Phase signature of topological transition in Josephson junctions

William Mayer1∗, Matthieu C. Dartiailh1∗, Joseph Yuan1,

Kaushini S. Wickramasinghe1, Alex Matos-Abiague2, Igor Zutic3, and Javad Shabani11Center for Quantum Phenomena, Department of Physics, New York University, NY 10003, USA

2Department of Physics & Astronomy, Wayne State University, Detroit, MI 48201, USA3Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA and

∗These authors contributed equally to this work

Topological transition transforms common su-perconductivity into an exotic phase of mat-ter, which holds promise for fault-tolerant quan-tum computing1,2. A hallmark of this topo-logical transition is the emergence of Majoranastates, quasiparticles that nonlocally store a sin-gle electron3. While two-dimensional semicon-ductor/superconductor heterostructures4–7 aredesirable platforms for topological superconduc-tivity, direct phase-measurements as the finger-print of the underlying topological transition areconspicuously missing. By embedding two gate-tunable Josephson junctions in a phase-sensitiveloop geometry, we measure for the first time aπ-jump in the superconducting phase across thejunction, a signature of the topological transi-tion. Furthermore, in epitaxial Al/InAs junctionswe observe closing and reopening of the super-conducting gap with increasing in-plane magneticfield, B‖, coincident with the π-jump. Theoreticalsimulations confirm this transition is topologicaland compatible with the emergence of Majoranastates. Remarkably, in each Josephson junction,this topological transition can be controlled bychanging the gate voltage. These findings revealversatile two-dimensional platforms for scalabletopological quantum computing.

Majorana bound states (MBS), which are their own an-tiparticles, are predicted to emerge as zero-energy modeslocalized at the boundary between a topological super-conductor and a topologically-trivial region3. MBS cannonlocally store quantum information and their non-Abelian exchange statistics allows for the implementationof quantum gates through braiding operations1. Thismakes them ideal candidates for robust qubits in fault-tolerant topological quantum computing2. Rather thanseeking elusive spinless p-wave superconductors requiredfor MBS, a common approach is to use conventional s-wave superconductors to proximity-modify semiconduc-tor heterostructures with suitable symmetries8.

Early MBS proposals were focused on one-dimensional(1D) systems such as proximitized nanowires and atomicchains9–13, where the observation of a quantized zero-bias conductance peak (ZBCP)14 provided the supportfor MBS. However, the inherent difficulties in the tech-nological implementation of the required networks andthe intrinsic instabilities of their 1D elements have mo-tivated the search for versatile 2D platforms using more

conventional devices such as Josephson junctions (JJs)and spin valves4–7,15–18. Recent experiments6,7 suggestthat planar JJs are particularly promising because theysupport topological superconductivity over a large pa-rameter range. The change between trivial and topolog-ical superconductivity, probed by ZBCP, is realized byapplying in-plane magnetic field, B‖, and biasing the su-perconducting phase between 0 and π. This is achievedby embedding the JJ in a loop6 or by using two strongly-asymmetric JJs7 in a superconducting quantum interfer-ence device (SQUID).

Since ZBCP could arise even without topological su-perconductivity, it is crucial to identify alternative sig-natures. A striking example is the closing and re-opening of the superconducting gap with an increasingmagnetic field that is simultaneously accompanied by aphase jump8–10,19,20. Direct phase measurements haveproven to be a powerful probe to elucidate unconven-tional superconductivity21. In materials with a muchhigher g-factor, (up to 100) a small Fermi velocity andsub-Tesla B-fields a Zeeman 0-π transition can originatefrom a strong carrier spin-polarization22. Trivial Zee-man/exchange 0-π transition can happen without spin-orbit coupling (SOC)23,24, but is then expected to beindependent of the B‖-direction, in contrast with thecase of topological superconductivity. A critical currentbehavior that could be consistent with gap reopeningwas reported17. However, accompanying phase measure-ments were absent and its wide JJ geometry would min-imize the topological protection.

In contrast to refs. 6, 7, in our JJs the phase is notbiased. Instead, it self-adjusts to minimize the under-lying free energy of the system in both the trivial andtopological phases. In this work, we demonstrate howtwo gate-tunable and nearly-symmetric JJs, forming aSQUID shown in Fig. 1a, provide a platform to realizethe first direct phase-sensitive measurements of a topo-logical transition. This provides complementary evidencefor the topological nature of our observation of a super-conducting gap reopening.

