Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires

In most naturally occurring superconductors, electrons with opposite spins are paired up to form Cooper pairs. This includes both conventional $s$-wave superconductors such as aluminum as well as high-$T_\text{c}$, $d$-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 progress. Instead, engineered platforms where $s$-wave superconductors are brought into contact with magnetic materials have shown convincing signatures of equal-spin pairing. 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 chain.

To probe spin pairing, one can split up a Cooper pair, separate the two electrons and measure their spins. The process to split a Cooper pair is known as crossed Andreev reflection (CAR) [10][11][12]. In this process, the two electrons end up in two separated non-superconducting probes (Figure 1a), each of these normal (N) probes collecting a single elementary charge, e. Alternative processes exist such as normal Andreev reflection (AR), with a 2e charge exchange between a single normal probe and the superconductor (S), and elastic co-tunneling (ECT), with 1e charge from one normal probe crossing the superconductor and ending up in the other normal probe. AR does not allow to measure the separate spins and thus this process needs to be suppressed.
Following the approach of previous Cooper pair splitting studies [13][14][15][16][17][18], we realize this by using quantum dots (QDs) with large charging energies that only allow for 1e transitions. This suppresses 2e-AR to ∼ 5% of the total current in each junction (see Figure ED2). The remaining CAR and ECT processes are sketched in Figure 1b. In ECT, 1e is subtracted from one QD and added to the other, whereas in CAR, an equal-sign 1e charge is either added or subtracted simultaneously to each QD. We will use this difference to distinguish ECT from CAR. Besides charge detection, QDs can be configured to be spin-selective in a magnetic field [19,20]. Figure 1c illustrates that ECT involves equal spin states in both QDs, whereas CAR from a singlet Cooper pair requires opposite spin states. Interestingly, these rules of spin combinations can be relaxed in the presence of inhomogeneous magnetic fields or spin-orbit interaction, both of which allow the possibility for triplet pairing [21][22][23][24][25][26][27][28][29]. For instance, spin-orbit (SOC) coupling can rotate an opposite-spin configuration into an equal-spin pair. In this report, we first demonstrate charge measurements, as illustrated in Figure 1b, followed by spin-selective detection of ECT and CAR, which sets us up to detect CAR with equal-spins when spin precessions are induced by SOC.

Charge filtering
The device and the measurement setup are illustrated in Figure 1d. A short segment of an InSb nanowire is proximitized by a thin Al shell, which is kept The energy diagram in Figure 1e illustrates that ECT requires alignment of the QD levels (µ LD = µ RD ), both positioned within the transport window defined by the bias voltages V L and V R . We restrict the bias settings to V L = −V R for ECT unless mentioned otherwise. In Figure 1e, the transport window is thus defined by −eV L > µ LD = µ RD > −eV R . To study co-tunneling processes which only occupy a higher-energy intermediary state virtually, the QD excitations and bias voltages are kept within the induced superconducting gap, i.e., lower in energy than any state in the hybrid (see Figure ED9). We define current to be positive when flowing from N into S for both sides, implying that ECT yields opposite currents, I L = −I R . On the other hand, CAR requires anti-symmetric alignment between the two QD levels, µ LD = −µ RD [17], to satisfy overall energy conservation, as shown in Figure 1f. We restrict bias settings to V L = V R for CAR unless specified. Thus, the transport window in  Figure ED2) and are strong along a straight line with a positive slope. Using the lever arm of QD gates extracted in Figure ED1, we find this line to be µ LD = µ RD . In panel e, we set V L = V R = 150 µV and similarly observe I L ≈ I R along a straight line with a negative slope where µ LD = −µ RD . Several features in these data allow us to attribute the origin of these sub-gap currents to CAR and ECT instead of competing transport processes. The non-local origin of the measured currents, expressed by the (anti-)symmetric energy requirement on both QDs and current correlation, rules out local Andreev reflection. The bias and QD energies being kept lower than any sub-gap bound state excludes resonant tunneling into and out of them. The only mechanisms known to us that can explain these observations are CAR and ECT [30].
In Figure ED2 we extract from this measurement Cooper pair splitting visibilities of 91% and 98% for the left and right QDs, respectively. Their product of 90%, for the first time realized, exceeds the minimum value of 71% required for a Bell test [16]. The high efficiency of Cooper pair splitting reported in this   work compared to previous reports relies on having a hard superconducting   gap in the proximitized segment and on having multiple gates for each QD, allowing control of the chemical potential of QDs independently from QD-lead couplings. Both requirements are enabled by recent advancements in the fabrication technique [31]. The dashed lines in Figure 2d,e indicate the boundaries of the transport window, as illustrated with corresponding colors surrounding the grey area in Figure 2b,c. For convenience, we introduce the correlated current I corr ≡ sgn(I L I R ) |I L I R |, plotted in Figure 2f,g for the corresponding ECT and CAR measurements. This product is finite only when currents through both junctions are nonzero, allowing us to focus on features produced by ECT or CAR (see Figures ED10 and ED11). Its sign directly reflects the dominant process: ECT being negative and CAR positive.

