Abstract
Three decades after the prediction of chargevortex duality in the critical vicinity of the twodimensional superconductorinsulator transition (SIT), one of the fundamental implications of this duality—the charge BerezinskiiKosterlitzThouless (BKT) transition that should occur on the insulating side of the SIT—has remained unobserved. The dual picture of the process points to the existence of a superinsulating state endowed with zero conductance at finite temperature. Here, we report the observation of the charge BKT transition on the insulating side of the SIT in 10 nm thick NbTiN films, identified by the BKT critical behavior of the temperature and magnetic field dependent resistance, and map out the magneticfield dependence of the critical temperature of the charge BKT transition. Finally, we ascertain the effects of the finite electrostatic screening length and its divergence at the magnetic fieldtuned approach to the superconductorinsulator transition.
Introduction
In the early 1970s, Vadim Berezinskii^{1}, Michael Kosterlitz, and David Thouless^{2,3} introduced the idea of a topological phase transition in which pairs of bound vortex excitations unbind at a critical temperature T_{BKT}. The nature of the BerezinskiiKosterlitzThouless (BKT) transition is different from the standard phase transitions described by the Landau paradigm. There is no symmetry breaking associated with the onset of the order parameter, but instead there is a change in the behavior of the twopoint correlation function. At T < T_{BKT} it decays algebraically, changing to exponential for T > T_{BKT}. Thin superconducting films soon became the principal experimental realization for studying the BKT transition^{4,5}. However, detailed investigations of the physics of the superconductorinsulator transition (SIT) in Josephson junction arrays (JJA) led to the prediction that the chargeanticharge unbinding BKT transition should appear at finite temperature on the insulating side of the SIT^{6,7,8}. Using the framework of gauge theory, Diamantini et al.^{9} demonstrated that in the planar JJA, the vortexcharge duality leads to a zerotemperature quantum phase transition between a superconductor and its mirror image, a state with zero conductance, which they termed a superinsulator. The physical origin of a superinsulating state is the charge confinement due to the logarithmic interaction between the charges in twodimensional (2D) systems^{6,10,11}. In disordered superconducting films, the charge confinement on the insulating side of the SIT results from the divergence of the dielectric constant ε in the critical vicinity of the transition. The logarithmic interaction holds over distances \(d < r\mathop{ < }\limits_{ \tilde {}}{\rm{\Lambda }}\simeq \varepsilon d\), where d is the thickness of the film and Λ is the electrostatic screening length^{11}. This parallels the logarithmic interaction between vortices on the superconducting side, which causes the vortex bindingunbinding topological BKT transition into the superconducting state at finite temperature T = T_{VBKT}^{1,2,3}. Accordingly, logarithmic interactions between charges on the insulating side of the SIT is expected to give rise to a classical charge BKT transition into the superinsulating state with the conductance going to zero at a finite temperature T = T_{CBKT}^{10,11,12}. Applied magnetic fields can tune the SIT with high resolution, offering a window into unexplored electronic functionalities. In the critical vicinity of the SIT on the superconducting side the system should possess superinductance^{13}, and at the insulating side the system is expected to be a supercapacitor due to a diverging dielectric constant^{11}. This calls for a thorough study of the highly resistive state that terminates twodimensional superconductivity at the quantum critical point whose nature remains a subject of intense research^{10,14,15,16,17,18,19}.
Existing experimental data on JJA^{20,21,22}, superconducting wire networks^{23}, InO_{ x }^{19}, andTiN films^{18,24} support the picture of the dual vortexcharge BKT transitions and corresponding formation of the mirror superconductingsuperinsulating states. While there have been numerous experimental hallmarks of superinsulating behavior, the evidence for the charge BKT transition, with its characteristic criticality (see Eq. (1) below), has remained elusive. To address this challenge, we examine a NbTiN film, which is expected to combine the high stability of TiN films with the enhanced superconducting transition temperature T_{ c } of NbN films, due to a larger Cooper pairing coupling constant as compared to TiN. We thus expect that T_{CBKT} is likewise enhanced, opening a wider window for observing critical behavior.
Sample preparation
To grow suitable NbTiN films, we employed the atomic layer deposition (ALD) technique based on sequential surface reaction stepbystep film growth. The fabrication technique is described in detail in the Supplemental Material. This highly controllable process provides superior thickness and stoichiometric uniformity and an atomically smooth surface^{25,26} as compared to chemical vapor deposition, the standard technique used to grow NbTiN films^{27}. We used NbCl_{5}, TiCl_{4}, and NH_{3} as gaseous reactants; the stoichiometry was tuned by varying the ratio of TiCl_{4}/NbCl_{5} cycles during growth^{28}. The superconducting properties of these ultrathin NbTiN films were optimized by utilizing AlN buffer layers grown on top of the Si substrate^{29}. NbTiN films of thicknesses d = 10, 15, and 20 nm were grown, varying only the number of ALD cycles (240, 420, and 768 cycles, respectively), with all other parameters of the ALD process held constant. We show in Fig. 1(a) a highresolution transmission electron microscopy (HRTEM) image of the crosssection of the 10 nm thick NbTiN film. It reveals that both the AlN buffer layer and the NbTiN have a finedispersed polycrystalline structure. Presented in Fig. 1(b) is a plan view of a large area containing many crystallites. The densely packed crystallites have different orientations and are separated by atomically thin intercrystallite boundaries. A statistical analysis of the image finds the average crystallite size to be approximately 5 nm. The electron diffraction data for the film are shown in Fig. 1(c). The clearly seen rings confirm a polycrystalline structure. The analysis of the diffraction data along the direction [220] of the Si substrate displayed in Fig. 1(d) reveals that the NbTiN crystallites have the same rocksalt crystal structure as both NbN and TiN. Using Vegard’s law, we find that our NbTiN film is an approximately 7:3 solid solution of NbN and TiN (See Supplemental Material for further information on the fabrication and measurement techniques, films structural, superconducting and transport parameters, and I–V characteristics).
