## Introduction

In Josephson junctions1, the Current-Phase relationship (CPR) is given by the first Josephson equation2, IJ(φ) = I0 sin (φ + φ0). Time-reversal and spatial parity symmetries3, Px,Py,Pz impose the equality IJ(φ → 0) = 0 and so only two states for the phase shift φ0 are possible. φ0 = 0 in standard junctions and φ0 = π in presence of a large Zeeman field, as obtained in hybrid superconducting-ferromagnetic junctions4,5,6 or in large g-factor materials under magnetic field7,8.

To observe an anomalous phase shift φ0 intermediate between 0 and π, both time-reversal and parity symmetries must be broken9,10,11,12,13,14,15,16,17,18,19,20,21,22. This can be obtained in systems with both a Zeeman field and a Rashba spin-orbit coupling term $${\mathrm{H}}_{\mathrm{R}} = \frac{\alpha }{\hbar }({\mathbf{p}} \times {\mathbf{e}}_z).\sigma$$ in the Hamiltonian11,20, where α is the Rashba coefficient, ez the direction of the Rashba electric field and σ a vector of Pauli matrices describing the spin. Physically, these terms lead to a spin-induced dephasing of the superconducting wave-function.

This anomalous phase shift is related to the inverse Edelstein effect observed in metals or semiconductors with strong spin-orbit coupling. While the Edelstein effect consists in the generation of a spin polarization in response to an electric field23, the inverse Edelstein effect24, also called spin-galvanic effect, consists in the generation of a charge current by an out-of-equilibrium spin polarization. These two magneto-electric effects are predicted also to occur in superconductors as a consequence of a Lifshitz type term in the free energy25,26. Thus, in a superconductor with a strong Rashba coupling, a Zeeman field induces an additional term in the supercurrent. In Josephson junctions this term leads to the anomalous phase shift20.

Several designs of Josephson junctions leading to an anomalous phase shift have been proposed theoretically where the Zeeman field can be obtained from an applied magnetic field14,16,20 or by using a magnetic element11. These designs include the use of atomic contacts12, quantum dots13,27, nanowires17,18, topological insulator19,28,29,30, diffusive junction20, magnetic impurity21, ferromagnetic barrier11, and diffusive superconducting-ferromagnetic junctions with non-coplanar magnetic texture31.

Experimentally, the anomalous phase shift φ0 can be detected in a Josephson Interferometer (JI) through measurements of the CPR. Anomalous phase shifts have been identified recently in JIs fabricated from the parallel combination of a normal ‘0’ and ‘π’ junction22 that breaks the parity symmetries.

Because of its large g-factor g = 19.532 and large Rashba coefficient, Bi2Se3 is a promising candidate for observing the anomalous Josephson effect due to the interplay of the Zeeman field and spin-orbit interaction. In this topological insulator33,34, the effective Rashba coefficient of the topological Dirac states is about α 3 eVÅ35, while the Rashba coefficient of the bulk states, induced by the broken inversion symmetry at the surface, has a value in the range 0.3–1.3 eVÅ as measured by photoemission36,37.

As detailed in refs. 20,26 the amplitude of the anomalous phase depends on the amplitude of the Rashba coefficient α, the transparency of the interfaces, the spin relaxation terms such as the Dyakonov-Perel coefficient and whether the junction is in the ballistic or diffusive regime. At small α, the anomalous phase is predicted proportional to α3, at large α, it should be proportional to α.

In the ballistic regime11 and for large α, the anomalous phase shift is given by $$\varphi _0 = \frac{{4E_{\mathrm{Z}}\alpha L}}{{(\hbar v_{\mathrm{F}})^2}}$$ for a magnetic field of magnitude B and perpendicular to the Rashba electric field, where $$E_{\mathrm{Z}} = \frac{1}{2}g\mu _BB$$ is the Zeeman energy, L is the distance between the superconductors and vF is the Fermi velocity of the barrier material. For the Rashba spin-split conduction band with α ≈ 0.4 eVÅ, vF = 3.2×105 ms−1 and junction length L = 150 nm, a magnetic field B = 100 mT generates an anomalous phase $$\varphi _0 \simeq 0.01 \pi$$, while for Dirac states35 with vF = 4.5×105 ms−1, $$\varphi _0 \simeq 0.005 \pi$$.

In the diffusive regime, the expected anomalous phase shift has been calculated in ref. 20. For weak α, highly transparent interfaces and neglecting spin-relaxation, the anomalous phase shift is given by the relation:

$$\varphi _0 = \frac{{\tau m^{ \ast 2}E_{\mathrm{Z}}(\alpha L)^3}}{{3\hbar ^6D}}$$
(1)

where τ = 0.13 ps is the elastic scattering time, $$D = \frac{1}{3}v_{\mathrm{F}}^2\tau = 40\,{\mathrm{cm}}^2\,{\mathrm{s}}^{ - 1}$$ is the diffusion constant and m* = 0.25 me is the effective electron mass38.

