Field-Tunable 0-π-Transitions in SnTe Topological Crystalline Insulator SQUIDs

The manifestation of spin-orbit interactions, long known to dramatically affect the band structure of heavy-element compounds, governs the physics in the surging class of topological matter. A particular example is found in the new family of topological crystalline insulators. In this systems transport occurs at the surfaces and spin-momentum locking yields crystal-symmetry protected spin-polarized transport. We investigated the current-phase relation of SnTe thin films connected to superconducting electrodes to form SQUID devices. Our results demonstrate that an assisting in-plane magnetic field component can induce 0-π-transitions. We attribute these findings to giant g-factors and large spin-orbit coupling of SnTe topological crystalline insulator, which provides a new platform for investigation of the interplay between spin-orbit physics and topological transport.

Topological states of matter are researched in a large variety, ranging from 1D nanowire systems with strong spin-orbit coupling 1,2 over 2D quantum spin Hall insulators 3,4 to 3D topological insulators [5][6][7] as the most common examples of this quickly emerging field. Topological crystalline insulators (TCI) 8 , with SnTe as a representative model material, constitute a new class of 3D materials within this widespread family, for which the topological properties are governed by mirror symmetries of the crystal lattice rather than time-reversal symmetry, giving rise to multiple Dirac surface channels with spin-momentum locking [9][10][11] . A fundamental and common interest in these states of matter is based upon the interplay of spin-polarized surface channels and superconducting pairing, ever since the possible realization of formerly elusive topological superconductivity in hybrid systems of such materials and common s-wave superconductors was predicted 12 . One of the particularly enticing prospects of this is the conceptual implementation of topological quantum computing 13 , which is enabled by non-Abelian and delocalized quasiparticle excitations commonly referred to as Majorana zero modes 14 . The scope of possible effects in such structure, related to unconventional pairing and phase relations, has been recently extended because of a more complete picture of the role of spin-orbit coupling in low-dimensional electron systems, most notable the similarities between Rashba-type spin splitting 15 and topological spin-momentum locking. In this context, the impact of Zeeman fields has been used particularly as a driving force between trivial and unconventional regimes in theoretical proposals [16][17][18] as well as experimental demonstrations [19][20][21] . This approach is fuelled by the experimental consent that the contributions of topological surface states are difficult to isolate from non-depleted bulk channels in common transport measurements. The latter are hence often dominated by trivial bulk characteristics [22][23][24] . Accordingly, both the realization of finite-momentum Cooper pairing 19,21 and more demonstratively the occurrence of fractional 20,25 and half-flux 19,20 offset of Josephson junctions have been reported under applied in-plane magnetic field in related systems.
It is well established that the current-phase relationship (CPR) of a superconductor-insulator-superconductor (SIS) junction is sinusoidal in nature, following ϕ = ϕ I I ( ) sin( )