The films were lithographically patterned into bars and resistivity measurements were performed at subKelvin temperatures in helium dilution refrigerators (see the details of the sample geometry and measurement technique in (See Supplemental Material for further information on the fabrication and measurement techniques, films structural, superconducting and transport parameters, and I–V characteristics)). The temperature dependences of the resistance, R(T), given as resistance per square, are shown in Fig. 2(a) over four decades in temperature. Upon cooling in zero magnetic field, all three films undergo a superconducting transition that manifests as a severe resistance drop. The superconducting transition temperature, T_{ c }, is determined by the inflection point of R(T) and marks the temperature at which thermodynamically stable Cooper pairs appear. For the most highly disordered sample, this precedes the point at which global phase coherence, and hence zero resistivity, occurs^{30}. Instead, with decreasing film thickness (and increasing disorder), spatial fluctuations of the superconducting gap become increasingly important, giving rise to the formation of selforganized structures of superconducting islands in a nonsuperconducting environment^{31,32}. The temperature T_{ c } decreases with decreasing film thickness and consequent increasing sheet resistance.
Results
The resistances of all three films exhibit peaks at temperatures just above T_{ c }, with the peak amplitudes increasing as the thickness decreases. Similar trends were observed in the parent compounds TiN^{33} and NbN^{34} near the SIT and were attributed to quantum contributions to conductivity due to weak localization and electronelectron interaction effects. The sheet resistance of the thinnest film achieves a maximum of 17.56 k\({\rm{\Omega }}/\square \); notably, this well exceeds the quantum resistance R_{ Q } = h/(2e)^{2} = 6.45 k\({\rm{\Omega }}/\square \) which was widely believed to be the upper boundary for the existence of superconductivity in two dimensions^{35,36,37}. A similar peak of 29.4 k\({\rm{\Omega }}/\square \), well above R_{ Q }, was seen in TiN^{38}.
Focusing on the behavior of the thinnest film (d = 10 nm), where the degree of disorder permits fully tuning the sample with experimentallyaccessible magnetic fields, we note first that the global coherent superconducting state is not achieved at lowest temperatures. Instead, the behavior of the zero field resistivity suggests that the film falls into the Bose metallic state^{39}, featuring a finite density of free vortices. The appearance of such vortices in the absence of a magnetic field is a fundamental feature of the BKT transition, which revolves around thermal fluctuations inducing vortexantivortex pairs. Figure 2(b) presents a set of magnetoresistance curves, R(B), taken at different temperatures below T_{ c } = 0.33 K determined by the inflection point of R(T) at zero magnetic field. All of the R(B) data presented below were measured at voltages V = 100 μV, i.e. in the lowvoltage response regime (see the discussion in the end of the “Data analysis” section below). Prominent features of these magnetoresistance curves, especially profound at lowest temperatures, are the crossing point at very low field B_{SIT} = 0.015 T that marks the SIT (discussed below) and the sharp kink at some temperaturedependent magnetic field above which the resistance increases extremely quickly as a function of field. The field associated with the kink shifts to lower fields with decreasing temperature. Inspecting the magnetoresistance behavior in the large field interval, one sees that it shows inherently nonmonotonic behavior marked by a colossal insulating peak, see Fig. 2(c). Importantly, these peaks develop at magnetic fields for which the thicker films are still fully superconducting, i.e. the field B_{max}, where the maximum is observed, is well below the upper critical field B_{c2}. This suggests that the indefinite growth of R(B) at low temperatures/magnetic fields, as well as the peak in the resistance at higher temperatures in the 10 nm film, is an implication of Cooper pairing. The inset in Fig. 2(c) shows that the position of the maximum of the resistance peaks moves to lower fields with decreasing temperature. There is a kink in the B_{max}(T) dependence at \(T\simeq {T}_{c}\), with the slope decreasing significantly when passing to T > T_{ c }. Extrapolating the data to T → 0 shows that B_{ max }(T) shifts to nearly zero field upon decreasing the temperature. Taken together, this indicates that the mechanism that drives the system into the strongly localized state overpowers the effect of the suppression of the Cooper pairing by the magnetic field, which would be expected to diminish the resistance of the Cooper pair insulator^{40}. Note here that the mere nonmonotonicity of magnetoresistance cannot be taken as an indication of the SIT. For example, vortices crossing the superconducting wires under the applied current may also lead to large amplitude oscillations of the magnetoresistance.
