Bereziskii-Kosterlitz-Thouless transition in the Weyl system \ce{PtBi2}
Arthur Veyrat, Valentin Labracherie, Rohith Acharya, Dima L. Bashlakov, Federico Caglieris, Jorge I. Facio, Grigory Shipunov, Lukas Graf, Johannes Schoop, Yurii Naidyuk, Romain Giraud, Jeroen van den Brink, Bernd Büchner, Christian Hess, Saicharan Aswartham, Joseph Dufouleur
BBereziskii-Kosterlitz-Thouless transition in the Weyl sys-tem PtBi Arthur Veyrat , Valentin Labracherie , Rohith Acharya , Dima L. Bashlakov , FedericoCaglieris , , Jorge I. Facio , Grigory Shipunov , Lukas Graf , Johannes Schoop , , Yurii Naidyuk ,Romain Giraud , , Jeroen van den Brink , , Bernd Büchner , , Christian Hess , , , SaicharanAswartham & Joseph Dufouleur , *† Leibniz Institute for Solid State and Materials Research (IFW Dresden), Helmholtzstraße 20, D-01069 Dresden, Germany B. Verkin Institute for Low Temperature Physics and Engineering, NASU, 47 Nauky Ave., 61103Kharkiv, Ukraine CNR-SPIN, Corso Perrone 24, 16152 Genova, Italy Department of Physics, TU Dresden, D-01062 Dresden, Germany Université Grenoble Alpes, CNRS, CEA, Grenoble-INP, Spintec, F-38000 Grenoble, France Center for Transport and Devices, TU Dresden, D-01069 Dresden, Germany Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, D-42097Wuppertal, Germany
Symmetry breaking in topological matter became, in the last decade, a key concept in con-densed matter physics to unveil novel electronic states. In this work, we reveal that bro-ken inversion symmetry and strong spin-orbit coupling in trigonal PtBi lead to a Weyl * Corresponding author. † E-mail: [email protected] a r X i v : . [ c ond - m a t . s up r- c on ] J a n emimetal band structure, with unusually robust two-dimensional superconductivity in thinfims. Transport measurements show that high-quality PtBi crystals are three-dimensionalsuperconductors ( 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) mKand highly-anisotropic critical fields. Our results reveal a Berezinskii-Kosterlitz-Thoulesstransition with T BKT (cid:39) mK and with a broadening of Tc due to inhomogenities inthe sample. Due to the very long superconducting coherence length ξ in PtBi , the vortex-antivortex pairing mechanism can be studied in unusually-thick samples (at least five timesthicker than for any other two-dimensional superconductor), making PtBi an ideal platformto study low dimensional superconductivity in a topological semimetal.Keywords: Weyl semimetals, superconductivity, low dimensionality, Berezinskii-Kosterlitz-Thouless
Two-dimensional (2D) superconductivity has attracted a lot of attention for more than eighty yearssince the discovery of the superconducting properties of Pb and Sn thin films . In two dimen-sions, low energy fluctuations prevent the spontaneous breaking of continuous symmetries at anyfinite temperature . Nevertheless, a quasi-long range correlation of an order parameter can de-velop at low temperature. Such an ordered phase remains very fragile and can be easily destroyedby the presence of topological point defects like vortices. The transition between the quasi-long2ange ordered phase and the disordered phase is called the Bereziskii-Kosterliztz-Thouless (BKT)transition . Below the critical temperature T BKT , pairing between vortex and antivortex allows aquasi-long range order. At higher temperature, the larger entropy of unbound topological defectsprevails and the dissociation of vortex-antivortex pairs leads to a disordered phase . Such a tran-sition should take place in a 2D superconductor, i.e. when the superconducting phase coherencelength ξ becomes larger than the thickness d of the superconducting film ( ξ > d )
11, 12 .The experimental evidence of a BKT transition remains generally very challenging due tothe sensitivity of the ordered phase to any structural disorder
13, 14 . Nevertheless, BKT transitions in2D superconductors have been identified in different samples such as in low disordered evaporatedthin films
15, 16 or thin films grown by molecular beam epitaxy , exfoliated superconductors ,oxide heterostructures , field effect transistors
