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A two-site Kitaev chain in a two-dimensional electron gas


Artificial Kitaev chains can be used to engineer Majorana bound states (MBSs) in superconductor–semiconductor hybrids1,2,3,4. In this work, we realize a two-site Kitaev chain in a two-dimensional electron gas by coupling two quantum dots through a region proximitized by a superconductor. We demonstrate systematic control over inter-dot couplings through in-plane rotations of the magnetic field and via electrostatic gating of the proximitized region. This allows us to tune the system to sweet spots in parameter space, where robust correlated zero-bias conductance peaks are observed in tunnelling spectroscopy. To study the extent of hybridization between localized MBSs, we probe the evolution of the energy spectrum with magnetic field and estimate the Majorana polarization, an important metric for Majorana-based qubits5,6. The implementation of a Kitaev chain on a scalable and flexible two-dimensional platform provides a realistic path towards more advanced experiments that require manipulation and readout of multiple MBSs.

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Fig. 1: Device, model and CSDs.
Fig. 2: Tuning Γe and Γo with magnetic field angle.
Fig. 3: Electrostatically tuning to the Majorana sweet spot.
Fig. 4: Majorana sweet spots in varying magnetic field.

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Data availability

Raw data and analysis scripts for all presented figures are available via Zenodo at (ref. 37).


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We thank M. Leijnse, G. Katsaros, G. Wang, N. van Loo, A. Bordin and F. Zatelli for valuable discussions and for providing comments on the manuscript. The experimental research at Delft was supported by the Dutch National Science Foundation (NWO) and a TKI grant of the Dutch Topsectoren Program. A.M.B. acknowledges NWO (HOTNANO) for their research funding. C.-X.L. acknowledges subsidy by the Top consortium for Knowledge and Innovation programme (TKI). S.G. and M.W. acknowledge financial support from the Horizon Europe Framework Program of the European Commission through the European Innovation Council Pathfinder grant no. 101115315 (QuKiT).

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Authors and Affiliations



S.L.D.t.H. fabricated the devices. Q.W. and I.K. contributed to the device design and optimization of fabrication flow. Measurements were performed by S.L.D.t.H., Q.W. and P.K. Numerical analysis was provided by S.L.D.t.H., A.M.B. and C.-X.L, under the supervision of M.W. MBE growth of the semiconductor heterostructures and the characterization of the materials was performed by D.X. and C.T. under the supervision of M.J.M. The manuscript was written by S.L.D.t.H., Q.W. and S.G., with inputs from all coauthors. T.D. and S.G. supervised the experimental work in Delft.

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Correspondence to Srijit Goswami.

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Extended data figures and tables

Extended Data Fig. 1 Characterization of QDs and ABS spectroscopy for device A.

a–b, Coulomb diamond measurements of the left and right QDs, in the regime used for measurements presented in Fig. 4. Charging energies are estimated to be 1.1 meV in each QD. The lever arm of the left and right dot is estimated to be α ≈ 0.11. c–d, Zoomed in views of figures (a) and (b) in a smaller energy range. Due to a large tunnel coupling between the QDs and the hybrid section, YSR-states form in the sub-gap spectrum of the QDs. Tunneling spectroscopy around the charge degeneracy points in (a-b) reveal clear sup-gap features within the Coulomb diamonds.As VQDL and VQDR are tuned, the sub-gap features form an eye-shape feature enclosing the doublet charge occupation. This behavior is typical for YSR-states with large charging energies28. e, Crossed Andreev reflection and elastic co-tunneling require the presence of extended ABSs. Local Gl and non-local conductance Gnl of the hybrid region are measured via tunnelling spectroscopy and their identical energy dependence as a function of VABS highlights that ABSs extend across the entire hybrid section. Comparable behavior was observed in a wide VABS range from 0 to − 1 V. The measurement presented in Fig. 1 is taken at the VABS with the eye-shaped crossing. f, ABS spectroscopy as a function external magnetic field at VABS = −623 mV. The effect of splitting of the doublet state can be observed at low fields. A g-factor of 5.5 is extracted by linear fitting of the lowest sub-gap states (dashed line) in Extended Data Fig. 1d.