Spin blockade at zero magnetic field
Spin-degenerate orbital levels can each be occupied with two electrons with opposite spins. Figure 3a shows the charge stability diagram measured with negative biases on both N leads. We label the charge occupations relative to the lower-left corner, with some unknown but even number of electrons in each QD. Increasing the gate voltages V LD and V RD increases the occupation of left and right QD levels one by one from (0,0) to (2,2). In between charge transitions, the occupation is fixed with possible spin configurations as indicated in Figure 3a. At charge degeneracies, I corr is generally non-zero. However, the correlated current is very weak at the (0, 0) ↔ (1, 1) transition compared to the other three. This can be understood as a CAR-mediated spin-blockade,  recombine into a Cooper pair. However, whenever the QDs are both occupied with the same spin, CAR is suppressed and thereby blocks the transport cycle.
Note that SOC in InSb is known to not lift this blockade [32,33]. Figure 3c also shows a similar ECT-mediated spin-blockade when applying anti-symmetric biases to the N leads. This effect is intimately related to the well-known Pauli

Spin filtering
At finite magnetic field B, the four charge degeneracies in Figure 3a can become bipolar spin filters [19,20]. This requires the Zeeman energy in the QDs to exceed the bias voltage, electron temperature, and hyperfine interaction, yet remain smaller than the level spacing of the QDs. Under these conditions, we use ↑ / ↓ (along the applied B direction) to denote the two spin-split QD eigenstates. Only ↓ electrons are transported across a QD at the 0 ↔ 1 transition and only ↑ electrons at 1 ↔ 2. Figure 4b illustrates the consequence of spin filtering for CAR processes, namely a complete suppression for parallel spins. The opposite is expected for ECT with only spin-conserved tunneling being allowed. We first apply B = B y = 100 mT, in the plane of the substrate and perpendicular to the nanowire. The four panels in Figure 4c  and ↓↑ with ECT biases +− and −+. The observation of spin conservation suggests spin is a good quantum number. Thus, any spin-orbit field in the InSb nanowire, B SO (including both possible Rashba and Dresselhaus SOC), must be parallel, or nearly parallel, to B y . In this case, CAR provides a coupling mechanism only for an opposite-spin configuration in the two QDs. We note that the exact B SO direction as measured by suppression of equal-spin CAR or opposite-spin ECT depends on gate settings and the device used (e.g., Figure ED6). We have measured directions within 20 • of being perpendicular to the nanowire axis but its angle with the substrate plane can range from 0 to 60 • . This observation is consistent with the expectation of B SO being perpendicular to the nanowire axis for both Rashba and Dresselhaus SOC [36][37][38].
To quantify the observation that CAR is anti-correlated with the spin alignment of the QDs, we perform a spin correlation analysis [39,40] similar to that in Ref [41], which analogously reports reduced CAR amplitudes when QD spins are parallel compared to anti-parallel. The results, presented in Figure  and ↓↑ in ECT. The observed CAR coupling for ↑↑ and ↓↓ is interpreted as a measure of the equal-spin coupling between the QDs. In Figure ED4, we show that these observations do not qualitatively depend on the magnitude of |B| as long as spin polarization is complete (above ∼ 50 mT).
To further investigate the field-angle dependence, we measure CAR and ECT while rotating |B| = 100 mT in the plane of the substrate, see Figure 4g.
For this measurement, we apply a ±100 µV bias voltage only on one side of the device while keeping the other bias zero. This allows us to measure both CAR and ECT without changing the applied biases, as can be understood from the same basic principles outlined in Figure 1 (see Figure ED6 for details of this measurement scheme and the affiliated data repository for plots of the raw data). We take the maximum value of each I corr scan at a particular bias