29, 30 and more recently in encapsulated structuresbased on one or several exfoliated monolayers of van der Waals materials, which are not super-conducting in their bulk form . 2D superconductivity and BKT transitions are of particularinterest when they take place in materials with brocken inversion symmetry ( I ) and strong spin-orbit coupling, where the electron spin degeneracy of Bloch states is lifted. This may lead tounconventional superconducting states, such as a mixed singlet and triplet superconductivity ,or a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) finite momentum pairing . It can also enablethe realization of a Weyl semimetal phase, which, coupled to superconductivity, constitutes an idealplatform to study unconventional superconducting states or topological superconductivity .In this context, PtBi is of particular interest. Beyond its very large magnetoresistance3easured in the hexagonal and pyrite crystal structures, the spin-orbit coupling together withthe broken I are responsible for a variety of interesting electronic properties including a strongRashba-like spin splitting , triply degenerated points , sixfold fermion near the Fermi level andnon-trivial topology in the monolayer limit . In this work, in addition to the unveiling of Weylpoints at 48 meV above the Fermi energy, we characterize the superconducting transition of amacrosopic crystal of trigonal PtBi , from which we infer a long coherence length of 55 nm. Re-markably, we show that the superconductivity persists in a 60 nm thin exfoliated sample in whichits 2D nature is evidenced and for which a Berezinsky-Kosterlitz-Thouless (BKT) transition takesplace. 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 reportedso far , making PtBi a prime candidate to study low dimensional superconductivity. Among thefew Weyl semimetals showing a superconducting transition under ambiant pressure , PtBi is,together with MoTe , the only one exhibiting such a 2D superconductivity. We studied the electronic structure of trigonal PtBi in the space group P31m(Figure 1.a) based on the crystal structure reported in Ref. 67 and focused on the consequencesof broken I . We performed fully-relativistic density-functional calculations treating the spin-orbitcoupling in the four component formalism, as implemented in Ref. 68. The energy bands alongthe path indicated in Figure 1.b is presented in Figure 1.c. Similar to previous works
55, 67 , thebandstructure indicates a semimetallic character with several bands crossing the Fermi energy,4enerating various electron and hole pockets. Broken I opens up the possibility of accidentalcrossing 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 twelveWeyl nodes, lying 48 meV above the Fermi energy. Six of the nodes are related by combinationsof the three-fold rotation and reflection symmetries, while the remaining six nodes are connectedto the former by time-reversal symmetry (Figure 1 b).Usually, Weyl nodes in the bulk electronic structure induce open Fermi arcs in the surfaceelectronic structure. A calculation based on a semi-infinite slab along the 0001 direction shows thatthe surface Fermi energy contours in PtBi present a strong sensitivity to the surface termination(see SI-1). In particular, while the Weyl nodes place within the projection of the bulk Fermisurface, clear spectral weight connecting opposite chirality Weyl nodes can be observed, which ismore intense for a Bi -terminated surface. Magnetoresistance.
To investigate the properties of PtBi , high quality single crystals oftrigonal PtBi were grown via self-flux method as described in details in Ref. 67. We first contactedbulk single crystals of PtBi in a four probe configuration using silver wires and epoxy. A crystalwas mounted on a piezo driven rotator and measured under magnetic field down to liquid heliumtemperature. When cooled down, the resistance decreases such that the residual resistance ratioreaches about 130 (see SI-2). At low temperature, we measured the magnetoresistance for differentangles of the magnetic field ranging between -30° and 120°with 0° corresponding to an out-of-plane field and 90° to an in-plane field. 5 strong anisotropy of the magnetoresistance is measured when tilting the magnetic fieldwith respect to the orthogonal orientation and a maximum is reached for a tilt angle θ ∼ ± B = 15 T (Figure 1.d and 1.e), oneorder of magnitude smaller than the magnetoresistance reported in Ref. 55, but larger than what isreported in Ref. 69. For such angles, the magetoresistance is almost linear and large Shubnikov-deHaas oscillations appear at low temperature. They rapidly disappear when the angle deviates fromoptimized values, as reported for stoichiometric PtBi . K. Asingle sharp peak emerges at a magnetic frequency of
T, in very good agreement with the mainpeak measured in Ref. 55. These oscillations are therefore attributed to a single pocket, the crosssection of which is . nm − large, in the direction perpendicular to the magnetic field ( θ = 20 °).This corresponds to a typical wave vector k ∼ . nm − . Bulk superconductivity.