Extended Data Fig. 2 Characterization of device B.

a, Scanning electron micro-graph of device B, used to obtain the measurements presented in Fig. 2. Scale bar shown is 100 nm. b–c, Coulomb diamond measurements of the left and right QDs. Charging energies are extracted to be 1.4 meV. The lever arm αN is extracted to be about 0.11 for each QD. This lever arm is used for the extraction in Fig. 2d. d–e, To validate the direction of BSO and to show the connection between interactions in strongly coupled QDs and underlying ECT and CAR processes, we first measure ECT and CAR currents in the weakly coupled dots, as detailed in29. With BBSO, measurements of CAR (d) and ECT (e) show the typical blockades for same-spin and opposite-spin charge configurations respectively. In f–g, with BBSO, the spin non-conserving ECT and CAR processes are observed to be revived. h, Measuring CAR and ECT rates as a function of magnetic field angle θ shows the currents for the spin non-conserving processes are indeed smoothly controlled and become suppressed when θ = 0. This supports the interpretation that BBSO when B is perpendicular to the 1-D channel. i–j, Next, the QDs are operated with higher tunnelling rates between the QDs and the SC, to enable strong couplings. Similar to Fig. 1e-f, CSDs are obtained in the strongly interacting regime, taken with the verified BBSO and BBSO respectively.

Extended Data Fig. 3 Extended dataset of Fig. 2.

a, CSDs measured at various magnetic field angles θ between 0° and 90°, used to extract the data shown in Fig. 2d. b, Example of the extraction process. For each obtained CSD, VQDL and VQDR are converted to energies μL and μR using lever arms obtained in Extended Data Fig. 2. Next, the conductance is extracted along a μL = − μR or μL = μR line. c, Two Gaussian peaks are fitted to extract the separation between the two avoided crossings, from which the quantity \(\sqrt{| {\varGamma }_{{\rm{o}}}-{\varGamma }_{{\rm{e}}}| }\) is obtained (plotted in Fig. 2d).

Extended Data Fig. 4 Characterisation of YSR-states in QDs of Fig. 3.

To complement the data in Fig. 3, Extended Data Fig. 5 and Extended Data Fig. 10, we measure the sub-gap states in QDL and QDR (see Extended Data Fig. 1). Using this, we obtain the lever arms of VQDL and VQDR on the YSR-state energies (denoted αYSR) (see Methods). a–b, Sub-gap spectroscopy of QDL and QDR at Bz = 0 mT. From the slopes of the states upon crossing VL, VR = 0, we estimate αYSR ≈ 0.045. Applying an external magnetic field lowers the energy of ABSs in the hybrid region, as a result of Zeeman splitting. This in turn will affect the YSR-spectrum of the QDs, due to increased hybridization between the QDs and the ABS. c–d, Measuring sub-gap spectroscopy of QDL and QDR at Bz = 225 mT for the same settings as in (a–b) shows indeed the effective lever arm here decreases to αYSR ≈ 0.028.

Extended Data Fig. 5 Full conductance spectra at the sweet spot upon detuning VQDL and VQDR.

A comparison between numerically calculated conductance and measured conductance in support of Fig. 3, measured at B = 225 mT. Presented results show the evolution of Gl and Gnl for four different cases: (a–b) detuning VQDL, (c–d) detuning VQDR, (e–f) detuning both VQDL and VQDR simultaneously along a diagonal path and (g–h) detuning both anti-diagonally. For each case, we find the behavior of both Gl and Gnl is well described by the numerical results.

Extended Data Fig. 6 Extended dataset for Fig. 3.

a, Sets of CSDs obtained while varying VABS in the range presented in Fig. 3. The range of VQDL and VQDR is constant for each measurement. The slight drift of the avoided crossing upon varying VABS is owed to cross-capacitance between VABS and the potential of the QDs.

Extended Data Fig. 7 Energy diagrams detailing non-local transport.

