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Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires

Abstract

In most naturally occurring superconductors, electrons with opposite spins form Cooper pairs. This includes both conventional s-wave superconductors such as aluminium, as well as high-transition-temperature, d-wave superconductors. Materials with intrinsic p-wave superconductivity, hosting Cooper pairs made of equal-spin electrons, have not been conclusively identified, nor synthesized, despite promising progress1,2,3. Instead, engineered platforms where s-wave superconductors are brought into contact with magnetic materials have shown convincing signatures of equal-spin pairing4,5,6. Here we directly measure equal-spin pairing between spin-polarized quantum dots. This pairing is proximity-induced from an s-wave superconductor into a semiconducting nanowire with strong spin–orbit interaction. We demonstrate such pairing by showing that breaking a Cooper pair can result in two electrons with equal spin polarization. Our results demonstrate controllable detection of singlet and triplet pairing between the quantum dots. Achieving such triplet pairing in a sequence of quantum dots will be required for realizing an artificial Kitaev chain7,8,9.

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Fig. 1: Transport processes, device and energy diagrams.
Fig. 2: CAR and ECT.
Fig. 3: Spin blockade of CAR and ECT at zero magnetic field.
Fig. 4: Spin-resolved CAR and ECT.

Data availability

Raw data presented in this work and the processing/plotting codes are available at https://zenodo.org/record/5774828.

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Acknowledgements

This work has been supported by the Dutch Organization for Scientific Research (NWO) and Microsoft Corporation Station Q. We also acknowledge a subsidy from Top Consortia for Knowledge and Innovation (TKl toeslag) and support from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 828948, project AndQC. We thank G. de Lange, S. Bergeret, V. Golovach, J. Robinson, M. Aprili, C. Quay, D. Loss and J. Klinovaja for discussions.

Author information

Authors and Affiliations

Authors

Contributions

G.W., G.P.M., N.v.L. and A.B. fabricated the devices. G.W., T.D., S.L.D.t.H. and A.B. performed the electrical measurements. G.W. and T.D. designed the experiment and analysed the data. G.W., T.D. and L.P.K. prepared the manuscript with input from all authors. T.D. and L.P.K. supervised the project. C.-X.L. developed the theoretical model and performed numerical simulation with input from M.W. S.G., G.B. and E.P.A.M.B. performed InSb nanowire growth.

Corresponding author

Correspondence to Tom Dvir.

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Extended data figures and tables

Extended Data Fig. 1 QD characterization in device A.

a, Coulomb blockade diamonds of the left QD. Superimposed dashed lines represent a model with charging energy 2.1 meV, Δ = 250 μV and lever arm 0.4. VLD shown here is different from other measurements of this resonance due to a drift in one tunnel barrier gate during the process of the experiment. b, Coulomb blockade diamonds of the right QD. Superimposed dashed lines represent a model with charging energy 2.75 meV, Δ = 250 μV and lever arm 0.435. In both QDs, no subgap current is visible, indicating QDs are weakly coupled to S and retain their charge states. c, Current through the left QD at VL = 500 μV measured against gate voltage and magnetic field along the nanowire, Bx. Spin-degenerate orbitals Zeeman-split in opposite directions while 0 < Bx < 0.5 T and cross around 0.5 T when Zeeman energy becomes greater than the level spacing ~1.2 meV (see Extended Data Fig. 3 for g-factor extraction). The orbital used in Figs. 3, 4 is the pair of resonances marked by grey dashed lines at B = 100 mT. d, Current through the right QD at VR = 500 μV. The orbital used in Figs. 34 is outside the measured range in this plot immediately to the right. All QD resonances we investigated behave similarly including those in Fig. 2, which are selected because of high data resolution and Cooper pair splitting efficiency.

Extended Data Fig. 2 More analysis on data presented in Fig. 2, including Cooper pair splitting efficiency extraction at B = 0.

a,b, ECT IL, IR and averaged currents. Top panel shows signals between the horizontal grey lines averaged over VRD. Right panel shows signals between the vertical grey lines averaged over VLD. Almost no background current is visible unless both dots participate in transport. c, IL + IR of the ECT measurement is almost 0, verifying IL = IR in most of the phase space except when both QDs are at zero energy and charge selection no longer plays a role. d,e, CAR IL, IR and averaged currents, similar to a,b. Using ηL ≡ (1 − IL,BG/IL,max) where the background IL,BG is taken as the average current when VRD is off-resonance and VLD is on-resonance in the right panel of d, we obtain a Cooper pair splitting visibility of 91.3% for the left junction. Similarly, the right junction has splitting visibility ηR = 98%. This gives combined visibility ηLηR = 89.5%. f, IL − IR of the CAR feature is almost 0, verifying IL = IR except a small amount of local Andreev current in the left QD, manifesting as a vertical feature independent of VRD near VLD = 210.8 mV.