The JJs based on epitaxial Al/InAs (see Methods) areengineered to support high-interfacial transparency androbust proximity-induced superconductivity in InAs4.Both junctions (1, 2) of the SQUID are W=4 µm wideand L=100 nm long, while the area of the SQUID loop is25 µm2. This yields junctions in the short limit and witha mean free path on the same order as their length [seeSupplementary Information (SI)]. Using a vector mag-

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Fig. 1: Topological SQUID. a, False-colour SEM imageof a SQUID composed of two 4 µm wide JJ with a gap of100 nm. The central area is about 25 µm2 and each junctionis independently gateable. The x-direction is colinear to thecurrent flow in the junctions. b, Measurement of the junctiondifferential resistance at zero flux as a function of the gatevoltage applied on JJ2. At V 2

g = 0 V both JJs can carry asupercurrent, below V 2

g < −5.5 V, JJ2 behaves like an opencircuit. c, Spectrum of the junction without and e, with thephase-bias of π. d, Predicted critical current of a junction inthe presence of an in-plane field along y.

net, we apply B‖ along an arbitrary axis defined by θ,as indicated in Fig. 1a, and impose a phase differencebetween the two JJ. The versatility of our setup can beseen from the measured SQUID differential resistance asa function of an applied bias current, I, and gate volt-age V 2

g in Fig. 1b. The critical current, Ic, at whichthe SQUID acquires a finite resistance, decreases and be-comes constant for V 2

g < −5.5 V, indicating that JJ2 isfully depleted. This shows that each JJ can be studiedindividually. Additional data demonstrating that we canoperate this device either as a SQUID or as a single JJare presented in the SI.

With B‖ along the y-direction, tight-binding simula-tions predict a single junction, without a phase bias, willundergo a topological phase transition at finite in-planemagnetic field, as plotted in Fig. 1c. A hallmark of thistransition is the (partial) closing and reopening of thesuperconducting gap8,19,20 manifested as a minimum incritical current shown in Fig. 1d. Above that closing,the system is in a topological phase dominated by chiralp-type superconductivity, as shown in SI. This is in con-trast with a phase-biased junction6,7 which, at π-phase,can exhibit a topological phase at arbitrary low By, asshown in Fig. 1e.

In Figs. 2a and b, we present the By-dependence of

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Fig. 2: Reopening of superconducting gap in magneticfield. Measurement of the differential resistance of JJ1 asfunction of an applied in-plane field along the y-axis at twodifferent gate voltages a, V 1

g = −1.5 V, b V 1g = 1.4 V. In both

cases, JJ2 is depleted (V 2g = −7 V) and does not participate

in the transport. At high gate b, a closing and reopening ofthe superconducting gap is observed around 600 mT for JJ1.c, Zero-bias differential resistance of JJ2 as a function of theapplied in-plane field and the gate voltage. V 1

g is set to -7V.

the JJ1 critical current at two different gate voltages: aV 1g = −1.5 V and b V 1

g = 1.4 V. In Fig. 2a, at lower V 1g

and thus at a lower density, we observe a trivial mono-tonic decrease of Ic with By. Remarkably, at higher V 1

g

in Fig. 2b, we see a striking difference where the super-conducting gap closes and reopens around By = 600 mT,in agreement with the tight-binding results from Fig. 1d.Above that gap closing, we measure Ic ∼ 20 nA, consis-tent with the gap reopening and topological transition.After the gap reopens the resistance vanishes for I < Ic.In contrast, after gap closing, only non-zero resistancewas measured in ref. 18.

By considering Thouless energy, ET = (π/2)~vF /L,where vF is the Fermi velocity and L is the gap ofthe junction, one may expect to reach the topologicalphase more easily at low density (smaller gate voltages)since the transition has been predicted to occur aroundEZ ∼ ET /2, where EZ is the Zeeman energy19. How-ever, this neglects the vF -mismatch between the Al andInAs regions25, and SOC26, which can both change withdensity27. The topological transition occurs when thegap closes at ky = 0. For B‖ along the y directionspin is conserved at ky = 0 and SOC can be effec-tively gauged away yielding a SOC-independent topolog-ical condition9,10,19. However, SOC is still needed for a

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Fig. 3: Phase signature of topological transition from SQUID interferometry. a,b, SQUID oscillations for By =100 mT a and 850 mT b for different V 2

g and V 1g = −2 V. The dashed lines indicate the position of the maximum at V 2

g = −4 Vused as a phase reference. The stars mark the position of the maximum of the oscillation. The solid orange lines are best fits tothe SQUID oscillations used to extract the period and the phase-shift between different V 2

g values. c, Ground-state phase (redsolid line) and calculated SQUID phase-shift (blue and green dashed lines), see SI, as a function of the in-plane field calculatedusing a tight-binding model. A phase jump of about π occurs at the field corresponding to the closing of the gap identified inFig. 1c. d, Phase difference between the SQUID oscillation at V 2

g = −4V and the oscillation at a different value as a function ofBy. The solid lines correspond to linear fits of the data for By ≤ 450 mT. e, Phase shift from which the linear By-contributionhas been subtracted to highlight the phase jump occurring for the two higher V 2

g values.

sizable topological gap and MBS localization.

