Introduction

Superinsulators1,2,3, see ref. 4. for a review, are materials hosting a macroscopic quantum state which is a mirror image of superconductivity. In superinsulators, the effective electromagnetic action is compact quantum electrodynamics (QED) at strong coupling5. It is the compactness of the gauge group that entails the presence of magnetic monopoles, see ref. 6. for a review, which are instantons7 in two spatial dimensions (2D) and particles in three spatial dimensions (3D). Correspondingly, the superinsulating state is a plasma of (non-relativistic) magnetic monopole instantons in 2D3 and a magnetic monopole condensate in 3D8. In these states, electric fields are constrained into flux tubes between Cooper pairs and Cooper holes. These flux tubes are dual to Abrikosov vortices in superconductors, with the difference that “electric vortices” can end on internal sources. An elementary excitation made of an electric flux tube connecting a Cooper pair and a Cooper hole is the purely electric equivalent of a strong interaction pion, with Cooper pairs playing the role of quarks. The flux tubes are confining strings preventing the separation of charges and leading to an infinite resistance, even at finite temperatures3,8, for the same reason that quarks can never be extracted from pions and other hadrons. Hence the name “superinsulators”. Magnetic monopoles and confining strings, which were just abstract objects allowed in compact versions of quantum electrodynamics (QED)5 become experimentally observable physical objects in superinsulators1,2,3,4.

Superinsulation is destroyed when the string tension vanishes at either sufficiently high temperature or at large applied voltage. The corresponding disappearance of superinsulation is a phase transition characterized by the temperature scaling of the resistance. These temperaure scalings are different in two-dimensional (2D) and 3D (bulk) superinsulators. In a 2D superinsulator, the temperature scaling assumes the classical Berezinskii-Kosterlitz-Thouless (BKT) form9,10,11, whereas in 3D it changes to a Vogel–Fulcher–Tamman (VFT) scaling12, exactly as for the dual type-III superconductors13.

Superinsulators have been detected in superconducting films of various materials2,14,15,16,17, one of which was thick enough to show hints of a 3D behavior14. All these films showing superinsulation appear to have an emergent granular structure18,19: they are arrays of superconducting granules connected by Josephson tunneling junctions. This implies that these materials are effective Josephson junction arrays (JJA), see20 for a review, which are the metamaterial for which superinsulation has been first predicted1. In all these films the superinsulating state was obtained by driving them across the superconductor-to-insulator transition (SIT)2, see4 for a review, by decreasing the films thickness, or by applying a magnetic field or a gate voltage.

Here we report the observations obtained for a material which is by itself a bulk (3D) superinsulator at low temperatures, so that there is no need to drive it across a transition from a superconductor in order to observe superinsulation. This superinsulating metamaterial is obtained by nanopatterning a slab of NbTiN by punching a regular array of holes into it. As we now show, this is the first bulk superinsulator at temperatures below a critical temperature of 0.31∘K, which is more than five times higher than the typical critical temperature in superinsulating films.

Superinsulation has not yet been observed in JJA. As was pointed out in ref. 21., this is happening since, in presently available arrays, having typically about 500 to 1000 islands in their very largest realizations, it is difficult to simultaneously obtain large enough electric screening lengths due to the islands’ self-capacitances and accomodate an entire electric string. The structure presented in this paper is essentially a regular JJA with 10 million islands and a very large electric screening length due to the large dielectric constant of NbTiN. As originally predicted1, superinsulation immediately appears in such a large regular structure. If disorder would be the root cause of the diverging resistance of superinsulators, this phenomenon should not be observed in the present regular and ordered nanopatterned materials. The fact that, instead, it is clearly observed in these regular and ordered systems is the final proof that superinsulation has nothing to do with disorder.