When cooled down to very low temperature in a dilution fridge, theresistance decreases down to T ∼ K and remains nearly constant down to the superconductingtransition at a critical temperature T c ∼ mK (see Figure 2.a and methods for the definition ofthe critical temperature, current and field). At T = 100 mK, the zero-resistance state is destroyedby the application of a mT large in-plane magnetic field, corresponding to the low temperaturein-plane critical field. In macroscopic structures, the critical field is found to be almost isotropic(see SI-3). Point contact measurements (see Methods) indicate that T c increases likely due to localpressure similar to MoTe (see inset of Figure 2.a and SI-4): the differential resistance ( dV /dI )6f bulk samples shows a zero-bias drop of almost 50 % at . K, at about . × T c . The drop, isstill visible at temperatures up to . K, corresponding to an almost fourfold increase from zero-pressure measurements.To further characterize the superconducting state, we measured the temperature dependencyof the critical field B c . In the absence of any confinement, the Ginzburg-Landau theory in threedimensions (3D) predicts a linear temperature dependence of B c for T close to T c , B c ( T ) = Φ πξ (cid:16) − TT c (cid:17) , (1)with Φ = h/ e the superconducting flux quantum ( h is the Planck constant and e the electroniccharge) and ξ the superconducting coherence length at T = 0 . The result is plotted in Figure 2.bwhere a linear dependence of the in-plane critical field with temperature is measured over a widerange of temperature below T c . A linear fit in the temperature range allowed us to extract a largecoherence length in PtBi , with ξ =
56 nm.
Two-dimensional superconductivity.
The long ξ measured in a macroscopic crystal andthe 2D van der Waals layered nature of PtBi open the way to a mechanical exfoliation down tothicknesses lower than ξ in order to observe a transition from 3D to 2D superconductivity. To thisend, a 57 nm thick and few tens of micron large exfoliated flake was contacted and measured downto 100mK (see Methods).Like the bulk samples, it exhibits a superconducting transition, with a drop in the resistanceto zero at T c = 370 mK, almost a factor two lower than the critical temperature of the parent7acroscopic crystal. To further characterize the superconducting state, we measured the differ-ential resistance ( dV /dI ) as a function of a DC-current and for different temperatures below andabove T c (Figure 4.a and SI-6). For T (cid:29) T c , dV /dI remains constant and equal to the equilibriumnormal state resistance R N . For T (cid:46) T c , the superconducting transition takes place and a gapopens up leading to a vanishing dV /dI at zero bias. At high DC currents, dV /dI recovers its valueof the normal state R N . The transition occurs at a critical current I c ∼ µ A. The value of I c isalmost temperature independent well below the critical temperature and it slowly vanishes whenthe temperature approches T c (see Figure 3.a and SI-8). The critical current depends linearly onthe magnetic field for both an in-plane and an out-of plane magnetic field (see SI-7).The critical field B c is found to be strongly reduced for a perpendicular field (along the c -axis). In this direction, B c is about one order of magnitude smaller than the value measured in themacrostructure ( B c, ⊥ ∼ 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 effectin the ab − plane. Nevertheless, we measured two different linear behaviors of B c, ⊥ , crossingeach other at about 335 mK (see also SI-9 for different set of contacts). The low temperaturelinear dependence corresponds to ξ = 180 nm and a T c of 392 mK, substantially larger than the T c (cid:39) mK measured. The high temperature behavior corresponds to T c = 360 mK, in goodagreement with the measured value, and to ξ = 120 nm. An explanation of this unexpectedcrossover is still lacking so far.When tilting the magnetic field by 90 °, we can extract the temperature dependence of B c , (cid:107) B c of amacroscopic structure. This value remains below the Pauli paramagnetic limit
70, 71 B p (cid:39) . T c (cid:39)
