Evolution of 4π-Periodic Supercurrent in the Presence of an In-Plane Magnetic Field

In the presence of a 4π-periodic contribution to the current phase relation, for example in topological Josephson junctions, odd Shapiro steps are expected to be missing. While missing odd Shapiro steps have been observed in several material systems and interpreted in the context of topological superconductivity, they have also been observed in topologically trivial junctions. Here, we study the evolution of such trivial missing odd Shapiro steps in Al–InAs junctions in the presence of an in-plane magnetic field Bθ. We find that the odd steps reappear at a crossover Bθ value, exhibiting an in-plane field angle anisotropy that depends on spin–orbit coupling effects. We interpret this behavior by theoretically analyzing the Andreev bound state spectrum and the transitions induced by the nonadiabatic dynamics of the junction and attribute the observed anisotropy to mode-to-mode coupling. Our results highlight the complex phenomenology of missing Shapiro steps and the underlying current phase relations in planar Josephson junctions designed to realize Majorana states.

The carrier density n of W1 and W2's 2DEG is measured in a Van-der-Pauw geometry. For W1 and W2, the density is shown to be n = 5.97 × 10 11 cm −2 and n = 8.24 × 10 11 cm −2 , respectively. JJ2 has an additional dielectric Al 2 O 3 layer deposited using atomic layer deposition which is not present for JJ1. The presence of the Al 2 O 3 layer is expected to further increase n 1 for JJ2. The higher intrinsic n in W2 and the additional increase in n due to the Al 2 O 3 layer indicates that JJ2 is expected to have significantly stronger SOC effects compared to JJ1.

Additional Measurements Device characterization
In Fig. S1 and Fig. S2, we present some additional DC measurements on JJ1 and JJ2. For JJ1, the VI characteristic in Fig. S1a shows agreement between the forward and backward sweeps at T = 30 mK indicating the absence of hysteresis. All the measurements presented on JJ1 are performed at T = 30 mK. On the other hand, as seen in Fig. S2, JJ2 exhibits hys- VI characteristic in the forward and backward directions at a T = 30 mK where the junction is hysteretic and b at T = 800 mK where there is no hysteresis observed. c Differential resistance as a function of current bias and out-of-plane magnetic field B ⊥ . The inset illustrates the direction of the field with respect to the junction. d Differential resistance as a function of current bias and applied gate voltage V g . For JJ2, the measurements presented in c and d as well as all the measurements presented in the manuscript are performed at T = 800 mK to avoid the effects of hysteresis.
teresis at T = 30 mK but not at T = 800 mK. Thus, for JJ2, all the measurements presented are performed at T = 800 mK to avoid hysteresis effects.
As a function of out-of-plane magnetic field, the critical current exhibits a conventional Fraunhofer-like pattern as seen in the differential resistance map in Fig. S1b and Fig. S2c. Due to flux-focusing caused by the Meissner effect and London penetration, 2,3 the effective area of the junction as calculated from the periodicity of the pattern is expected to be larger than the nominal geometry.
In-plane field dependence Figure S3: Angle anisotropy of critical field for JJ1. Differential resistance as a function of current bias and in-plane magnetic field B θ at different values of θ. The critical field B θ c , field at which supercurrent disappears, is seen to decrease as the angle is swept from θ = 0 • to θ = 90 • . Figure S4: Angle anisotropy of critical field for JJ2. Differential resistance as a function of current bias and in-plane magnetic field B θ at different values of θ.
The in-plane field data presented in Fig. S3 and Fig. S4 show an angle anisotropy in the critical field B θ c . This behavior has been reported in other epitaxial Al-InAs Josephson junctions and discussed in Ref. 2 A possible explanation is the induction of a flux dipole in the junction due to the bending of field lines around the contact edges. 2 This adds an additional out-of-plane field component close to the superconducting contacts affecting an electron flowing in the in-plane direction. This effect arises only when there is a component of the in-plane field along the current, thus restricting the critical field in the θ ̸ = 0 direction and most significantly at θ = 90 as evident from the data presented. Fig. S5 and Fig. S6 show a complete set of the Shapiro measurements for JJ1 for different in-plane field values and angles. In Fig. S7, we show how Q 12 evolves as a function of frequency at zero and finite field. The data shows an decrease in f 4π at B 0 • = 200 mT while the ratio f 4π /f J doubles. Fig. S8 through Fig. S13 show a complete data set of Shapiro measurements for JJ2 at f = 3.5 GHz for different in-plane field values and angles at three different gate voltage values. A similar data set for f = 2 GHz is shown in Fig. S14 and Fig. S15.