To gain insight into the nature of the magneticfieldinduced states, we examine R(T) at different magnetic fields. Figure 3(a) displays for the 10 nm thick film the fanlike set of magnetoresistance curves, characteristic of the magnetic fieldinduced SIT. The crossing point (B_{SIT}, R_{ c }) in Fig. 2(b) now corresponds to nearly temperature independent R(T) at B = 0.015 T, separating the superconducting and insulating behavior. Two important comments are in order here. First, the field of the crossing point B_{SIT} = 0.015 T is two orders of magnitude lower than the upper critical field B_{c2}. This distinguishes it from the crossing point displayed by the thicker film (d = 15 nm) occurring at 11 T and resulting from the quantum contributions to conductivity from superconducting fluctuations^{41,42} as well as similar crossing points in materials such as InO_{ x }^{43}. Second, the resistance at the SIT is R_{ c } = 4.7 kΩ, close but not equal to the quantum resistance 6.45 kΩ.
Replotting these data as log R vs. 1/T curves in Fig. 3(b), we see that the behavior of the resistance in the entire temperature range cannot be reduced to the Arrhenius temperature dependence with a single activation energy. Qualitatively similar behavior has been observed in thin films of the parent compound TiN^{18}, as well as InO_{ x } films^{19} and patterned Josephson Junction Arrays^{20}. Instead of simple Arrhenius behavior, there is a complicated evolution of the resistance curves with increasing magnetic field. While at low fields the log R(1/T) dependence exhibits hyperactivation, i.e. faster than thermally activated growth^{18}, at larger fields, the log R(1/T) curves exhibit a kink and bend down with decreasing temperature. We contrast this to the behavior of InO_{ x } thin films^{19}, where hyperactivation was likewise observed, but is consistent with an EfrosShklovskii model followed by a crossover to a Mott hopping regime^{44}. As shown in Fig. 3(c), the NbTiN film exhibits a pronounced departure from EfrosShklovskii behavior for B ≤ 8 T.
In order to illuminate the physics governing the R(T) behavior, we replot the lowfield data as the conductances, G = 1/R, vs. temperature in Fig. 3(d). We see in the conductance curves an insulating analogue of the drop to zero of the resistance at the onset of superconductivity. In the dual mirror picture of the conductance, we thus see the transition of the system into a superinsulating state characterized by zero conductance at finite temperature. This suggests that we can write the conductance in the generic form ln G ∝ −a/(T − T^{*})^{α} for the finite temperature zero conducting state. Using T ^{*} as an adjusting parameter, we find that α = 0.48 ± 0.03 gives the best fit to the experimental data. This is consistent with α = 1/2 corresponding to critical BKT behavior obtained by a detailed renormalization group analysis^{45} of the system with infinite screening length:
where T_{CBKT} replaces T^{*}, and b is a constant of order unity. In Fig. 3(e) we plot G vs. (T/T_{CBKT} − 1)^{−1/2} for fields 0.12 and 0.25 T. The correct choice of the only adjustable parameter, T_{CBKT}, for each field (shown in the legend for Fig. 3(d)), produces a linear dependence, allowing the determination of b as the slope of the respective lines. The dashed lines in Fig. 3(d) correspond to Eq. (1). A close inspection of the fits reveals that while the conductance curves at B = 0.12 and 0.25 T precisely follow the formula over six decades, the 0.50 T curve displays a slight departure (Fig. 3(d)), indicating that the assumption of critical CBKT behavior combined with an infinite screening length has started to break down.
These results establish a superinsulator as a confined low temperature charge BKT phase of the Cooper pair insulator. In this phase, vortices form a Bose condensate that completely blocks the motion of the Cooper pairs.
Data analysis
To proceed further with the analysis, we choose the value of T_{CBKT} for every isomagnetic G(T) curve and plot in Fig. 4(a) the conductance normalized by its value at temperature T = 4T_{CBKT} vs. the normalized temperature T/T_{CBKT}(B). The corresponding fielddependent charge BKT transition temperature T_{CBKT} is shown in Fig. 4(b). Remarkably, the complex diversity of the R(T) curves (exhibiting subactivation at high magnetic field and hyperactivation at small field in Fig. 3(b)) rescaled onto a set of uniform curves that collapse at T/T_{CBKT} > 2.5. Moreover, the fielddependent evolution of the curve shapes, including the change in concavity, now reduces simply to a successive deviation from the universal curve: the higher the magnetic field at which the curve is measured, the higher the T/T_{CBKT} ratio at which the given curve departs from the universal envelope. We stress that the above normalization procedure does not presume any special temperature dependence of G(T). The choice of the temperature at which the normalizing value of the conductance is taken is somewhat arbitrary, as seen from the quality of the collapse over the 2.5–5 range in T/T_{CBKT}.
We now describe the overall G(T, B) behavior using a two dimensional Coulomb gas model^{5} that generalizes the Kosterlitz formula (1) by incorporating a selfconsistent solution to the effects of electronic screening. The conduction is controlled by the density of free charge carriers, G ∝ n_{ f }, i.e. the conductance is proportional to the inverse squared mean distance between the carriers. In the critical BKT region, n_{ f } is the 2D density of the unbound charges, which is related to the correlation length λ at which the unbound charges appear via the equation:
where λ_{ c } is the smaller of the bare electrostatic screening length of the film, Λ, or the lateral linear dimension of the film. The screening length defines the maximal spatial scale of logarithmic charge interactions in the film, V(r) ∝ ln(r/Λ) for r_{0} < r < Λ. Since \({r}_{0}\approx d\ll {\rm{\Lambda }}\), the film is effectively 2D with respect to the Coulomb charge interactions over a broad range of length scales.