To test these theoretical predictions, we fabricated single Josephson junctions and JIs from Bi2Se3 thin films of 20 quintuple layers thick, ~20 nm, grown by Molecular Beam Epitaxy and protected by a Se layer, see Supplementary Note 1 and Supplementary Figure 1. As described in Supplementary Figure 2, these junctions are in the diffusive regime. From the measurement of the relative phase shift between two JIs with different orientations of the Josephson junctions with respect to the in-plane magnetic field, we observed unambiguously the anomalous phase-shift predicted by Eq. (1).

## Results

### Current-phase relationship and Shapiro steps

The JI shown in Fig. 1a consists of two junctions in parallel of widths W1 = 600 nm and W2 = 60 nm, respectively. The phase differences φ1 and φ2 for the two junctions are linked by the relation $$\varphi _1 - \varphi _2 = 2\pi \frac{\phi }{{\phi _0}}$$, where ϕ = BzS is the magnetic flux enclosed in the JI of surface S, Bz is a small magnetic field perpendicular to the sample, i.e. along ez, and ϕ0 is the flux quantum. In this situation, the Zeeman energy is negligible and oriented along the Rashba electric field, which implies that φ0 = 0. As the critical current $$I_{{\mathrm{c}}_1}$$ is much higher than $$I_{{\mathrm{c}}_2}$$, then φ1 = π/2 and $$I_{\mathrm{c}} = I_{{\mathrm{c}}_1} + I_{{\mathrm{c}}_2}{\mathrm{cos}}(\omega B_z)$$ with ω = 2πS/ϕ039. Thus, a measurement of the critical current Ic as function of Bz provides a measure of the current $$I_{{\mathrm{c}}_2}$$ as function of φ2, i.e. the CPR. From the voltage map as a function of current I and Bz, shown Fig. 1b, the critical current Ic is extracted when the voltage across the device exceeds the value Vswitch = 4 μV, as shown in Fig. 1c. We find that the CPR displays a conventional sinusoidal form IJ = Ic sin (φ), as shown by the fit in Fig. 1d. Furthermore, under microwave irradiation, the JI displays a conventional, 2π periodic, Shapiro pattern, as shown Fig. 2, and detailed discussion in Supplementary Note 2.

### Asymmetric Fraunhofer pattern with in-plane magnetic field

Figure 3b–d show resistance maps dV/dI of a single junction, Fig. 3a, as function of current I and Bz for different values of an in-plane magnetic field, By. Figure 3e shows the corresponding critical current curves. A Fraunhofer pattern is observed with the first node located at $$B_0 \simeq 1.2\,{\mathrm{mT}}$$. This value is consistent with the theoretical value $$B_0 = \frac{{\phi _0}}{{W(L + 2\lambda _z)}}$$, using the effective magnetic penetration depth λz = 175 nm and taking into account flux-focusing effects, see Supplementary Note 3. While for By = 0, the Fraunhofer pattern is symmetric with respect to Bz, this pattern becomes asymmetric upon increasing the amplitude of By. This evolution is shown in the critical current map as a function of Bz and By in Fig. 3f. We observed a much less pronounced asymmetry when we apply the magnetic field in the x direction as shown in Supplementary Figure 5. Similar behavior has been observed recently in InAs40 interpreted as the consequence of a combination of spin-orbit, Zeeman and disorder effects. As described in ref. 3, the generation of an anomalous phase shift requires breaking all symmetry operations U leaving UH(φ)U = H(−φ), where H is the full Hamiltonian of the system including spin-orbit interactions. These symmetry operations are shown in Table 1 together with the parameters breaking those symmetries. This table shows that for a system with a finite spin-orbit coefficient α, finite By is sufficient to generate an anomalous phase shift. However, additional symmetry operations U leaving UH(Bz, φ)U = H(−Bz, φ) must be broken to generate an asymmetric Fraunhofer pattern, as shown in Supplementary Table 1. In addition to non-zero values for α and By, disorder along y direction, i.e. non-zero Vy, is required to generate an asymmetric Fraunhofer pattern. AFM images, as in shown in Supplementary Figure 6g, show that the MBE films present atomic steps. Due to the dependence of the Rashba coefficient on film thickness35, phase jumps along the y direction of the junction can be produced by jumps in the Rashba coefficient and explains the polarity asymmetry of the Fraunhofer pattern. As detailed in Supplementary Note 4, using a simple model, the asymmetric Fraunhofer pattern measured experimentally can be simulated, as shown in Fig. 3g.

