Abstract
Majorana zero modes—quasiparticle states localized at the boundaries of topological superconductors—are expected to be ideal building blocks for faulttolerant quantum computing^{1,2}. Several observations of zerobias conductance peaks measured by tunnelling spectroscopy above a critical magnetic field have been reported as experimental indications of Majorana zero modes in superconductor–semiconductor nanowires^{3,4,5,6,7,8}. On the other hand, twodimensional systems offer the alternative approach of confining Majorana channels within planar Josephson junctions, in which the phase difference φ between the superconducting leads represents an additional tuning knob that is predicted to drive the system into the topological phase at lower magnetic fields than for a system without phase bias^{9,10}. Here we report the observation of phasedependent zerobias conductance peaks measured by tunnelling spectroscopy at the end of Josephson junctions realized on a heterostructure consisting of aluminium on indium arsenide. Biasing the junction to φ ≈ π reduces the critical field at which the zerobias peak appears, with respect to φ = 0. The phase and magneticfield dependence of the zeroenergy states is consistent with a model of Majorana zero modes in finitesize Josephson junctions. As well as providing experimental evidence of phasetuned topological superconductivity, our devices are compatible with superconducting quantum electrodynamics architectures^{11} and are scalable to the complex geometries needed for topological quantum computing^{9,12,13}.
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The data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
This work was supported by Microsoft Corporation, the Danish National Research Foundation, the Villum Foundation, ERC Project MUNATOP, CRC183 and the Israeli Science Foundation. We thank E. O’Farrell, M. Hell, K. Flensberg and J. Folk for discussions.
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Nature thanks Kaveh Delfanazari and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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Contributions
S.G., C.T., T.W., R.K., G.C.G. and M.J.M. developed and grew the InAs/Al heterostructure. A.F., A.M.W. and A.C.C.D. fabricated the devices. A.F., A.M.W. and A.C.C.D. performed the measurements with input from C.M.M. and F.N. Data analysis was done by A.F., A.M.W. and E.P.M. F.S., A.K., E.B. and A.S. developed the theoretical model and carried out the simulations. C.M.M. and F.N. conceived the experiment. All authors contributed to interpreting the data. The manuscript was written by A.F., A.M.W., C.M.M. and F.N. with suggestions from all other authors.
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Correspondence to Charles M. Marcus or Fabrizio Nichele.
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Extended data figures and tables
Extended Data Fig. 1 Calculated spectra for a symmetric device.
a, Topological phase diagram as a function of the Zeeman energy \({E}_{{\rm{Z}}}\) and the 2DEG chemical potential µ for two values of the phase bias \(\phi \) = 0, π, calculated from the tightbinding Hamiltonian for JJ1 with infinite length (see Methods) and symmetric superconducting leads. The curves indicate the critical value of \({E}_{{\rm{Z}}}\) above which the system is tuned into the topological phase. b, Topological phase diagram as a function of \({E}_{{\rm{Z}}}\) and \(\phi \) for different values of µ, as indicated by the horizontal ticks in a. The diagrams were calculated for a junction with width W_{1} = 80 nm, superconducting lead width W_{S1} = 160 nm, induced gap ∆_{L,R} = 150 µeV and Rashba spin–orbit coupling constant α = 100 meV Å. The length of the junction L_{1} was assumed to be infinite in order to obtain a welldefined topological invariant, as described in the Methods. c–g, Calculated energy spectra as a function of \(\phi \) for different values of the Zeeman energy. The spectra were obtained for the same parameters used in a and b, except for L_{1} = 1.6 µm. For this left–right symmetric junction, the topological transition can occur at \({E}_{{\rm{Z}}}\) = 0 for \(\phi \) = π and specific values of chemical potential. We calculate the spectra and Majorana wavefunctions at this finetuned chemical potential µ = 79.25 meV (corresponding to the purple curve in b). For the chosen parameters, the system undergoes a topological transition at \({E}_{{\rm{Z}}}\) = 0 for \(\phi \) = π and at \({E}_{{\rm{Z}}}\) = 0.12 meV for \(\phi \) = 0. The lowestenergy subgap states are shown in red and indicate two Majorana zero modes at the edges of the junction in the topological regime. As a function of \({E}_{{\rm{Z}}}\) these states first reach zero energy at \(\phi \) = π and progressively extend in phase. At high values of \({E}_{{\rm{Z}}}\), the Majorana modes oscillate around zero energy owing to their hybridization, caused by the finite size of our system. This is particularly evident at \(\phi \) = π, where the induced gap is minimized and the coherence length is maximized. h, i, Probability density \({\left\Psi \right}^{2}\) of the Majorana wavefunction calculated as a function of the spatial directions x and y in JJ1 for \({E}_{{\rm{Z}}}\) = 0.13 meV and \(\phi \) = 0, π. The x coordinate extends in the width direction including the superconducting leads (\(2{W}_{{\rm{S}}1}+{W}_{1}\)) = 0.4 µm, with x = 0 indicating the centre of the junction, while y is the coordinate along the length of the junction. The Majorana wavefunctions are localized in the y direction at the edges of the junction when the lowest energy states in the spectrum are close to zero energy. In the x direction, the Majorana modes are delocalized below the superconducting leads, owing to our geometry having \({W}_{{\rm{S}}1}\ll {\xi }_{{\rm{S}}}\).