Extended Data Fig. 3 QD excitation spectra measured using methods previously described53, from which we extract QD g-factor, level spacing and SOC.

a,b, Left QD excitation spectra evolving under B applied along y (a) and x (b) for the spin-up ground state. Grey lines mark the field value at which the data in the main text are taken. The observation that opposite-spin excited states cross each other in a means spin is conserved, implying BSO in the QD and B point along the same direction, that is, y. Opposite-spin states in b, by contrast, anti-cross due to SOC. The quantities needed to calculate the opposite-spin admixture weight (level spacing δ, Zeeman splitting EZ and spin–orbit level repulsion gap 2〈HSO〉 can be directly read from b (see Supplementary Information for details). b shows the largest value of spin–orbit level repulsion that we have measured in the QDs, which is used as an upper-bound estimation for the effect of SOC in QD in Supplementary Information. The Zeeman-splitting slopes yield g = 45, that is, Zeeman energy gμBB = 260 μeV at B = 100 mT. c,d, Left QD excitation spectra under B along y and x for the spin-down ground state. The g-factor and level spacing are similar to those in a,b (as seen in data above 0.3 T) but the spin–orbit level repulsion is smaller. e. Right QD excitation spectrum under B along x for the spin-up ground state. Anti-crossings of similar widths to d can be observed, although interpretation of the spectrum lines is less clear. No good data could be obtained for the y direction and the spin-down ground state. dI/dV in all panels is calculated by taking the numerical derivative after applying a Savitzky–Golay filter of window length 5 and polynomial order 1 to the measured current. The measurements shown here were conducted using different QD orbitals than those used in Figs. 34. The obtained magnitude of the SOC should be taken as an estimate rather than a precise value.

Extended Data Fig. 4 B dependence of CAR and ECT amplitudes of device A.

Measurements of CAR and ECT at 4 × 4 spin and bias combinations similar to those in Fig. 4g are performed as functions of B, both when B = ByBSO and when B = BxBSO. At around B = 50 mT, Zeeman energy exceeds the applied bias voltage of 100 μV and transport across QDs becomes spin polarized. The equal-spin CAR and opposite-spin ECT amplitudes no longer substantially depend on B at higher fields.

Extended Data Fig. 5 Theoretical calculations of CAR and ECT amplitudes at finite B, from which we extract the SOC strength in the hybrid segment.

See Supplementary Information and ref. 45 for details. ad, CAR and ECT amplitudes (proportional to currents) at hybrid-segment μ = 6.3 meV for the four spin combinations when B is rotated in-plane. Dashed lines are the average of each curve. The ratio between ↑↑ CAR to ↑↓ CAR is taken as a proxy of the triplet spin component over singlet in the following panels. e, Numerical (solid) and analytical (dashed) calculations of angle-averaged ↑↑/↑↓ CAR ratio are shown in the vicinity of three quantized levels in the hybrid segment (see Supplementary Information and ref. 45 for details). Variation is small throughout the numerically investigated ranges and all are close to the analytical result, signalling that the triplet component estimation is insensitive to the exact chemical potential assumed in the theory. f, Dependence of the triplet component on the SOC strength α for a length as in device A (200 nm), numerically calculated at three representative chemical potentials together with the analytical result. In Fig. 4g, triplet/singlet ratios defined here range from ~0.1 to ~0.25. This puts the estimation of α in the range of 0.11 to 0.18 eV Å, in agreement with reported values in literature (0.1 to 0.2 eV Å)46,47. g, Dependence of the triplet component on the SOC strength α for a length as in device B (350 nm), numerically calculated at three representative chemical potentials together with the analytical result. Similar comparison with data in Extended Data Fig. 6 yields estimations of α in the range of 0.05 to 0.07 eV Å. The weaker SOC could be attributed to the higher VPG used here (0.4 V for device B compared to 0 V for device A) weakening the inversion-symmetry-breaking electric field.

Extended Data Fig. 6 Anisotropic CAR and ECT reproduced in device B.

