Bereziskii-Kosterlitz-Thouless transition in the Weyl system PtBi2

semimetal band structure, with unusually robust two-dimensional superconductivity in thin ﬁms. Transport measurements show that high-quality PtBi 2 crystals are three-dimensional superconductors ( T c (cid:39) 600 mK) with an isotropic critical ﬁeld ( B c (cid:39) 50 mT). Remarkably, we evidence in a rather thick ﬂake (60 nm), exfoliated from a macroscopic crystal, the two-dimensional nature of the superconducting state, with a critical temperature T c (cid:39) 370 mK and highly-anisotropic critical ﬁelds. Our results reveal a Berezinskii-Kosterlitz-Thouless transition with T BKT (cid:39) 310 mK and with a broadening of Tc due to inhomogenities in the sample. Due to the very long superconducting coherence length ξ in PtBi 2 , the vortex-antivortex pairing mechanism can be studied in unusually-thick samples (at least ﬁve times thicker than for any other two-dimensional superconductor), making PtBi 2 an ideal platform to study low dimensional superconductivity in a topological semimetal.


Wuppertal, Germany
Symmetry breaking in topological matter became, in the last decade, a key concept in condensed matter physics to unveil novel electronic states.In this work, we reveal that broken inversion symmetry and strong spin-orbit coupling in trigonal PtBi 2 lead to a Weyl 1 Introduction Two-dimensional (2D) superconductivity has attracted a lot of attention for more than eighty years since the discovery of the superconducting properties of Pb and Sn thin films 1 .In two dimensions, low energy fluctuations prevent the spontaneous breaking of continuous symmetries at any finite temperature [2][3][4][5][6] .Nevertheless, a quasi-long range correlation of an order parameter can develop at low temperature.Such an ordered phase remains very fragile and can be easily destroyed by the presence of topological point defects like vortices.The transition between the quasi-long range ordered phase and the disordered phase is called the Bereziskii-Kosterliztz-Thouless (BKT) transition [7][8][9] .Below the critical temperature T BKT , pairing between vortex and antivortex allows a quasi-long range order.At higher temperature, the larger entropy of unbound topological defects prevails and the dissociation of vortex-antivortex pairs leads to a disordered phase 10 .Such a transition should take place in a 2D superconductor, i.e. when the superconducting phase coherence length ξ becomes larger than the thickness d of the superconducting film (ξ > d) 11,12 .
In this context, PtBi 2 is of particular interest.Beyond its very large magnetoresistance measured in the hexagonal 52 and pyrite 53 crystal structures, the spin-orbit coupling together with the broken I are responsible for a variety of interesting electronic properties including a strong Rashba-like spin splitting 54 , triply degenerated points 55 , sixfold fermion near the Fermi level 56 and non-trivial topology in the monolayer limit 57 .In this work, in addition to the unveiling of Weyl points at 48 meV above the Fermi energy, we characterize the superconducting transition of a macrosopic crystal of trigonal PtBi 2 , from which we infer a long coherence length of 55 nm.Remarkably, we show that the superconductivity persists in a 60 nm thin exfoliated sample in which its 2D nature is evidenced and for which a Berezinsky-Kosterlitz-Thouless (BKT) transition takes place.To our knowledge, and excluding the case of effective 2D layered superconductors 58,59 , the flake is five times thicker than any superconducting films exhibiting a BKT transition reported so far 60 , making PtBi 2 a prime candidate to study low dimensional superconductivity.Among the few Weyl semimetals showing a superconducting transition under ambiant pressure [61][62][63][64][65] , PtBi 2 is, together with MoTe 2 66 , the only one exhibiting such a 2D superconductivity.