Discussion
The oscillating CAR signals in Figure [38]. Such SOC also exists in our InSb-based QDs and can lead to nominally ↑ QD eigenstates having a small ↓ component [43]. In Figure ED3 and Supplementary Material, we quantify this effect and argue that the oppositespin admixture is too small to explain the measured amplitude of the ECT and CAR anisotropy.
The superconducting pairing in the hybrid segment itself is predicted to hold a triplet component due to SOC as well [44]. The shape and amplitude of our observed oscillations allow comparison with a theory adopting this assumption [45], resulting in an estimated spin-orbit strength in the hybrid section between 0.11 and 0.18 eV ·Å for Device A and 0.05 to 0.07 eV ·Å for Device B (see Figure ED5). This estimation agrees with reported values in the literature [46,47]. While the existence of triplet pairing component in the hybrid is thus consistent with our results, it is not the only possible explanation. During the tunneling process between the QDs, the electrons traverse through a bare InSb segment, whose SOC could also result in spin precession [36,38]. Both scenarios, however, support an interpretation of spin-triplet superconducting coupling between the QDs necessary for construction of a Kitaev chain [8].
Finally, we remark that the role of the middle Al-InSb hybrid segment of our devices in electron transport has not been discussed in this work. Figure ED9 shows that this segment hosts discrete Andreev bound states due to strong confinement in all three dimensions and these states are tunnel-coupled to both N leads. The parallel theoretical work modelling this experiment [45] shows that these states are expected to strongly influence CAR and ECT processes upon variation of the gate voltage underneath the hybrid segment. The experimental observations of the gate tunability of CAR and ECT is presented in Ref. [48].

Conclusion
In conclusion, we have measured CAR and ECT in an N-QD-S-QD-N device with and without spin filtering. For well-defined, specific settings consistent with our expectations, we observe Cooper pair splitting for equal spin states in the QD probes. These observations are consistent with the presence of a triplet component in the superconducting pairing in the proximitized nanowires, which is one of the building blocks for a topological superconducting phase [49,50]. More generally, our results show that the combination of superconductivity and SOC can generate triplet CAR between spin-polarized QDs, paving the road to an artificial Kitaev chain [7][8][9]. The realization of a Kitaev chain further requires increasing the coupling strength between QDs to allow the formation of a hybridized, extended state. This is confirmed in a parallel work where the QDs are driven to the strong coupling regime [51].

Device characterization and setup
The main device, A, and the measurement setup are illustrated in Figure 1d.
An InSb nanowire is in ohmic contact with two Cr/Au normal leads. The center is covered with a 200 nm-wide thin Al film. Device A has a 2Å, sub-monolayer Pt grown on top, which increases the magnetic-field compatibility [52]. Device B presented in Figures  We note that screening due to the presence of multiple metallic gates and a superconducting film in between diminishes cross-coupling between V LD and V RD .

Device fabrication
Our hybrid-nanowire devices are fabricated on pre-patterned substrates, following the shadow-wall lithography technique described in Refs [31,53] and specific details in the supplementary information of Ref [52].

Transport measurements
Devices A and B are cooled down in dilution refrigerators with base temperature ∼ 20 mK, equipped with 3D vector magnets and measured using standard voltage-biased dc circuits illustrated in Figure 1. No lock-in technique is used except Figure ED9. Current amplifier offsets are calibrated using known

Device tune-up
The tuning of our device, in particular the QDs, is done as follows. First, we form a single barrier between N and S by applying a low voltage on the gate closest to S on each side. We then perform local and non-local tunnel spectroscopy of the hybrid segment and locate a V PG range in which a hard gap is observed at low energies and extended Andreev bound states are observed at high energy (see Figure ED9). Having located a desired value of V PG , we form a second barrier in each junction by applying a lower voltage on the gates closest to the N leads. The confined region between the two barriers thus becomes a QD. We characterize the QDs by measuring its current above the superconducting gap, applying |V L |, |V R | > ∆/e as a function of V LD , V RD and applied magnetic field (see Figure ED1c,d). We look for a pair of resonances that correspond to the filling of a single non-degenerate orbital. This is indicated by two resonances separated by only the charging energy at zero field and their linear Zeeman splitting when B > 0. We finally measure CAR and ECT between the two QDs (as discussed in Figure 2). We optimize the measurement by controlling the gates separating the QDs from S to balance low local Andreev current (lowering gate voltage) with high signal-to-noise ratio (raising gate voltage). Having reached a reasonable balance, we characterize again the QDs ( Figure ED1).