Effect of microwave frequency range limitations
To observe missing Shapiro steps in the presence of I 4π , a microwave frequency below f 4π is required. For both devices, we perform measurements at the lowest frequency possible to maximize the effects of I 4π . The RF signal is applied through an antenna in the puck that is close to the sample. Due to this coupling, the effective power radiated on the device changes from frequency to frequency. For the measurements, we choose frequencies where the entire Shapiro map is visible in our accessible power range. The lower limit of this accessible frequency range is primarily set by the weak coupling of the microwave antenna to the device at low frequencies.
The data presented in Fig.  2 of the manuscript shows a decrease in f 4π with the application of an in-plane magnetic field consistent with a suppression of I 4π . In Fig. S8, we present Shapiro maps at f = 3.4 GHz for JJ2 at V g = 0V where the first step is seen to fully reemerge around B   Fig. S15. This data shows that our study is not limited by the accessible frequency range of our setup. In the manuscript, we focus mainly on the f = 3.4 GHz data because it is difficult to extract accurate information from the f = 2 GHz data due to the size of the Shairo steps and low coupling.
Extracting Q 12 from the experimental Data To quantify how strong the first Shapiro step is, we calculate a ratio between the first and second Shapiro steps similar to the analysis done in Ref. 4 The strength of a Shapiro step can be quantified by binning (in fractions of hf 2e ) the voltage distribution of the V-I curve and creating a histogram of bin counts as a function of power. Horizontal linecuts through the colormap of the voltage distribution (i.e., Fig.  1d and 1e in the manuscript) present the amplitude of the Shapiro steps in terms of bin counts. A high bin count corresponds to a welldefined Shapiro step, while a low count corresponds to a suppressed or missing step. As the power is increased, Shapiro steps start to become more visible while the amplitude of the supercurrent (n = 0) decreases and eventually vanishes. At higher power, an oscillatory pattern corresponding to Bessel functions emerges. The power range considered for this analysis is always the low power regime which is the power range below the vanishing of the supercurrent. A ratio Q n,n+1 = sn s n+1 , where s n is the max step size (bin count) of step n with n ∈ Z can be calculated. A ratio of first to second step, Q 12 < 1, corresponds to a suppression of the first step while the reemergence of the first step with Q 12 ≳ 1. At low frequencies, the Shapiro steps are not, well-defined and the Shapiro steps can have s n that is not significantly greater than the "background". Thus, even when the first step is missing, Q 12 doesn't reach a value close to 0 at low in-plane fields.
For a specific range of B 0 • , Fig. S5 and Fig. S8 show a slight asymmetry around the current bias axis in terms of the position of the Shapiro steps. This asymmetry is not seen in the B 90 • case shown Fig. S6 and Fig. S9. The current bias as a function of B 0 • plots pre-     B 90 • co is seen to be 150 mT.        Shapiro maps. We note that even in the presence of this asymmetry, the calculated values of Q 12 from the positive and negative sides of the current bias are quantitatively similar.

Theoretical Analysis
In this section, we qualitatively examine the Andreev bound states (ABSs) spectrum of a wide Josephson junction with an applied finite magnetic field. We discuss the primary features that allow for a robust (topologically trivial) 4π-periodic supercurrent to arise and describe the mechanisms by which a magnetic field can suppress the 4π-periodic supercurrent, causing the reappearance of odd Shapiro steps.