660 mT. Secondly, the temperature dependence of B c , (cid:107) is not linear. Rather, it follows the 2Dphenomenological Ginzburg-Landau theory valid for ξ > d where d is the thickness of the su-perconductor: B c , (cid:107) ( T ) = Φ √ πξ (0) d (cid:114) − TT c , (2)Taking into account a misalignment of 1.1°, it is possible to fit the temperature dependence of B c , (cid:107) (see SI-11). A very good agreement between the fit and the experimental data is obtained(Figure 3.b) and yields the values for T c = 363 mK, ξ = 90 nm and d = 52 nm. The thicknessis in very good agreement with the value measured by atomic force microscopy d AFM =
57 nm,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. Thevalue of ξ is between the value found in macroscopic structure and that given by the temperaturedependence of B c , ⊥ . The discrepancy between the values of ξ extracted from B c , ⊥ ( T ) and from B c , (cid:107) ( T ) are reasonable and might result from in-plane anisotropy as already measured in MoTe .The angular dependence B c ( θ ) with θ being the angle between the magnetic field and theperpendicular axis of the flake, can give further evidence of the low dimensionality of the super-9onducting state. As shown by the Tinkham model, and contrary to the 3D anisotropic mass modelderived from the Ginzburg-Landau theory, the 2D nature of the superconductivity is indicated bya cusp-like peak in B c ( θ ) at θ = (cid:18) B c ( θ ) cos( θ − θ ) B c , ⊥ (cid:19) + (cid:18) B c ( θ ) sin( θ − θ ) B c , (cid:107) (cid:19) = 1 , and (3) (cid:12)(cid:12)(cid:12)(cid:12) B c ( θ ) cos( θ − θ ) B c , ⊥ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) B c ( θ ) sin( θ − θ ) B c , (cid:107) (cid:19) = 1 , (4)where θ stands for the misalignment angle. In Figure 3.c, a very sharp peak of B c is measured at θ (cid:39) ° with a ratio between the largest and the smallest critical field that can be as large as .As it can be seen in the inset in the Figure 3.c, the sharpness of the peak allows us to measure veryprecisely θ and we found θ = 1 . ° in our sample. Due to the very strong anisotropy measured inthe sample, such a minimal misalignment induces a reduction of about % of B c at θ = 0 °and themisalignment has to be taken into account as mentioned above. Importantly, a cusp-like angulardependence can be evidenced in the inset in Figure 3.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 fromthe 3D model can be seen in the inset. BKT Transition.
Despite the large thickness of our PtBi sample, the angular and tem-perature dependences of B c give clear evidence of 2D superconductivity, opening the way to theobservation 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 ) . 10s shown in Figure 4.a, due to the finite critical current, the V ( I ) characteristics becomenon-linear at low bias and low temperature, with a temperature-dependent power law as: V ∝ I a ( T ) with a ( T ) = 1 + πJ S ( T ) /T, (5)where J S is the superfluid density and the exponent a is equal to 1 at T (cid:38) T c , as well as at highcurrent bias ( I (cid:29) I c ) for T (cid:46) T c . In a BKT transition, a ( T ) at I ∼ I c slowly increases when T decreases below T c and it largely exceeds 1 at temperatures much below the BKT temperature T BKT < T c . At the BKT transition, πJ S ( T BKT ) /T BKT = 2 so that a ( T BKT ) = 3 . Consideringthe V ( I ) characteristics measured in our flake above and below T c , we observe a non-ohmicity( a ( T ) > ) when T (cid:46) T c (see Figure 4.a and 4.b). In a logarithmic plot, the V ( I ) shows a cubicpower law for T BKT (cid:39) mK (Figure 4.a). A confirmation of the BKT transition temperaturecan be done by plotting a ( T ) = d log V /d log I as a function of the bias DC-current as shown inFigure 4.b where a ( T ) reaches 3 for T = T BKT (cid:39) mK.In order to confirm that a BKT transition occurs in our sample, we focused on the temperaturedependence of the resistance of our exfoliated thin film. Following Halperin and Nelson theory ,Benfatto et al. took, in absence of inhomogeneities and size effect, a temperature dependence ofthe resistance for T (cid:38) T BKT given by R ( T ) /R N = 1 / [1 + (∆ σ/σ N )] with ∆ σσ N = 4 A (cid:34) sinh (cid:32) α (cid:114) T c − T BKT T − T BKT (cid:33)(cid:35) (6)where A is a number of the order of unity and α is the scale of