Abstract
Cooper pairs occupy the ground state of superconductors and are typically composed of maximally entangled electrons with opposite spin. In order to study the spin and entanglement properties of these electrons, one must separate them spatially via a process known as Cooper pair splitting (CPS). Here we provide the first demonstration of CPS in a semiconductor twodimensional electron gas (2DEG). By coupling two quantum dots to a superconductorsemiconductor hybrid region we achieve efficient Cooper pair splitting, and clearly distinguish it from other local and nonlocal processes. When the spin degeneracy of the dots is lifted, they can be operated as spinfilters to obtain information about the spin of the electrons forming the Cooper pair. Not only do we observe a near perfect splitting of Cooper pairs into oppositespin electrons (i.e. conventional singlet pairing), but also into equalspin electrons, thus achieving triplet correlations between the quantum dots. Importantly, the exceptionally large spinorbit interaction in our 2DEGs results in a strong triplet component, comparable in amplitude to the singlet pairing. The demonstration of CPS in a scalable and flexible platform provides a credible route to study onchip entanglement and topological superconductivity in the form of artificial Kitaev chains.
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Introduction
Coupling two normal leads to a superconductor can give rise to nonlocal transport processes directly involving both leads. Two oppositespin electrons from a Cooper pair in the superconductor can be split into the leads via a process known as Cooper pair splitting (CPS). The dominant transport mechanism that gives rise to CPS is crossed Andreev reflection (CAR), whereby a higher order process allows two electrons to be injected simultaneously into the superconductor to form a Cooper pair. Additionally, a single electron can tunnel through the superconductor from one lead to the other through a process known as elastic cotunnelling (ECT). The ability to control these processes has important implications for two distinct fields. Firstly, efficient CPS can be used to generate spatially separated entangled electrons, that can be used to perform a Bell test^{1,2,3,4,5,6}. Secondly, in the context of topological superconductivity, it has been shown that CAR and ECT are crucial ingredients required to implement a Kitaev chain^{7} using quantum dotsuperconductor hybrids^{8,9}.
CPS has been studied in various mesoscopic systems coupled to superconductors, such as semiconductor nanowires^{10,11,12,13}, carbon nanotubes^{14,15}, and graphene^{16}. Quantum dots (QDs) are generally added between the leads and the superconductor. The charging energy of the QDs ensures that electrons forming a Cooper pair preferentially split into separate dots, rather than occupying levels in the same dot. This results in correlated electrical currents at the two normal leads. It has thus far been challenging to independently measure the relevant virtual processes (i.e. ECT and CAR) and isolate them from local processes, such as normal Andreev reflection or direct tunnelling via subgap states. In a set of recent studies on hybrid nanowires, it was shown that these challenges could be overcome to create a highly efficient Cooper pair splitter^{17} and to realize a minimal Kitaev chain^{18}. A key idea is that the QDs were coupled via extended Andreev bound states (ABSs) in the semiconductorsuperconductor hybrid^{19,20,21}, rather than the continuum above the superconducting gap. Therefore, by controlling the ABS energy with electrostatic gates, it was possible to tune the relative amplitudes of ECT and CAR. These developments pave the way for more advanced experiments, where the geometrical constraints of 1D systems will pose restrictions on the complexity of possible devices. An ideal platform to overcome these restrictions are semiconductor 2DEGs. Not only do they offer flexibility in device design, but also serve as a scalable platform to create and manipulate topologically protected Majorana bound states in artificial Kitaev chains.
We demonstrate here for the first time the observation of Cooper pair splitting in a 2D semiconductor platform. This is achieved by coupling two quantum dots via a hybrid proximitized section in an InSbAs 2DEG. By applying an external magnetic field, we polarize the spins of the QDs, allowing us to use them as spinfilters. This, in combination with highly efficient CPS, allows us to accurately resolve the spin of the electrons involved in CAR and ECT. The large spinorbit coupling in our 2DEGs, in combination with the device dimensions, results in significant spin precession for the electrons. Importantly, we show that this leads to strong equalspin CAR currents that are of similar amplitude to the conventional oppositespin processes. Through rotation of the magnetic field angle relative to the spinorbit field, we show that the ECT and CAR processes can be tuned to equal amplitudes, satisfying a key requirement for realizing a Kitaev chain in semiconductorsuperconductor hybrids.