Results
Band structure.We studied the electronic structure of trigonal PtBi 2 in the space group P31m Magnetoresistance.To investigate the properties of PtBi 2 , high quality single crystals of trigonal PtBi 2 were grown via self-flux method as described in details in Ref. 67.We first contacted bulk single crystals of PtBi 2 in a four probe configuration using silver wires and epoxy.A crystal was mounted on a piezo driven rotator and measured under magnetic field down to liquid helium temperature.When cooled down, the resistance decreases such that the residual resistance ratio reaches about 130 (see SI-2).At low temperature, we measured the magnetoresistance for different angles of the magnetic field ranging between -30°and 120°with 0°corresponding to an out-ofplane field and 90°to an in-plane field.
A strong anisotropy of the magnetoresistance is measured when tilting the magnetic field with respect to the orthogonal orientation and a maximum is reached for a tilt angle θ ∼ ±20°.
Close to these angles, the magnetoresistance reaches 3000% at B = 15 T (Figure 1.d and 1.e), one order of magnitude smaller than the magnetoresistance reported in Ref. 55  of bulk samples shows a zero-bias drop of almost 50% at 1.55 K, at about 2.5 × T c .The drop, is still visible at temperatures up to 2.9 K, corresponding to an almost fourfold increase from zeropressure measurements.
To further characterize the superconducting state, we measured the temperature dependency of the critical field B c .In the absence of any confinement, the Ginzburg-Landau theory in three dimensions (3D) predicts a linear temperature dependence of B c for T close to T c , with Φ 0 = h/2e the superconducting flux quantum (h is the Planck constant and e the electronic charge) and ξ the superconducting coherence length at T = 0.The result is plotted in The critical field B c is found to be strongly reduced for a perpendicular field (along the caxis).In this direction, B c is about one order of magnitude smaller than the value measured in the macrostructure (B c,⊥ ∼ 6mT).The temperature dependence of B c,⊥ is linear (Figure 3.b in blue), like in the case of the macroscopic crystal.This was expected since there is no confinement effect in the ab− plane.Nevertheless, we measured two different linear behaviors of B c,⊥ , crossing each other at about 335 mK (see also SI-9 for different set of contacts).The low temperature linear dependence corresponds to ξ = 180 nm and a T c of 392 mK, substantially larger than the T c 360 mK measured.The high temperature behavior corresponds to T c = 360 mK, in good agreement with the measured value, and to ξ = 120 nm.An explanation of this unexpected crossover is still lacking so far.
When tilting the magnetic field by 90 °, we can extract the temperature dependence of B c, (within a 1.1 °misalignment as shown below).Two important differences can be noticed with respect to the 3D case.Firstly, the amplitude of the critical field is about 30 times larger when pointing into the plane than perpendicularly to it.Taking the full angular dependence into account, so as to correct the misalignment, the ratio between in-plane and out-of-plane critical fields amounts up to 57.The in-plane critical field reaches 240 mT, four times larger than the B c of a macroscopic structure.This value remains below the Pauli paramagnetic limit 70,71 B p 1.84T c 660 mT.Secondly, the temperature dependence of B c, is not linear.Rather, it follows the 2D phenomenological Ginzburg-Landau theory 72 valid for ξ > d where d is the thickness of the superconductor: Taking into account a misalignment of 1.1°, it is possible to fit the temperature dependence of indicating that superconductivity cannot be attributed to surface states but rather to bulk states.
Indubitably, this temperature dependence points to the 2D nature of the superconductivity.The value of ξ is between the value found in macroscopic structure and that given by the temperature dependence of B c,⊥ .The discrepancy between the values of ξ extracted from B c,⊥ (T ) and from B c, (T ) are reasonable and might result from in-plane anisotropy as already measured in MoTe 2 66 .
The angular dependence B c (θ) with θ being the angle between the magnetic field and the perpendicular axis of the flake, can give further evidence of the low dimensionality of the super-conducting state.As shown by the Tinkham model, and contrary to the 3D anisotropic mass model derived from the Ginzburg-Landau theory, the 2D nature of the superconductivity is indicated by a cusp-like peak in B c (θ) at θ = 90°(corresponding to an in-plane field).The 3D and 2D angular dependences are respectively given by where θ 0 stands for the misalignment angle.In Figure 3.c, a very sharp peak of B c is measured at θ 90°with a ratio between the largest and the smallest critical field that can be as large as 57.
As it can be seen in the inset in the As shown in Figure 4.a, due to the finite critical current, the V (I) characteristics become non-linear at low bias and low temperature, with a temperature-dependent power law 73 as: where J S is the superfluid density and the exponent a is equal to 1 at T T c , as well as at high where A is a number of the order of unity and α is the scale of the vortex-core energy, which may deviate from its regular value α = 1.The fit of the experimental data can reproduce the temperature dependence but gives T BKT 200 mK, much below the value given by the V (I) characteristics.
Fixing T BKT to 310 mK, as given by V (I) measurements, no reasonable fit could be obtained (see the red dashed line in Figure 4.c for the best fit).
A model that accounts for spatial inhomogeneities was proposed in Ref. 73, where a Gaussian distribution of the critical current, or equivalently, of T BKT , is introduced in the Halperin-Nelson model.We have now with b 2α T BKT /(T c − T BKT ).This model fits very well the experimental data even for T BKT fixed at 310 mK, the value given by the V (I) characteristics (see Figure 4.c) and we found δ 24 mK, a minimal spread of the transition temperature.The parameter b and A are respectively 1.1 and 13, some typical values expected from the theory 73 .From the value of b, one can calculate the superconducting (BCS) transition temperature T c , assuming that the vortex-core energy takes its conventional value (α = 1).We found T c = T BKT × (1 + 4/b 2 ) = 410 mK which corresponds to about 70% of the macroscopic crystal value, a reasonable decrease of T c for an exfoliated flake.