Results
The measured device consists of a 40 nm layer of SnTe and 30 nm Ta electrodes on top, shaped as a SQUID ring of ≈ 4 μm 2 loop area with 300 nm arms and 100 nm gaps forming the junctions as shown in Fig. 1a. A schematic of the device in the coordinate system of the magnetic field is shown in Fig. 1b. The temperature dependence of the critical current is measured (Fig. 2a) and fitted with the law with ∝ [35][36][37] .  (Fig. 2b). Tantalum is chosen as a superconducting electrode in our structure because it provides strong induced superconductivity and high critical currents through the SnTe weak-link. High in-plane critical fields are necessary to induce 0-π transitions. The strong spin-orbit coupling of Ta does not influence the observed effects because the switching characteristic is determined by the SnTe weak-link. The latter is justified by the difference between the critical temperature of SnTe (T C SnTe ≈ 900 mK) and the critical temperature of Ta (T C Ta ≈ 2.5 K). The thermal avalanche from the switching at low temperatures causes transitions of the nearby superconducting structures into a resistive state as well which leads to resistance values not solely containing the weak link. In order to limit the disturbing influence of large thermal hysteresis effects, further measurements are performed at elevated temperatures ≥ T 500mK. From measurements close to ≈ T 900mK c SnTe , we deduce normal state resistance ≈ Ω R 5 n of the weak links used in the relation above, which corresponds well to supporting resistivity measurements of an SnTe Hall bar structure (see Supplementary Information). The extracted normal state resistance of our devices demonstrates multichannel transport. Therefore, there is contribution from the bulk states of the TCI composition as well similar to other reports 41,42 . Our previous investigations on the bare SnTe devices have shown the manifestation of the TCI surface states through weak-antilocalization measurements 24 .
Subsequently, the CPR response to magnetic field is probed. Application of an out-of-plane magnetic field B z generates a SQUID modulation whose periodicity corresponds well to the spatial dimensions of the device for purely 2π-dominated transport (Fig. 2b). The absence of any 4π-periodic contributions in DC measurements of topological matter is a commonly reported effect, which results from bulk-shunting and therefore poisoning of surface states 22-24 on long measurement time scales. A less-dissipative approach is provided by RF measurements, where the 4π-effect expresses itself as the vanishing of odd-integer Shapiro steps, as recently demonstrated for strained 3D topological insulator HgTe 43 , HgTe quantum wells 44  The CPR shows strongly reduced SQUID modulation depth (Fig. 2b). Most trivially, such behaviour can stem from asymmetric junctions with ≠ I I c1 c2 , but our fabrication scheme should yield reasonably symmetric devices, for which asymmetry arises only microscopically. Here, the effect is therefore attributed to strong kinetic effects in the SnTe/Ta hybrid, which also explains the triangular shape of the CPR 45  Here, the large β k arises as consequence of both the strong proximity-coupling with large critical currents and the large L k in our devices. There is, however, no reason to assume the 0-π-effect originates from the large kinetic inductance and the Ta, which constitutes a common (type-1) s-wave superconductor. As the critical temperature of the SnTe weak links is approached, the critical current and hence β k decreases. Indeed, we observe that close to ≈ T 900mK c SnTe the CPR shows the classical cosine-like flux dependence of the SQUID. More details on the CPR and R n can be found in the Supplementary Information.
Significant altering of the conventional modulation pattern is observed when the device is subject to an additional in-plane magnetic field B x (see Fig. 3). A linear drift in the (B x , B z )-plane is corrected (here and for all following images containing in-plane fields) by rotation, as this constitutes the impact of non-perfect sample alignment within the magnetic field axes, which is confirmed by measurements in different field directions and a repeated measurement during a second cool down.
For ≈ B 155mT x (and similarly for negative fields of approximately the same magnitude), a drastic change of this regime is observed, with repeated field-induced transitions between a ϕ = 0 SQUID and a ϕ = π SQUID, as emphasized by the black arrows. Similar transitions are also observed in other weak links with strong spin-orbit coupling materials 19,20,46 .
The transition at ≈ B 220mT x is shown in Fig. 4. Notably, the switching of the phase does not occur abruptly in B x , but takes shape in a finite range of ≈ 4 mT. The transition thus comes along with fractional flux periodicity. Particularly, the occurrence of a distinct half integer flux quantum state is stressed (within the range of green contours), which serves as a fingerprint of the boundary of a 0-π-transition. Such an effect has been theoretically predicted in closely related systems of Rashba-type spin-orbit coupled superconductors 37,47 . A closer look at the evolution of the transitional regime reveals the repeated occurrence of 0-π-transitions with similar spacing in field ∆B x SQ for the first 4 transitions, as shown in Fig. 5a. We attribute this behavior to distinct 0-π-switches of the two weak links, which are patterned nominally symmetrical and should hence obey the same physics. However, they exhibit slightly different onset fields and spacing ∆B x JJ due to microscopic patterning-induced and growth-related asymmetry. When one junction switches to a π-regime, the overall SQUID exhibits a π-shift, and when the second junction switches to a π-regime, the SQUID recovers a 0 state. This gives rise to the following transition pattern → π → π π → π → (0, 0) (0, ) ( , ) ( , 0) (0, 0) for the two junctions, as shown schematically in Fig. 5b. For larger values, the B x superconductivity starts to be strongly suppressed as the critical field is approached. x , a non-linear shift of the CPR is observed, attributed to microscopic asymmetry. For | | ≥ B 155mT x , a transitional regime is reached, where continuous field-induced 0-π-transitions occur, whose positions are indicated with black arrows. . The curves are vertically offset for clarity. The in-plane field induces a continuous 0-π-transition in the CPR, as indicated by the black arrow. The transition comes along with fractional flux states and, in particular, a half-integer periodicity is pointed out (green contours).
Scientific RepoRts | (2019) 9:1987 | https://doi.org/10.1038/s41598-018-38008-1 As discussed by Hart et al. 19 , structural inversion symmetry (SIA), bulk inversion symmetry (BIA) and Zeeman effect coupling (ZEC), possibly modified by random phase distribution, can all lead to a spatially varying order parameter,. We should point out that there is pronounced phase drift in some regions between two complete 0-π transitions; for instance at ≈ B {153 mT, 220mT, 272mT, 308mT} x . Random phase distribution might be responsible for a skewed Fraunhofer pattern and this is a viable explanation for the observed regions of phase drift in our SnTe-SQUIDs 19 .
While SIA and BIA lead to order parameter oscillations along the junction and perpendicular to the junction respectively, ZEC causes isotropic in-plane shift of the Cooper pair momentum 19 . Therefore, we argue that the behaviour of our system is dominated by the Zeeman coupling as our observed effect is similar in different in-plane field directions (see Supplementary Information). Nevertheless, I c (B y , B z ) is not entirely the same as I c (B x , B z ) and hence we assume that there is a SIA contribution which is responsible for this small mismatch.
It has been shown that DC SQUIDs with two purposefully asymmetric weak links with strong spin-orbit interaction exhibit 0-π-transitions as a function of the applied in-plane field in HgTe/HgCdTe 19,41 , Bi nanowire-based devices 20 or BiSb topological semimetal 48 .
According to Hart et al. 19 , the onset magnetic field for a 0-π-transition in a single junction dominated by Zeeman coupling is , where we substitute the Thouless energy extracted from the fit in Fig. 2a. This gives an estimate for the g-factor of SnTe of g ≈ 24 − 35 for the two junctions, respectively. The second 0-π-transition is predicted 19 to be at =

