In particular, in 2D, once superconductivity is destroyed by disorder, an insulating ground state is expected to emerge, as a result of a direct superconductortoinsulator quantum phase transition. In such systems, no metallic state is theoretically expected to survive to the slightest disorder. Here we map out the phase diagram of amorphous NbSi thin films as functions of disorder and film thickness, with two metallic phases in between the superconducting and insulating ones. These two dissipative states, defined by a resistance which extrapolates to a finite value in the zero temperature limit, each bear a specific dependence on disorder. We argue that they originate from an inhomogeneous destruction of superconductivity, even if the system is morphologically homogeneous. Our results suggest that superconducting fluctuations can favor metallic states that would not otherwise exist.
Introduction
In 2D systems, disorder induces quantum interferences between electronic wave functions, eventually leading to their localization. No matter how weak the disorder, Anderson localization prevents the diffusion of the electronic motion so that no 2D metal can exist^{1}. However, in the presence of competing orders, new electronic ground states can prevail^{2}. In the case of thin disordered superconducting films, when Cooper pairing, competing with Coulomb repulsion, is not sufficient to establish a longrange coherence between localized states, an insulator is expected to arise from the superconducting ground state. This SuperconductortoInsulator Transition (SIT) is either explained by amplitude or phase fluctuations of the superconducting order parameter, and conventional theories do not allow for any intervening metallic state^{3}. Experimentally however, dissipative behaviors have been observed^{4,5,6,7,8,9,10,11}. Several hypotheses have been put forward to explain these, amongst which: vorticesinduced dissipation^{4,5}; a coupling to a dissipative bath^{6}; the existence of a quasi2D metal^{12}; the competition between Josephson coupling and the charging energy^{13}; important Coulomb interactions^{14}. Moreover, in the vicinity of the SIT, the focus has also recently been set on fluctuations as a possible cause for inhomogeneous electronic phases^{15,16}. The nature and origin of such metalliclike behaviors is thus still debated. Whether they live on at T = 0 and constitute one of the system’s ground states is also an important issue that needs to be solved. In order to gain more insight on these questions, a systematic quantitative analysis of these dissipative phases would be extremely profitable.
Thin metal alloy films constitute particularly wellsuited systems for this study. Indeed, in compounds such as aNb_{x}Si_{1−x} (aNbSi) which we consider, disorder can be progressively increased either by a reduction of the sample thickness or by a variation in stoichiometry, but also through a thermal treatment^{17,18}. In this letter, we present the phase diagram of aNbSi thin films across the disorderinduced SIT, along with the characterization of corresponding phases. Once superconductivity is destroyed, we surprisingly identify two metallic regimes – that we have called “Metal 1” and “Metal 2”. The insulating phase subsequently appears for films of normal conductivity smaller than e^{2}/h. This suggests that, in this system, the SIT and the MetaltoInsulator Transition (MIT) happen successively. The Metal 1 can be interpreted as emerging from superconducting fluctuations, whereas the Metal 2 should be of fermionic nature.
In this study, we have considered 40 samples with compositions x from 8.5 to 18.5%, thicknesses d from 4 to 50 nm, and submitted to heat treatments at temperatures θ_{ht} from 70 to 250 °C. The effects of these disorderchanging parameters (x, d, θ_{ht}) on the properties of superconducting aNbSi thin films have been described elsewhere^{17}. We will here examine the transport features of the films as a function of σ_{N}, the normal state conductivity at T = 500 mK, taken as a measure of disorder. All resistances (conductances) will be plotted in units of the quantum of resistance (conductance) h/e^{2} (e^{2}/h). The films are continuous, uniformly disordered and amorphous up to θ_{ht} = 500 °C. The deposition and heat treatment procedures are described in the Methods section. When superconducting, the samples are either 2D or in the quasi2D limit, i.e. , where ξ is the superconducting coherence length. The films were measured using standard low frequency transport measurements at very low temperature (see Methods).
All films fall into one of four categories, identified by their low temperature transport characteristics. We will identify each one on a specific example – a 23nmthick aNb_{13.5}Si_{86.5} film that has been thermally treated –. We will then quantitatively analyze each ground state and show how their features evolve with disorder.