the vortex-core energy, which maydeviate from its regular value α = 1 . The fit of the experimental data can reproduce the temperaturedependence but gives T BKT (cid:39) mK, much below the value given by the V ( I ) characteristics.11ixing T BKT to 310 mK, as given by V ( I ) measurements, no reasonable fit could be obtained (seethe 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 Gaussiandistribution of the critical current, or equivalently, of T BKT , is introduced in the Halperin-Nelsonmodel. We have now R ( T ) R N = 1 √ πδ (cid:90) exp (cid:32) − ( t − T BKT ) δ (cid:33) × A (cid:34) sinh (cid:32) b (cid:114) tT − t (cid:33)(cid:35) − dt (7)with b (cid:39) α (cid:112) 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 (cid:39) mK, a minimal spread of the transition temperature. The parameter b and A are respectively . and , some typical values expected from the theory . From the value of b , one can calculate thesuperconducting (BCS) transition temperature T c , assuming that the vortex-core energy takes itsconventional value ( α = 1 ). We found T c = T BKT × (1 + 4 /b ) = 410 mK which corresponds toabout 70% of the macroscopic crystal value, a reasonable decrease of T c for an exfoliated flake. The role of inhomogeneities in the BKT transition can be seen in both the V ( I ) characteristics aswell as in the R ( T ) dependence. In the latter, as discussed above, the tail of R ( T ) can be very wellfitted by a Gaussian distribution of the BKT transition temperature resulting from spatial inhomo-geneities of the superconducting flake. The fit gives reasonable values of the different physicalparameters. A consequence of inhomogeneities is the lack of universal jump of the superfluid den-12ity 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 . Inhomogeneitiessmooth out the temperature dependence of a ( T ) . This is indeed what we measured in our samplewhere the value of a ( T ) slowly increases when the system goes through the BKT transition andno 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 ofcontact pairs with T BKT = 310 − mK (see SI-10).We also note that the ratio B c , (cid:107) /B c , ⊥ is found to be very large, up to 57, which is unusualfor such thick superconducting nanostructures ( d (cid:39) nm). This large ratio can be attributedfirst to the very long coherence length measured in our nanostructures ( ξ > d ) and second to thechange of slope measured in B c , ⊥ ( T ) with a discontinuity of dB c , ⊥ /dT at T (cid:39) mK. Sucha discontinuity is present for all the contact pairs measured in this nanostructure. The very largeratio measured and more particularly the large value of B c , (cid:107) (as compared to k B T c ) for such a largethickness makes exfoliated flakes of PtBi single crystal a very promising candidate to study the2D superconductivity beyond the Pauli limit
70, 71 , provided that the thickness is reduced further inorder 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 dimensionalsuperconductivity sheds new light on PtBi , a material that was previously known for its verylarge linear magnetoresistance, its large spin-orbit coupling and the prediction of triply degeneratepoints in its band structure. Together with MoTe , it appears to be the only semimetal exhibiting13eyl physics and 2D superconductivity. Beyond topological superconductivity, the very strongspin-orbit coupling coupled to the superconducting transition is of particular interest in the searchfor unconventional pairing in superconductors and might favor, for instance, the presence of FFLOsuperconducting states.The evidence of a BKT transition in a nm thick flake makes of PtBi a remarkable system.This very large thickness is highly unusual for the observation of 2D superconductivity. To ourknowledge and excluding the case of layered superconductors, our flake is 5 times thicker than thethickest 2D superconductor reported so far . The origin of the reduced dimensionality cannot beattributed to any surface effect or to any layered nature of PtBi since we established the 3D natureof the superconductivity in macroscopic single crystals and since the effective superconductingthickness given by the fit of B c , (cid:107) ( T ) compares very well with the geometrical