Electric fields that arise from voltages below a critical voltage value do not penetrate superinsulators. This is a phenomenon dual to the usual Meissner effect for magnetic fields in superconductors, see refs. 4,16 for a review. However, when sufficient energy is provided by external electric and magnetic fields, electric pions, which are thin strings of electric flux with  ± charges at their ends, can be excited in the interior of the bulk sample. These electric strings move under the influence of the external magnetic field in the plane perpendicular to the magnetic field and arrange themselves under the influence of their interactions and the applied voltage in such a way that local electric polarization configurations arise in this plane. We show how, in absence of external electric fields, polar nematic orders, see22 for a recent review, arise as a consequence of the competition between interactions of electric strings and the effect of a magnetic field. When an in-plane electric field is added, the induced polar orders become more complex. We observe the resulting effects by measuring configurations of the local, magnetic- and electric-field-induced voltages arising at various locations in the sample and in different directions. These configurations form only for ranges of applied magnetic field and voltage where the superinsulating state exists and electric pions can be excited, thus confirming their origin.

Results

The nanopatterned materials

The NbTiN films, having the thickness of 10 nm, are grown using atomic layer deposition (ALD) on a SiO2/Si substrate at a temperature 350∘. The structure of the films is of a polycrystalline nature with the grain size being about 5 nm, as determined with high resolution electron microscopy. To carry out the transport measurements, the films are first patterned by means of a conventional UV lithography and plasma etching into bridges of 100 μm wide and with 100 μm and 250 μm distances between the voltage probes. Then, making use of electron lithography and subsequent plasma etching, a square lattice of holes with the diameter  ≈ 40 nm and the period a = 100 nm is created, covering an area 100 × 2000 μm2 for the first film, see Fig. 1, and an area 100 × 1000 μm2 for the second film. This corresponds to 10, or 5 million lattice units for the whole surface of the sample, respectively. The experiment is performed in the dilution refrigerators. The measurements of the temperature T and the magnetic field B dependencies of the resistance and differential resistance are carried out using a standard two-probe low-frequency ac/dc technique at the frequency 3 Hz with an ac voltage 100 nV. For all measurements, we use lock-in amplifiers SR865a and sub-femtoampèremeter Keythley 6430 and nanovoltmeter Agilent 34420a. The magnetic field is applied perpendicular to the film surface.

Fig. 1: Perforated film.
figure 1

Image of a section of one of the perforated NbTiN films, collected by scanning electron microscope.

Full size image

Discussion of experimental results

In Fig. 2 we present the conductance dependencies on temperature for the two perforated films, obtained in two different dilution refrigerators. The initial film is a superconductor, with the critical transition temperature TC = 1.8 K. The conductance of the perforated films, on the contrary, shows superinsulating behavior by approaching 0 instead of diverging. At low temperatures, noise is observed due to the current measurement circuit approaching its limit. The conductance of superinsulators completely vanishes below a finite critical temperature, see Methods. The precise value of the critical temperature is determined by a BKT scaling in 2D- and by a VFT scaling in 3D systems12, with the two behaviors differing by their distinct powers in the exponential scaling law, see Methods for details. Although the width of the perforated films is only 10 nm, the 3D VFT conductance scalings with transition temperatures TVFT = 0.31 K for the 2 mm long sample and TVFT = 0.275 K for the 1 mm sample are clearly favored, indicating that these are bulk superinsulators.

Fig. 2: Temperature dependence of the conductance.
figure 2

Temperature dependence of the conductances of the two nanoperforated films and fits by BKT and VFT scalings. The excellent fits of the experimental data to the VFT scalings confirm that the nanoperforated samples are bulk superinsulators. Also shown are the predicted \({T}_{{\rm{dev}}}^{{\rm{th}}}\) temperatures where finite-size effects are expected to cut-off the VFT fits for infinite samples and the corresponding experimental \({T}_{{\rm{dev}}}^{\exp }\) deviation temperatures. Note, again, the excellent match to the experimental data.

Full size image

As in the case of superconducting films, the scaling is of course cut off by the finite size effects23,24. Due to the higher power in the scaling exponent though, these finite size effects are more important for 3D systems than for 2D ones. The finite size effects can be estimated by cutting off the exponential increase of the correlation length when approaching the transition at the smallest dimension of the sample, in this case 0.1 mm, see “Methods”. The so obtained estimates predict deviations from the VFT scalings at a temperature Tdev ≈ 0.61 K for the 2 mm film and at a temperature Tdev ≈ 0.55 K for the 1 mm film, which are again in excellent agreement with what is measured in the experiment.