The low temperature transport properties of the 23nmthick aNb_{13.5}Si_{86.5} film are represented in Fig. 1a. The asdeposited film is superconducting, with a welldefined zero resistance state below the critical temperature T_{c} = 50 mK. As is usual, the superconducting phase presents a positive Temperature Coefficient of Resistance () at low temperature. After thermal treatment, σ_{N} decreases, signaling an enhanced effective disorder. Up to θ_{ht} = 140 °C, the low temperature TCR stays positive, but the sheet resistance R_{min} measured at T = 10 mK is finite. For θ_{ht} = 110 °C, the resistance even saturates at low temperature. The sample then is in a phase that we shall call Metal 1. For θ_{ht} > 140 °C, the sample is characterized by a negative TCR and a finite R_{min}, corresponding to a phase we have named Metal 2. All finite resistances measured at our lowest temperature have been checked to be intrinsic and not due to experimental artefacts (see Methods). At even larger disorder, the system eventually becomes insulating. This phase can be described by a negative TCR and a diverging low temperature resistance (not reached in the data set shown in Fig. 1a).
Let us analyze each phase more quantitatively. First, the superconducting state is characterized by two temperature scales. Starting from the high temperature regime, the resistance abruptly drops at T_{c0}, defined by the temperature at which the derivative is maximum . For low disordered films, this energy scale corresponds to the characteristic superconducting energy (see Methods and supplementary materials). The resistance then becomes null below the superconducting critical temperature T_{c}, indicating the establishment of global phase coherence. The evolution of these two characteristic temperatures with disorder is shown in Fig. 1b for 23nmthick samples. At low disorder, the two energy scales evolve in a similar manner. However, at a first critical conductivity σ_{c1}, T_{c} drops to zero, signaling the destruction of the macroscopic phase coherence, while T_{c0} continues to be finite. σ_{c1} therefore marks the transition between the superconducting and the Metal 1 phases. As will be seen in the phase diagram (Fig. 4), σ_{c1} depends linearly on the thickness d and, in the strictly 2D limit, extrapolates to σ_{Q} = 4e^{2}/h. In the Metal 1 phase, the R(T) characteristics abruptly drop below T_{c0}, but, while always maintaining a positive TCR, extrapolate to a finite value at T = 0. We have estimated this residual resistance to be close to R_{min}, the resistance measured at T = 10 mK. The evolution of T_{c0}, R_{min}, and of the sign of the TCR – all three characteristic of the Metal 1 phase – with disorder is given in Fig. 1b for the 23nmthick aNb_{13.5}Si_{86.5} film. Although, in this phase, T_{c0} no longer represents the superconducting critical temperature, its continuous evolution from the superconductor into the Metal 1 regime suggests the importance of superconducting fluctuations in this latter state. As can be seen, R_{min} rapidly increases with increasing disorder, until, at a second critical conductivity σ_{c2}, the three quantities reflect a change of regime: the evolution of the residual resistance slows down, T_{c0} extrapolates to zero and the TCR changes sign. For all measured samples, the modifications in the behavior of the TCR, of R_{min} and of T_{co} simultaneously occur at σ_{N} = 1/R_{min}, i.e. for a sample which would have a constant conductivity from T = 10 mK to T = 500 mK. However σ_{c2} is not universal and varies with the film thickness as can be seen in Fig. 4.
Let us now specifically consider the Metal 2 phase. Figure 2a shows the evolution of the transport characteristics of a 5nmthick aNb_{13.5}Si_{86.5} film with disorder. All curves show the same qualitative features: from the high temperature regime, the resistance first progressively increases as is expected on the insulating side of the SIT. However, below a characteristic temperature T_{sat}, the concavity of the R(T) curve changes, while maintaining a negative TCR, and levels off at low temperature at R_{min}. T_{sat} can then be defined as the highest temperature at which R_{min} is attained within 1%. This temperature is characteristic of the energy scale at which the saturation of the resistance sets in. The evolution of T_{sat} with disorder is plotted in Fig. 2b for the 5nmthick aNb_{13.5}Si_{86..5} film: as disorder is increased, the temperature range over which shrinks, while the product R_{min} σ_{N} becomes larger, showing a progressive weakening of the metallic state. In Fig. 2c is plotted the reduction of the residual conductivity σ_{min} = 1/R_{min} as the disorder level is increased, for all samples in the Metal 2 regime. The evolution is universal, thus showing that the sole knowledge of normal state conductivity σ_{N} fully determines the value of σ_{min}, whatever the experimental disordertuning parameter (x, θ_{ht} or d). This second metallic regime vanishes close to .