Analysis of the structure of the obtained CAR and ECT patterns
Fitting the data in Figure 2d and Figure 2e to a theory model [45] (see supplementary information) yields QD-QD coupling strengths on the order of electron temperature. Such weak tunnel coupling does not alter the QD eigenstates significantly and allows us to operate QDs as good charge and spin filters. We further notice that finite ECT and CAR currents can be observed when both QDs are within the transport window but not on the diagonal lines dictating energy conservation. Since they appear only on one side of the (anti-)diagonal line corresponding to down-hill energy relaxation, these currents result from inelastic processes involving spontaneous emission and are thus non-coherent. We note that the data shown in Figure 2d and Figure 2e are taken at different gate settings than the rest of the paper and are selected because of high data resolution and Cooper pair splitting efficiency. The (anti-) diagonal resonance line and the strongly (anti-)correlated currents are generic to all QD orbitals that we have investigated.

Role of the Pt layer
Another source for SOC in our Device A could come from the Pt sub-atomic top layer, although we have not found evidence for this in previous studies [52].  Current through the left QD at V L = 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 Figure ED3 for g-factor extraction). The orbital used in Figures 3 and 4 is the pair of resonances marked by grey dashed lines at B = 100 mT. d. Current through the right QD at V R = 500 µV. The orbital used in Figures 3 and 4 is outside the measured range in this plot immediately to the right. All QD resonances we investigated behave similarly including those in Figure 2, which are selected because of high data resolution and Cooper pair splitting efficiency.     Figure 4g; see caption of the lower panels for details. e.-g. Selected views of Icorr at three representative angles (marked with boxes of the corresponding color as dashed lines in panels a-d). These measurements are performed at V L = 70 µV, V R = 0 because the right QD allows significant local Andreev current at finite bias due to one malfunctioning gate. This measurement scheme, which is also employed in Figure 4g, allows us to measure both ECT and CAR without changing the bias. Inset in panel c illustrates when CAR and ECT processes occur using V L < V R = 0 as an example. Following the same analysis in Figure 1, we measure ECT when −eV L < µ LD = µ RD < 0 and CAR when −eV L < µ LD < 0 < µ RD = −µ LD < eV L . 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.  Figure ED6 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. the QD energies are kept below the lowest-lying excitation of the middle hybrid segment at all times to avoid sequential tunneling into and out of it. The g-factor of the superconductingsemiconducting hybrid state is seen to be 21 from this plot, smaller than that in QDs. b. g RL ≡ dI R /dV L . The presence of nonlocal conductance corresponding to this state proves this is an extended ABS residing under the entire hybrid segment, tunnel-coupled to both sides. We note that this is the same dataset presented in another manuscript [52] 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.   either, because of SOC [35]. Figure ED3 shows that along y, the spin-orbit field direction of the QD, spin is a good quantum number and excited states belong- The amount of spin-up mixed into the nominally spin-down QD level is ⟨H SO ⟩/(δ−E Z ) according to first-order perturbation theory [35], where 2⟨H SO ⟩ is the spin-orbit level repulsion gap between the two mixing orbitals, δ the orbital level spacing between them and E Z the Zeeman splitting. All energies can be directly read out from bias spectroscopy of the QD excitation energies as a function of B ( Figure ED3b,d). The largest value of ⟨H SO ⟩/(δ − E Z ) we measured is seen in Figure ED3b, where the spin-orbit level repulsion is 0.4 meV and the difference between Zeeman splitting and level spacing is 1.6 meV. We thus observe no more than 0.2/1.6 = 12.5% of the opposite admixture at B = 100 mT, namely the nominally |↑⟩ state becomes modified as |↑⟩+0.125 |↓⟩ via perturbation. If for each QD, the nominally spin-up level has a spin composition of α η |↑⟩ + β η |↓⟩ and the nominally spin-down is β η |↑⟩ − α η |↓⟩ where η = L, R and coefficients are normalized, the spin-flipping probability produced by SOC in QD is |α L β R +α R β L | 2 . Using the worst-case-scenario numbers extracted from Figure ED3b

g -factor anisotropy
InSb nanowire QDs have anisotropic g-factors [43]. For essentially the same reason as presented in [43], g anisotropy is not a plausible cause of the observed spin blockade and lifting behavior. In summary, an anisotropic g-tensor has three principal axes and rotating B in one plane generally encounters two of them. Along both axes, the spins in the dots are aligned and blockade is complete. This results in two peaks and dips in 180 • of rotation instead of one, inconsistent with our observed periodicity.

Theoretical modelling
We present here the theoretical model of the experimental data presented in this work. Details of the model can be found in the parallel theory work [45].