The nontrivial evolution of the superconducting gapand topological transition show similar behaviour in bothJJ1 and JJ2. Figure 2c presents the zero-bias differentialresistance of JJ2 as a function V 2

g and By. At the largest

V 2g , the transition occurs at ∼ 500mT and moves towards

higher By, as V 2g is decreased. Below V 2

g = −1.5 V, noevidence of any transition remains. The lower-magneticfield transition in JJ2 compared to JJ1 can be attributedto small variation of junction properties, for example,lower supercurrent and the corresponding induced gap.

While the observed non-monotonic dependence of Icwith By is consistent with a transition to topologicalsuperconductivity, phase-sensitive measurements with aSQUID could independently confirm this scenario. How-ever, it is generally difficult to avoid arbitrary field offsetsbetween measurements. Here, following the approach de-scribed in ref. 28, we use the gate tunability of our deviceto measure the phase offset between the oscillations ob-

served at different gate voltage but acquired during asingle Bz sweep. Using SQUID interferometry, wecan identify a topological transition by setting JJ1 atV 1g = −2 V as the reference junction. At this gate volt-

age, JJ1 does not show a topological transition at any By.The resulting SQUID oscillations in JJ2 reveal some cru-cial differences between By = 100mT and 850mT, shownrespectively in Figs. 3a and 3b, for various V 2

g . From theresults in Fig. 2c, we expect that JJ2 would never reachthe topological regime at 100 mT. Indeed, in Fig. 3a, weonly observe a small phase-shift which we attribute to theinterplay between the Zeeman splitting and SOC23. Athigher By in Fig. 3b, there is a larger phase-shift betweenV 2g = −3 V and -4 V than in Fig. 3a, consistent with the

expected linear increase in By23. However, comparing

V 2g = −1 V and higher-gate values, one can note that a

phase-shift of about π occurs.

Our tight-binding calculations, presented in Fig. 3c,reveal that a phase jump is a signature expected for the

4

topological transition with the emergence of MBS, givenin SI. As shown, both the ground-state phase (the phaseminimizing the energy in the absence of current) and thephase-shift maximizing the current through the SQUIDexhibit a phase jump when the topological transition oc-curs.

In Fig. 3d we present the phase-shift between the ref-erence scan performed at V 2

g = −4 V and subsequentgate values. At low By, below the topological transition,we observe that the phase is linear in By, as indicatedby the solid lines corresponding to linear fits to the val-ues below 450 mT. The increase in slope with V 2

g can

be attributed to the increase of SOC28. For V 2g = −3

V and −1 V, the linear trend holds over all By. How-ever, for V 2

g = 1 V, 2 V and 3 V, a jump can be observedaround 550 mT, followed by another linear portion. Toseparate these effects, we subtract the linear componentextracted from low-By fits (see Methods). The correctedphase in Fig. 3e reveals a phase jump with magnitudenear π, around the superconducting gap closing. This isa strong evidence for existence of topological phase tran-sition consistent with theoretical calculations in Fig. 3c.The phase jump rules out orbital effects that could leadto a minimum in the critical current.

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Fig. 4: In-plane magnetic-field anisotropy of the gapclosing. Differential resistance of JJ1 as function of the biascurrent and the y-component of B‖, applied at an angle θwith respect to the y-direction as depicted in Fig. 1a.

A distinct feature of the observed topological transi-tion is its interplay of SOC and B‖. In our system, thetopological regime is expected when B‖ is along the y-direction, i.e. θ = 0. We test this by probing the gap-closing in a tilted B‖, away from By. In Fig. 4 we show

zoom-ins of the gap-closing of JJ1 at V 1g = 1.4 V at dif-

ferent angles. As θ is increased, the gap-closing is weak-ened. Similarly, SQUID data at θ = 10, shown in SI,display a reduced phase-shift, which may indicate a re-

duced topological gap. In contrast to smaller angles, atθ = 20 the gap decreases monotonically which suggeststhat s-wave order prevails and no transition is observed.Unlike the insensitivity of trivial Zeeman 0−π transitionsto the in-plane field rotation, our strong dependence onthe B‖-direction is further evidence for a topological 0-πtransition.

By embedding Al/InAs Josephson junctions in anearly-symmetric SQUID loop, we are able to measuretwo distinct signatures of a topological phase transition,the reopening of the superconducting gap and thecoincidental π-jump of the superconducting phase. Ourfindings demonstrate the emergence of a topologicalphase in the system. In addition to By, the top gatevoltage is shown to be an efficient control knob formanipulating the topological phase transition. Thisoffers a scalable platform for detection and manipulationof Majorana bounds states for development of complexcircuits capable of fault-tolerant topological quantumcomputing. The versatility of this two-dimensionalgeometry and SQUID sensing may also advance stud-ies of MBS using magnetic textures for topologicalsuperconductivity29,30 and support other exotic statesthat can be probed by phase-sensitive signatures31.

Data Availability The data that support the findingsof this study are available within the paper and its Sup-plementary Information. Additional data are availablefrom the corresponding author upon request.