For films such that σ_{N} < e^{2}/h, transport properties are consistent with an insulator typical of the SIT: as can be seen in Fig. 3a, they are well described by a phenomenological Arrhenius law: R(T) = R_{0} exp (T_{0}/T). The evolution with disorder of the characteristic temperature T_{0} and the corresponding R_{0} is displayed in Fig. 3b. Like σ_{min} in the previous Metal 2 phase, T_{0} decreases universally as σ_{N} increases, whatever the disordertuning parameter. Both characteristic parameters concurrently vanish at a third critical conductivity σ_{c3} = (1.8 ± 0.8)e^{2}/h.
Although the microscopic nature of these dissipative regimes is unclear for now, one can formulate some hypotheses. The continuous evolution of T_{c0} from the superconducting phase into the Metal 1, where its value remains finite, assesses the importance of superconducting fluctuations in this first metallic regime. This argument is also supported by the thicknessdependence of all relevant quantities (T_{c0}, R_{min}, σ_{c1}, and σ_{c2}) in this phase. Indeed, a previous work had evidenced the specific role of d, as opposed to θ_{ht} or x, on the superconducting properties of aNbSi films^{17}. This distinct role is also found in the Metal 1 phase, but not in the Metal 2 or insulating regimes. The Metal 1 could then correspond to a phase where short living Cooper pairs survive locally to the destruction of global superconductivity. In the literature, one can find both theoretical predictions and experimental evidence of such an inhomogeneous electronic phase with nonzero pairing amplitude, coming from the destruction of superconductivity via phase fluctuations^{8,9,13,15,19}. However these works considered systems with builtin inhomogeneities and cannot, rigorously, be applied to our system. Moreover, and although numerous experimental and theoretical works dealt with the possibility of a metallic state as a result of the interaction between disorder and superconductivity^{4,5,6,7,8,9,10,11,12,13,15,16}, to the best of our knowledge, there has been no prediction of two distinct dissipative regimes in morphologically homogeneous systems at zero magnetic field. In the Metal 2 phase, by contrast, the transport characteristics seem to be only governed by the normal conductivity σ_{N}. This signals a transition or a crossover to a quasiparticulesdominated transport regime. In this phase, the saturation of the resistance at low temperature could be explained by a parallel channel of conduction shortcircuiting the localized fermions^{20}. The origin or nature of this additional channel, whether of quantum metal or incoherent Cooper pairs, is still to be determined, but also supports the inhomogeneous nature of electronic transport in this system.
All our results can be summarized in the phase diagram shown in Fig. 4, where the different regimes are represented as a function of disorder and film thickness. As can be seen, at lower thicknesses, the superconducting state occupies a larger portion of the phase diagram. Although counterintuitive, this reflects how enhanced superconducting fluctuations may actually help the establishment of global phase coherence. In the d → 0 limit, Metal 2 vanishes, as could be expected in the case of a fermionic metal^{1} where the least amount of disorder is sufficient to localize the electrons at low enough temperature. The insulating phase sets in at the critical conductivity , independently of the film thickness. This level of disorder corresponds to the IoffeRegel criterion at which the MIT is expected in 3D^{21,22}. This criterion has also been observed in 2D electron gases at the interface of semiconductor heterostructures^{23}. The most surprising feature of this phase diagram is the appearance of the Metal 1 phase, which vanishes in the 3D limit but survives in the strictly twodimensional limit, insofar as our results can be extrapolated. It grows from superconductivity and accompanies its evolution down to the lowest thicknesses. This intriguing dissipative state shows how quantum fluctuations do not only destroy ordering but may also significantly contribute to the appearance of new electronic phases.
Methods
Sample deposition and heat treatment procedure
aNbSi films have been prepared at room temperature and under ultrahigh vacuum (typically a few 10^{−8} mbar) by electron beam codeposition of Nb and Si, at a rate of the order of 1 Å.s^{−1}. The evaporation rates of each source were monitored in situ by a dedicated set of piezoelectric quartz crystals in order to precisely monitor the composition and the thickness of the films during the deposition. These were also corroborated ex situ by Rutherford backscattering spectroscopy (RBS) measurements.
The films were deposited onto sapphire substrates coated with a 25nmthick SiO underlayer designed to smooth the substrate surface. The samples were subsequently protected from oxidation by a 25nmthick SiO overlayer. Similar films have been measured to be continuous, amorphous, and structurally nongranular at least down to a thickness d = 25 nm^{17}.