In large samples, showing the infinite-system behavior, all vestiges of superconductivity are lost, since the system is in a different, superinsulating phase. When the sample size is decreased and finite-size effects become more important, though one can still observe a dip in the resistance at the original superconducting temperature T = 1.8 K. This is shown in Fig. 3 for a 0.1 mm nanopatterned film.

Fig. 3: Temperature dependence of the resistance of a 0.1 mm small film.
figure 3

Temperature dependence of the resistance of a 0.1 mm nanopatterned film, showing a residual dip in correspondence of the superconducting critical temperature of the original, non-patterned film.

Full size image

Below the VFT transition temperature, the nanoperforated system is in its superinsulating phase. In Figs. 4 and 5 we show the absolute and differential conductances measured at T = 60 mK, deep in the superinsulating state. This confirms the vanishing of the conductance independently of the applied voltage (and magnetic field) in a range ∣V∣ < Vcr(B). This is the electric Meissner state of the superinsulator.

Fig. 4: Conductance in the superinsulating state.
figure 4

Magnetic field and voltage dependence of the conductance of the nanoperforated sample, measured at T = 60 mK, below the superinsulating transition.

Full size image
Fig. 5: Differential conductance in the superinsulating state.
figure 5

Magnetic field and voltage dependence of the differential conductance of the nanoperforated sample, measured at T = 60 mK, below the superinsulating transition.

Full size image

Electric fields do not penetrate a superinsulator in its electric Meissner state. However, contrary to the dual superconductors, small open electric flux tubes with  ± charges at their ends can be excited4,16 if the energy provided by the external fields is sufficient. These small strings, whose gap and dimension are characterized by their string tension, are the purely electric equivalent of strong interaction pions.

When external electromagnetic fields F μν of a sufficient strength are applied to a superinsulating sample, they create a four-tensor current of strings J μν, comprising, e.g., a vector density J 0i for an electric field E i in the plane and a tensor current J12 for a magnetic field B3 perpendicular to it, see Methods. Let us consider first just a magnetic field B3 = B in the third direction. When such a magnetic field is applied, electric strings are excited. Due to their short-range antiferroelectric interaction, these strings tend to anti-align themselves along a nematic director n. The external magnetic field, however, tends to push anti-aligned strings apart, see “Methods”. As a consequence, when the density of strings is not too high, this external interaction wins the competition and it is energetically favorable to have aligned strings moving all in the same direction under the influence of the external orthogonal magnetic field, thereby maintaining a large distance and a small interaction energy among them. The equilibrium configuration is thus a steady current of aligned electric strings, which amounts to a ferroelectric polar nematic order, detected as the polarization pattern shown in Fig. 6. When the magnetic field increases, however, the density of strings becomes large. In this case, the interstring interaction becomes the dominant one and an antiferroelectric nematic order is favored, which does not give rise to polarization effects. The increase and subsequent decrease of the induced voltage by increasing the magnetic field is thus explained by the excitation of electric string with a ferroelectric nematic order at low densities and an antiferroelectric nematic order at high densities.

Fig. 6: Polar nematic order induced by a magnetic field.
figure 6

Magnetic field dependence of the induced voltage, measured at T = 60 mK deep in the superinsulating state.

Full size image

Note that in zero magnetic field a small induced voltage is observed. We expect this to be caused by loops of electric fields in the ground state. The superinsulating ground state is a condensate of magnetic monopoles, see Methods, whose magnetic field becomes squeezed in the plane in the case of thin films. In this case, the monopoles become tunneling events in which vortices appear and disappear in the ground state. Such events correspond to time-varying magnetic fields and are accompanied by electric field loops. In the bulk of the sample these typically cancel out but they can give rise to polarization effects on the boundaries of finite systems, especially when these are connected to measuring devices. Note that this ground state effect is not affected by an applied constant magnetic field which does not create electric field loops. The only effect of a constant magnetic field is to excite electric strings. Of course, the excitation of a small amount of electric strings by the finite temperature can also not be excluded. In Fig. 6 we present the in-plane voltages induced by a magnetic field perpendicular to the perforated film at T = 60 mK, deep in the superinsulating state.

