Supplementary Information for Controlled Finite Momentum Pairing and Spatially Varying Order Parameter in Proximitized HgTe Quantum Wells

Junctions were fabricated using HgTe/HgCdTe heterostructures grown in the [001] crystal direction (the z direction), composed as shown in Supplementary Figure 1. Wafer I contained an 8 nm quantum well with an electron density of 13.5 × 10/cm and a mobility of 390, 000 cm/Vs. Wafer II contained a 7.8 nm quantum well with an electron density of 2.9 × 10/cm and a mobility of 790, 000 cm/Vs. Josephson junctions fabricated on these wafers were aligned at varying angles with respect to the [110] and [11̄0] cleavage edges of the crystal, but we do not know which is which in our samples. Therefore, we can only specify that the angular alignment corresponds to a rotation angle θ with respect to the [100] crystal axis, modulo π/2. Although we do not know which principal axis θ is referenced to experimentally, our model predicts the same results when θ is referenced to either. The x and y axes always lie respectively perpendicular and parallel to the direction of current flow in devices (see Figure 1 of main text). Throughout this supplement, devices are referred to in the following manner. Device A was fabricated by depositing aluminum leads onto a mesa etched into Wafer I, and was oriented at an angle θ = π/4. Devices B, C, and D were concurrently fabricated by depositing aluminum leads on Wafer II, and were

In the measurements of critical current presented here, increasing the external field in either the x or y direction results in a decrease of the maximum critical current (Supplementary Figure 3a, b). This decrease occurs more rapidly for B y , where the critical current becomes too small to reliably measure when B y exceeds 0.44 T. B x , however, must exceed 1.1 T before critical currents become immeasurably small.
Upon normalization of the interference pattern at each separate value of parallel field, the asymmetry between the two directions becomes more pronounced (Supplementary Figure 3c, d). At all values of B x , the shape of the Josephson interference remains essentially unaffected. By contrast, as B y increases the critical current splits into two separate maxima which occur at larger values of |B z |. In Supplement XI we model the effect of the in-plane field B y , which we expect to induce a finite x component of the pairing wavevector that grows linearly with B y . In junctions with finite length and with B z = 0, this pairing momentum in the x direction leads to oscillations in the order parameter which are most pronounced near the ends of the junction. As a consequence, we expect that as B y increases, the maximum critical currents in our junctions will occur at values of |B z | that grow linearly with B y . With only the parallel field B x present, however, the induced pair momentum is expected to lie along the y direction. In this case no such wavevector in the x direction is observed.

IV. Josephson Junctions Rotated with Respect to the Crystal
Measurements of the differential resistance were performed on junctions oriented at different angles with respect to the crystal lattice, in order to determine whether bulk inversion asymmetry (BIA) plays a significant role in the momentum acquired by Cooper pairs. As previously mentioned, these devices A-E have orientation θ = π/4, 0, π/2, π/4, and π/4 respectively. Devices A-D use aluminum leads, while device E uses niobium leads. For each set of the devices, we explored behaviors resulting from a parallel magnetic field applied in the x direction or in the y direction, which in the previous section were shown for device D to differ.

DOI: 10.1038/NPHYS3877
Even as the angle θ varies among devices, the manner in which superconductivity evolves due to the applied field B x remains qualitatively unchanged at high density (Supplementary Figure 4). Devices B-D, fabricated on a single piece of wafer, show quantitative agreement in the value of B x at which a superconducting node appears. Devices A and E were separately fabricated on Wafer I, and show slight quantitative differences but nevertheless the same shape. The appearance of these nodes in the interference evolution, with no dependence on the crystal orientation, signals that structural inversion asymmetry (SIA) dominates the behavior of our quantum wells in the electron-doped regime (Supplement VI-IX).
With the parallel field applied in the y direction and at high density, the interference pattern splits, forming a 'V' shape as B y increases that is qualitatively identical for all values of θ (Supplementary Figure 5). The slope of the two arms of the 'V' varies among devices, but is similar for devices B-D which were fabricated concurrently. The most dramatic effect is seen for device E with 130 nm thick niobium leads, in which the slope is approximately 7 times smaller than the other devices.