Acknowledgements. This work is supported byDARPA Topological Excitations in Electronics (TEE)program.

Authors contributions. W.M and M.C.D. fabri-cated the devices and performed the measurements withJ.S. providing input. J.Y., K.S.W. and J.S. designed andgrown the epitaxial Al/InAs heterostructures. W.M. andM.C.D. performed data analysis and A.M.-A. developedthe theoretical model and carried out the simulations.J.S. conceived the experiment. All authors contributedto interpreting the data. The manuscript was written byW.M, M.C.D., A. M.-A., I.Z. and J.S. with suggestionsfrom all other authors.

Methods

Growth and fabrication. The Josephson junction(JJ) structure is grown on semi-insulating InP (100) sub-strate. This is followed by a graded buffer layer. Thequantum well consists of a 4 nm layer of InAs grown on a6 nm layer of In0.81Ga0.25As. The InAs layer is capped bya 10 nm In0.81Ga0.25As layer to produce an optimal inter-face while maintaining high two-dimensional electron gas(2DEG) mobility27. This is followed by in situ growthof epitaxial Al (111). Molecular beam epitaxy allowsgrowth of thin Al films where the in-plane critical fieldcan exceed 2 Tesla4. JJs fabricated on the same waferexhibit highly-transparent interface between the super-

5

conducting layer and the 2DEG32. Additional character-ization data are provided in the SI.

Devices are patterned by electron beam lithographyusing PMMA resist. Transene type D is used for wetetching of Al and a III-V wet etch (H2O : C6H8O7 :H3PO4 : H2O2) is used to define deep semiconductormesas. We deposit 50 nm of AlOx using atomic layerdeposition to isolate gate electrodes. Top gate electrodesconsisting of 5 nm Ti and 70 nm Au are deposited byelectron beam deposition.

Measurements. The device has been measured inan Oxford Triton dilution refrigerator fitted with a 6-3-1.5 T vector magnet which has a base temperature of7 mK. All transport measurements are performed usingstandard dc and lock-in techniques at low frequencies andexcitation current Iac = 10 nA. Measurements are takenin a current-biased configuration by measuring R=dV/dIwith Iac, while sweeping Idc. This allows us to find thecritical current at which the junction or SQUID switchesfrom the superconducting to resistive state. It should benoted we directly measure the switching current, whichcan be lower than the critical current due to effects ofnoise. For the purposes of this study we assume they areequivalent.

The two junctions show small variations in normal re-sistance, R1

n = 102 Ω, R2n = 110 Ω and critical current,

I1c = 4.4 µA, I2c = 3.6 µA, measured V 1g = V 2

g = 0 V.Our characterization of JJs in B‖ shows that both havethe same critical magnetic field Bc ≈ 1.45 T, for thin-filmAl, independent of the B‖ direction. In contrast, bothjunctions have strong anisotropy of Ic with B‖ directionas shown in SI. These findings are consistent with theprevious measurements on JJs based on InAs 2DEG5.

Phase extraction. To extract the phase plotted inFig. 3 we fit the SQUID oscillations. To model theSQUID pattern, we sum the contributions of two JJs witha phase difference and maximize the current with respectto the sum of the phases. At low field the current-phaserelation (CPR) of our junction has saw-tooth like profile(see Fig. 3a, at -4V the asymmetry of the critical currentis such that the SQUID oscillation is dominated by the

small current JJ CPR), which can be described as:

I(φ) = Icsinφ√

1− τ sin2 φ/2, (1)

where φ is the phase across the junction and τ is aneffective junction transparency.

We model the envelope as a sinc cardinal function.Since all the measurements have been carried out on asingle device, we can use a single field to phase conver-sion factor to fit all our data and a single Fraunhoferenvelope period. In addition, at a given parallel field, weassume the transparencies of the junctions to be equaland independent of the gate, and we use a single value forthe Fraunhofer field offset and the critical current of thejunction whose gate is kept fix. Finally, only the criticalcurrent and phase of the junction whose gate is tuned arefitted for each trace. The data analysis underlying ourfitting procedure is similar to that from ref. 28.

The phase of the oscillations at each gate voltage rel-ative to the reference is extracted from the fits by com-puting the difference in the phase offsets extracted foreach trace. The phase difference is wrapped in the −π,π range. We can then unwrap it to obtain a monotonicevolution as a function of the gate voltage. This corre-sponds to the result one would obtain by following theorange stars in Fig. 3b. As discussed in ref. 23, the lin-ear contribution of phase can be related to the interplaybetween SOC and Zeeman splitting. We fit the low-fielddata (B < 450 mT) in Fig. 3d and subtract this contri-bution to achieve corrected phase information as shownin Fig. 3e.

Using this fitting procedure provides a systematic ex-traction of the phase. However, we note that similarconclusions could be drawn from simply pin-pointing themaximum of the oscillation at each field and gate. Inparticular, the π-phase jump is directly visible in theraw data when the gate voltage drives the topologicaltransition.