Conclusion
We have demonstrated assisted, reproducible 0-π transitions in SnTe-based SQUIDs in this article. The observed four transitions correlate well with theoretical prediction for induced Cooper pair momentum due to Zeeman coupling in the strong spin-orbit coupling material SnTe. The experimentally determined onset fields and field spacings between the 0-π transitions show remarkably close scaling agreement with the theoretical predictions and we have extracted g ≈ 30 for our SnTe weak links. We believe that the observation of this unconventional effect will further fuel the interest in the integration of SnTe topological crystalline insulator in superconducting devices with new functionalities. Such field-tunable devices can be crucial components in proposed future topological quantum computing schemes.

Methods
SnTe films of 40 nm thickness were grown by co-sputtering of Sn and Te on Si/SiO 2 substrates in similar fashion to the previously presented work 24 and the respective Supplementary Information. Here we use 30 nm films of chemically more stable Ta superconductor to proximity-induce Cooper pairing in the SnTe. Patterning of the samples is done with a positive-resist lift-off process for the Ta rings and a subsequent negative-resist argon milling process controlled by secondary ion mass spectroscopy to remove the excess SnTe and hence reduce the width of the weak links and stray current contributions. (0,0) (π,0) ( π,π) (0,π) (0,0) SQUID (0) (π)

Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.