Relating λ_{ c } and the density of the unbound charges through the Poisson equation, we derive, following^{5}, a selfconsistent equation for λ:
where \({{}\kern0.5em Z}\) is a constant. As λ_{ c } → ∞, Eqs (2) and (3), reduce to Eq. (1), with b = 2 ln(1/\({{}\kern0.5em Z}\)).
We verify this model against the data by fitting a selfconsistent solution of Eqs (2) and (3) to the measured isomagnetic G(T) (see representative curves in Fig. 4(a)); the only adjustable parameter in these fits is the screening length λ_{ c }. The field evolution of the derived λ_{ c } values is shown in Fig. 4(c). Thus, we see that the twodimensional Coulomb gas model fully describes the complex diversity of the experimental data, including both the BKT critical behavior and the deviation from criticality, using only one adjustable parameter.
A signature of the BKT transition is a currentvoltage curve of the form I ∝ V^{α} at T ≤ T_{BKT}, with α experiencing a jump from α = 1 to α = 3 at T = T_{BKT}. For our NbTiN films, Fig. 5 displays a set of I–V curves measured as a function of temperature for magnetic field B = 0.6 T (I–V curves for additional fields are shown in (See Supplemental Material for further information on the fabrication and measurement techniques, films structural, superconducting and transport parameters, and I–V characteristics)). In this field, the electrostatic screening length is approximately equal to the lateral linear size of the film (see Fig. 4(c)). T_{BKT} determined from I(V) coincides with T_{BKT} determined from R(T).
Discussion and Conclusion
We now discuss the implications of identifying the nature of the Cooper pair insulator as a twodimensional neutral Coulomb plasma of excessive/deficit Cooper pairs, each carrying the charge ±2e, and analyze further the parameters of this Cooper pair plasma. We note first the apparent divergence of λ_{ c } upon decreasing B. That λ_{ c } depends on B enables its identification as the electrostatic screening length in disordered films, λ_{ c } = Λ = εd. Accordingly, its divergence upon decreasing B corresponds to the divergence of the dielectric constant upon approach to the SIT, with a cutoff of the divergence as λ_{ c } approaches the lateral system size. The value of the magnetic field, B_{SIT} = 0.015 T, where the SIT occurs, is determined by the crossing point of the resistive curves, as seen in Fig. 2(b), whose appearance is a hallmark of the fielddriven SIT. In the vicinity of the fielddriven SIT the corresponding correlation length diverges as \({(B{B}_{{\rm{SIT}}})}^{\tilde{\nu }}\)^{35}, as do the relevant physical quantities. To analyze the character of the divergence we plot λ_{ c }/r_{0} vs (B − B_{SIT})/B_{SIT} (Fig. 6(a)), where we see that at the lowest fields λ_{ c } ∝ (B − B_{SIT})^{−ν}, with ν = 0.51 ± 0.02 and ν = 2.44 ± 0.04 further from the transition. In strongly disordered films, one can expect the SIT to be governed by a percolation transition (see^{11} and references therein), so it is natural to analyze the scaling behavior in the framework of a random JJA or a random inductorcapacitor network (ICN), with the magnetic field moving the system away from the percolation transition at B = B_{SIT}. At large field, the screening length λ_{ c } is not large, and hence the electric behavior of the network is dominated by the nodes’ capacitances to ground C_{0}^{11,40}. Numerical analysis of the ICN in the vicinity of the transition in the large C_{0} limit^{46} yields ν = 2.52, in good agreement with our experimental findings. This establishes the percolative behavior near the transition and suggests that the magnetic field indeed acts as the parameter destroying the superconducting bonds on approach to the SIT from the superconducting side. We thus expect that the superfluid density should scale as ρ_{ s } ∝ (B_{SIT} − B)^{ν}. In the limit of C_{0} → 0, i.e. in the region of large λ_{ c }^{40}, the duality principle^{11,35,36} implies that on the insulating side the dielectric constant scales as \(\varepsilon \sim {\rho }_{s}^{1}\). The exponent ν = 0.53 in the dual limit was found in^{47}, in agreement with our experimental result of ν = 0.51 ± 0.02.
Our finding is in a full accord with the polarization catastrophe paradigm built on the divergence of ε as function of the carrier concentration at the approach to the quantum metalinsulator transition^{48,49}. In our experiment, it is B − B_{SIT} that plays the role of the deviation of the carrier concentration from its critical value.