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Supplementary information

I. ADDITIONAL EXPERIMENTAL RESULTS

A. Single Josephson junction

Josephson junctions (JJ) fabricated on epitaxialAl/InAs heterostructures have recently shown highly-transparent contacts with product of IcRn exceeding theparent superconducting gaps. Figure S1(a) shows thematerials stack of our Josephson junctions with epitaxialAl/InAs heterostructure. The separation between super-conducting contacts, L, in this work is 100 nm, coherencelength, ξ0 = ~vF /(π∆) is 770 nm, the dirty coherencelength ξ0,d =

√ξ0le is near 400 nm and the mean free

path, le, is 200 nm in this structures at zero-gate voltage.The condition for short ballistic is le L which cannotbe satisfied here since le is only 200 nm. In general our100 nm devices are quasi-ballistic le > L32.

0

0.4

0.8

1.2

dirty coherence length

L (m

m)

La

InAs (4 nm)

Al

In0.81Ga0.19As (10 nm)

In0.81Al0.19As Si

Al AlOx

Ti/Au

mean free path

b

This work

coherence length

Long

diff

usiv

esh

ort

diffu

sive

shor

t ba

llist

ic

In0.81Ga0.19As (6 nm)

Fig. S1: Single JJ. a, Schematic of material structure, de-vice dimensions and its length, L. b, Regime in which thejunction is expected to be as a function of L. Red triangleshows the JJs used in this work. Coherence length, dirty co-herence length, and mean free path of the sample at zero-gatevoltage are also shown.

B. Operation of the SQUID as a single junction

The device described through the paper is a SQUIDwhose both Josephson junctions (JJ1, JJ2) can be gatedindependently. In Fig. S2, we illustrate how we can tran-sition between a the SQUID regime, in which fast SQUIDoscillations are clearly visible atop the Fraunhofer pat-tern of the junctions, to b single junction regime, inwhich SQUID oscillations are completely absent but wepreserve the Fraunhofer pattern of the junction which isnot depleted.

−1 0 1 2 3Perpendicular field (mT)

0

2

4

0

20

40

Diff

eren

tial r

esis

tanc

e (Ω

)

Bia

s cu

rren

t (µA

)

b

Vg1 = 0 V

Vg2 = -7 V

0

2

4

6

0

20

40

Diff

eren

tial r

esis

tanc

e (Ω

)a

Vg1 = 0 V

Vg2 = 0 V

Bia

s cu

rren

t (µA

)

−1 0 1 2 3Perpendicular field (mT)

Fig. S2: SQUID to single junction transition. a,SQUID oscillations of the device when both JJs are not gatedand in the absence of in-plane field. b, Equivalent scan whenJJ2 is fully depleted by applying V 2

g = −7 V, which reducesto a Fraunhofer pattern. This figure also appears in the maintext of ref.28.

C. Fraunhofer pattern in the presence of a parallelmagnetic field

The application of an in-plane magnetic field leads toa reduction of the critical current of the JJs and a distor-tion of the Fraunhofer pattern as illustrated in Fig. S3.

-1.0 0.0 1.0Perpendicular field (mT)

0

100

200

300

400Parallel Field By 500 mT

0

40

80

120

Diff

eren

tial r

esis

tanc

e (Ω

)

-1.0 0.0 1.0Perpendicular field (mT)

0

100

200

300

400Parallel field Bx 250 mT

0

40

80

120

Diff

eren

tial r

esis

tanc

e (Ω

)

a b

Bia

s cu

rren

t (nA

)

Bia

s cu

rren

t (nA

)Fig. S3: Fraunhofer pattern of JJ1 in the presenceof an in-plane field. a Fraunhofer pattern when applying250 mT along the x-direction i.e. parrallel to the current.b Fraunhofer pattern when applying 500 mT along the y-direction. This figure also appears in the SI of ref.28

The change in the critical current of the JJ appears tostrongly depend on the direction of the applied in-planefield. In Fig. S3, the amplitude of the critical current issimilar in both plots, but the magnitude of the appliedmagnetic field is twice as large in the y-direction a com-pared to the x-direction b. For both directions of thefield, the Fraunhofer pattern appears asymmetric whichis not the case in the absence of the in-plane field, as il-lustrated in the main text. The observed distortions aresimilar for both orientation of the field.

When comparing those data to the ones presented inthe main text, one can notice that the width of the firstnode is about twice as small. We attribute this effect,which is also visible in the SQUID oscillations, to thetransition out of the superconducting state of the indiumlayer at the back of the sample. The transition occursaround 30 mT and does not otherwise impact our study.

8

D. Gap closing at finite magnetic field driven bygate voltage

As illustrated in Fig. 2 of the main text, one can drivethe system from the trivial to the topological state at a fi-nite field by increasing the gate voltage, resulting in bothan increase of the electronic density and in an increaseof spin-orbit coupling (SOC) strength. In Fig. S4, wepresent the superconducting gap closing and reopeningas a function of the gate voltage applied to the junctionin the presence of a parallel field of 750 mT applied alongthe y-axis.