CAR and ECT resonant currents
In the Main Text, the resonant current flowing through the N-QD-S-QD-N hybrid system is measured. Such CAR and ECT resonant currents have the forms [45] respectively. Here Γ DL is the total QD-N coupling strength, i.e. sum of left-QD-left-N coupling and right-QD-right-N coupling. Γ CAR and Γ ECT are the effective CAR and ECT coupling between the two QDs. These couplings depend on the properties of the middle hybrid segment and on the spin polarization in the QDs, but do not depend on the energies of the QDs. Thus in the (ε l , ε r )-plane, the resonant current assumes a Lorentzian shape with the broadening being Γ DL and reaching the maximum value along ε l = ±ε r for ECT and CAR, respectively. Thereby the maximum current for CAR and ECT are where a = CAR or ECT [45]. Here P CAR/ECT are the transition probability of electrons between two dots: P CAR is the probability of two electrons of a Cooper pair tunneling from the middle superconductor to the two separate dots and P ECT is the probability of a single electron tunneling from the left to the right QD.
Fitting the correlated currents in Figure 2 to Equation (1), we obtain total QD-N tunnel couplings of 57 (panel d) and 29 µeV (panel e) and QD-QD couplings of 9.6 (panel d) and 3.8 µeV (panel e).

QD-QD coupling strengths
The Bogoliubov-de-Gennes (BdG) Hamiltonian for the one-dimensional semiconductor-superconductor hybrid nanowire is

Triplet Cooper pair splitting
where Ψ(x) = [c ↑ (x), c ↓ (x), c † ↑ (x), c † ↓ (x)] ⊺ is the Nambu spinor, τ α and σ α are Pauli matrices acting on the Nambu and spin space, L is the length of the hybrid nanowire, m * is the effective mass of InSb, µ is the chemical potential, α R is the strength of Rashba spin-orbit coupling, E Z is half Zeeman spin splitting, θ is the angle between the spin-orbit field and the Zeeman field, and ∆ ind is the induced superconducting pairing strength in the nanowire. In the limit of confinement in all three dimensions as is the case with Device A, ∆ ind can be read as the lowest energy that the discrete superconducting-semiconducting hybrid states reach. We define the direction of the Zeeman field to be always along σ z , thus it is the spin-orbit field that rotates in our reference frame.
The tunneling Hamiltonian between the hybrid nanowire and the QDs is where d r,σ and d l,η are the annihilation operators of the spin-polarized state in QD-l and QD-r. Here we assume that the single-electron tunneling is spin-conserving, and that there is a single QD level with a particular spin polarization (along σ z ) near the Fermi energy of the normal QDs. That is, the dot state in the left QD has spin-σ and that in the right QD has spin-η. In the tunneling limit, i.e., t l,r ≪ ∆ ind , the effective CAR or ECT tunneling between the two dots is well described by the second-order virtual process ⟨H T H −1 0 H T ⟩ where ⟨·⟩ denotes the ground state of the hybrid nanowire. For example, the effective CAR coupling between spin-η state in QD-l and spin-σ state in QD-r is Γ CAR,ησ = t l t r ⟨c † η (0)H −1 where [u m↑ (x), u m↓ (x), v m↑ (x), v m↓ (x)] ⊺ is the Bogoliubov wavefunction of the m-th eigenstate with excitation energy E m > 0. Thereby for the resonant current flowing through the normal QDs with particular spin polarizations, we have I max CAR,ησ ∝ (Γ CAR,ησ ) 2 ∝ P CAR,ησ = |a ησ | 2 .
For the numerical simulation, we first discretize the Hamiltonian in Eq. (3) into a tight-binding Hamiltonian using KWANT [57]. We then get the eigenenergies and eigenfunctions by diagonalizing the Hamiltonian and calculate the probabilities using Eqs. (5) and (7).

Analytical formula of angular oscillation amplitudes
The main use of the numerical simulations is to compare with the measured up-up/up-down CAR amplitudes to obtain an estimation of the SOC strength α R . Ref [45] also contains derivations of an analytical expression that directly relates α R to the angle-averaged up-up/up-down ratio of CAR amplitudes: P CAR,↑↑ P CAR,↑↓ = sin 2 (k so L) 2 − sin 2 (k so L) (9) where k so = mα R /ℏ 2 and L is device length. This expression is insensitive to microscopic details of the wavefunction in the hybrid produced by varying chemical potential, disorder and inhomogeneous α R . It depends only on the nanowire length and the averaged spin-orbit length. The ratio is no greater than one and reaches the maximum when the nanowire length is a half-integer multiple of the spin-orbit length. As we can see in Figure ED5, the difference between numerical simulations and Eq. (9) is indeed small. The deviations mostly originate from finite Zeeman field not captured by the analytical formula. Thus, this simple expression provides us with a way of extracting the spin-orbit constant α R directly from the anisotropic CAR/ECT measurements without any other knowledge than device length and effective electron mass.