Having determined the screening length and T_{CBKT} as a function of magnetic field, we now can make an independent crosscheck on the 2D Coulomb nature of the superinsulator. Shown in Fig. 6(b) are Arrhenius plots of the normalized conductance vs. T_{CBKT}/T at various magnetic fields, highlighting the thermally activated behavior at low temperatures, T < T_{CBKT}. Note that Eqs (2) and (3) describe conductance only at T > T_{CBKT}, shown as a shaded rectangle. The field dependent activation temperatures T_{0} are presented in the inset to Fig. 6(b). When the typical size of the unbound pair becomes comparable to Λ, the interaction ceases to be logarithmic and the conductance is dominated by thermodynamically activated free charges. The onset of this activated behavior marks the point at which the correlation length becomes comparable to the screening length. Thus, the lowtemperature tails in G(T) are expected to be exponential and to depart from the BKT criticality curve. Theoretical calculations^{10,40} and simulations^{50} of thermally activated hopping transport in a 2D insulator with logarithmic Coulomb interactions between the charge carriers yield an activation temperature \({T}_{0}\simeq {T}_{{\rm{CBKT}}}\,\mathrm{ln}({\lambda }_{c}/{r}_{0})\). In the same inset we present our experimental values of T_{CBKT} ln(λ_{ c }/r_{0}) at the same fields; these indeed appear remarkably close to the independently determined T_{0} in accord with the theoretical expectations. This correspondence validates the 2D Coulomb logarithmic interaction between charges at distances not exceeding the screening length. Similarly, exponential lowtemperature tails in the resistance were observed in JJA on the superconducting side of the SIT. The tails appeared below the vortex BKT transition temperature where the applied magnetic field introduced the excess unbound vortices^{51}.
We now can resolve the longstanding open question in the study of the SIT: the origin of the giant peak in the magnetoresistance^{17,19,52,53,54}. It arises from the combination of the dielectric constant rapidly decaying with the increase of the magnetic field and the nonmonotonic behavior of T_{CBKT}. In order to gain insight into the behavior of the latter, we employ the model of JJA, an array of superconducting granules connected with Josephson links, which is a longstanding model for a critically disordered superconducting film^{55}. In the parent compound NbN, for example, disordered films were seen to have a selfinduced granular structure with the elemental cell having characteristic size ~1–2\(\xi \)^{56}. It also should be noted that the relevant disorder is electronic rather than structural, and can emerge from structurally homogenous films, particularly when in close proximity to a SIT^{57}. The origin of the nonmonotonic behavior of T_{CBKT} can be explained by recalling that the energy gap of the Cooper pair insulator in JJA, Δ_{ c }(B), is suppressed by the Josephson coupling E_{ J } between the neighboring granules, Δ_{ c }(B) = Δ_{ c }(0)[1 − AE_{J}(B)/E_{ c }]^{40}, where E_{ c } is the Coulomb energy of a single granule and A is a constant. The Josephson coupling is maximal at zero field and, in the irregular JJA, has the minimum at the frustration factor f = 1/2^{58}, where f ≡ BS/Φ_{0}, S is the average area of the JJA elemental cell and Φ_{0} is the magnetic flux quantum. Accordingly, the effective Coulomb energy acquires the maximum at f = 1/2, i.e. the nonmonotonic behavior of T_{CBKT} reflects the nonmonotonic behavior of E_{J} as a function of the magnetic field. The nonmonotonic nature of E_{J} also can be directly observed in I–V curves, where the activation voltage varies nonmonotonically with field (see SI). This enables us to estimate the parameters of the system as follows. The observed maximum in T_{CBKT} at B ≈ 4 T (Fig. 4(b)) implies that the average area of an elemental cell of our selfinduced granular structure, S ≈ 260 nm^{2} and, hence, the linear size of the elemental cell \(\sqrt{S}\approx 16\) nm ≈ 3.5ξ, where ξ = 4.5 nm is the superconducting coherence length of the NbTiN film (see SI). Interestingly, this correlates with the analogous estimates for TiN, where \(\sqrt{S}\approx 4\xi \) was observed^{59}. The described nonmonotonic behavior is accompanied by the overall suppression of the superconducting gap by the increasing magnetic field. The latter eventually would suppress the superconducting gap in Cooper pair droplets and hence Δ_{ c }, resulting in a further drop of the resistance. Then, the Cooper pair insulator ends up as a metal^{38}.
By comparison to NbTiN, the behaviors complying with the formation of the superinsulating state were observed in other materials at very low temperatures. In TiN films the superinsulator appeared at 40 mK^{24}. More recently, the finite temperature zeroconductance state in InO_{ x } was reported at \(T\mathop{ < }\limits_{ \tilde {}}35\) mK^{19}. The temperature dependence of the conductance in InO_{ x } was found to follow the socalled VogelFulcherTamman dependence, σ ∝ exp[−const/(T^{*} − T)]^{60,61,62}. This, however, can be viewed as a manifestation of the same BKT physics, but in a more disordered system^{63}.
The Kosterlitzlike exponential behavior described by Eq. (1) also appears in the framework of manybody localization (MBL) theory^{64,65}. One can thus ask if the observed critical behavior of the NbTiN films should instead be considered in the context of MBL. A detailed comparison of BKT and MBL physics is given in^{63}, but we briefly note that our data and analysis unambiguously evidences the primary role of the long range two dimensional logarithmic Coulomb interactions between charges which are a platform for BKT physics. By contrast, the original MBL model^{64,65} was constructed for one dimensional systems in the strict absence of the long range interactions. While recent work has proposed the extension of MBL to a model with long range interactions^{66}, it is not clear whether the exponential critical behavior would survive in the systems with interactions. Even more important is that in the MBL framework, taking the system deeper into the insulator (i.e. moving away from the SIT) via increasing magnetic field and/or disorder is expected to enhance localization and hence critical behavior. As discussed above, we observe the opposite trend: increasing magnetic field shrinks the screening length and narrows the window in which the criticality is seen, consistent with the expectations of a BKT system. In sum, this indicates that at present MBL does not offer the appropriate framework for describing the superinsulating state and its related criticality.