Extended Data Fig. 2 Nonsymmetric device spectra.
Calculated topological phase diagrams and energy spectra for a left–right asymmetric junction (here the asymmetry is introduced by having ∆_{L} \(\ne \)∆_{R}). As explained in the Methods, the left–right symmetry may be broken by disorder^{27,28}, different geometric sizes of the superconducting leads, or different coupling of the 2DEG to the superconductors on the two sides of the junctions. a, Topological phase diagram as a function of \({E}_{{\rm{Z}}}\) and µ for \(\phi \) = 0, π, calculated from the tightbinding Hamiltonian for JJ1 with infinite length (see Methods). b, Topological phase diagram as a function of \({E}_{{\rm{Z}}}\) and \(\phi \) for different values of µ, as indicated by the horizontal ticks in a. The diagrams were calculated for a junction with W_{1} = 80 nm, W_{S1} = 160 nm, leftinduced gap ∆_{L} = 150 µeV, rightinduced gap ∆_{R} = 100 µeV and α = 100 meV Å. c–g, Calculated energy spectra as a function of \(\phi \) for different values of the Zeeman energy. The spectra were obtained for the same parameters used in a and b, except for L_{1} = 1.6 µm. Note that the gap for the finetuned chemical potential µ = 79.25 meV, which closes at \(\phi \) = π and \({E}_{{\rm{Z}}}\) = 0 for a left–right symmetric junction (see Extended Data Fig. 1a, c), now becomes nonzero, and is approximately \( {{\Delta }}_{{\rm{L}}}{{\Delta }}_{{\rm{R}}}\). As a result, the topological transition for \(\phi \) = π occurs at finite Zeeman field. For the chosen parameters, the system undergoes a topological transition at \({E}_{{\rm{Z}}}\) = 0.02 meV for \(\phi \) = π and at \({E}_{{\rm{Z}}}\) = 0.1 meV for \(\phi \) = 0. The lowest subgap states are shown in red and indicate two Majorana zero modes at the edges of the junction in the topological regime. The behaviour of the calculated Majorana modes is qualitatively consistent with that of the observed zerobias peaks in tunnelling conductance. h, i, Probability density \({\left\Psi \right}^{2}\) of the Majorana wavefunction calculated as a function of the spatial directions x and y in JJ1 for \({E}_{{\rm{Z}}}\) = 0.13 meV and \(\phi \) = 0, π.
Extended Data Fig. 3 Transport spectroscopy in transverse field for device 1.
a–h, Differential conductance G as a function of the magnetic flux Φ threading the SQUID loop and source–drain bias, V_{sd}, measured at different values of the transverse magnetic field B_{t} (applied in plane orthogonally to the junction) in device 1. Several ABSs enter the gap without sticking to zero energy. The induced gap collapses at B_{t} ≈ 360 mT.
Extended Data Fig. 4 Quantum point contact characterization and stability of the zerobias peak.
a, G as a function of V_{sd} and QPC voltage V_{qpc} at zero field in device 1. b, Differential conductance at zero source–drain bias, G(V_{sd} = 0 mV), versus averaged differential conductance at finite source–drain bias, G(V_{sd} > 0.4 mV). The green line is the theoretically predicted conductance in an Andreevenhanced QPC, \({G}_{{\rm{S}}}=2{G}_{0}\frac{{G}_{{\rm{N}}}^{2}}{{\left(2{G}_{0}{G}_{{\rm{N}}}\right)}^{2}}\) (ref. ^{38}), where \({G}_{{\rm{S}}}\) is the subgap conductance, \({G}_{{\rm{N}}}\) is the abovegap conductance and \({G}_{0}=2{e}^{2}/h\) is the quantum of conductance. No fitting parameters have been used. c, G as a function of V_{sd} and V_{qpc} at parallel field B_{} = 780 mT and \(\phi \) ≈ 0.8π for gate voltages V_{1} = −110 mV and V_{top} = −35 mV. d, G as a function of V_{sd} and V_{top} at B_{} = 600 mT and \(\phi \) ≈ 0 for V_{1} = −118.5 mV and V_{qpc} = −2.366 mV. In both c and d, the ZBP is robust against variation of the abovegap conductance of about one order of magnitude. e, f, G as a function of V_{sd} and B_{}, for different values of \(\phi \) in device 1. The plots have been reconstructed from measurements similar to those shown in Fig. 2 of the main text. For \(\phi \) ≈ π, a ZBP forms at B_{} = 0.35 T, whereas for \(\phi \) = 0 it appears at B_{} = 575 mT. The ZBP at \(\phi \) ≈ π oscillates and moves away from zero energy as the field is increased.