From the above measurements one can conclude that the basic differences in interference as B x or B y is increased have little to do with the orientation of the crystal lattice. The most striking difference is found among the data with the magnetic field oriented along the y direction, in which the thickness of the leads correlates to the slope of the interference splitting. This behavior, which results from magnetic flux penetrating the area dL formed by the length L of the junction and the height difference d between the center of the quantum well and the center of the leads, is modeled in Supplement XI.
V. Evolution of Interference Lobes as the Parallel Magnetic Field B x Increases, in a Device with θ = π/2 In the previous section, it was observed that at particular values of the parallel magnetic field B x , nodes of suppressed superconductivity occur in our Josephson junctions. Additionally, this evolution of the Josephson interference with the parallel field B x does not depend on the device orientation with respect to the crystal. The appearance of nodes with lack of θ dependence already suggests that the effect of bulk inversion asymmetry (BIA) in our devices is small, and that structural inversion asymmetry (SIA) dominates (Supplement VI-IX). Still, the evolution of critical currents, in particular the maximum critical current of interference lobes occurring at nonzero perpendicular field, can provide further evidence that BIA is weak. When θ = π/2, in the limit of strong BIA these side lobe maximum critical currents are expected to grow as the parallel field B x is increased from zero ( text). However, if SIA is strong, these critical currents are expected to monotonously decay as the parallel field increases up to the first node.
In Device C, oriented at θ = π/2, we study the evolution of the first three side lobes adjacent to the central lobe (Supplementary Figure 6a). A measurement of the critical currents of these lobes indicates that all lobes are largest when B x = 0 T (Supplementary Figure 6b). We extract the maximum critical current for each side lobe, plotted in Supplementary Figure 6c for each lobe. All critical currents are largest when B x = 0 T, and all decay monotonously until becoming indistinguishable from zero. This evolution of the side lobe critical currents provides additional evidence consistent with weak BIA in our devices.

VI. Four-Band Model and Spin-Orbit Effects in the Quantum Well
A four-band model has been developed starting from k · p theory, and subsequently used to describe the topology of HgTe quantum wells [2]. Here we adopt an elaboration on this model, where bulk inversion asymmetry (BIA), structural inversion asymmetry (SIA), and coupling to an external magnetic field are included. The four bands originate in the s− and p−like bands of the underlying crystals, so that the basis states are written as |E1, m J = +1/2 , |H1, m J = +3/2 , |E1, m J = −1/2 , and |H1, m J = −3/2 .
In this notation, E1 refers to electron-like states with angular momentum 1/2, while H1 refers to holelike states with angular momentum 3/2. The Hamiltonian describing the system is then [3, 4, 5]: where and  In the remaining case, the effect of dominant BIA with no SIA is investigated in Supplementary   Figure 7g-i, for an angle θ th = 0 and up to a Fermi energy of 20 meV. Similarly to the case of strong SIA, with no external magnetic field present the bands are spin-split at nonzero wavevectors. However, the texture of spins at the Fermi surfaces displays tetrahedral symmetry in this case. As a result, when parallel magnetic field is present in the x (y) direction, the two Fermi surfaces shift oppositely in the x (y) direction. This shifting is orthogonal to the shifts present with strong SIA, and does not agree with our interference measurements on devices aligned with θ = 0 or π/2. Furthermore, the shift direction rotates by π/2 as the angle θ th becomes π/4. This prediction that the direction of induced Cooper pair momentum should depend on θ th is also inconsistent with our results.
In the main text, we model the evolution with density of the in-plane g-factorg, the Fermi velocity v F , and the pair momentum shift ∆k ≈gµ B B x / v F , under the assumption that BIA is absent and SIA is dominant (equivalent to a perpendicular electric field of 10 mV/nm). At each density, a particular value of the magnetic field satisfies the condition ∆kW = π/2, leading to a node in the induced superconductivity in the junction. We find that the evolution with density of this nodal magnetic field value agrees with our model at high densities. In niobium and aluminum devices respectively, the value of the nodal magnetic field is consistent with values of ofg/v F that are approximately 1.9 and 1.4 times greater than those expected theoretically. Using different values for the electric field only weakly modifies this conclusion. For example, if the SIA was instead equivalent to a perpendicular electric field of 40 mV/nm, we would find values of ofg/v F approximately 2.1 and 1.5 times greater than the theoretical expectation, only a ∼ 10% difference.