To summarize, our findings clearly establish the finite temperature superinsulating state in NbTiN as the low temperature charge BKT phase of the Cooper pair insulator. We demonstrate superinsulating behavior in a new material with a substantially higher transition temperature of nearly 200 mK, allowing for the first time a detailed characterization of behavior of the system both above and below T_{CBKT} and its evolution in a wide range of magnetic fields.
Methods
The fabrication is built upon the Atomic Layer Deposition technique. The structure of films grown on Si substrates with AlN buffer layers was investigated using a JEOL4000EX electron microscope operated at 400 kV, with a pointtopoint resolution of 0.16 nm and a line resolution of 0.1 nm. The details of sample fabrication, analysis and measurements are given in SM.
Data availability
The authors declare that all relevant data supporting the findings of this study are available within the article and its supplementary information file.
References
 1.
Berezinskii, V. L. Destruction of longrange order in onedimensional and twodimensional systems having a continuous symmetry group I. Classical systems. Sov. Phys. JETP 32, 493 (1970).
 2.
Kosterlitz, J. M. & Thouless, D. J. Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory). J. Phys. C: Solid State Phys. 5, L124 (1972).
 3.
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in twodimensional systems. J. Phys. C: Solid State Phys. 6, 1181 (1973).
 4.
Mooij, J. E. In Percolation, Localization and Superconductivity. eds Goldman, A. M. & Wolf, S. A. (Plenum Press, New York and London) 325 (1984).
 5.
Minnhagen, P. The twodimensional Coulomb gas, vortex unbinding, and superfluidsuperconducting films. Rev. Mod. Phys. 59, 1001 (1987).
 6.
Fazio, R. & Schön, G. Charge and vortex dynamics in arrays of tunnel junctions. Phys. Rev. B 43, 5307 (1991).
 7.
Mooij, J. E. et al. Unbinding of ChargeAnticharge Pairs in TwoDimensional Arrays of Small Tunnel Junctions. Phys. Rev. Lett. 65, 638 (1990).
 8.
Goldman, A. M. The BerezinskiiKosterlitzThouless Transition in Superconducors. In 40 years of the BerezinskiiKosterlitzThouless theory, 135 World Scientific (2013).
 9.
Diamantini, M. C., Sodano, P. & Trugenberger, C. A. Gauge theories of Josephson junction arrays. Nuclear Physics B 474, 641 (1996).
 10.
Vinokur, V. M. et al. Superinsulator and quantum synchronization. Nature 452, 316 (2008).
 11.
Baturina, T. I. & Vinokur, V. M. Superinsulatorsuperconductor duality in two dimensions. Annals of Physics 331, 236 (2013).
 12.
Anderson, P. W. Lectures on amorphous systems. III condensed matter in Les Houches Lectures NorthHolland (1979).
 13.
Bell, M. T., Sadovskyy, I. A., Ioffe, L. B., Kitaev, A. Yu & Gershenson, M. E. Quantum Superinductor with Tunable Nonlinearity. Phys. Rev. Lett. 109, 137003 (2012).
 14.
Kowal, D. & Ovadyahu, Z. Disorder induced granularity in an amorphous superconductor. Solid State Comm. 90, 783 (1994).
 15.
Sambandamurthy, G., Engel, L. M., Johansson, A., Peled, E. & Shahar, D. Experimental evidence for a collective insulating state in twodimensional superconductors. Phys. Rev. Lett. 94, 017003 (2005).
 16.
Steiner, M. & Kapitulnik, A. Superconductivity in the insulating phase above the fieldtuned superconductorinsulator transition in disordered indium oxide films. Physica C 422, 16 (2005).
 17.
Baturina, T. I., Mironov, A. Yu, Vinokur, V. M., Baklanov, M. R. & Strunk, C. Localized superconductivity in the quantumcritical region of the disorderdriven superconductorinsulator transition in TiN thin films. Phys. Rev. Lett. 99, 257003 (2007).
 18.
Baturina, T. I., Mironov, A., Yu., Vinokur, V. M., Baklanov, M. R. & Strunk, C. Hyperactivated Resistance in TiN Films on the Insulating Side of the DisorderDriven SuperconductorInsulator Transition. JETP Lett. 88, 752 (2008).
 19.
Ovadia, M. et al. Evidence for a finitetemperature insulator. Sci. Rep. 5, 13503 (2015).
 20.
Kanda, A. & Kobayashi, S. Precursor of Charge KTB Transition in Normal and Superconducting Tunnel Junction Array. J. Phys. Soc. Japan 64, 19 (1995).
 21.
Kobayashi, S., Kanda, A. & Yamada, R. Charge KosterlitzThouless transition in twodimensional arrays of small tunnel junctions. Jpn. J. Appl. Phys. 34, 4548 (1995).
 22.
