Asymmetric Little–Parks oscillations in full shell double nanowires

Little–Parks oscillations of a hollow superconducting cylinder are of interest for flux-driven topological superconductivity in single Rashba nanowires. The oscillations are typically symmetric in the orientation of the applied magnetic flux. Using double InAs nanowires coated by an epitaxial superconducting Al shell which, despite the non-centro-symmetric geometry, behaves effectively as one hollow cylinder, we demonstrate that a small misalignment of the applied parallel field with respect to the axis of the nanowires can produce field-asymmetric Little–Parks oscillations. These are revealed by the simultaneous application of a magnetic field perpendicular to the misaligned parallel field direction. The asymmetry occurs in both the destructive regime, in which superconductivity is destroyed for half-integer quanta of flux through the shell, and in the non-destructive regime, where superconductivity is depressed but not fully destroyed at these flux values.


I. LITTLE-PARKS OSCILLATIONS IN 5 DEVICES
Figure S1a-e shows data and fit of Little-Parks oscillations in five different devices. Parameters used for the fit can be found in Table S1 as well as their corresponding errors. The error bars are rough estimates changing only one parameter at a time, while keeping the other three fixed. All devices are fitted with the Little-Parks model with good agreement. Figures S1a,b correspond to Device 1 and 2 analyzed in the main text. To obtain a good fit, the parameter A ⊥ is chosen as a fitting parameter. In order to get an insight of the origin of A ⊥ , we investigate this parameter versus junction length for the five devices shown in Fig. S1f. However, no correlation between the two parameters is found as expected [S1]. Figure S1g presents the correlation between the d * /ξ ratio and the switching current at the first half-integer flux quantum I Φ=Φ 0 /2 sw . Here, d * is the diameter of a cylinder with the same area as the two hexagonal nanowires (effective diameter) and ξ the coherence length extracted from the fits. It is observed that devices with small ratio exhibit destructive Little-Parks (Devices 2 and 3), while those with larger ratio exhibit non-destructive Little-Parks oscillations, in agreement with theoretical work [S2] and previous experiments in nanowires [S3, S4], even though our device cross section is ellipsoid and not circular.  S1. Model parameters used to fit data all devices in Fig. S1. From left to right, coherence length (ξ), effective perpendicular flux parameter (A ⊥ ), effective parallel flux area (A ), and ratio of shell thickness (ts) to effective single cylinder diameter (d * ).
Device ξ (nm)  . Dependence of critical current on field angle and perpendicular field (a,b) Colormaps of differential conductance, dV /dI, plotted as a function of bias current, I, and (a) angle between the vector magnetic field B and coil X, ϕ, and (b) magnetic field applied perpendicular the double nanowires with misalignment θ = 4.4 • , B θ ⊥ . In (a), the critical current shows clear modulations and it reaches its minimum (maximum) when φ is such that θ = 90 (θ = 0), reflecting the phenomenological finding that the area threaded by parallel magnetic flux is smaller than the effective perpendicular parameter (A < A ⊥ ). The critical perpendicular field found from the measurement in (b) is significantly smaller than the parallel critical field (0.1 T against 0.95 T), supporting the data and interpretation in (a). The asymmetry against ±B θ ⊥ (ϕ = 2.2, 5.34 rad) in both (a) and (b) is attributed to coil remanence. The data was fitted with a calculation of the critical current, Ic (dashed lines), using the single hollow superconducting cylinder model. Reasonably good fits are obtained, with the fit quality decreased due to asymmetries. Data obtained from Device 1.
Here we analyze the dependence of I c as a function of a rotating magnetic field of magnitude B r = 0.1T, showing that the two parameters A ⊥ and A defined in the main text have to be different in order to have a modulating critical current versus a rotating magnetic field. Figure S2a shows such a measurement from Device 1, where the black dashed-line is the model we use based on Eqs. 1,2 of the main text. As the applied magnetic field is weak, we have n = 0 and the last term of Eq. 2 (α ) is negligible. As a result, α and α ⊥ would be identical if the two parameters are the same. A calculation from Ref. [S1] for a solid cylinder found a factor of 2 between the parameters A and A ⊥ (A ⊥ /A =2), while we experimentally find a factor of A ⊥ /A ≈ 2.5 − 5 for our elliptical hollow cross-sectional (13 nm shell) full-shell nanowires agreeing with the calculation on A ⊥ being larger than A . Theoretical work on our specific system is needed to understand if these findings are consistent with theory. Figure S2b presents an I − B measurement of Device 1 as a function of dV /dI at an angle perpendicular to nanowire axis.The black dashed-line is the fit, showing good agreement to the data, using the same parameters as in Fig. S2a and in Fig. 2 of the main text. Figure S3 presents Little-Parks oscillations of I c and T c of Device 3 to justify Eq. 4 of the main text. The relation between the zero-field and finite-field critical current and critical temperature noted in Eq. 4 of the main text is verified in the data. Explicitly, at B θ = 0 we measure I sw (B θ = 0) = −24 µA and T c (B θ = 0) = 1.3 K, and at B θ such that

III. TEMPERATURE DEPENDENCE OF LITTLE-PARKS OSCILLATIONS
Negative values of current are chosen to capture the switching current as the sweep direction is from positive to negative. For these values we verify the relation between I c /I c0 and T c /T c0 in Eq. 4 of the main text within 5%, assuming that I sw = I c . The lack of additional lobes beyond the ones at Φ Φ 0 = ±1 is due to a large angle misalignment deduced from the fit in Fig. S1d, which effectively reduces the critical magnetic field of the device.

IV. MAGNETIC FIELD REMANENCE
In this section we discuss the finite magnetic field remanence from the X and Z coils that were used to probe the Little-Parks effect in our devices. Figures S4a,b show measurements of I c as a function of B x swept upwards and downwards respectively, as noted by the red arrow. Note that the maximum value of I c shifts by ≈ 10 mT between the two panels. Figures S4c,d exhibit the same effect for the Z coil B z . This effect explains the small inconsistencies -which are mostly visible near zero magnetic field -between datasets, when the magnetic field sweep direction changes.  S4. Evidence for finite remanence in coils X and Z. (a,b) Colormaps of differential conductance, dV /dI, plotted as a function of field in (a,b) coil X, Bx and (c,d) coil Z, Bz and positive bias current, I. Arrows on the horizontal axis indicate field direciton. The pattern of destruction of superconductivity is asymmetric and hysteretic on the direction of Bx and Bz, with a field-direction dependent small offset of < 20 mT seen in both cases.