Gat

e vo

ltage

Vg2 (

V)

Diff

eren

tial r

esis

tanc

e (Ω

)

-200 -100 0 100 200

-4

-2

0

2

4

-5.0 -2.5 0.0 2.5 5.0-6

-5

-4

-3

-2

-1

0

Bias current (µA) Bias current (nA)

In-plane field By = 0 mT In-plane field By = 750 mT

0

50

100

150

200a b

Fig. S4: Gate-driven topological transition. a, Mea-sured JJ2 differential resistance as a function the applied gatevoltage in the absence and b, presence of in-plane field of 750mT along the y-direction. The JJ 1 is depleted by applyingV 1g = −7 V.

E. Phase jump across the gap closing: Magneticfield applied at θ = 10

We observe in Fig. 4 that the partial closing of thesuperconducting gap survives up to an angle of θ ∼ 10,away from the y-direction. We present in Fig. S5 themeasured phase jump through the transition. While aphase jump can be observed, its magnitude is reducedcompared to the perfectly aligned (θ = 0) situation.

II. THEORETICAL CALCULATIONS

Theoretical simulations of a single JJ were performedby using the Bogoliubov-de Gennes (BdG) Hamiltonian,

H =

[p2

2m∗− µS +

α

~(pyσx − pxσy) + V0(x)

]τz

− g∗µB2

B · σ + ∆(x)τ+ + ∆∗(x)τ− , (2)

where p is the momentum, µS the chemical potentialin the superconducting (S) region, α is the RashbaSOC strength, B is the external magnetic field, and

In-plane field By (mT)

Pha

se-s

hift

(ref

eren

ce v

alue

Vg2 =

-4 V

)

Vg2 = -3.0 V

Vg2 = -1.0 V

Vg2 = 1.0 V

Vg2 = 3.0 V

0 100 200 300 400 500 600 700In-plane field By (mT)

0

0.5π

π

1.5π

2π

Cor

rect

ed p

hase

-shi

ft

b

a Magnetic field applied along θ = 10 ˚

0 100 200 300 400 500 600 700 800

0

0.5π

π

1.5π

Vg2 = -3.0 V

Vg2 = -1.0 V

Vg2 = 1.0 V

Vg2 = 3.0 V

Fig. S5: Phase jump in the presence of a misalignedfield a, Phase difference between the SQUID oscillation atV 2g = −4V and the oscillation at a different value as a function

of the applied in-plane field along the y direction. The field isapplied with a 10 angle with respect to y-direction. The solidlines are linear fits to the data for By ≤ 450 mT. b, Phaseshift from which the contribution linear in the magnetic fieldhas been subtracted to highlight the phase jump.

m∗ = 0.03 m0 and g∗ = 10 are the electron effec-tive mass and effective g-factor in InAs, respectively.The function V0(x) = (µS − µN )Θ(L/2 − |x|) describesthe changes in the normal (N) region chemical poten-tial (µN ) due to the application of the gate voltage,while ∆(x) = ∆ei sgn(x)φ/2Θ(|x| −W/2) accounts for thespatial dependence of the superconducting gap ampli-tude and the corresponding phase difference (φ). Theτ -matrices are the Nambu matrices in the electron-holespace and τ± = (τx ± τy)/2. The eigenvalue problem forthe BdG Hamiltonian is numerically solved by using afinite-difference scheme on a discretized lattice as imple-mented in Kwant33, with a lattice constant a = 10 nm.The calculated eigenenergies (En) are then used to com-pute the free energy34,35,

F = −2kBT∑En>0

ln

[2 cosh

(En

2kBT

)]. (3)

and the supercurrent,

I(φ) =2e

~dF

dφ. (4)

9

The ground state phase (φGS) is the phase that mini-mizes the free energy and the critical current (Ic) corre-sponds to the maximum of the supercurrent with respectto the phase, i.e. Ic = maxφ I(φ).

The temperature and magnetic-field dependencies ofthe superconducting gap are taken into account by usingthe BCS relation,

∆(T,B) ≈ ∆(T, 0)

√1−

[B

Bc(T )

]2, (5)

where

∆(T, 0) ≈ ∆0 tanh

[1.74

√TcT− 1

]. (6)

Here ∆0 = 1.74kBTc with Tc as the superconductor crit-ical temperature. The temperature dependence of thecritical magnetic field is approximated as,

Bc(T ) = Bc

[1−

(T

Tc

)2], (7)

where Bc is the critical magnetic field at zero tempera-ture.