When an additional voltage is applied in plane, one can measure the non-local induced voltage Vnl at a different location on the sample. These measurements are shown in Figs. 7 and 8 as a function of magnetic field for a temperature of T = 60 mK and of temperature for a magnetic field B = 3.5 T. They show that the polar nematic order vanishes when the applied voltage exceeds the critical value for superinsulation and confirm thus the origin of this phenomenon.

Fig. 7: Induced non-local voltages.
figure 7

Magnetic field and applied voltage dependencies of induced polarization voltages at different positions of applied voltage probes. Temperature of measurements is T = 60 mK.

Full size image
Fig. 8: Induced non-local voltages.
figure 8

Temperature and applied voltage dependencies of induced polarization voltages at different positions of applied voltage probes, measured with an applied perpendicular magnetic field of 3.5 T.

Full size image

When both a perpendicular magnetic field and an in-plane electric field are present, complex polarization patterns can be detected in the film plane, depending on where the voltage is applied and where the non-local voltage is measured. In Figs. 9 and 10 we present such non-local polarization patterns. The first couple of subscripts on the applied voltage V indicates the probes, the second couple the contacts where external voltage is applied. These measurements were done at a temperature T = 160 mK, below the superinsulating transition.

Fig. 9: Induced non-local voltages.
figure 9

a–d Magnetic field and applied voltage dependencies of induced polarization voltages at different positions of voltage probes, indicated by the first subscripts on the induced non-local voltages. Applied voltage contacts are indicated by the second subscripts on the induced non-local voltages. Last number of second subscript is ground. Temperature of measurements is T = 160 mK. e Schematic measuring circuit.

Full size image
Fig. 10: Induced non-local voltages.
figure 10

a–d Magnetic field and applied voltage dependencies of induced polarization voltages at different positions of voltage probes, indicated by the first subscripts on the induced non-local voltages. Applied voltage contacts are indicated by the second subscripts on the induced non-local voltages. Last number of second subscript is ground. Temperature of measurements is T = 160 mK. e Schematic measuring circuit.

Full size image

Note that in all these measurements, a magnetic field of at least ~2–3 T is required to produce appreciable induced voltages. Since the length scale ℓ associated with a magnetic field B, in a model where 2e is taken as the unit charge, is given by

$$\ell=\sqrt{\frac{\hslash }{2eB}}=\frac{18}{\sqrt{B(Tesla)}}nm\,,$$
(1)

this means that, in this sample, the electric pion excitations have a dimension smaller than  ≈ 13–10 nm, which is essentially the width of the perforated films, see “Methods”. The fundamental string excitations of the model, thus, feel the third dimension and this is why these perforated films show 3D bulk superinsulation.

Methods

Magnetic monopoles and superinsulation

Superinsulation occurs in granular materials, where the granularity can be either emergent18,19 or synthetic, as in JJA20 or in the present nanopatterned metamaterial. Whatever case is realized, such a system can be modeled as an effective JJA with random couplings. One might thus be lead to think that disorder-induced localization is the origin of the vanishing conductance. This has been already disproved by several experiments, even on films with irregular granularity16,17. In the present nanopatterned material, however, the granularity is perfectly regular and ordered. If disorder would be the root cause of the vanishing conductance, this phenomenon should not take place in these nanopatterned materials. But it clearly does! This unambiguously excludes disorder as the origin of superinsulation.

The point is that the circulation of the local order parameter phase on neighboring granules can lead to the formation of coreless vortices and that there is nothing preventing these vortices to end inside the system so that their ends form a magnetic monopole-antimonopole pair. Such configurations must be included in the electromagnetic effective action because the gauge group in a lattice-like granular system is compact5. The presence or absence of magnetic monopoles becomes, thus, only a dynamical issue depending on the value of the effective coupling constant inside the material8. And these monopoles can condense, leading to superinsulation as electric confinement, without any role of disorder.