The asdeposited films are considered to have experienced a thermal treatment at θ_{ht} = 70 °C, due to heating during the deposition process. This was confirmed by lowtemperature measurements of the conductivity^{17}. Subsequent heat treatments have been performed for 1 h, under a flowing N_{2} atmosphere. To prevent any thermal shock, the samples were then slowly cooled back down to room temperature. TEM measurements were performed on similar films that were annealed in situ from θ_{ht} = 70 °C to 700 °C: no structural nor composition changes have been observed up to an annealing temperature of θ_{ht} = 500 °C^{17}.
Estimation of the superconducting coherence length
A minimal estimate of the superconducting coherence length can be derived from the critical temperature T_{c,bulk} and normal conductivity σ_{N, bulk} of a bulk sample of same stoichiometry, using Gor’kov’s development of the GinzburgLandau theory in the dirty limit: with D = 0.6 cm^{2}.s^{−1}, extracted from critical field measurements^{24}. For x = 13.5% (T_{c, bulk} = 146 mK), we have ξ = 43 nm and for x = 18% (T_{c, bulk} = 965 mK), ξ = 17 nm^{17}. All the considered samples have d < ξ and can therefore be considered to be 2D from the point of view of superconductivity.
Transport measurements
Transport measurements were carried out down to 10 mK in a dilution refrigerator, using a resistance measurement bridge and standard ac lockin detection techniques. The applied bias has been checked to be sufficiently low to be in the ohmic regime, or, when superconducting, below the critical current. All electrical leads are thermalized at the mixing chamber and at the different stages of the cryostat. They are also filtered from RF at room temperature.
To ensure the intrinsic nature of the low temperature dissipative regimes, we have checked that the measured residual resistances R_{min} do not depend on the sample area, nor on its volume or geometry. Note that these dissipative states have also been observed in the case of a single sample that has been progressively annealed, and measured in the same measurement environment.
Determination of T_{c0}
The characteristic temperature T_{c0} is defined by the temperature at which the derivative is maximum . This criterion agrees well with the temperature extracted from superconducting fluctuations analysis close to the superconducting transition. The comparison of the two methods is presented in the supplementary material.
Estimation of R_{min}
We have taken R_{min} = R(T = 10 mk) as an estimate of the residual resistance at zero temperature.
For samples that present resistance saturations at low temperature (both in the “Metal 1” and “Metal 2” regimes), the extrapolation of the resistance at zero temperature is equal (within 1%) to R_{min}. For samples that have not reached the saturating temperature (sample with θ_{ht} = 130 °C in Fig. 1a for example), equating the zerotemperature residual resistance to R_{min} is an approximation. The corresponding error can be estimated by taking the low temperature slope of the R(T) curve and extrapolate it to T = 0. The error has been found to be smaller than 2% (except for θ_{ht} = 130 °C: 18%).
Estimation of σ_{c3}
σ_{c3} can estimated by two different methods:

Either by fitting the T_{0}(σ_{N}) curve of Fig. 3b. This gives σ_{c3} ≈ 1.3.

Or by taking the median point between the last metallic sample () and the first insulating one () which gives σ_{c3} = 1.8 ± 0.8.
It is the latter criterion we have considered to draw the phase diagram in Fig. 4 since the corresponding error bars encompass the first two criteria.
Additional Information
How to cite this article: Couëdo, F. et al. Dissipative phases across the superconductortoinsulator transition. Sci. Rep. 6, 35834; doi: 10.1038/srep35834 (2016).
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Acknowledgements
This work has been partially supported by the ANR (grant No. ANR06BLAN0326) and by the Triangle de la Physique (grant No. 2009019TTSI2D). We thank L. Benfatto, K. S. Tikhonov and M.V. Feigel’man for stimulating discussions.
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L.B. prepared the aNbSi samples. F.C., O.C., A.D., V.H. and C.M.K. carried out the experiments. F.C. and C.M.K. analyzed the data. F.C. and C.M.K. wrote the manuscript and V.H., O.C. and L.D. commented on it. C.M.K. and L.D. initiated this work.
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Couëdo, F., Crauste, O., Drillien, A. et al. Dissipative phases across the superconductortoinsulator transition. Sci Rep 6, 35834 (2016). https://doi.org/10.1038/srep35834
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