thickness measuredby AFM. Apart from promising properties of 2D superconductor that might be observed in thinnerstructures, such as the increase of B c , (cid:107) well beyond the Pauli limit, the large thickness is also anindication of a very robust signature of the usually fragile 2D superconducting state. PtBi mightbe therefore a simple and privileged platform for studying 2D superconductivity and the BKTtransition. The calculation of the Weyl nodes and of the surface Fermi surface were per-formed based on a tightbinding model obtained by constructing Wannier functions with the pro-14ective technique implemented in the FPLO code. The model includes the orbitals Bi 6p, Pt 6s andPt 5d.To evaluate the robusness of the Weyl nodes, we also performed calculations for the crystalstructure reported in Ref. 74. These confirm the presence of Weyl nodes albeit, in this case, theyare 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 zvdWin our structural model yields the Weyl nodes at 79 meV indicating that, in fact, zvdW controls toa large extent the Weyl node energy. Point contact measurements.
The contacts where made by mechanically touching the edgeof the sample with the wire of the noble metal. The differential resistance (dV/dI) of the contactswas recorded via lock-in technique in a quasi four probe configuration. Latter was achieved bysoldering one pair of a current-potential wiring to the copper sample clamp and another pair to thecounter electrode (Ag, Au or Cu).
Measurements techniques.
For the superconductivity in macroscopic crystals, DC sourceswere used in a delta mode, a method particularly adapted to the measurement of low resistancesamples. The measurements of nanostructures were done using standard lock-in amplifier tech-niques at low frequency ( f <
Hz).
Nanostructure fabrication.
The exfoliation was made on Si/SiO substrates, with nm15hickness of oxide, to enhance optical contrast. The exfoliated structures were then contacted bystandard electron-beam lithography procedure, with Cr-Au contacts. The structure we measuredfor this work, which can be seen in the supplementary information, has a thickness of 60nm, for alateral size of about 10 µ m. Definition of the critical parameters.
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Acknowledgements
DLB and YuGN acknowledge funding by Volkswagen Foundation and are grateful for supportby the National Academy of Sciences of Ukraine under project Φ We present below the temperature dependence of the resistivity of a single macroscopic crystalexhibiting a residual resistance ratio of about 130.
S3 - Isotropy of the critical field in a macroscopic structureS4 - Point contact measurements for different magnetic fields
As for the temperature, point-contact measurements show an enhanced local superconductivity inmagnetic field likely due to local pressure, with indications of superconductivity at magnetic fieldsup to 1T. This value is six times larger than its bulk pressure value ( ≈ mT). The Figure 8shows the enhancement of B c at T = 1 . K under the point contact for a magnetic field appliedalong the c-axis. The differential resistance ( dV /dI ) shows a zero-bias drop of almost 50 % at 0 Tand . K. At this temperature, a deviation from the ohmicity is observed at magnetic field up to1 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 fieldaligned in the a-b plane of the nanostructure. Measurements are done at T (cid:39) mK and withan AC current of 5 µ A. The perpendicular field can be swept up to ±
6T (the main coil of the 3D28agnet system) whereas the parallel field could be swept only up to ± S6 - dV/dI for different contact pairs