6004002000

1.6

1.2

0.8

0.4

0.0

B (T

)

E / Δ x (nm)

y (μ

m)

-0.5 0.0 0.5

4

3

2

1

0

1.0

0.8

0.6

0.4

0.2

a b

0.0

Fig. S6: Simulation of Majorana bound states. a, Mag-netic field dependence of the low-energy ground-state spec-trum of a JJ. The energy gap closes and reopens at the fieldvalue for which φGS starts to shift from zero to nearly π (seeFig. 3c in the main text), indicating a topological phase tran-sition and the emergence of Majorana bound states (MBS).The red lines indicate the evolution of finite-energy states intoMBS inside the topological gap. b, Probability density of theMBS in the JJ for B = 0.7 T (see black dashed line in a). Theprobability density, which has been normalized to its maxi-mum value, clearly indicated the formation of MBS localizedat the end of the junction. The green dashed lines indicatethe edges of the normal region.

For the numerical simulations we used µS = µN =0.5 meV, ∆0 = 0.23 meV, α = 10 meV nm, and

Bc = 1.6 T. For the calculation of the critical current,we used periodic boundary conditions along the junctionand assumed kBT = 0.3∆0.

Figure S6a presents the magnetic-field dependence ofthe low-energy ground-state spectrum, i.e. the spectrumcalculated when the phase difference across the junctionequals φGS and the corresponding free energy is mini-mized. At low field, φGS ≈ 0, the JJ is in the topolog-ically trivial states with no MBS. As the field increases,the energy gap closes and reopens at a field of about0.5 T, indicating a topological phase transition in whichfinite-energy states evolve into MBS (red lines) residinginside the topological gap. The topological transition isaccompanied by a shift in the ground-state phase fromzero to a value close to π (see Fig. 3c in the main text).In Fig. S6b, we plot the normalized probability densityof the lowest energy states at B = 0.7 T. The localizationof these zero-energy states at the ends of the junction isa clear indication of the MBS formation. The degree oflocalization depends on both material properties36 anddevice geometry37. The green dashed line marks theboundary between the middle area of the junction whichis not in contact with the superconductor and the outerregions.

The topological transition in our system is the resultof the change of the parity of the number of occupiedstates. When the states are spin degenerate, the sys-tem is topologically trivial because the number of occu-pied states remain even independently of the phase value.The magnetic field breaks the spin degeneracy leading togap closings at certain values of the phase. The systemundergoes a topological phase transition at the gap clos-ings, where the parity of the number of occupied stateschanges. In agreement with previous calculations19, wefind that the gap closing is sensitive to both ϕ and By, asillustrated in Fig. S7. A topological phase transition oc-curs when gap closing regions (yellow lines) are crossed.Thus, by increasing By the system transits from trivialto topological and back to trivial state.

In-plane field By (T)0.0 0.4 0.8 1.2

1.5π

0

0.5π

π

2π

ϕ

1.6

-0.2

-0.4

-0.6Trivial Trivial

Topological

E0/Δ

Fig. S7: Zero-temperature phase diagram in the φ−By

plane. The energy gap closes when the largest energy (E0)in the negative side of the spectrum E(ky = 0) approacheszero (yellow lines).

In the absence of SOC, a Zeeman field (or exchangesplitting) alone can still lead to gap closings accompa-

10

nied by minima of the critical current and 0 − π phasejumps23. However, such signatures do not correspond toa topological phase transition. According to the Altland-Zirnbauer classification38, the topological superconduc-tivity in JJs is related to the symmetry class BDI (if B isparallel to the S/N interfaces) or D (if B is perpendicularto the S/N interfaces). In the absence of SOC the systemhas an additional U(1) symmetry, namely the invarianceunder spin rotations around the direction of the Zeemanfield and belongs to the symmetry class A. Therefore,in the absence of SOC the system is non-topological .Physically, this can be understood by noting that in theabsence of SOC the excitation gap in a spin-polarizedsystem (due to Zeeman interaction or exchange splitting)with s-wave proximity pairing is zero and the spectrumbecomes gapless. Therefore, the 0 − π phase jumps andcritical current minima in JJs, caused by a Zeeman (orexchange) field but zero SOC have a trivial characterand do not imply transition into the topological super-conducting state. In such a case both the phase and crit-ical current are independent of the magnetic field direc-tion. In contrast, our experimental results show a strongdependence on the direction of the magnetic field (seeFig. 4) and cannot be associated with a trivial Zeeman0−π transition at zero SOC. Furthermore, we have clearexperimental evidence of a sizable SOC in our system27.

Considering a JJ with mirror symmetry, the systembelongs to the symmetry class BDI or D in dependenceof whether the magnetic field is parallel or perpendicularto the junction. The dependence of the phase and criticalcurrent on the magnetic field direction originates from achange in the system symmetry class as the magneticfield is rotated in the plane of the JJ.