Discussion
The role of inhomogeneities in the BKT transition can be seen in both the V (I) characteristics as well as in the R(T ) dependence.In the latter, as discussed above, the tail of R(T ) can be very well fitted by a Gaussian distribution of the BKT transition temperature resulting from spatial inhomogeneities of the superconducting flake.The fit gives reasonable values of the different physical parameters.A consequence of inhomogeneities is the lack of universal jump of the superfluid den-sity J(T ) and of a(T ) when T → T BKT .In an ideal homogeneous sample, the coefficient a(T ) should jump discontinuously at T BKT from 1 for T > T BKT to 3 for T ≤ T BKT .Inhomogeneities smooth out the temperature dependence of a(T ) 73 .This is indeed what we measured in our sample where the value of a(T ) slowly increases when the system goes through the BKT transition and no discontinuity of a could be evidenced (??.b).Importantly, the fit of R(T ) gives δ = 24 mK, a value that compares very well with the spatial distribution of T BKT measured in different sets of contact pairs with T BKT = 310 − 340 mK (see SI-10).
We also note that the ratio B c, /B c,⊥ is found to be very large, up to 57, which is unusual for such thick superconducting nanostructures (d 60 nm).This large ratio can be attributed first to the very long coherence length measured in our nanostructures (ξ > d) and second to the change of slope measured in B c,⊥ (T ) with a discontinuity of dB c,⊥ /dT at T 335 mK.Such a discontinuity is present for all the contact pairs measured in this nanostructure.The very large ratio measured and more particularly the large value of B c, (as compared to k B T c ) for such a large thickness makes exfoliated flakes of PtBi 2 single crystal a very promising candidate to study the 2D superconductivity beyond the Pauli limit 70,71 , provided that the thickness is reduced further in order to weaken or suppress any orbital depairing effect on the superconductivity.
The discovery of both the non-trivial topology of the band structure and the low dimensional superconductivity sheds new light on PtBi 2 , a material that was previously known for its very large linear magnetoresistance, its large spin-orbit coupling and the prediction of triply degenerate points in its band structure.Together with MoTe 2 66 , it appears to be the only semimetal exhibiting Weyl physics and 2D superconductivity.Beyond topological superconductivity, the very strong spin-orbit coupling coupled to the superconducting transition is of particular interest in the search for unconventional pairing in superconductors and might favor, for instance, the presence of FFLO superconducting states.
The evidence of a BKT transition in a 60 nm thick flake makes of PtBi 2 a remarkable system.
This very large thickness is highly unusual for the observation of 2D superconductivity.To our knowledge and excluding the case of layered superconductors, our flake is 5 times thicker than the thickest 2D superconductor reported so far 60 .The origin of the reduced dimensionality cannot be attributed to any surface effect or to any layered nature of PtBi 2 since we established the 3D nature of the superconductivity in macroscopic single crystals and since the effective superconducting thickness given by the fit of B c, (T ) compares very well with the geometrical thickness measured by AFM.Apart from promising properties of 2D superconductor that might be observed in thinner structures, such as the increase of B c, well beyond the Pauli limit, the large thickness is also an indication of a very robust signature of the usually fragile 2D superconducting state.PtBi 2 might be therefore a simple and privileged platform for studying 2D superconductivity and the BKT transition.To evaluate the robusness of the Weyl nodes, we also performed calculations for the crystal structure reported in Ref. 74.These confirm the presence of Weyl nodes albeit, in this case, they are found at higher energy, 96 meV.Noteworthy, between these two crystal structures, the values of a and c differ in less than 0.1% while differences in the Bi coordinates lead to a van der Waals gap (zvdW) 3% smaller in our refinement.A third calculation based on an artificially enlarged zvdW in our structural model yields the Weyl nodes at 79 meV indicating that, in fact, zvdW controls to a large extent the Weyl node energy.
Point contact measurements.The contacts where made by mechanically touching the edge of the sample with the wire of the noble metal.The differential resistance (dV/dI) of the contacts was recorded via lock-in technique in a quasi four probe configuration.Latter was achieved by soldering one pair of a current-potential wiring to the copper sample clamp and another pair to the counter electrode (Ag, Au or Cu).
Measurements techniques.For the superconductivity in macroscopic crystals, DC sources were used in a delta mode, a method particularly adapted to the measurement of low resistance samples.The measurements of nanostructures were done using standard lock-in amplifier techniques at low frequency (f < 200 Hz).
Nanostructure fabrication.The exfoliation was made on Si/SiO 2 substrates, with 290 nm thickness of oxide, to enhance optical contrast.The exfoliated structures were then contacted by standard electron-beam lithography procedure, with Cr-Au contacts.The structure we measured for this work, which can be seen in the supplementary information, has a thickness of 60nm, for a lateral size of about 10µm.
Definition of the critical parameters.The critical temperature T c is defined in this work as the temperature for which the resistance becomes half of the resistance in the normal state R N .
Similarly, the critical field B c and the critical current respectively.