Finally, we note that the BHZ model is an approximation based on 8-band k ·p theory, which neglects all quantum well subbands except for the lowest electron-like (E1) and heavy hole-like (H1) sub-bands.
This approximation is most valid when all other sub-bands are well-separated energetically from these two bands, which occurs for quantum well widths near 6.3 nm [5]. For the quantum wells discussed in this work, whose widths were near 8 nm, the second hole sub-band H2 actually becomes closer in energy to the E1 band than the H1 band. This could lead to corrections to the in-plane g-factors and dispersion relations in the quantum well. Furthermore, the sub-band energies calculated from k · p theory in [5], on which the BHZ model is based, do not include the effect of a parallel magnetic field.
Modifications to the full 8-band Kane model under a parallel magnetic field could also carry through to change the dispersion relations and g-factors in the quantum well. It is possible that considering a full 8-band k · p model could help explain the disagreement at low electron density between our data and our theory based on the BHZ model ( Figure 4 of the main text).

VII. Model of a Two-Dimensional Electron Gas Contacted by Superconducting Leads
In the following sections, we model the coupling of superconducting leads to our quantum well. We consider a geometry in which a two-dimensional electron gas (2DEG) is contacted by a pair of superconducting leads with a controlled phase difference between them, and we seek to calculate the maximum supercurrent that can be carried between the strips. The following is a more complete derivation of the pair propagator F used in the main text to carry out this goal. We assume a Hamiltonian where H 0 is the Hamiltonian for the 2DEG in the absence of the superconductor, and H 2 is the coupling between the superconductors and the 2DEG, described by a pairing Hamiltonian of the form Here is an operator which annihilates a singlet pair of electrons in the 2DEG at the point (x, y), while the pair potential ∆(x, y) is a complex number that depends on the phase of the superconductor and the tunneling amplitude at that point.
We assume that the contacts between the 2DEG and the superconductors occur at the edges of the superconductors, located at y = 0 and y = W , so that we may write with −L/2 < x < L/2. We assume that the magnitude of the coupling is constant along each lead, but the phase will vary if there is a perpendicular magnetic field B z = 0. We choose a gauge where the vector potential points in the x direction, with A x = −B z (y − W/2), so that the vector potential vanishes along the midline of the 2DEG. If the superconducting strips have identical widths W SC , then the couplings λ j will have the form with j = 1, 2. The phases in equation (7) are determined by the condition that there should be no net current flow along the length of the superconducting strips, so the phase gradient in each superconductor should be canceled by the vector potential along the center line of the superconductor.

DOI: 10.1038/NPHYS3877
We assume that B z is sufficiently weak that the cyclotron radius R c = k F /eB z is large compared to W (typically in our devices R c ≈ 10 µm, which is large compared to W = 800 nm). In this case, we may ignore the effect of B z on the trajectories of electrons in the 2DEG. Moreover, since we have chosen the vector potential to vanish along the midline of the 2DEG, an electron crossing from y = 0 to y = W will acquire no net phase due to the vector potential. We also ignore, for the moment, any orbital effects of the parallel field B || . Thus the 2DEG Hamiltonian H 0 will include the Zeeman coupling to B || , as well as the spin-orbit coupling, but will not include terms due to the magnetic field in the kinetic energy.
To lowest order in the couplings λ j , the portion of the total energy that depends on the phase difference between the two superconducting leads can be written in the form: where Ψ(x, y) 1 is the order parameter at point (x, y) induced by the superconductor j = 1. In turn, this may be written in the form where F is the propagator from point (x 1 , 0) to point (x, y) for an induced Cooper pair. We will determine the form of F in the following section.
As a first approximation, we may ignore the fact that there are boundaries of the 2DEG at x = ±L/2 and that electrons will be reflected at these boundaries (either specularly or diffusely, and possibly with a spin flip). Similarly, we ignore the possibility of single-particle reflection at y = 0 or y = W , where the superconducting leads touch the 2DEG. Furthermore, we assume that the electron density is constant in the 2DEG, and we ignore any interactions between electrons in the 2DEG. We also ignore scattering by impurities inside the 2DEG. Then the propagator F may be calculated for an infinite, translationally invariant 2DEG, where the momentum of each electron is a good quantum number. We believe that corrections due to reflections at the boundaries will have quantitative effects but will not affect qualitatively the form of our results. Modeling of critical current including specular reflections at mesa edges will be discussed in Supplement X and XI.

