Non-local entanglement is a key ingredient to quantum information processing. For photons, entanglement has been demonstrated1, but it is more difficult to observe for electrons. One approach is to use a superconductor, where electrons form spin-entangled Cooper pairs, which is a natural source for entangled electrons. For a three-terminal device consisting of a superconductor sandwiched between two normal metals, it has been predicted that Cooper pairs can split into spin-entangled electrons flowing in the two spatially separated normal metals2,3,4,5, resulting in a negative non-local resistance and a positive current–current correlation6,7. The former prosperity has been observed8,9, but not the latter. Here we show that both characteristics can be observed, consistent with Cooper-pair splitting. Moreover, the splitting efficiency can be tuned by independently controlling the energy of the electrons passing the two superconductor/normal-metal interfaces, which may lead to better understanding and control of non-local entanglement.
Entanglement of electrons may arise in the spatial degree of freedom (orbital entanglement) or the spin degree of freedom (spin entanglement). Recently, orbital entanglement in a fermionic Hanbury Brown and Twiss two-particle interferometer was observed using current cross-correlation measurements10,11, but further investigation is still required to verify the entangled states12. Spin-entanglement has been predicted to exist at the superconductor/normal-metal (SN) interface2,3 and can be understood in the context of Andreev reflection13, in which a low-energy electron in the normal metal impinges on the SN interface and a hole is retroreflected whereas a Cooper pair is created in the superconductor.
When two normal metals are coupled to a superconductor with spatial separation comparable to the superconducting coherence length (ξS), roughly the size of a Cooper pair, it is predicted that electrons in the two normal metals can also be coupled by means of a non-local analogue of Andreev reflection called crossed Andreev reflection6,7 (CAR). As a Cooper pair splits into two coupled electrons with opposite spin orientation that are then injected into the two normal-metal leads, instantaneous currents of the same sign are generated across the two SN interfaces, giving rise to a negative non-local resistance as well as a positive current–current correlation between the SN junctions. Previous experimental attempts focused on non-local resistance measurements8,9,14,15, but the observation of CAR is complicated by another non-local process called elastic cotunnelling, in which electrons in the normal-metal leads tunnel across the superconductor with the help of Cooper pairs, resulting in a positive non-local resistance and a negative current–current correlation. The electron keeps its spin orientation during the tunnelling. In terms of the conductance, theoretical studies have found that these two non-local processes tend to cancel each other exactly in the lowest order approximation in the tunnelling limit16, but experimental results and further theoretical investigation show that exact cancellation does not occur if the normal-metal leads are ferromagnetic8, if the NS interfaces are highly transparent14,17 or if there is strong Coulomb interaction9,18. For tunnelling junctions it was shown that Coulomb interaction may lead to a transition from elastic cotunnelling to CAR with increasing voltage bias9,18.
The noise cross-correlation measurement is an especially powerful tool to probe the correlation between charge carriers in mesoscopic systems19; for example, recently it has been used for probing orbital entanglement of electrons from independent sources11. For a three-terminal NSN device, it was also predicted that current–current correlation can be used to directly probe CAR and elastic cotunnelling without the drawbacks of the resistance measurement20,21. However, despite a considerable amount of theoretical work, no experimental observation has been reported. Below we describe both resistance and noise cross-correlation measurements, and in each type of measurement a tunable CAR is observed. It is worth noting that observation of a positive correlation alone is not proof of CAR and Cooper pair splitting22,23, but the observation of both a negative non-local resistance and a positive cross-correlation for our devices provides compelling evidence of CAR, which involves spin-entanglement according to current theoretical understanding.
In Fig. 1 we show a scanning electron micrograph of a hybrid device, as well as a schematic of the measurement circuit (see the Methods section for sample fabrication and measurements). Two normal-metal/insulator/superconductor (NIS) junctions are formed between Al and Cu leads, with room-temperature resistances in the range of 10–20 kΩ. The advantage of using tunnelling junctions instead of junctions with highly transparent interfaces is the following. First, the large tunnelling resistance ensures that there is little leakage current from one junction to the other; that is, currents from different sides flow only to the superconducting lead (ground), so the bias across the two junctions can be varied independently. Second, for a highly transparent interface, elastic cotunnelling is expected to dominate over CAR at all temperatures and at energies below the gap voltage14,17, whereas for tunnelling junctions CAR can dominate elastic cotunnelling at high bias9,15,18. Third, the Coulomb interactions between the electrons in the normal metals may lead to a ‘dynamical’ Coulomb blockade effect (DCB), which prevents both electrons from being injected into the same normal-metal lead and thus facilitates the separation of the two spin-entangled electrons3. Note that the DCB in the lead may favour elastic cotunnelling instead of CAR because the number of electrons in the superconductor changes by two for CAR (ref. 18).