S1 -Fermi arcs S2 -Temperature dependence
We present below the temperature dependence of the resistivity of a single macroscopic crystal exhibiting a residual resistance ratio of about 130.
S3 -Isotropy of the critical field in a macroscopic structure

S4 -Point contact measurements for different magnetic fields
As for the temperature, point-contact measurements show an enhanced local superconductivity in magnetic field likely due to local pressure, with indications of superconductivity at magnetic fields up to 1T.This value is six times larger than its bulk pressure value (≈ 150 mT).The Figure 8 shows the enhancement of B c at T = 1.55 K under the point contact for a magnetic field applied along the c-axis.The differential resistance (dV /dI) shows a zero-bias drop of almost 50% at 0 T and 1.55 K.At this temperature, a deviation from the ohmicity is observed at magnetic field up to 1 T, far beyond its bulk value.

S5 -Magnetoresistances of the nanostructure
The Figure 9 shows the magnetoresistances along the c-axis of the exfoliated crystal and for a field aligned in the a-b plane of the nanostructure.Measurements are done at T 100 mK and with an AC current of 5 µA.The perpendicular field can be swept up to ± 6T (the main coil of the 3D magnet system) whereas the parallel field could be swept only up to ± 2T.As for the macroscopic single crystal, the magnetoresistance is found to be almost linear and it is larger for a perpendicular field.