The compact Euclidean effective action determining the properties of the granular material follows from the standard general relation and is (we use natural units c = 1, ℏ = 1, ε0 = 1)

$$Z =\mathop{\sum }\limits_{\{{m}_{\mu }\}}\int{\mathcal{D}}{A}_{{{\mu }}}\,{{\rm{e}}}^{-S}\,,\\ S = \mathop{\sum }\limits_{x}\frac{1}{4{f}^{2}}{\left({F}_{{{\mu \nu }}}-2\pi {S}_{\mu \nu }\right)}^{2}\,,$$
(2)

where x denotes the granule sites, Fμν = dμAν − dνAμ is the usual electromagnetic field tensor, the integers mμ = (1/2)KμαβSαβ describe the conserved magnetic monopole currents, Kμαβ being the proper gauge-invariant lattice version of the curl operator ϵμναβ∂ν (see e.g., ref. 4)., and the dimensionless coupling f is the effective strength of the Coulomb interaction in the material. The integer magnetic currents mμ are the only physical degrees of freedom contained in the plaquette integers Sμν. Non-relativistic effects can be incorporated by an effective speed of light smaller than c. If the unit of an electric charge is 2e, magnetic charge is quantized in units of 2π/2e = π/e. For simplicity, we have presented here the partition function at zero temperature; a finite temperature can be incorporated by considering a finite lattice of length 1/kBT in the Euclidean time direction, with periodic boundary conditions for our bosonic fields, see e.g., ref. 4.

Integrating out the gauge field Aμ one obtains the model formulated solely in terms of magnetic monopole currents

$$Z =\,{Z}_{0}\cdot \mathop{\sum }\limits_{\{{m}_{\mu }\}}{{\rm{e}}}^{-S}\,,\\ S =\,\frac{{\pi }^{2}}{{f}^{2}}\mathop{\sum }\limits_{x}{m}_{\mu }\frac{1}{-{\nabla }^{2}}{m}_{\mu }\,,$$
(3)

where Z0 is the partition function of the non-compact model, describing normal Mott insulators, and − 1/∇2 is a short-hand notation for the Poisson Green’s function G(x), defined by

$$-{\nabla }^{2}G({\bf{x}})={\delta }^{2}({\bf{x}})\,.$$
(4)

Therefore monopoles interact via a Coulomb potential in a 4D Euclidean space. For weak coupling they are suppressed and we have a Mott insulator, for strong coupling they Bose condense5, see ref. 4. for a review. In this state, electric charges are confined by a linear potential generated by an electric flux tube between the charges and holes3,4, and we have a superinsulator.

Vogel–Fulcher–Tamman scaling

Superinsulators turn into Mott insulators when the temperature is raised enough, so that the formation of a condensate of monopoles is suppressed. In 2D, this is a BKT transition9,10,11, characterized by the BKT scaling of the resistance

$$R(T)\propto {{\rm{e}}}^{\frac{{b}_{2}}{\sqrt{| T-{T}_{{\rm{BKT}}}| }}}\,.$$
(5)

Below this transition, the resistance is infinite, and the system, therefore, is a dual mirror of superconductors, which demonstrate a zero resistance below this BKT transition. This is what distinguishes superinsulators from Mott insulators: in superinsulators, the resistance is infinite in an entire finite temperature domain. In 3D, the physics is the same but the resistance scaling is different12,

$$R(T)\propto {{\rm{e}}}^{\frac{{b}_{3}}{| T-{T}_{{\rm{VFT}}}| }}\,.$$
(6)

This behavior is known as a Vogel–Fulcher–Tamman (VFT) scaling and is the hallmark of 3D bulk superinsulators.

If the voltage applied to a superinsulator is below a critical threshold voltage Vc1, the current does not flow across the system. A current can be passed through a superinsulator by applying a voltage above a critical threshold voltage Vc1. In this case, the electric flux tubes connecting the charges of the opposite signs and preventing their further separation are stretched to the extent that they touch the boundary of the sample. Monopoles are still at work but the flux tube is long enough so that current can pass through the system. This is the electric mixed state16, the state dual to the magnetic mixed state in type-II superconductors, in which the applied magnetic field is above a critical value Hc1 and penetrates the sample in a form of Abrikosov vortices. Analogously, in superinsulators, the electric fields are expelled entirely from superinsulators as long as applied voltages remain below Vc1, we refer to this phenomenon as to the electric formation of a Meissner state4,16. At the intermediate voltages, Vc1 < V < Vc2, electric strings form within a superinsulator. And, finally, when the applied voltage is above a second critical voltage Vc2, superinsulation is destroyed entirely.