1.0

0.8

I c/Ic(0

)

0.0

0.2

0.5

0.0 0.4 0.8 1.2 1.6B (T)

θ = 0 ˚θ = 30 ˚θ = 50 ˚θ = 70 ˚θ = 90 ˚

Fig. S8: Magnetic-field dependence of the critical cur-rent for different orientations of the in-plane magneticfield. As the magnetic-field orientation deviate from beingparallel to the junction, the minimum of the critical currentoccurs at larger magnetic field amplitudes and eventually dis-appears, indicating the suppression of the topological phasetransition. This trend is in qualitative agreement with theexperimental observations.

In Fig. S8, we present the numerical results of the im-

pact of the field misalignment on the gap closing. Whenthe magnetic field reaches a critical value of about 1.15 T,as estimated from Eq. (7) at T = 0.3∆0/kB , supercon-ductivity is suppressed, independent of the orientation ofthe in-plane field. Consequently, all the curves in Fig. S8collapse at B ≈ 1.15 T. However, as the magnetic fieldmisalignment increases, the minima of the critical cur-rent occurs at larger magnetic fields and eventually disap-pear. The suppression of the topological phase transitionwhen the magnetic field misalignment is large enough isin qualitative agreement with the experiment. However,the simulation appears far less sensitive to the field mis-alignment than what was experimentally observed sincethe gap reduction persists for an angle θ of 50. We be-lieve this discrepancy originates from normal reflection atthe ends of the S regions, whose size in the theoretical cal-culations differs from the actual experimental sample. Inthe theoretical simulations of the critical current, whichassume a system 600 nm long in the x-direction and in-finite in the y-direction, while the actual experimentaldimensions are about 2 µm along x and 4 µm along y.Numerical simulations of the critical current for such alarge system become extremely challenging.

For the theoretical calculation of the phase-shift asmeasured by the SQUID, the SQUID supercurrent wastaken as:

It = Ic1sin(φ1 − φ0)√

1− τ sin[(φ1 − φ0)/2]2+ I2(φ2). (8)

The first contribution with τ characterizing the junc-tion transparency describes JJ1, which is kept in thetopologically trivial phase, while the second contribution,describing JJ2, is numerically computed as describedabove. The phases φ1 and φ2 corresponding, respectively,to JJ1 and JJ2 are related to the flux piercing the SQUID,φ1 − φ2 = 2πΦ/Φ0, where Φ0 is the flux quantum. Thephase φ0 represents the anomalous contribution linearin the in-plane magnetic field By. By maximizing thetotal current It with respect to φ2 we obtain the fluxdependence of the critical current and the correspond-ing phase-shift and extract the linear contribution. Thetheoretical phase-shift is shown in Fig. 3c.

In addition to material properties, the size and ge-ometry of the junctions play an important role for thecreation of MBS. Since the number of subgap Andreevbound states (ABS) increases with the junction length [Lin Fig. 1(a)], the lowest ABS are pushed down in energy,reducing the topological gap that protects the MBS19,39.The upper limit for the topological gap scales as 1/L2.Therefore, short junctions (as the ones considered herewith L = 100 nm) are desirable for realizing well local-ized and protected MBS.

The junction geometry of our device [see Fig. 1(a)] withLW and magnetic field parallel to the S/N interfaces(i.e., along the y direction) supports the formation oftwo zero-energy MBS localized away from each other, asshown in Fig.S6(b). However, in the wire limit considered

11

in ref. 23, with small W and LW , the two vacuum/Nedges are so close from each other that the MBS fuseinto a finite energy excitation and no MBS is expected

to remain. MBS can still form in nanowire JJs if the Sand N regions are long enough and the magnetic field isalong the wire9,40.

33. Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal,X. Kwant: a software package for quantum transport.New J. Phys. 16, 063065 (2014).

34. Bardeen, J., Kummel, R., Jacobs, A. E. & Tewordt, L.Structure of vortex lines in pure superconductors. Phys.Rev. 187, 556–569 (1969).

35. Beenakker, C. W. J. Three “universal” mesoscopicJosephson effects. In Transport Phenomena in MesoscopicSystems, 235–253 (Springer, 1992).

36. Haim, A. & Stern, A. Benefits of weak disorder in one-dimensional topological superconductors. Phys. Rev. Lett.122, 126801 (2019).

37. Laeven, T., Nijholt, B., Wimmer, M. & Akhmerov, A. R.Enhanced proximity effect in zigzag-shaped Majorana

Josephson junctions. arXiv:1903.06168.38. Altland, A. & Zirnbauer, M. R. Nonstandard symme-

try classes in mesoscopic normal-superconducting hybridstructures. Phys. Rev. B 55, 1142–1161 (1997).

39. Scharf, B., Pientka, F., Ren, H., Yacoby, A. & Han-kiewicz, E. M. Tuning topological superconductivity inphase-controlled Josephson junctions with Rashba andDresselhaus spin-orbit coupling. Phys. Rev. B 99, 214503(2019).

40. San-Jose, P., Prada, E. & Aguado, R. ac Josephson effectin finite-length nanowire junctions with Majorana modes.Phys. Rev. Lett. 108, 257001 (2012).