S6 -dV/dI for different contact pairs
Although the critical current is almost independent of the set of contacts considered, the exact shape of the differential resistance dV /dI(I dc ) depends on the set considered.Some contacts shows a single peak shape whereas others have several peaks, as seen in the Figure 10.Moreover, this feature does not depend on the contacts used for current bias, as shown in the Figure 10 and

S7 -Critical current versus magnetic field
We measured the dependence of the dV /dI(I dc ) with the magnetic field.In both directions B ⊥ and B , the critical current as defined in the main text (R(I c ) = R N /2) is found to be proportional to the magnetic field with a larger sensitivity when the field is applied along the perpendicular direction.The results are indicated in Figure 10 for an in-plane magnetic field and in Figure 11 for an out-of-plane magnetic field.The diamond shape of I c (B) in both direction indicates the proportionality of I c with B.
All the dV /dI show a similar diamond shape like, including the multi-peaked differential resistance.7) in the main text.The ac current was 1 µA and the measurements were done by increasing the temperature by steps of 10 mK with a total stabilization time longer than one hour so that the sample is very well thermalized.

(Figure 1
.a) based on the crystal structure reported in Ref. 67 and focused on the consequences of broken I.We performed fully-relativistic density-functional calculations treating the spin-orbit coupling in the four component formalism, as implemented in Ref. 68.The energy bands along the path indicated in Figure 1.b is presented in Figure 1.c.Similar to previous works 55, 67 , the bandstructure indicates a semimetallic character with several bands crossing the Fermi energy, generating various electron and hole pockets.Broken I opens up the possibility of accidental crossing of bands at isolated points (Weyl nodes).A search for Weyl nodes between bands N and N + 1, where N is the number of valence electrons per unit cell, yields the existence of twelve Weyl nodes, lying 48 meV above the Fermi energy.Six of the nodes are related by combinations of the three-fold rotation and reflection symmetries, while the remaining six nodes are connected to the former by time-reversal symmetry (Figure 1 b).Usually, Weyl nodes in the bulk electronic structure induce open Fermi arcs in the surface electronic structure.A calculation based on a semi-infinite slab along the 0001 direction shows that the surface Fermi energy contours in PtBi 2 present a strong sensitivity to the surface termination (see SI-1).In particular, while the Weyl nodes place within the projection of the bulk Fermi surface, clear spectral weight connecting opposite chirality Weyl nodes can be observed, which is more intense for a Bi 2 -terminated surface.

Figure 2
.b where a linear dependence of the in-plane critical field with temperature is measured over a wide range of temperature below T c .A linear fit in the temperature range allowed us to extract a large coherence length in PtBi 2 , with ξ = 56 nm.Two-dimensional superconductivity.The long ξ measured in a macroscopic crystal and the 2D van der Waals layered nature of PtBi 2 open the way to a mechanical exfoliation down to thicknesses lower than ξ in order to observe a transition from 3D to 2D superconductivity.To this end, a 57 nm thick and few tens of micron large exfoliated flake was contacted and measured down to 100mK (see Methods).Like the bulk samples, it exhibits a superconducting transition, with a drop in the resistance to zero at T c = 370 mK, almost a factor two lower than the critical temperature of the parent macroscopic crystal.To further characterize the superconducting state, we measured the differential resistance (dV /dI) as a function of a DC-current and for different temperatures below and above T c (Figure4.aand SI-6).For T T c , dV /dI remains constant and equal to the equilibrium normal state resistance R N .For T T c , the superconducting transition takes place and a gap opens up leading to a vanishing dV /dI at zero bias.At high DC currents, dV /dI recovers its value of the normal state R N .The transition occurs at a critical current I c ∼ 22 µA.The value of I c is almost temperature independent well below the critical temperature and it slowly vanishes when the temperature approches T c (see Figure3.a and SI-8).The critical current depends linearly on the magnetic field for both an in-plane and an out-of plane magnetic field (see SI-7).