Results
Device and characterization
Devices are fabricated on an InSbAs 2DEG with epitaxial aluminum grown by molecular beam epitaxy. This material has been established to have a low effective mass, high gfactor and large spinorbit coupling^{22,23}. Figure 1a, b illustrate the device structure together with the threeterminal measurement circuit. The two depletion gates (pink) define a quasi1D channel of about 150 nm, contacted on each side with normal leads. The middle of the channel is proximitized via a 150 nmwide aluminium strip (green), which is kept electrically grounded. Quantum dots on the left and right are created using the finger gates (blue) and the ABS energy is controlled by the central ABS gate (purple). The biases V_{L} and V_{R} applied to the left and right leads can be varied independently. The currents I_{L} and I_{R} in the left and right leads are measured simultaneously. We define a positive current as the flow of electron charge from the leads to the superconductor.
First, the two innermost finger gates are used to define tunneling barriers on either side of the hybrid region. Figure 1c, d show the measured local conductance G_{RR} = \(\frac{d{I}_{{{{{{{{\rm{R}}}}}}}}}}{d{V}_{{{{{{{{\rm{R}}}}}}}}}}\) and nonlocal conductance G_{LR} = \(\frac{d{I}_{{{{{{{{\rm{L}}}}}}}}}}{d{V}_{{{{{{{{\rm{R}}}}}}}}}}\) as a function of the ABS gate voltage V_{ABS}. The induced gap in the hybrid section is found to be Δ_{ind} ≈ 220 μeV. The correspondence between G_{RR} and G_{LR} shows the presence of an extended discrete ABS in the proximitized section. The observed signswitching in the nonlocal signal is typical for an extended ABS probed in a threeterminal measurement^{24,25,26}. Next, two quantum dots are created on either side of the proximitized section. Their electrochemical potentials are controlled by applied voltages V_{QDL} and V_{QDR}. The charge stability diagrams of both QDs (Fig. 1e, f) show Coulomb diamonds with clear evenodd spacing. The pair of Coulomb peaks show linear splitting as a function of magnetic field, indicative of a spindegenerate single orbital level (Fig. S1). The superconducting gap Δ_{ind} is clearly visible at the charge degeneracy points, indicative of a weak coupling to the proximitized region^{27,28}. Charging energies of QDL and QDR are 1.9 meV and 1.4 meV respectively, much larger than the induced superconducting gap.
CAR and ECT
For CAR, an electron from each of the two leads is simultaneously transferred to the superconductor via an extended ABS to form a Cooper pair (Fig. 2a). This should therefore result in positively correlated currents in the leads (I_{L} = I_{R}). For ECT (Fig. 2b), an electron from the left or right lead tunnels to the opposite lead via the hybrid section, which should thus give rise to negatively correlated currents (I_{L} = − I_{R}). As we will show below, by controlling the QD levels and voltage biases, it is possible to distinguish currents arising from ECT and CAR. Such measurements are shown in Fig. 2c, d. Here V_{QDL} and V_{QDR} are each tuned close to a selected charge degeneracy point and the currents I_{L} and I_{R} are simultaneously measured. The large charging energies of the dots ensure that each lead strongly prefers accepting or donating a single electron. We further ensure that the applied biases are lower in energy than any subgap states in the hybridized region, such that local transport is suppressed. To demonstrate CAR, we set V_{L} = V_{R} = − 120 μV and sweep V_{QDL} and V_{QDR}. A finite current is observed only along a line with negative slope, for both I_{L} and I_{R} (Fig. 2c). Furthermore, the currents are equal both in magnitude and sign (Fig. 2e). Converting the gate voltages to electrochemical potentials (μ_{L}, μ_{R}), we confirm that CAR mediated transport occurs when μ_{L} = −μ_{R} (Fig. S4c). This is consistent with the requirement that the energies of the electrons forming the Cooper pair must be equal and opposite. To demonstrate ECT, we apply biases with opposite polarity (V_{L} = −V_{R} = −120 μV). Unlike CAR, a finite current is observed only along a line with positive slope (Fig. 2d). This is consistent with energy conservation during ECT, which demands that μ_{L} = μ_{R}. Furthermore, the currents are now equal in magnitude, but opposite in sign (Fig. 2f). Note that when biasing only V_{L} or V_{R} and grounding the other lead, both ECT and CAR become visible in the charge stability diagram (Fig. S2).