In CSDs presented in Fig. 3, a clear sign change is observed when changing from the ΓO > ΓE regime to the ΓO < ΓE regime. This can be understood by considering the possible transport cycles that underlie the measured non-local conductance. a, When ΓO > ΓE, Gnl is observed to be negative in the measured CSDs (see Fig. 3c). c, When ΓO < ΓE, the same measurements yield a positive Gnl (see Fig. 3a). Horizontal and vertical dashed lines indicate μR = 0 and μL = 0 respectively. The state of the uncoupled system is labelled in each quadrant. b,d, In such CSD measurements, zero-bias transport can take place when the odd and even ground states are degenerate. For non-local transport to occur, the system can accept a hole/electron from one lead, and relax non-locally to its original state by either (b) donating a hole/electron to the opposite lead, giving rise to negative Gnl, or (d) accept a hole/electron from the opposite lead, giving rise to positive Gnl. The preferred path is dictated by the quadrant in μL, μR space where the odd-even degeneracy occurs. e, When μL, μR > 0 or μL, μR < 0, the former path is expected to dominate and the resulting Gnl will be negative. f, When μL > 0 and μR < 0 or vice versa, the latter path is expected to dominate and resulting Gnl will be positive.

Extended Data Fig. 8 Numerical analysis of Egap and Edet.

Numerical calculations supporting the results presented in Fig. 4. Through the procedure detailed in Methods, Majorana sweet spots are obtained and analysed for fields between 0 mT and 300 mT. a, Field evolution of GRR line-traces at each sweet spot, showing the excitations above the ZBPs gradually increasing in energy and then saturating at ± 30 μeV. b, Field evolution of GRR line-traces when QDR detuned by 3Δind, showing the excited states increase linearly in energy. From calculations in (a) and (b), Egap and \({E}_{\det }\) are obtained, given by the energy between the lowest even-parity state and second-lowest odd-parity state. c, Extraction of Egap (solid) and the Majorana polarization (dashed), for different values of the tunneling parameter t. In each case tso = 0.4t. Larger tunnel coupling results in larger hybridization between ABSs, in turn lowering the MP at a specific magnetic field. d, Extraction of \({E}_{\det }\) for various values of detuning μR. In each case the slope at low fields corresponds to 2Ez (dashed grey line). The larger the detuning of μR, the longer this holds. The dashed black line shows the energy of the ABS EABS. At large detuning \({E}_{\det }\) will increase linearly with 2Ez, until becoming of comparable EABS becomes the lowest energy scale.

Extended Data Fig. 9 Raw datasets for Fig. 4h.

Obtained ‘sweet spots’ at magnetic fields between 0 mT and 300 mT. a–i, CSDs and tunnelling spectroscopy are measured at each sweet spot, where VQDR is detuned. From these measurements \({E}_{\det }\) and Egap are extracted, as described in the main text and in Methods.

Extended Data Fig. 10 Extended datasets supporting Fig. 4h.

Reproduction of the main results from Fig. 4, using the orbitals shown in Fig. 3. Data was obtained at 6 different field values B between 0 and 250 mT. At each field VABS is adjusted to tune to the sweet spot. a, Extraction of \({E}_{\det }\) and Egap, similar to the analysis presented in Fig. 4l. From a linear for of \({E}_{\det }\) a g-factor of 5.7 is estimated. b, Numerically obtained \({E}_{\det }\) and Egap, using parameters tuned to compare to (a). At 250 mT, an estimate of M ≈ 0.9 is obtained. Extrapolation for comparison to Fig. 4l yields M ≈ 0.92 at 300 mT. c,d, Waterfall plots highlighting the line-traces used to extract the data in (a). e–j, Raw datasets of CSDs and tunnelling spectroscopy measurements, from which (a-d) is extracted. Datasets at 150 mT and 225 mT datasets are repeated from Fig. 3 and Extended Data Fig. 5 respectively.

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ten Haaf, S.L.D., Wang, Q., Bozkurt, A.M. et al. A two-site Kitaev chain in a two-dimensional electron gas. Nature 630, 329–334 (2024).

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