B
c, (seeSI-11).A very good agreement between the fit and the experimental data is obtained (Figure3.b)and yields the values for T c = 363 mK, ξ = 90 nm and d = 52 nm.The thickness is in very good agreement with the value measured by atomic force microscopy d AFM =57 nm,

Figure 3 .
c, the sharpness of the peak allows us to measure very precisely θ 0 and we found θ 0 = 1.1°in our sample.Due to the very strong anisotropy measured in the sample, such a minimal misalignment induces a reduction of about 25% of B c at θ = 0°and the misalignment has to be taken into account as mentioned above.Importantly, a cusp-like angular dependence can be evidenced in the inset in Figure3.c, which also presents 2D (red) and 3D (blue) fits of B c (θ).Whereas the 2D model fits very well the experimental data, a clear departure from the 3D model can be seen in the inset.BKT Transition.Despite the large thickness of our PtBi 2 sample, the angular and temperature dependences of B c give clear evidence of 2D superconductivity, opening the way to the observation of a BKT transition.The signatures of such a phase transition can be measured in V (I) characteristics at different temperatures and in the temperature dependence of the resistance R(T ).
Figure 4.b where a(T ) reaches 3 for T = T BKT 310 mK.
The calculation of the Weyl nodes and of the surface Fermi surface were performed based on a tightbinding model obtained by constructing Wannier functions with the pro-jective technique implemented in the FPLO code.The model includes the orbitals Bi 6p, Pt 6s and Pt 5d.

Figure 1 :
Figure 1: (a) Crystal structure of trigonal PtBi 2 .(b) Brillouin zone.Green (light blue) points correspond to Weyl nodes of positive (negative) chirality.(c) Bandstructure along the path indicated in panel (b), which includes one of the Weyl nodes located at 48meV above the Fermi energy.(d) Magnetoresistance of a single crystal for different tilted angles θ of the field, θ being the angle between the magnetic field and the out-of-plane direction.(e) Angular dependence of the magnetoresistance at B = 15 T. (f) Fast Fourier Transform of the Shubnikov-de Haas oscillations in (d), for T=5K and θ = 20°.

Figure 2 :
Figure 2: (a) The temperature dependence of the resistance of a single crystal for T ≤ 46 K with a superconducting transition measured at T c 600 mK.The inset shows the differential resistance dV /dI as a function of the DC voltage in point contact measurements at different temperatures between 1.55 K and 3.1 K, indicating the onset of the superconductivity at about 2.5 K under point contact.(b) the temperature dependence of B c (grey points) with the linear fit from Equation (1), leading to ξ = 56 nm at very low temperature.

Figure 3 :
Figure 3: (a) Differential resistance dV /dI of the nanostructure as a function of the DC current at temperatures ranging regularly from 100 mK to 400 mK (the curves are shifted for better visibility).I c was found to be about 20 µA.(b) temperature dependencies of the perpendicular (B c,⊥ ) and planar (B c, ) critical fields with their fit to the 2D theoretical model, including the misalignment measured in (c).(c) the angular dependence of the critical field at T =100 mK shows a sharp peak with a maximum corresponding to θ = 91.1 °, indicating a 1.1°misalignment.Due to the sharpness of the peak, the θ = 0°is reduced by about 25% with respect to the true parallel critical field.Inset: the experimental data points (grey circles) of B c (θ) at T = 100 mK are fitted with the 3D Ginzburg-Landau model (blue line) and the 2D Tinkham model (red line).A better agreement is found for the 2D model.