Importantly, for both CAR and ECT we observe no notable current when the bias and energy conditions are not met, indicating that unwanted local processes are strongly suppressed. In combination with strongly correlated currents, this suggests a relatively large signaltonoise ratio of the CPS process. To characterize this, we calculate the CPS efficiency and visibility (Fig. S4). Following^{15,17}, we obtain a combined CPS efficiency above 90%, on par with the highest previously reported values^{15,17}. Applying a larger bias that exceeds the subgap state energy (but is still below Δ_{ind}) results in additional local, noncorrelated signals which only depend on a single QD (Fig. S3) and significantly reduce the CPS efficiency.
To systematically characterize the CAR and ECT measurements, we calculate the correlated current \({I}_{{{{{{{{\rm{corr}}}}}}}}}\equiv {{{{{{{\rm{sgn}}}}}}}}({I}_{{{{{{{{\rm{L}}}}}}}}}{I}_{{{{{{{{\rm{R}}}}}}}}})\cdot \sqrt{{I}_{{{{{{{{\rm{L}}}}}}}}}{I}_{{{{{{{{\rm{R}}}}}}}}}}\) (Fig. 2c, d)^{17}. It is nonzero only when I_{L} and I_{R} are both nonzero and thus highlights features mediated by ECT and CAR. Furthermore the sign of I_{corr} clearly distinguishes CAR (always positive) from ECT (always negative).
Zero field spin blockade
In the absence of a magnetic field, the orbital levels of the QDs are spindegenerate. Therefore, if the dot has an even number of electrons, the first electron to occupy the next orbital (a transition denoted as 0 ↔ 1) can be either spinup or spindown. However, to add the second electron (1 ↔ 2), the Pauli exclusion principle requires it to have an opposite spin. The effect of this spinfilling rule leads to a blockade of transport, which depends on the nature of the underlying process.
We first focus on ECT in the ( − ,+) bias configuration, denoting that a negative bias is applied to the left lead and a positive bias is applied to the right lead (Fig. 3b). When the QDs are tuned to the (0 ↔ 1, 1 ↔ 2) transition, a situation can arise where the left QD is occupied with e.g. a spinup electron (coming from the left lead), whereas the right QD can only accept a spindown electron (since the spinup state has already been occupied). At this point transport from left to right is blocked, analogous to the wellknown Pauli blockade in double quantum dots^{29}. This spin blockade is clearly seen when the QDs are tuned over successive charge transitions. In Fig. 3c we see that the ECT current is suppressed for the (0 ↔ 1, 1 ↔ 2) transition. Reversing the bias polarities to (+, − ), a similar blockade is observed for the (1 ↔ 2, 0 ↔ 1) transition, as expected (see Fig. 3e).
In the (−, −) configuration, only CAR mediated transport can occur and we find a suppression in CAR current for the (0 ↔ 1, 0 ↔ 1) transition. This is a direct consequence of the Cooper pairs in an swave superconductor having a singlet pairing. Thus, for transport to occur, each QD must donate an electron of opposite spin in order to create a singlet Cooper pair in the superconductor. Transport is therefore blocked when both dots are occupied by electrons with the same spin (Fig. 3a). Finally, in the (+, +) configuration a blockade is expected for the (1 ↔ 2, 1 ↔ 2) transition (Fig. 3d), as observed in the measurements. Qualitatively similar measurements of spin blockade for CAR and ECT are presented for another device (Fig. S8). We note that a finite amount of current remains for each blockaded transition, indicating the presence of a spinrelaxation mechanism in our system. The hyperfine interaction is one such mechanism that can lift the Pauli blockade^{30,31}. We confirm this by applying a magnetic field to suppress the spinmixing due to the hyperfine interaction, and find that 35 mT is sufficient to fully suppress the remaining current (Fig. S5).