Extended Data Fig. 5 Tunnelling spectroscopy at lower tunnelling transmission in devices 1 and 2.
a–j, Results obtained for device 1. k–t, Results obtained for device 2. The devices were intended to be lithographically identical. a, c, e, g, i, G as a function of Φ threading the SQUID loop and V_{sd} measured at different values of B_{} in device 1 (W_{1} = 80 nm). The QPC was tuned to reduce the abovegap conductance by a factor of about 3 with respect to the one measured in the regime presented in the main text. At zero field, the subgap conductance is suppressed. Colour extrema have been saturated. b, d, f, h, j, Conductance line cuts versus V_{sd} taken at \(\phi \) = 0, π, as indicated by red and black ticks in a, c, e, g, i. The grey dashed lines indicate V_{sd} = 0. k, m, o, q, s, G as a function of Φ threading the SQUID loop and V_{sd} measured at different values of B_{} in device 2 (W_{1} = 80 nm). Colour extrema have been saturated. l, n, p, r, t, Conductance line cuts versus V_{sd} taken at phase bias \(\phi \) = 0, π, as indicated by red and black ticks in k, m, o, q, s. The grey dashed lines indicate V_{sd} = 0.
Extended Data Fig. 6 Measurement of the third harmonic of the current.
a, G as a function of V_{sd} and Φ at B_{} = 850 mT measured with an excitation amplitude V_{ac} = 3 µV in device 1. b, Numerical second derivative of the conductance \(G^{\prime\prime} \left({V}_{{\rm{sd}}}\right){\left.=\left({\partial }^{2}G/\partial {V}^{2}\right)\right}_{{V}_{{\rm{sd}}}}\) as a function of V_{sd} and Φ calculated from the data shown in a. c, Third harmonic of the current \({I}_{3\omega }\) versus V_{sd} and Φ measured by the lockin amplifier using an excitation V_{ac} = 15 µV, as explained in the Methods. To increase the signaltonoise ratio, the amplitude of the excitation has been chosen to be greater than the temperaturelimited fullwidth at halfmaximum of a Lorentzian feature, that is, V_{ac} ≈ 3.5 k_{B}T, where \({k}_{{\rm{B}}}\) is the Boltzmann constant and T ≈ 40 mK is the electron temperature in our devices. Most of the features present in b are reproduced in c. d, e, Line cuts of G as a function of V_{sd} taken at \(\phi \) = 0, π as indicated by the ticks in a. f, \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\) as a function of Φ: a positive value of \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\) indicates a ZBP in G. See Methods for further details.
Extended Data Fig. 7 Zerobias peak stability in device 2 (W_{1} = 80 nm).
a, G as a function of Φ threading the SQUID loop and V_{sd} measured at different values of B_{} in device 2 at V_{1} = −191 mV. b, Third harmonic of the current \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\) measured by the lockin amplifier at zero bias as a function of V_{1} and Φ for different values of B_{}. \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\propto G^{\prime\prime} \left({V}_{{\rm{sd}}}=0\right)\) \(={\left.\left({\partial }^{2}G/\partial {V}^{2}\right)\right}_{{V}_{{\rm{sd}}}=0}\), as shown in the Methods and Extended Data Fig. 6. A positive value of \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\) corresponds to a ZBP in conductance as a function of V_{sd}. As B_{} is increased, the ZBP expands in phase and in V_{1} range, consistent to what is observed in Fig. 3 of the main text for device 1. c, \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\) measured by the lockin amplifier at zero bias as a function of B_{} and Φ for different values of V_{1}. A positive value of \({I}_{3\omega }\left({V}_{{\rm{sd}}}=0\right)\) corresponds to a ZBP in conductance as a function of V_{sd}. The critical field at which the ZBP first appear is minimized at φ ≈ π. The behaviour of the ZBP is tuned by V_{1} and is qualitatively consistent with the topological phase diagrams shown in Extended Data Fig. 2a, b.