Figure 4 :
Figure 4: (a) V (I) characteristics for different temperatures below T BKT and above T c in a logarithmic scale.The dashed line stands for V ∝ I 3 .(b) the same experimental data are plotted as d log V /d log I(I) = a = 1 + πJ S /T and the dashed line indicate the limit for which a(T BKT ) = 3.Both inset indicate that T BKT 310 mK.(c) the temperaure dependence of the resistance of a PtBi 2 from the normal regime (T 500 mK) down to T = 250 mK < T BKT .The Halperin-Nelson theory (red dashed line) with T BKT = 310 mK is not able to fit the experimental data.Accounting for inhomogeneities, the fit with Benfatto's fomulae is in very good agreement with the experimental data for T BKT = 310 mK as found from the out-of-equilibrium measurements.The fit leads to T c = 412 mK and δT BKT = 23 mK.We note that we do not fit R(T )/R N here, rather the equivalent quantity R(T )/R(500mK) = R(T )/R N × R N /R(500mK).

Figure 6 :
Figure 6: Temperature dependence of the resistivity of a macroscopic crystal between T = 300 K and 4.2 K.

Figure 7 :
Figure 7: Magnetoresistance of a macroscopic crystal of PtBi 2 with the magnetic field applied outof-plane (B ⊥ ) or along the two perpendicular in-plane directions (B ,0 and B ,90 ) measured at low temperature (T ∼ 100 mK).For the sake of clarity, the different curves are shifted.

Figure 8 :
Figure 8: Differential resistance dV /dI as a function of the DC voltage in point contact measurements at different magnetic fields between 0 T and 2 T at 1.55 K, indicating the onset of the superconductivity below 1 T for T = 1.55 K.

Figure 9 :
Figure 9: The magnetoresistances at T ∼ 100 mK for an in-plane and out-of-plane magnetic fields are shown in red and blue respectively.A picture of the sample can be seen in the graph together with the connection configuration for the measurements of the magnetoresistances.The typical size of the sample is about 10 µm.

Figure 10 :
Figure 10: dV /dI(I dc ) measured at T 100 mK for different set of contacts.The ac current is 1 µA and the differential resistance remains constant for higher dc-current.The contact configuration is indicated in the picture with the colors corresponding to the color of the curves.

Figure 11 :
Figure 11: Similar measurements as shown in ?? with different contacts used for both current bias and voltage probes.

Figure 12 :
Figure 12: (a) Mapping of dV /dI(I dc ) for different parallel magnetic field measured for −75µA ≤ I dc ≤ 75µA and with I ac = 1µA.(b) Parallel magnetic field dependency of the critical current as defined in the text for the same contacts as for (a).The contacting configuration is indicated in the inset of (b).

Figure 13 :
Figure 13: (a) Mapping of dV /dI(I dc ) for perpendicular parallel magnetic field measured for −75µA ≤ I dc ≤ 75µA and with I ac = 1µA.(b) Perpendicular magnetic field dependency of the critical current as defined in the text for the same contacts as for (a).The contacting configuration is the same as in Figure 12 and is indicated in the inset of (b).

Figure 15 :
Figure 15: Temperature dependence B c,⊥(T ) for different set of contacts indicated in the inset.The blue points are the data points already shown in the main text.The black points are the data points for a set of contact where the discontinuity of the slope is clearly visible, similar to the one reproduced in the main text whereas the red points are the points for which no discontinuity could be measured because of the too low accuracy of the determination of B ⊥ close to T c .

Figure 16 :
Figure 16: The R(T ) measured for different set of contacts indicated in the inset and fitted by Equation (7) in the main text.The ac current was 1 µA and the measurements were done by increasing the temperature by steps of 10 mK with a total stabilization time longer than one hour so that the sample is very well thermalized.

Table 1 :
Results of the temperature dependence of the resistance from the normal regime down to below the BKT transition.The value of T BKT is fixed by non-equilibrium measurements and α is assumed to be equal to unity whereas b, A and δ are the free parameters of the fit.T c is calculated based on the value of b and α. contact T BKT (mK) parameter b parameter A α δ (mK) T c (mK)