Singlet and triplet ECT/CAR
The spin degeneracy of the QD levels is lifted by applying a magnetic field, allowing us to operate them as spinfilters (Fig. S1). When the Zeeman splitting exceeds ∣eV_{L}∣, ∣eV_{R}∣, only spinup (↑) electrons are involved in transport at a (0 ↔ 1) transition and only spindown (↓) at a (1 ↔ 2) transition. In the absence of spinorbit coupling, CAR is only expected to occur when both QDs are tuned to host electrons with opposite spin. The opposite applies to ECT, where a current is only expected when the QDs are tuned to receive electrons with equal spin.
As shown in Fig. 4b, when an inplane field of 150 mT is applied along B_{y} (i.e. perpendicular to the channel), CAR current is only present in the quadrants where the electrons have opposite spins (↑↓ and ↓↑) and completely suppressed for the equalspin (↑↑ and ↓↓) configuration. Similarly, no current is detected for oppositespin ECT, while transport is allowed for equalspin ECT. This spindependent transport indicates that the direction of the spinorbit field B_{SO} is along B_{y}, making spin a good quantum number. This is also consistent with the expected Rashba spinorbit interaction in a quasi1D channel with momentum along the zdirection and electric field perpendicular to the 2DEG plane. Applying the magnetic field perpendicular to B_{SO} (i.e. along B_{z}), a spinup electron may acquire a finite spindown component, due to the spinorbit interaction. The consequence of this can be seen in Fig. 4c, where we now observe sizeable currents for equalspin CAR and oppositespin ECT. The full evolution of the spinspecific ECT and CAR currents can be obtained by performing an inplane rotation of the magnetic field (Fig. 4d). The averaged amplitudes of equalspin CAR and oppositespin ECT currents 〈I_{corr}〉 are found to oscillate smoothly between full suppression at θ ≈ 90^{∘} and 270^{∘} (B∥B_{SO}), and their maximum strength at θ ≈ 0^{∘} and 180^{∘} (B⊥B_{SO}). This result does not depend on a specific choice of orbitals in the QDs (Fig. S6).
The ability to accurately resolve the spin of the electrons in CPS is particularly relevant in the context of entanglement witnessing. An important metric capturing this, is the spin crosscorrelation^{3,4}. As described in^{32}, we calculate the spin crosscorrelation from the measured currents as:
and plot it for both CAR and ECT as a function of θ (Fig. 4e). I^{ij} corresponds to the average correlated current 〈I_{corr}〉 associated with each spin configuration, where i, j ∈ {↑, ↓}. C = ±1 when there is a perfect correlation or anticorrelation between the spins of electrons entering the QDs. In contrast, C = 0 when the probabilities of equalspin and oppositespin transport become equal. When B∥B_{SO} we obtain a value of C = −0.96 for CAR, demonstrating a nearly perfect singlet pairing between the QDs. Similarly, for ECT C = +0.93 is obtained. When B⊥B_{SO}, C reaches close to 0 for both CAR and ECT, stressing that the triplet component can be tuned to be of similar magnitude to the conventional singlet pairing.
In conclusion, we have used quantum dotsuperconductor hybrids to demonstrate highly efficient Cooper pair splitting in a twodimensional semiconductor platform. Using spinpolarized quantum dots, we performed spinselective measurements of ECT and CAR and showed that the strong spinorbit interaction in ternary 2DEGs results in comparable strengths of singlet and triplet correlations between the quantum dots. Finally, through magnetic field rotations, we showed that it is possible to obtain equal amplitudes of ECT and CAR, establishing 2DEGs as an ideal platform to study Majorana bound states in artificial Kitaev chains.