Electric strings and their equations of motion

In the superinsulating state, the fundamental excitations are strings, representing thin electric flux tubes see ref. 4. for a review. Contrary to Abrikosov vortices in superconductors, these strings are open, with a Cooper pair and a Cooper hole at their ends. Their gap and dimension are governed by the string tension σ, which is the string equivalent of mass for particles. Such open electric flux tubes can be thought of as the electric equivalent of strong interaction pions.

As the dynamics of charged particles is encoded in a four-vector current jμ, describing their world-lines and coupled to a vector gauge potential Aμ, the dynamics of strings is encoded in an antisymmetric tensor current Jμν, describing their world-sheets and coupled directly to the electromagnetic fields Fμν, which are the fundamental fields4,16; there is no more a vector potential, except to describe the coupling of the string endpoints, described again by a current jμ = ∂ν Jμν encoding the world-sheet boundaries.

The equation of motion for strings in the superinsulating state is given by

$${J}^{\mu \nu }\propto {F}^{\mu \nu }+\ldots \,,$$
(7)

which is the electric equivalent of the London equations jμ ∝ Aμ for a superconductor (as in this case the … denote higher-order in derivatives terms)4,16. The charged endpoints follow the usual electromagnetic laws.

When an applied external magnetic field B3 = B becomes sufficiently strong to excite one pion per unit volume, ∣B∣ > O(σ), this creates a tensor current J12, which can be understood as the current in direction 1 of a unit vector charge in direction 2 minus the current in direction 2 of a unit vector current in direction 1, where the vector charge represents the electric field. A rigid string thus moves as a whole in the direction E × B and, correspondingly, the Lorentz force on the endpoint charges stretches the string in the direction of E, making it even more rigid. Correspondingly, when a sufficiently strong electric field E i, i = 1, 2 is applied in the plane perpendicular to the magnetic field, this creates static strings with vector density J 0i in the interior of the sample16.

Electric strings interact with each other via a short-range, Yukawa-type potential4. As for magnetic flux tubes in supercoundcutors, parallel electric strings of the same polarity repel each other, while electric strings of opposite polarity attract each other.

Finite-size effects

As usual, phase transitions in real samples are broadened due to finite-size effects. We can estimate the temperature where these deviations start to show up by considering the correlation length of the VFT transition. This is given by refs. 4,12

$$\xi=\xi (T=\infty )\,{{\rm{e}}}^{\frac{{b}_{3}}{| T-{T}_{{\rm{VFT}}}| }}\,,$$
(8)

where ξ(T = ∞) ≈ ℓ, with ℓ being the ultraviolet cutoff, which in our case is the effective lattice spacing of the nanopatterned material4. At the transition, the correlation length diverges exponentially, in analogy to the 2D BKT case but with a different power in the exponent. However, this divergence is clearly cut off at the sample size L in real samples. One can estimate the temperature at which this deviation shows up by setting ξ = L in the above equation. This gives

$${T}_{{\rm{dev}}}={T}_{{\rm{VFT}}}+\frac{{b}_{3}}{{\rm{\ln }}(L/\ell )}\,.$$
(9)

For the 2 mm sample, TVFT = 0.31°K, b3 = 2.05°K, L = 0.1 mm, and ℓ ≈ 100 nm, which gives Tdev ≈ 0.61°K. For the 1 mm sample, TVFT = 0.275°K, b3 = 1. 9°K, with L = 0.1 mm, and ℓ ≈ 100 nm, as for the other sample: this gives Tdev ≈ 0.55°K. Note that, due to the larger power in the exponential scaling, finite-size effects are much more important for VFT than for BKT scaling.