VIII. Derivation of a General Formula for the Pair Propagator F
As discussed previously, the portion of the total energy that depends on the phase difference between the two superconducting leads can be expressed, to leading order in the coupling constants λ j , in terms beyond lowest order and should take into account the finite widths of the superconducting contacts and changes in the local chemical potential resulting from the coupling to the superconductor. The effect of these corrections may be described by introduction of an amplitude for normal reflection where the 2DEG meets the edge of the superconductor, as well as a renormalization of the amplitude for Andreev reflection, which will clearly have an effect on the overall magnitude of the coupling between the two superconductors and therefore on the maximum critical current. Since we do not know the precise strength of the coupling between the 2DEG the superconductors in the first place, we are not interested in this overall magnitude of the critical current, but rather in its dependence of the critical current on the parameters of the system, such as direction and strength of the magnetic field, the sample geometry, and the electron density.
A potential concern for our analysis is that beyond the lowest order in perturbation theory, one should include processes where an electron can undergo multiple reflections between the two superconductors before it is absorbed in an Andreev process at one side or the other. However, processes involving multiple reflections will fall off faster with W than the processes included in equation (8), particularly if one takes into account disorder at the superconducting interface. Consequently, we feel justified in neglecting such processes here.
If one uses equation (8) to calculate the critical current, one finds that dependences of the critical current on system parameters such as the direction and strength of the magnetic field arise from interference between contributions to the integral from different points x and x 1 , which will be particularly sensitive to variations of the phase of F (x, x 1 , W ) as a function of these variables. We remark that, strictly speaking, calculations beyond lowest-order perturbation theory may lead to dependences of the amplitude for Andreev reflections at the boundaries on the angle of incidence and on the electron energy that differ somewhat from the lowest-order results, which would give, in turn, corrections to the space dependence of the integrand in equation (8) 2DEG with momentum k is given by the 2 × 2 matrix where B j are the x and y components of the in-plane magnetic field,g is an effective g-factor,k ≡ k/k, andk i S ij σ j is the spin-orbit coupling term, which we assume to be small compared to the Fermi energy.
We have here assumed a single electron band, and assumed that band structure is isotropic in the absence of spin-orbit coupling. We can then write with η = ±1. Here kη are the two eigenvalues, and Pk η are projection matrices given by withβ ≡ β/ β . We next define a 2 × 2 matrix function Then, letting r = (x − x 1 , y), the pair propagator F (x, x 1 , y) may be expressed as where T indicates the matrix transpose.
We are interested in the situation where k F r 1, and | | E F . Then the integration over the direction of k is dominated by regions close to the end points where k is either parallel or antiparallel to r, and the expression for g( r, ) may be approximated by where P η± is equal to Pk η , withk = ±r ≡ ± r/r and When we substitute the expression for g in formula (15) for F ( r), we may ignore the terms proportional to e ±2ik F r , as these rapidly oscillating terms will give vanishing contribution to the energy if the width of the contacts between the 2DEG and the superconducting strips are large compared to 1/k F .
Performing the integrals over and in the remaining terms, one obtains the result

IX. Special Cases and Limiting Forms of the Pair Propagator
Here we discuss several special cases which lead to limiting forms for the pair propagator F ( r). The above expressions (equations (18) and (19)) may be simplified in the limit where the Zeeman energy is small compared to the spin-orbit energy splitting. When B || = 0, we find thatβ(r) = −β(−r), so that N ηη = δ ηη . Furthermore, when η = η , we see that the exponent in equation (18) is equal to zero, so F will have no oscillations as a function of r. If B || is nonzero but still small compared to the spin-orbit splitting, it remains a good approximation to set N ηη = δ ηη . In the exponent, however, we have An important example is the case of pure SIA spin-orbit coupling, where the matrix S has the Rashba form, S ∝ iτ y , where τ y is a Pauli matrix. In this case we may write In the case of pure BIA coupling, the matrix S is ∝ τ z , in our coordinate system. We may again write the phase accumulation in the form (21), but now the direction of ∆ k depends on the directions of B || relative to the crystal axes.