Discussion
The demonstration of singlet and triplet correlations with Cooper pair splitters in 2DEGs paves the way for more advanced experiments to study entanglement and topological superconductivity. An interesting open question relates to the underlying mechanism that allows for strong triplet CAR in these devices. One possibility is for two equalspin electrons to form a normal swave Cooper pair, due to spin precession in the tunnel barriers. Another path is that an induced pwave superconducting pairing arises in the hybrid section, such that two equalspin electrons form a Cooper pair. In order to distinguish these possibilities, we propose to create quasi1D channels that are bent (rather than straight), resulting in different spinorbit directions in each arm of the Cooper pair splitter^{3,33}. Such devices are easily implemented in 2DEGs where any arbitrary shape of the channel can be realized simply by altering the design of the depletion gates. Given the high fidelity spin correlation we have demonstrated here, such devices could also be used to detect entanglement by performing a Bell test with electrons from a Cooper pair^{3}.
Finally, the recent realization of a minimal Kitaev chain^{18} opens up several possibilities to systematically study Majorana bound states (MBSs). In this regard the 2DEG platform is again particularly suitable. It readily allows for extending these measurements to multisite QD chains, whereby the flexibility of the 2DEG would allow for the simultaneous measurement of density of states at the edges and in the bulk. Furthermore, one could use these chains to perform tests of nonAbelian exchange statistics via braiding experiments^{34,35}, which necessarily require a 2D platform.
Methods
Fabrication
Device 1 (main text) and Device 2 (supplementary) were fabricated using techniques described in detail in ref. ^{36}. A narrow aluminum strip is defined in an InSbAsAl chip by wet etching, followed by the deposition of two normal Ti/Pd contacts. After deposition of 20 nm AlOx via atomic layer deposition (ALD), the two depletion gates are evaporated. Following a second ALD (20 nm AlOx) Ti/Au gates are evaporated in order to define the QDs and tune the ABS energy.
Measurements
All measurements are performed in a dilution refrigerator with a base temperature of 20 mK. Magnetic fields are applied using a 3D vector magnet. The alignment of the magnetic field with respect to the device is expected to be accurate within ± 5^{∘}. Transport measurements are performed in DC using a threeterminal setup, where the aluminum is electrically grounded (Fig. 1b). Current amplifier offsets are determined by the average measured current when both dots are in Coulomb blockade. CAR and ECT processes can be observed over a wide range of V_{ABS} voltages. Once a V_{ABS} setting was found with both strong CAR and ECT currents, it was kept at a constant value throughout the rest of the measurements. Further care was taken to implement the same orbitals in both QDs for all presented measurements in the main text. The mismatch between exact V_{QDR} and V_{QDL} values at which ECT and CAR are observed is due to gate instabilities, causing a drift of charge degeneracy points over a period of time. Therefore, the field rotation measurement in Fig. 4e was performed multiple times. No quantitative difference was observed between measurements. Presented data was selected due to high stability of the QDs over the course of the measurements.
Overall, we have measured four fully functional devices at the time of writing this manuscript, all of which have produced highly efficient CAR and ECT mediated by extended ABSs. For three of these devices we have performed magnetic field rotations and observed angledependent oscillations of ECT and CAR currents.
Data availability
Raw data and analysis scripts for all presented figures are available at https://doi.org/10.5281/zenodo.7311374.
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Acknowledgements
We thank T. Dvir, G. Wang, C. X. Liu, M. Wimmer, C. Prosko, P. Makk, R. Aguado, D. Xu, and L. P. Kouwenhoven for valuable discussions and for providing comments on the manuscript. The research at Delft was supported by the Dutch National Science Foundation (NWO) and a TKI grant of the Dutch Topsectoren Program. The work at Purdue was funded by Microsoft Quantum.
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Q.W. and S.L.D.t.H. fabricated and measured the devices. I.K. contributed to the device design and optimization of fabrication flow. 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 Q.W., S.L.D.t.H., and S.G., with inputs from all coauthors. S.G. supervised the experimental work in Delft.
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Wang, Q., ten Haaf, S.L.D., Kulesh, I. et al. Triplet correlations in Cooper pair splitters realized in a twodimensional electron gas. Nat Commun 14, 4876 (2023). https://doi.org/10.1038/s4146702340551z
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DOI: https://doi.org/10.1038/s4146702340551z
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