### Current-phase relationship with in-plane magnetic field

To unambiguously demonstrate that an anomalous phase shift φ0 can be generated by finite spin-orbit coefficient α and finite magnetic field By alone, a direct measurement of the CPR with in-plane magnetic field is required. To that end, we measured simultaneously two JIs, oriented as sketched in Fig. 4a, differing only by the orientation of the small junctions with respect to the in-plane magnetic field.

The CPRs for the two JIs are measured as function of a magnetic field making a small tilt θ with the sample plane, which produces an in-plane By = B cos (θ) and a perpendicular Bz = B sin (θ) magnetic field, as sketched in Fig. 4c. In this situation, the critical current for the reference JI changes as Ic cos (ωref B) with $$\omega _{{\mathrm{ref}}} = \frac{{2\pi S_{{\mathrm{ref}}}}}{{\phi _0}}{\mathrm{sin}}(\theta )$$ where Sref is the surface of the JI. For the anomalous JI, the critical current changes as Ic cos (ωB) with:

$$\omega = \frac{{2\pi S}}{{\phi _0}}{\mathrm{sin}}\theta + C_{\varphi _0}{\mathrm{cos}}(\theta )$$
(2)

where $$C_{\varphi _0} = \frac{{\tau m^{ \ast 2}g\mu _{\mathrm{B}}(\alpha L)^3}}{{6\hbar ^6D}}$$ in the diffusive regime.

In Eq. (2), the first term arises from the flux within the JI of area S, the second term arises from the anomalous phase shift $$\varphi _0 = C_{\varphi _0}B\,{\mathrm{cos}}\,(\theta )$$.

Figure 4b shows voltage maps for two different orientations θ, Fig. 4c. At low B, the two JIs are in-phase and become out-of-phase at higher magnetic field, indicating that the frequency ω of the anomalous JI is slightly larger than the reference JI. This is also visible on the critical current plot, Fig. 5a, extracted from these voltage maps. To see this more clearly, the average critical current, shown as a continuous line in Fig. 5a, is removed from the critical current curve and the result shown in Fig. 5b for the two JIs. On these curves, the nodes at π(2n + 1/2), n = 0,1,.., are indicated by large red (blue) dots for the reference (anomalous) curve. At low magnetic field, the two JIs are in-phase as indicated by the blue and red dots being located at the same field position. Upon increasing the in-plane magnetic field, the two JIs become out-of-phase with the anomalous JI oscillating at a higher frequency than the reference JI, as indicated by the blue dot shifting to lower magnetic field position with respect to the red dot. This increased frequency for the anomalous device is expected from Eq. (2) as a consequence of the anomalous phase shift. Supplementary Figure 7a,b shows additional data taken from negative to positive magnetic field, across zero magnetic field. A plot of the phase difference between the two JIs as function of in-plane magnetic field, shown in Supplementary Figure 7c, demonstrates that the two JIs are in-phase at zero magnetic field and reach a dephasing approaching about π/2 for an in-plane magnetic field of 80 mT.

One also sees that the oscillation period of both JIs increase with increasing B. As detailed in Supplementary Note 3, this is due to flux focusing that makes the effective area of the JIs larger at low magnetic field. As the effect of flux focusing decreases with the increasing penetration depth at higher magnetic field, the effective areas of the JIs decreases upon increasing the magnitude of the magnetic field and so the period of oscillations increases.

While the two JIs have been fabricated with nominally identical areas, to exclude that the observed difference in frequencies between the two JIs is due to a difference of areas, we plot in Fig. 5c, the frequency ratio $$\frac{\omega }{{\omega _{{\mathrm{ref}}}}}(\theta )$$ measured at different angles θ. Because each curve contains several periods Ti, the frequency ratio is obtained from the average between the N periods ratio as $$\omega /\omega _{{\mathrm{ref}}} = \frac{1}{N}\mathop {\sum}\nolimits_{i = 1}^N \frac{{T_{i,{\mathrm{ref}}}}}{{T_i}}$$, where Ti,ref and Ti are the ith oscillation period for the reference and for the anomalous device respectively. This method enables ignoring the flux focusing effect because the ratio is only taken between two periods measured at about the same magnetic field. We find that the experimental data follows the relation:

$$\frac{\omega }{{\omega _{{\mathrm{ref}}}}}(\theta ) = \frac{S}{{S_{{\mathrm{ref}}}}} + \frac{{C_{\varphi _0}\phi _0}}{{2\pi S_{{\mathrm{ref}}}{\mathrm{tan}}(\theta )}}$$
(3)

At large θ, this ratio is equal to the ratio of areas $$S/S_{{\mathrm{ref}}} \simeq 1$$, however, for small θ, this ratio increases as 1/tan(θ), indicating the presence of an anomalous phase shift φ0.

Another way of extracting the frequency is described in the Supplementary Note 5 and leads to the same result, as shown in Supplementary Figure 8.