Kanda, A. & Kobayashi, S. Effect of selfcapacitance on charge KosterlitzThouless transition in twodimensional arrays of small tunnel junctions. Physica C 227, 238 (1996).
 23.
Baturina, T. I. et al. Nanopatternstimulated superconductorinsulator transition in thin TiN films. Europhys. Lett. 93, 47002 (2011).
 24.
Kalok, D. et al. Nonlinear conduction in the critical region of the superconductorinsulator transition in TiN thin films. J. Phys: Conf. Ser. 400, 022042 (2012).
 25.
Lim, B. S., Rahtu, A. & Gordon, R. G. Atomic layer deposition of transition metals. Nature Materials 2, 749 (2003).
 26.
Driessen, E. F. C., Coumou, P. C. J. J., Tromp, R. R., de Visser, P. J. & Klapwijk, T. M. Strongly Disordered TiN and NbTiN sWave Superconductors Probed by Microwave Electrodynamics. Phys. Rev. Lett. 109, 107003 (2012).
 27.
Makise, K., Terai, H., Tominari, Y., Tanaka, S. & Shinozaki, B. Duality picture of Superconductorinsulator transitions on Superconducting nanowire. Sci. Rep. 6, 27001 (2016).
 28.
Proslier, T., Klug, J. A., Becker, N. C., Elam, J. W. & Pellin, M. J. Atomic Layer Deposition of Superconductors. ECS Transactions 41, 237 (2011).
 29.
Shiino, T. et al. Improvement of the critical temperature of superconducting NbTiN and NbN thin films using the AlN buffer layer. Supercond. Sci. Technol. 23, 045004 (2010).
 30.
Larkin, A. & Varlamov, A. Theory of Fluctuations in Superconductors. (Clarendon Press, Oxford, 2005).
 31.
Berezinskii, V. L. Destruction of Longrange Order in Onedimensional and Twodimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems. Sov. Phys. JETP 34, 1144 (1972).
 32.
Sacepe, B. et al. DisorderInduced Inhomogeneities of the Superconducting State Close to the SuperconductorInsulator Transition. Phys. Rev. Lett. 101, 157006 (2008).
 33.
Baturina, T. I. et al. Superconducting phase transitions in ultrathin TiN films. Europhys. Lett. 97, 17012 (2012).
 34.
Chand, M. et al. Phase diagram of the strongly disordered swave superconductor NbN close to the metalinsulator transition. Phys. Rev. B 85, 014508 (2012).
 35.
Fisher, M. P. A. Quantum Phase Transitions in Disordered TwoDimensional Superconductors. Phys. Rev. Lett. 65, 923 (1990).
 36.
Fisher, M. P. A., Grinstein, G. & Girvin, S. M. Presence of quantum diffusion in two dimensions: Universal resistance at the superconductorinsulator transition. Phys. Rev. Lett. 64, 587 (1990).
 37.
Goldman, A. M. SuperconductorInsulator Transitions. Int. J. Mod. Phys. B 24, 4081 (2010).
 38.
Baturina, T. I., Strunk, C., Baklanov, M. R. & Satta, A. Quantum Metallicity on the HighField Side of the SuperconductorInsulator Transition. Phys. Rev. Lett. 98, 127003 (2007).
 39.
Das, D. & Doniach, S. Existence of a Bose metal at T = 0. Phys. Rev. B 60, 1261 (1999).
 40.
Fistul, M. V., Vinokur, V. M. & Baturina, T. I. Collective CooperPair Transport in the Insulating State of JosephsonJunction Arrays. Phys. Rev. Lett. 100, 086805 (2008).
 41.
Baturina, T. I., Bentner, J., Strunk, C., Baklanov, M. R. & Satta, A. From quantum corrections to magneticfieldtuned superconductorinsulator quantum phase transition in TiN film. Physica B 359–361, 500 (2005).
 42.
Tikhonov, K. S., Schwiete, G. & Finkel’stein, A. M. Fluctuation conductivity in disordered superconducting films. Phys. Rev. B 85, 174527 (2012).
 43.
Ovadia, M., Kalok, D., Sacepe, B. & Shahar, D. Duality symmetry and its breakdown in the vicinity of the superconductorinsulator transition. Nat. Phys. 9, 415 (2013).
 44.
Sacepe, B. et al. Highfield termination of a Cooperpair insulator. Phys. Rev. B 91, 220508(R) (2015).
 45.
Kosterlitz, J. M. The critical properties of the twodimensional xy model. J. Phys. C: Solid State Phys. 7, 1046 (1974).
 46.
Loh, Y. L. & Karki, P. Dielectric and diamagnetic susceptibilities near percolative superconductorinsulator transitions. J. Phys.: Condens. Matter 29, 425901 (2017).
 47.
Granato, E. & Kosterlitz, J. M. SuperconductorInsulator Transition and Universal Resistance in JosephsonJunction Arrays in a Magnetic Field. Phys. Rev. Lett. 65, 1267 (1990).
 48.
Paalanen, M. A., Rosenbaum, T. F., Thomas, G. A. & Bhatt, R. N. Stress tuning of the metalinsulator transition at millikelvin temperatures. Phys. Rev. Lett. 48, 1284 (1982).
 49.
Bhatt, R. N. Dielectric function near a polarization catastrophe. Phil. Mag. B 50, 189 (1984).