Extended Data Fig. 8 Tunnelling spectroscopy in device 3 (W_{1} = 120 nm).
a–h, G as a function of Φ and V_{sd} measured at different values of B_{} in device 3 (W_{1} = 120 nm). In this device, spectroscopy was performed with a QPC forming a tunnel barrier between the top edge of JJ1 and a wide planar Al lead, following the approach of refs ^{6,7}. At B_{} = 0, the superconducting probe generates a fluxindependent gap \({{\Delta }}_{{\rm{p}}{\rm{r}}{\rm{o}}{\rm{b}}{\rm{e}}}^{\ast }\) ≈ 200 µeV added to the junction gap ∆ ≈ 100 µeV, together with the characteristic features of negative differential conductance, as visible in a. When a moderate parallel field is applied, the superconducting gap below the superconducting plane softens, creating a finite density of states at zero energy. This feature allows the Al plane to be used as an effective normal lead that can probe discrete states close to zero energy in the junction. At B_{} = 250 mT, we can see a complete phase modulation of ∆, indicating that the Al plane gap is already soft (see b). As the field is increased, two ABSs move towards zero energy, forming a ZBP first localized at phase bias φ ≈ π and then extending up to φ = 0, as shown in c–f. At higher fields, the induced gap collapses (g, h). The lower value of ∆ and critical field compared to that observed in devices 1 and 2 are presumably due to the larger width of the junction.
Extended Data Fig. 9 Behaviour of the Josephson critical current at B_{} = 0.
a, To investigate the behaviour of the Josephson current in our device, we measured the differential resistance R = dV/dI of the SQUID with a conventional fourprobe technique by applying an a.c. bias I_{ac} < 5 nA, superimposed on a variable d.c. bias I_{dc}, to one of the superconducting leads of the interferometer. During these measurements the QPC was pinched off at V_{qpc} = −5 V. The Josephson critical current of JJ1 can be measured independently by pinching off JJ2. b, Differential resistance R_{1} of JJ1 as a function of the I_{dc} and V_{1} measured in device 2. The region of zero resistance indicates that a dissipationless Josephson current due to Cooper pair transport is flowing through the junction. c, R_{1} as a function of I_{dc} and the outofplane field B_{⊥} displaying a characteristic Fraunhofer pattern, with a periodicity compatible with the area of JJ1 \({W}_{1}\times {L}_{1}\) ≈ 0.13 µm^{2}. For both the measurements displayed in a and c, JJ2 was pinched off by setting the gate voltage V_{2} = −1.5 V. d, When JJ2 was open (V_{2} = 0), the differential resistance of the SQUID R_{SQUID} showed periodic oscillations (periodicity of 250 µT, consistent with the area of the superconducting loop, ∼8 µm^{2}) superimposed to the Fraunhofer patterns of both junctions. The ratio between the critical currents of the junctions at zero field is extracted from the average value of the SQUID critical current and the semiamplitude of the SQUID oscillations, resulting in I_{c,2}(0)/I_{c,1}(0) = 5.2.
Extended Data Fig. 10 Josephson current revival in parallel field.
a–c, Differential resistance R_{1} of JJ1 as a function of B_{⊥} and B_{} measured in devices 2, 6 and 7 for I_{dc} = 0 and I_{ac} = 5 nA. All the devices are characterized by width W_{1} = 80 nm and length L_{1} = 1.6 µm, while the width of the superconducting leads W_{S1} (see Fig. 1a) is varied. Device 2 is characterized by W_{S1} = 160 nm, device 6 has W_{S1} = 500 nm and device 7 W_{S1} = 1 µm. In the case of W_{S1} = 160 nm (a), JJ1 becomes resistive at B_{} ≈ 1.1 T, and a supercurrent revival is observed above 2 T. The normal state transition of the epitaxial Al occurs at B_{} ≈ 2.4 T, where the junction resistance reaches a value of about 1 kΩ without any magneticfield dependence. When W_{S1} is increased, the supercurrent revivals occur at lower values of B_{} and show an evident periodicity of about 300 mT and about 150 mT for W_{S1} = 500 nm (b) and W_{S1} = 1 µm (c), respectively. d, e, R_{1} as a function of I_{dc} and B_{} measured in devices 6 and 7 for B_{⊥} = 0, as shown by the dashed lines in b and c. The Josephson current shows a clear Fraunhofer pattern due to orbital effects of the inplane field penetrating the proximitized 2DEG below the Al leads^{10}. The measurements were performed with the QPC and junction 2 pinched off.
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