## Discussion

A fit of the experimental data with Eq. (3), Fig. 5c, enables extracting the coefficient $$\frac{{C_{\varphi _0}\phi _0}}{{2\pi S_{{\mathrm{ref}}}}} = 41 \pm 5 10^{ - 5}$$. Using the expression of $$C_{\varphi _0}$$ in the diffusive regime given above, we calculate a spin-orbit coefficient α = 0.38 ± 0.015 eVÅ. This value of the Rashba coefficient is consistent with the value extracted from Rashba-split conduction band observed by photoemission measurements36,37. Table 2 gives the anomalous phase shift extracted from the critical current oscillations at the largest magnetic field about 80–100 mT. The phase shift is extracted from the magnetic field difference between the last nodes of the oscillations, indicated by blue and red dots on Fig. 5. At this largest magnetic field, we find an anomalous phase shift $$\varphi _0 \simeq 0.9\pi$$ for all three tilt angles θ. This shows that the anomalous phase shift depends only on the parallel component of the magnetic field as expected. This experimental value is compared with the theoretical values calculated in the ballistic regime, for the Rashba-split conduction states and Dirac states, and in the diffusive regime, for the Rashba-split conduction states. This table shows that the Dirac states provide only a phase shift of 0.005π and so cannot explain the experimental data. The table also shows that the Rashba-split conduction states provide a phase shift of only 0.01π in the ballistic regime while they provide an anomalous shift of 0.94π in the diffusive regime, close to the experimental value, confirming that the junctions are indeed in the diffusive regime and demonstrating the validity of the theory leading to Eq. (1).

A detailed look at Table 1 shows that the anomalous shift observed here must be the consequence of finite Rashba coefficient and in-plane magnetic field. While Table 1 shows that disorder alone Vy is sufficient to generate an anomalous phase shift, this disorder-induced anomalous phase shift should exist even at zero magnetic field and should not change with magnetic field. In contrast, as discussed above, we have seen that the two JIs are in-phase near zero magnetic field and become out of phase only for at finite magnetic field. Thus, this observation implies that disorder Vy is absent, which is plausible as the small Josephson junction is only 150 nm × 150 nm. In the absence of disorder Vy, Table 1 shows that the only way for an anomalous phase shift to be present is that the coefficient α be non-zero. Indeed, if α were zero, the first and third symmetry operations of Table 1 would not be broken even with finite By.

To summarize the result of this work, the simultaneous measurements of the CPR in two JIs making an angle of 90° with respect to the in-plane magnetic field enabled the identification of the anomalous phase shift φ0 induced by the combination of the strong spin-orbit coupling in Bi2Se3 and Zeeman field. This anomalous phase shift can be employed to fabricate a phase battery, a quantum device of intense interest for the design and fabrication of superconducting quantum circuits41,42.

## Methods

### Sample preparation

The Bi2Se3 samples were grown by Molecular Beam Epitaxy. The crystalline quality of the films was monitored in-situ by reflection high energy electron diffraction and ex situ by x-ray diffraction, and by post growth verification of the electronic structure though the observation of the Dirac cone fingerprint in angle-resolved photoemission spectra as described in ref. 43. Following growth, the samples were capped with a Se protective layer. The Josephson junctions are fabricated on these thin films with standard e-beam lithography, e-beam deposition of Ti(5 nm)/Al(20–50 nm) electrodes and lift-off. The Se capping layer is removed just before metal evaporation by dipping the samples in a NMF solution of Na2S. In the evaporator chamber, the surface is cleaned by moderate in-situ ion beam cleaning of the film surface before metal deposition. While for standard junctions, an aluminum layer 50 nm thick is usually deposited, we also fabricated junctions with 20-nm-thick electrodes to increase their upper critical field as required by the experiments with in-plane magnetic field. See Supplementary Figure 3 for a lateral sketch of the devices. After microfabrication, the carrier concentration is about 1019 cm−3 and the resistivity about 0.61 mΩ.cm, as shown in Supplementary Figure 2. A comparison between the normal state junction resistance of the order of 20–50 Ω and the resistivity of the films indicates negligible contact resistance, i.e. the junction resistance is due to the Bi2Se3 film between the electrodes.

### Measurements details

The values for the normal resistance and critical current values measured on 20 devices are found to be highly reproducible, demonstrating the reliability of our procedure for surface protection and preparation before evaporation of the electrodes. The devices are measured in a dilution fridge with a base temperature of 25 mK. The IV curves are measured with standard current source and low noise instrumentation amplifiers for detecting the voltage across the junctions. The measurement lines are heavily filtered with π filters at room temperature at the input connections of the cryostat. They are also filtered on the sample stage at low temperature with 1 nF capacitances connecting the measurements lines to the ground.