Although the critical current is almost independent of the set of contacts considered, the exactshape of the differential resistance dV /dI ( I dc ) depends on the set considered. Some contactsshows 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 andFigure 11. 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 (cid:107) , the critical current as defined in the main text ( R ( I c ) = R N / ) is found to be proportionalto the magnetic field with a larger sensitivity when the field is applied along the perpendiculardirection. The results are indicated in Figure 10 for an in-plane magnetic field and in Figure 11for an out-of-plane magnetic field. The diamond shape of I c ( B ) in both direction indicates theproportionality of I c with B .All the dV /dI show a similar diamond shape like, including the multi-peaked differentialresistance. 29 Data of the Figure 3.a presented in a two dimensional plot where the temperature dependency of themaximum of the dV /dI can be shown. Such a maximum is located at a dc current that correspondsalmost exactly to the critical current so that the temperature evolution of the maximum also standsfor the temperature dependency of I c . The measurements are done by increasing the temperatureby steps of 10 mK with a stabilization time of about 35 min between two dV /dI ( I dc ) sweeps sothat the sample is very well thermalized. S9 - temperature dependence of the critical field for another set of contacts
An example of the temperature dependence of B c , ⊥ is shown in black in the Figure 15 for a set ofcontacts indicating a discontinuous slope at about mK, similar to the contact indicated in themain text (reproduced in blue). A set of contacts for which the accuracy of the measurement of B ⊥ is not enough to resolve the discontinuity of the slope is also indicated in red. S10 - Temperature dependence of the resistance for different contacts with the fit to BKT
Three additional temperature dependences were measured and fitted with the BKT model for dif-ferent set of contacts for voltage probes, source and drain. The results are shown in the Figure 14.We summarize in the Table 1 the results obtained from the fit of the different experimental30ata with the Equation (7) in the main text. The value of the BCS critical temperature is calculatedbased on the parameter given by the fit of R ( T ) assuming α = 1 . The value of T BKT is fixed bythe analysis of the I ( V ) at different temperature as explained in the main text. We notice that thestandard deviation of T BKT is in very good agreement with the typical value obtained for δ . S11 - determination of B c ( θ, T ) In order to extract the angle and temperature dependences of B c , we replace in the solution of theEquation (4) the thermal depencency of B ⊥ ( T ) and B (cid:107) ( T ) given in the main text by Equation (2)and Equation (1) respectively. Such a solution is given by: B c ( θ, T ) = 12 B (cid:107) ( T ) B ⊥ ( T ) | cos θ | sin θ (cid:32)(cid:115) B ⊥ ( T ) B (cid:107) ( T ) sin θ cos θ − (cid:33) (8)where θ is the angle of the magnetic field with the perpendicular direction of the exfoliated flake.31able 1: Results of the temperature dependence of the resistance from the normal regime down tobelow the BKT transition. The value of T BKT is fixed by non-equilibrium measurements and α isassumed to be equal to unity whereas b , A and δ are the free parameters of the fit. T c is calculatedbased on the value of b and α .contact T BKT (mK) parameter b parameter A α δ (mK) T c (mK)main text 310 1.15 13.4 1 23.6 412blue 345 1.09 57.8 1 15.1 447red 340 1.14 25.1 1 16.1 450green 310 1.1 30.9 1 19.5 40432igure 1: (a) Crystal structure of trigonal PtBi . (b) Brillouin zone. Green (light blue) points cor-respond to Weyl nodes of positive (negative) chirality. (c) Bandstructure along the path indicatedin 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 anglebetween the magnetic field and the out-of-plane direction. (e) Angular dependence of the magne-toresistance at B = 15 T. (f) Fast Fourier Transform of the Shubnikov-de Haas oscillations in (d),for T=5K and θ = 20 °. 33igure 2: (a) The temperature dependence of the resistance of a single crystal for T ≤ K with asuperconducting transition measured at T c (cid:39) mK. The inset shows the differential resistance dV /dI as a function of the DC voltage in point contact measurements at different temperaturesbetween 1.55 K and 3.1 K, indicating the onset of the superconductivity at about 2.5 K under pointcontact. (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. 34igure 3: (a) Differential resistance dV /dI of the nanostructure as a function of the DC current attemperatures ranging regularly from 100 mK to 400 mK (the curves are shifted for better visibility). I c was found to be about 20 µ A. (b) temperature dependencies of the perpendicular ( B c, ⊥ ) andplanar ( B c, (cid:107) ) critical fields with their fit to the 2D theoretical model, including the misalignmentmeasured in (c). (c) the angular dependence of the critical field at T =