The formula for F ( r) also becomes simple in the case where the Zeeman energy is large compared to the spin-orbit splitting. In this case, the Fermi surface consists of two concentric circles, with spin that are uniformly aligned on each circle, either parallel or antiparallel to B || . In order to form a spin singlet, we must choose one electron from each Fermi circle. If we also require that the momenta be parallel or antiparallel to r, we see that the induced electron pair will have a total momentum equal to Thus we should find that the phase shift is independent of the direction of r.
These expectations may be confirmed using the formulas derived above. In the case where the Zeeman energy is large compared to the spin-orbit splitting, we find that β(k) is independent ofk, and

X. Reflections from the Sample Edges
Taking into account the effects of electron reflections from the ends of the sample, at x = ±L/2, we should rewrite the propagator F in a more general form as where F 0 is the function given by equation (18)  respectively. We assume that the length L is long enough that we can neglect the effects of electrons that scatter multiple times from opposite boundaries. Here we will assume that the boundaries at x = ±L/2 are represented by infinite potential barriers, which are perfectly smooth, so that electrons are specularly reflected with no change in spin. The symmetry of our problem will then be such that F 1 and F 2 have identical functional forms, so we need only find the form of F 1 . For convenience, we move the left boundary to the line x = 0, and we assume that the right boundary is located at x = ∞. Using similar reasoning to what we used in the translationally invariant case, we may write F 1 in the form where for j = 1, 2, withk Furthermore, we have We now turn to one particular example. In order to evaluate expression (24) for F 1 , we must first evaluate the trace over a product of projection matrices and σ y . In the case of strong SIA coupling and weak magnetic field, the trace simplifies, and we obtain the result tr P η1+ where sin θ = W/s. Furthermore for B || in the y direction, we find Thus, in the case of strong SIA and B || in the y direction, we find where ∆k =gµ B B y /v F , and the constant C is the same as in equation (18).

XI. Modeling Josephson Interference
Using the pair propagator F ( r) and equation (8), we can calculate the Josephson energy and critical current for our junctions. In the limit of either strong BIA or strong SIA, the Cooper pair momentum shift occurs at an angle α with respect to the x axis and the pair propagator is As previously noted, in this case pairing occurs internally to each Fermi surface. In the limit of weak spin-orbit coupling, the pair propagator instead takes the form Due to the opposite spin polarization of the two Fermi surfaces, pairing in this limit is expected to occur between Fermi surfaces, in contrast to the limit of large spin-orbit coupling.
The Josephson energy E is obtained in each limit by evaluating equation (8). By differentiating the Josephson energy with respect to the phase difference φ 1 (0) − φ 2 (0) we find the current-phase relation of the junction, which is then maximized with respect to the phase difference to obtain the critical current.
In the main text we consider only a parallel field along the x direction. In both aluminum and niobium-based devices we experimentally observe that superconductivity weakens as the parallel field B x increases, in contrast with the cosine dependence predicted by our theoretical model. We believe that this effect results from spatially inhomogeneous screening of the parallel field at the edges of superconducting leads. The superconductor repels the in-plane field and slight roughness at the edges results in a weak magnetic field along the z direction that is positive at some locations and negative at others. This screening leads to a spatially varying random component of the phase that grows linearly with the in-plane field. Hence, we introduce a random phase χ ∝ (R 1 ( random variables R 1 (x 1 ) and R 2 (x 2 ) correspond to screening of the parallel field at each interface. The modeled step size in x is 40 nm, with no correlations between adjacent positions. The random phase χ is uniformly distributed between zero and an upper bound whose absolute magnitude is equal to 15% of the maximum phase generated by the intrinsic momentum. With this randomness, the calculated critical currents diminish in magnitude as the in-plane field increases, in agreement with the experimental observation (shown in Figure 3c of the main text for the case of dominant SIA).