Device B is fabricated similarly except for the absence of the Pt layer to exclude it as a possible spin-flipping mechanism in the nanowire. ad, CAR and ECT amplitudes for four spin combinations when rotating B = 80 mT in the plane spanned by the nanowire axis and BSO (defined as the direction where equal-spin CAR and opposite-spin ECT are maximally suppressed). The BSO in this device points approximately 30° out of plane (insets: cross-section in a and top view in b). Inset in c: a sketch of the type of bias voltage configurations used in this measurement and in Fig. 4g; see caption of the lower panels for details. eg, Selected views of Icorr at three representative angles (marked with boxes of the corresponding colour as dashed lines in ad). These measurements are performed at VL = 70 μV, VR = 0 because the right QD allows considerable local Andreev current at finite bias due to one malfunctioning gate. This measurement scheme, which is also employed in Fig. 4g, allows us to measure both ECT and CAR without changing the bias. Inset in c illustrates when CAR and ECT processes occur using VL < VR = 0 as an example. Following the same analysis in Fig. 1, we measure ECT when −eVL < μLD = μRD < 0 and CAR when −eVL < μLD < 0 < μRD = −μLD < eVL. The main features of the main text data can be reproduced, including anti-diagonal CAR and diagonal ECT lines, strong suppression of opposite-spin ECT and equal-spin CAR along one fixed direction, and their appearance in perpendicular directions.

Extended Data Fig. 7 Device B characterization.

a, False-colour SEM image of device B prior to the fabrication of N leads. Green is nanowire, blue is Al and red are the bottom gates. Scale bar is 200 nm. The hybrid segment is 350 nm long. b,c, QD diamonds of the levels used on both sides at B = 0. d, Left QD bias spectroscopy under applied B = Bx and VLD = 357 mV along the nanowire axis. Level spacing 2.7 meV, g-factor 61 and spin–orbit anti-crossing 2〈HSO〉 = 0.25 meV can be extracted from this plot. dI/dV in this panel is calculated by taking the numerical derivative of the measured current. e,f, Left and right QD levels evolving under finite Bx. The levels used for taking the data in Extended Data Fig. 6 and the field at which they are taken are indicated by grey dashed lines. g-factor is estimated to be 26 for the right QD.

Extended Data Fig. 8 Spin correlation analysis of the data in Fig. 4g.

We define CAR↑↑ ≡ (Icorr) for the ↑↑ spin configuration and similarly for the others, as defined in Fig. 4g. The spin correlation for a given B direction is calculated as (CAR↑↑ + CAR↓↓ −  CAR↑↓ − CAR↓↑)/(CAR↑↑ + CAR↓↓ +  CAR↑↓ + CAR↓↑). Perfectly singlet pairing yields −1 spin correlation. The −0.86 correlation when BBSO is limited by the measurement noise level and can be improved by more signal averaging or more sophisticated analysis methods that are less sensitive to noise. When B points along other directions, the spin anti-correlation reduces as expected for non-singlet pairing.

Extended Data Fig. 9 B dependence of the energy spectrum in the middle hybrid segment of device A revealing a discrete Andreev bound state.

a, gLL ≡ dIL/dVL. White line indicates the bias range in which the experiments at finite B field were performed: the QD energies are kept below the lowest-lying excitation of the middle hybrid segment at all times to avoid sequential tunnelling into and out of it. The g-factor of the superconducting–semiconducting hybrid state is seen to be 21 from this plot, smaller than that in QDs. b, gRL ≡ dIR/dVL. The presence of nonlocal conductance corresponding to this state proves this is an extended Andreev bound state (ABS) residing under the entire hybrid segment, tunnel-coupled to both sides. We note that this is the same dataset presented in another manuscript51 (reproduced under the terms of the CC-BY Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0); copyright 2022, the authors, published by Wiley-VCH) where it is argued that the observed Zeeman splitting of this ABS also rules out the possibility of the Pt top layer randomizing spin inside the InSb nanowire.

Extended Data Fig. 10 Plotting of raw data used in Fig. 3 and Fig. 4c,f.

For other raw data, see the affiliated data repository (section ‘Data availability’). ah, IL, IR spanning the four joint charge degeneracies and under four N bias polarities at B = 0. Figure 3a, for example, is obtained by taking data from c and g and calculating their geometric mean at each pixel. The horizontal lines in IR are due to local Andreev processes carried only by the right junction. Since IL = 0 away from the joint charge degeneracies, these purely local currents do not appear in Icorr. ip, IL, IR spanning the four joint charge degeneracies and under four N bias configurations at B = By = 100 mT. qx, IL, IR spanning the four joint charge degeneracies and under four N bias configurations at B = Bx = 100 mT.

Supplementary information

Supplementary Information

A single PDF file including four sections: 1. Discussion of the spin–orbit coupling in the QDs and the possible effect it can have on the interpretation of the experiment. 2. Discussion of g-factor anisotropy in the QDs and the possible effect it can have on the interpretation of the experiment. 3. Theoretical modelling of the QD–QD system and extracting the CAR and ECT coupling strength from the QD linewidth. 4. Supplementary references.

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Wang, G., Dvir, T., Mazur, G.P. et al. Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires. Nature (2022). https://doi.org/10.1038/s41586-022-05352-2

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