100 mK shows a sharppeak with a maximum corresponding to θ = 91 . °, indicating a 1.1° misalignment. Due to thesharpness of the peak, the θ = 0 ° is reduced by about 25% with respect to the true parallel criticalfield. Inset: the experimental data points (grey circles) of B c ( θ ) at T = 100 mK are fitted with the3D Ginzburg-Landau model (blue line) and the 2D Tinkham model (red line). A better agreementis found for the 2D model. 35igure 4: (a) V ( I ) characteristics for different temperatures below T BKT and above T c in a loga-rithmic scale. The dashed line stands for V ∝ I . (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 (cid:39) mK. (c) the temperaure dependence of the resistance of a PtBi from the normal regime ( T (cid:39) mK) down to T = 250 mK < T BKT . The Halperin-Nelson the-ory (red dashed line) with T BKT = 310 mK is not able to fit the experimental data. Accounting forinhomogeneities, the fit with Benfatto’s fomulae is in very good agreement with the experimentaldata 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 equivalentquantity R ( T ) /R (500 mK ) = R ( T ) /R N × R N /R (500 mK ) .36igure 5: Surface Fermi surface corresponding to the charge neutrality point. Green (light blue)points correspond to the projection of Weyl nodes of positive (negative) chirality. Left: [001]Bi -terminated surface. Right: [001] Bi -terminated surface.37igure 6: Temperature dependence of the resistivity of a macroscopic crystal between T = 300 Kand . K. 38igure 7: Magnetoresistance of a macroscopic crystal of PtBi with the magnetic field applied out-of-plane ( B ⊥ ) or along the two perpendicular in-plane directions ( B (cid:107) , and B (cid:107) , ) measured at lowtemperature ( T ∼ mK). For the sake of clarity, the different curves are shifted.39igure 8: Differential resistance dV /dI as a function of the DC voltage in point contact mea-surements at different magnetic fields between 0 T and 2 T at 1.55 K, indicating the onset of thesuperconductivity below 1 T for T = 1 . K. 40igure 9: The magnetoresistances at T ∼ mK for an in-plane and out-of-plane magnetic fieldsare shown in red and blue respectively. A picture of the sample can be seen in the graph togetherwith the connection configuration for the measurements of the magnetoresistances. The typicalsize of the sample is about 10 µ m. 41igure 10: dV /dI ( I dc ) measured at T (cid:39) mK for different set of contacts. The ac current is1 µ A and the differential resistance remains constant for higher dc-current. The contact configura-tion is indicated in the picture with the colors corresponding to the color of the curves.42igure 11: Similar measurements as shown in ?? with different contacts used for both current biasand voltage probes. 43igure 12: (a) Mapping of dV /dI ( I dc ) for different parallel magnetic field measured for − µ A ≤ I dc ≤ µ A and with I ac = 1 µ A. (b) Parallel magnetic field dependency of the critical current asdefined in the text for the same contacts as for (a). The contacting configuration is indicated in theinset of (b). 44igure 13: (a) Mapping of dV /dI ( I dc ) for perpendicular parallel magnetic field measured for − µ A ≤ I dc ≤ µ A and with I ac = 1 µ A. (b) Perpendicular magnetic field dependency of thecritical current as defined in the text for the same contacts as for (a). The contacting configurationis the same as in Figure 12 and is indicated in the inset of (b).45igure 14: dV /dI ( I dc ) measured at different temperatures, for − µ A ≤ I dc ≤ µ A and with I ac = 1 µ A. 46igure 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 datapoints for a set of contact where the discontinuity of the slope is clearly visible, similar to the onereproduced in the main text whereas the red points are the points for which no discontinuity couldbe measured because of the too low accuracy of the determination of B ⊥ close to T c .47igure 16: The R ( T ) measured for different set of contacts indicated in the inset and fitted byEquation (7) in the main text. The ac current was 1 µµ