In general, the parallel field B || can be oriented anywhere in the plane, which modifies α accordingly in the case that spin-orbit coupling is strong. Additionally, loosening the constraint that B || lie parallel to x introduces an artifact wavevector q y ≈ 2πB || sin(β)d/Φ 0 , where d is the height difference between the centers of the quantum well and of the superconducting leads, and β is the angle between the parallel magnetic field and the x axis. This additional phase arises due to the magnetic flux penetrating the area dL formed between the leads and the quantum well due to this height difference. Importantly, no flux penetrates this area when the parallel component of magnetic field is only in the x direction, so that in this case the pair momentum is solely determined by the Zeeman coupling and the spin-orbit coupling.
The behavior of Josephson interference in our devices essentially involves different mechanisms when the parallel magnetic field lies in the x or y direction. With the above modeling it is clear that this difference is due to the dependence of q y on the magnetic field angle β, so that the data for the magnetic Athough the parallel magnetic flux dominates the response of devices to the field B y , with purely SIA it is still in principle possible in this direction to extract the intrinsic nature of spin-orbit coupling.
Since the wavevectors q y and ∆k add and subtract, the 'V' shape of supercurrent evolution contains two nearly identical slopes, which in our measurements are unobservable due to the concurrent decay of superconductivity. However, normalizing the theoretical critical current magnitude still reveals the possibility to determine the nature of spin-orbit coupling using this parallel field direction ( Supplementary   Figure 8e).
An additional characteristic common among the data sets is an asymmetry in the interference pattern upon inversion of one component of the applied magnetic field. In Supplementary Figure 8f we show an interference pattern measured on Device F with both positive and negative components B y and B z .
Here the data appears invariant under inversion of both components of the magnetic field, as we expect We may include specular reflections at the mesa boundaries in the model, as discussed in Supplement X.
With these contributions, the interference evolution is quantitatively modified (Supplementary Figure   8i). However, the 'V' shape of the interference evolution is still present, with each arm of the 'V' having the same slope as was obtained by ignoring edge reflections. Hence, we conclude that the contribution of specular reflections preserving the spin direction only quantitatively modifies the expected device behavior. We have not carried out calculations for other boundary conditions, such as diffuse reflection, but we expect that results in these cases would not be qualitatively different from the cases of specular reflection or no reflection at all.

XII. Evidence for the Transition to a π-Junction
In a conventional Josephson junction with no external magnetic field, the supercurrent I S is related to the phase difference ∆φ between the leads via the Josephson relation I S = I C sin(∆φ). Here I C is the critical current of the junction. When the induced order parameter oscillates in space, it is possible that this order parameter can have a different sign at the boundary of each superconducting lead. This modifies the current-phase relation by a phase shift of π, so that in such a junction I S = I C sin(π + ∆φ). These junctions are referred to as π-junctions, and were first explored in systems composed of a ferromagnetic layer sandwiched between two superconductors [6]. A simple experiment which provides evidence of the π phase shift consists of two junctions connected in parallel and sharing the same superconducting leads. If one of the junctions is conventional and the other is a π-junction, then if the junctions also have equal critical currents the total supercurrent must be zero in the absence of external magnetic flux B z .
This contrasts with the standard result for two conventional junctions in series, in which the maximum supercurrent is expected with B z = 0.
In our junctions, applying a finite magnetic field B x results in finite momentum pairing in the y direction. For a junction with width W , we expect that a π-junction should then be realized when corresponding to the situation where the induced order parameter has a single node inside the junction. To carry out the experiment described above requires that we realize both a π-junction and a conventional junction. To achieve this goal, we have fabricated a device in which a junction with dimensions 800 nm × 4 µm is wired in parallel with a junction having dimensions 200 nm × 2 µm (Supplementary Figure 9a). When the condition π is satisfied for the 800 nm junction, the 200 nm junction will still be in the conventional regime. This experiment therefore allows one to detect the π phase shift, and also to verify that the parallel field B x necessary to achieve the shift depends on the junction width W . All data presented on this device was collected at a temperature of 10 mK in the system discussed previously (Supplement I).