Tight Binding Model
To characterize the ABS spectrum, we use an effective tight-binding model that we implement using the Kwant package . 5 Consider a square lattice in the x-y plane with dimensions (2L sc + 2L n + 1) × (2w + 1) and lattice constant a. Let Ω n , Ω sL , and Ω sR be the set of lattice sites in the normal, left superconducting, and right superconducting regions, respectively; let Ω be the set of all lattice sites in the system. The tightbinding version of the Hamiltonian we use is is the creation (annihilation) operator for an electron at site i with coordinate (x i , y i ) and spin s, τ i and σ i are Pauli matrices in particle-hole and spin space, respectively. Here, µ is the chemical potential, t = ℏ 2 /(2m * a 2 ), with m * as the effective mass, is the hopping parameter (assumed to be uniform throughout the system), ℓ soc,n is the strength of SOC coupling in the normal region, ℓ soc,sc is the strength of SOC coupling in the superconducting region, ∆ Z,n (∆ Z,sc ) are the Zeeman splitting in the normal region (superconducting region), ∆ is the superconducting gap, and ϕ is the phase across the junction. For all the results presented in the main text and the following, we assume ℓ soc,sc = ∆ Z,sc = 0. We have verified that qualitatively the results do not change if we set ℓ soc,sc and ∆ Z,sc equal to a finite value less or equal to the ones used for the normal region. The parameters used in the calculations are given in Table 1.

Long junction modes and detachment gap
Consider a Josephson junction in the x-y plane where the weak link is in the region −L x /2 < x < L x /2 and −w/2 < y < w/2, and the current in the junction flows in the xdirection. Let the superconducting coherence length associated with a subband of index n y ∆ t a ℓ soc,n L n L sc w n 300 µeV 59 meV 4 nm 7.5 meV·nm 100 nm 1 µm 500 nm 4 × 10 11 cm −2 be ξ ny = ℏv (ny) We see that for some n y modes v (ny) F can be sufficiently small so that L x > ξ ny . We refer to these modes as "long junction" modes. 6 The ABSs corresponding to these modes are "detached" from the continuum of states associated with the superconducting leads of the junction. We denote this detachment gap from the continuum by δ. The number of long junction modes and their detachments gaps is generally sensitive to the details of the system, e.g. doping n and width w.
When the junction has high transparency, the gap at ϕ = π is sufficiently small to allow Landau-Zener transitions when the system is diabatically driven. 6-8 At the same time, the relatively large value of δ strongly suppresses Landau-Zener transition to the continuum. The presence of ABSs with a large detachment gap and a small gap at ϕ = π leads to the presence of a 4π-periodic supercurrent, I 4π , when the JJ is biased. This results in a topologically trivial junction to have both 2π-and 4πperiodic supercurrent channels when driven diabatically. 9,10 We expect that the suppression by the magnetic field of the 4π-periodic supercurrent due to the long junction (non-topological) modes is responsible for the reappearance of the odds Shapiro steps for B θ > B co .

Effect of magnetic field
The presence of a magnetic field suppresses the superconducting gap ∆ and spin-splits the ABSs' energy level. Both of these effects contribute to a reduction of the detachment gap δ of the long junction modes responsible for the 4π behavior. When δ → 0 Andreev midgap states which undergo a Landau-Zener transition from a ground to an excited state at ϕ(mod2π) = π will be lost to the continuum at ϕ(mod2π) = 0 before completing a 4πperiodic cycle, restoring the supercurrent to a 2π-periodicity. 11 So far, we have focused our attention on the effect of the magnetic field on the detachment gap. The experimental results, however, show that the reappearance of the odd-Shapiro steps cannot be explained simply by tracking the evolution of δ with B θ . To illustrate this point more clearly, for I 4π we can assume: where I c is the critical current of the junction, and P ϕ is the effective probability of a Landau-Zener transition at phase ϕ. For isolated ABS states, we can write where Γ is the energy gap at ϕ between the ABSs involved in the Landau-Zener transition, and |V ϕ | is the instantaneous voltage across the junction when the phase is equal to ϕ. For ϕ = π, Γ = 2∆(B θ )(1 − τ ), 12 where τ is the junction's transparency. For ϕ = 2π, Γ ≈ δ(B θ ). The dependence of I 4π on B θ arises from the dependence of I c and the energy gaps Γ at ϕ = π and ϕ = 2π. Using Eqs. (7), (8) we can write: From the experimental estimates of B co , we can conclude that in general the ratio I 4π (B θ ) Ic(B θ ) can grow with B θ . On the other hand, theoretically we have that δ is linearly suppressed with B θ , and that, in first approximation ∆(B θ ) = ∆(B θ = 0)[1 − B θ /B c ] 1/2 . These scalings imply that the ratio I 4π (B θ ) Ic(B θ ) should always decrease with B θ , a result at odds with the experimental observations.