 50.
Ortuño, M., Somoza, A. M., Vinokur, V. M. & Baturina, T. I. Electronic transport in twodimensional high dielectric constant nanosystems. Sci. Rep 5, 9667 (2015).
 51.
van der Zant, H. J., Rijken, H. A. & Mooij, J. H. The Superconducting Transition of 2D JosephsonJunction Arrays in a Small Perpendicular Magnetic Field. J. of Low Temp. Phys. 79, 289 (1990).
 52.
Dubi, Y., Meir, Y. & Avishai, Y. Nature of the superconductorinsulator transition in disordered superconductors. Nature 449, 876 (2007).
 53.
Nguyen, H. Q. et al. Observation of Giant Positive Magnetoresistance in a Cooper Pair Insulator. Phys. Rev. Lett. 103, 157001 (2009).
 54.
Baturina, T. I. et al. Quantumcritical region of the disorderdriven superconductorinsulator transition. Physica C 468, 316 (2008).
 55.
Tinkham, M. Introduction to Superconductivity. (McGrawHill, New York, 1996).
 56.
Kamlapure, A. et al. Emergence of nanoscale inhomogeneity in the superconducting state of a homogeneously disordered conventional superconductor. Sci. Rep. 3, 2979 (2013).
 57.
Ganguly, R. et al. Magnetic field induced emergent inhomogeneity in a superconducting film with weak and homogeneous disorder. Phys. Rev. B 96, 054509 (2017).
 58.
Hollen, S. M., Fernandes, G. E., Xu, J. M. & Valles, J. M. Jr. Collapse of the Cooper pair phase coherence length at a superconductortoinsulator transition. Phys. Rev. B 87, 054512 (2013).
 59.
Baturina, T. I. et al. Dual threshold diode based on the superconductortoinsulator transition in ultrathin TiN films. Appl. Phys. Lett. 102, 042601 (2013).
 60.
Vogel, H. The law of the relationship between viscosity of liquids and the temperature. Physikalische Zeitschrift 22, 645 (1921).
 61.
Fulcher, G. S. Analysis of recent measurements of the viscosity of glasses. Journal of the American Ceramic Society 8, 339 (1925).
 62.
Tamman, G. & Hesse, W. Die abhängigkeit der viscosität von der temperatur bie unterkühlten flüssigkeiten. Zeitschrift für Anorganische und Allgemeine Chemie 156, 245 (1926).
 63.
Sankar, S., Vinokur, V. M. & Tripathi, V. Disordered BKT transition and superinsulation. Phys. Rev B 97, 020507 (2018).
 64.
Gornyi, I. V., Mirlin, A. D. & Polyakov, D. G. Interacting Electrons in Disordered Wires: Anderson Localization and LowT Transport. Phys. Rev. Lett. 95, 206603 (2005).
 65.
Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting manyelectron system with localized singleparticle states. Ann. Phys. 321, 1126 (2006).
 66.
Nandkishore, R. M. & Sondhi, S. L. ManyBody Localization with LongRange Interactions. Phys. Rev. X 7, 041021 (2017).
Acknowledgements
This work was supported by the Ministry of Education and Science of the Russian Federation, by the Grant of the President RF, project No MK4628.2016.2 (A.Yu.M., S.V.P., M.V.B., and A.K.G.). The high resolution electron microscopy was performed with support of RSF, project No 142200143. The work at Caltech was supported by National Science Foundation Grant No. DMR1606858 (D.M.S. and T.F.R.). The work at Argonne was supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division (V.M.V., T.P., and visits of T.I.B.). T.I.B. also acknowledges support by the Alexander von Humboldt Foundation and from the Consejeria de Educacian, Cultura y Deporte (Comunidad de Madrid) through the talent attraction program, Ref. 2016T3/IND1839. A.Yu.M. and T.I.B. were also supported from the ArgonneUniversity of Chicago collaborative seed grant. We are delighted to thank E. Shimshoni for illuminating discusions.
Author information
Affiliations
Contributions
T.I.B., V.M.V., and T.F.R. conceived the project and initiated this work; the films were synthesized by T.P.; A.Yu.M., D.M.S., S.V.P., M.V.B., and T.I.B. carried out the experiments; A.K.G. performed high resolution electron microscopy; A.Yu.M., D.M.S., T.F.R., V.M.V., and T.I.B. analyzed the data. All authors discussed the results and contributed in writing the manuscript.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mironov, A.Y., Silevitch, D.M., Proslier, T. et al. Charge BerezinskiiKosterlitzThouless transition in superconducting NbTiN films. Sci Rep 8, 4082 (2018). https://doi.org/10.1038/s41598018224511
Received:
Accepted:
Published:
Further reading

Quantum magnetic monopole condensate
Communications Physics (2021)

Supercapacitance and superinductance of TiN and NbTiN films in the vicinity of superconductortoinsulator transition
Scientific Reports (2021)

Superconducting phase transitions in disordered NbTiN films
Scientific Reports (2020)

Direct probe of the interior of an electric pion in a Cooper pair superinsulator
Communications Physics (2020)

The SuperconductorSuperinsulator Transition: Sduality and the QCD on the Desktop
Journal of Superconductivity and Novel Magnetism (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.