Due to the screening of the parallel magnetic field by the superconducting leads (described in Supplement XI), we have fabricated our device such that all edges of the leads which lie along the y direction are far from the active areas of the device which contain the quantum well. If this were not the case, the screened field would penetrate the quantum well imhomogeneously, leading to unwanted interference. to the area of the 800 nm junction. In the simulation with no screened flux, when the parallel field B x exceeds ≈ 1.2 T, the critical current develops a sharp minimum at B z = 0. This critical current minimum results from the formation of a π-junction in the 800 nm section of the device. In the simulation which includes screened flux, increasing the parallel field B x leads to oscillations in the critical current, even at B z = 0. These oscillations arise because the screened flux penetrates the center of the SQUID loop formed by etching the hole between the two junctions. Here, when the parallel field B x exceeds ≈ 1.2 T, the formation of a π-junction in the 800 nm junction manifests as a π phase shift in the SQUID oscillations. This is due to the fact that when one arm of the SQUID loop is a π-junction, the condition for the maximum supercurrent shifts by 1/2 flux quantum. To illustrate this effect, a line trace of the simulated critical current at B z = 0 is plotted in Supplementary Figure 9d.
To determine whether the π phase shift is present in our device, we measured the differential re-    The critical current as a function of perpendicular magnetic field B z , as parallel components of the magnetic field are varied. The data presented here was taken using the device with aluminum leads presented in the main text (device D). (a, b) As the parallel magnetic field in either the x or y direction is increased, the magnitude of the maximum critical current decreases. This decrease occurs more rapidly in y direction than in the x direction. (c) Normalizing the Fraunhofer interference at each value of B x shows that the shape of the interference pattern remains essentially unaffected until it becomes immeasurably small. (d) In the B y direction, normalization reveals a dramatically different behavior of the Fraunhofer interference, where critical current maxima occur at higher values of B z as B y is increased. Concurrently, the weight of the critical current at B z = 0 mT decreases to 0. These observations match those deduced through measurements of the differential resistance as the parallel magnetic field varies in either the x or y direction. Therefore both differential resistance and critical current measurements reflect the same basic phenomenon. and (f) were separately fabricated. All junctions had aluminum leads except for in (f), where niobium leads were used. (b) For a junction oriented at θ = π/4 with respect to the crystal, the differential resistance is monitored as both the perpendicular field B z and the parallel field B x are altered. As B x increases, the position of nodes in the interference pattern does not change, but the interference gradually disappears. (c) A junction oriented at θ = 0 and (d) a junction oriented at θ = π/2 with respect to the crystal show qualitatively identical behavior. (e) A further junction aligned at θ = π/4 shows the same behavior, as previously presented in the main text. (f ) Also in the main text, the junction with niobium leads is oriented with θ = π/4 and shows interference which remains strongly weighted at B z = 0 T. The observations on aluminum devices are all consistent with dominant SIA in the quantum well. With the junction aligned such that θ = π/4, the differential resistance is monitored as a function of the perpendicular field B z and the parallel field B y . Increasing B y rapidly causes the weight of interference fringes to shift to larger B z values, forming a 'V' shape. The interference evolves more rapidly due to a parallel field in the y direction than in the x direction due to the fact that leads are spatially displaced in z with respect to the quantum well. (c) Orienting a junction at θ = 0 introduces no qualitative change to the behavior, as is also the case with (d) a junction oriented at θ = π/2. Junctions from the main text with (e) aluminum and (f ) niobium leads are presented, also displaying similar behavior. The enhanced scale of B z in the niobium-based junction is due to the increased thickness of the leads. An external field in both the x and z directions may be simulated, neglecting screening effects from the leads. In this case, application of the magnetic field B x leads to a pair momentum shift in the y direction. At B x = 1.2 T, the 800 nm junction transitions to a π-junction state. Above this nodal field, the critical current develops a minimum at B z = 0 due to the presence of both a conventional and π-junction in the device. (c) Including screening of the magnetic field B x by the aluminum leads, the simulation shows that increasing B x while keeping B z = 0 leads to oscillations in the critical current. These are due to flux penetrating the central hole in the device, and are essentially the critical current oscillations of a SQUID loop. (d) In this situation, the transition of the 800 nm junction to a π-junction is then expected to manifest as a phase shift in these SQUID oscillations. A linecut of the simulated critical current at B z = 0, as a function of B x , displays this shift. (e) The measured differential resistance of the device, with a DC current bias of 20 nA, shows a node structure which matches the predicted interference. (f ) Extracting the differential resistance at B z = 0 shows a periodic dependence on B x at low fields, with the predicted phase shift above B x = 1.2 T.