Multi-level Landau-Zener transitions
This simple analysis shows that the naive treatment based on the assumptions of wellisolated long junction modes is not adequate. This is not surprising if we consider that for the JJs studied, due to their large width, the ABSs are not well separated, see Fig S16, and that upon applying a Zeeman field, a high-transparency long junction mode will shift in energy and cross many other mid-gap Andreev bound states, as shown in Fig S17. For ABSs spectra like the ones shown in Fig S17, we have a complex multi-level Landau-Zener (MLZ) problem. With many coupled energy levels evolving over some time interval, the MLZ problem quickly becomes computationally prohibitive.  Table 1 and Zeeman splitting ∆ Z,n = 0.
Spectra like the ones shown in Fig. S16, S17 do not show a clear trend for the dependence on B θ of the gaps Γ(ϕ = π): the gaps at ϕ = π between occupied and unoccupied ABSs appear always to be small for high-transparency, wide, JJs like the ones studied. This suggests that the dependence on B θ of the effective P π might be attributed to the evolution of the overlaps between the Andreev bound states involved in Landau-Zener processes. To qualitatively evaluate the multi-mode coupling in the junction,  Table 1 and Zeeman splitting ∆ Z,n = 0.3∆ in the 0 • direction.
we calculate the overlap: for 0 < ϕ i < π and π < ϕ f < 2π and where |ψ n ⟩ is the Andreev bound state with eigenenergy E n . To determine whether long junction modes couple to short junction modes, we pick E n corresponding to an occupied long junction mode and calculate O mn for all 0 < E m < ∆. The collection of overlaps form a distribution of values between 0 and 1. In the absence of a Zeeman field, the occupied long junction mode has a large overlap with only one unoccupied mode with large δ, and a very small overlap with all the other modes. A bi-modal distribution of overlaps like this will have a large kurtosis. A large kurtosis indicates that occupied long junction modes have a high probability of transitioning, for ϕ ≈ π, to unoccupied long junction modes and so contribute to I 4π . A small kurtosis indicates that occupied long junction modes are equally likely to transition to most of the low-energy, unoccupied modes present at ϕ ≈ π, and therefore have a large likelihood of tunneling to one of the several "short junction" modes, modes with small δ, and not contribute to I 4π . We find that the kurtosis in part, depends on the values of ϕ i and ϕ f chosen. Despite such dependence, the kurtosis can be used to obtain qualitatively how an in-plane magnetic field affects I 4π and, therefore, the reappearance of odd-Shapiro steps. For instance, the dependence of the kurtosis on the direction θ of the magnetic field appears to explain qualitatively the dependence on θ of B co /B c measured in JJ2. To understand such dependence, we can consider the continuum model Hamiltonian, h(k x , k y ), for the normal region of the JJ: For wide junctions, long junction modes have Fermi momenta mostly along the y-direction. Taking ⃗ k long junction ≈ (0, k y ), we see that for the long junction modes the SOC polarizes the spins in the x-direction making the energy of these modes sensitive to the direction of the Zeeman splitting ∆ Z . Thus, in general, for wide junctions with strong SOC, the Andreev bound state spectrum will be affected differently by an in-plane magnetic field in the x-and y-direction. By calculating the ABS spectrum for different directions, and the corresponding eigenstates, we are able to obtain the dependence of the kurtosis on θ. Fig. S18 shows the values obtained for the kurtosis for different values of ϕ f (keeping fixed ϕ i ) for a junction with large SOC. The blue lines show the results for the case when B θ = 0. In this case, the kurtosis is very high, suggesting that the long junction modes undergo Landau-Zener transitions that contribute to I 4π . The orange and green lines show the results obtained when B θ ̸ = 0 such that the Zeeman energy is ∆ Z = 0.3∆ and θ = 90 • , orange lines, θ = 0 • , green lines. We see that B θ ̸ = 0 suppresses the kurtosis indicating an increase of the mixing of long junction modes and shortjunction modes, resulting in a suppression of I 4π . In agreement with the experiment, specifically the data shown in Fig. 4, the results of Fig. S18 show that the kurtosis is larger when θ = 90 • than θ = 0 • , suggesting that, when the anisotropic suppression of I c with B is taken into account, an in-plane field along the θ = 0 • direction causes a larger suppression of I 4π , than an in-plane field along the θ = 90 • , the direction of the current.