Our NIS junctions have no pin holes in the thin insulating layer, as indicated by good fitting of the current–voltage characteristics at different temperatures in Fig. 2a using the simple semiconductor model (see Supplementary Notes 1 for the details of the fitting). When there is no pin hole, current flowing across the NIS junction consists of well-separated events of quantum mechanical tunnelling of electrons (no temporal correlation and inelastic scattering). In this case, the power of the current fluctuation is proportional to the magnitude of the current. This so-called shot noise reflects the particle nature of the charge carriers19. Here the voltage noise power SV is measured and it is related to the current noise power SI by , with
where dV/dI is the differential resistance at bias voltage V, R is the resistance, kB is the Boltzmann constant, T is the temperature, e is the electron charge, and F is the Fano factor, which equals 1 for full shot noise19,24. At e V ≫kBT, the hyperbolic cotangent term approaches 1 and SI is proportional to I, the hallmark of shot noise, as shown in Fig. 2b. The fitted value F=1 again indicates that there are no pin holes. Below the superconducting transition temperature, Andreev reflection may lead to doubled shot noise near zero bias, as has been observed for short diffusive NS junctions with transparent interfaces25. However, for typical NIS junctions such as ours, the probability of Andreev reflection is very small26,27; thus, the subgap current resulting from Andreev reflection is also small. As the shot noise is proportional to the current, it is difficult to verify the doubled shot noise for NIS junctions28. As for CAR, although in principle its presence can also be inferred from shot noise (autocorrelation) measurements20, it is even harder because CAR is expected to be dominated by the Andreev reflection process14,20. Another practical issue for autocorrelation measurements is that the extrinsic current noise from the measurement set-up (back action) can be much larger than the intrinsic shot noise.
Fortunately, with the non-local configuration shown in Fig. 1, the cross-correlation between VA and VB is only determined by non-local processes, and it is much less affected by any extrinsic current noise, as there is no circuit segment shared between the two channels except a short piece of superconducting Al wire, along which no voltage drop is expected to occur. In addition, the large differential resistance near zero bias (see Fig. 2) leads to a large non-local voltage–voltage correlation signal, which greatly helps the observation of the small non-local current–current correlation. However, the large differential resistance here also limits the bandwidth used for the cross-correlation measurement because of the capacitance of the electrical wiring (see the Methods section), which prevents us from making a reliable measurement below about 0.2 K.
Figure 3 shows the voltage noise power measured by cross-correlation, SV, for our device at three different temperatures: 0.4 K, 0.3 K and 0.25 K. For NIS junctions in the absence of the Coulomb interaction in the leads, it was shown that when e|V |, kBT≪Δ, where Δ is the superconducting gap, the current–current correlation between junction A and junction B is20
where GQ=2e2/h; ACAR(AEC) is the amplitude of CAR(elastic cotunnelling), determined by the distance between the two SN junctions and the properties of the interfaces. For simple tunnelling junctions, ACAR=AEC is predicted16. We note that SCAR and SEC have a similar form to the shot noise in equation (1). As the precise value of ACAR is not known, a normalized SAB is plotted in Fig. 3 to compare with experimental results. It is easy to see that by setting VA=VB, the elastic cotunnelling term in the brackets of equation (2) reduces to a bias-independent term 2kBT AEC, and the CAR term is maximized. As current sources instead of voltage sources were used in the experiment, current–voltage characteristics from the semiconductor model (see Fig. 2) were used to convert SI(VA,VB) to SV(IA,IB) for comparison.
At all three temperatures, when the bias of the two junctions is of the same polarity, there is a clearly positive correlation, consistent with CAR. When the bias of the two junctions is of opposite polarity, equation (2) predicts a negative SV, whereas in the experiment, SV changes from slightly positive to slightly negative as the temperature decreases, suggesting a reduced AEC, which indicates that the assumption ACAR=AEC is oversimplified. As VA,VB→0, although SAB is predicted to be 2kBT(ACAR−AEC), the experimental data show a positive correlation at 0.4 K that evolves to a sharp negative dip at 0.25 K, which is again not consistent with the assumption ACAR=AEC.
To understand this discrepancy, the Coulomb interactions in the leads need to be considered3,18,21. We note that the charging effect of solitary junctions has a strong temperature and bias dependence29. As the Coulomb energy associated with charging the superconducting lead and charging the normal-metal leads changes, ACAR and AEC also change and this may lead to a non-monotonic temperature and bias dependence of SAB. Note that here the junction capacitance is much smaller than that of the planar NISIN devices studied before9. As the charging energy is EC=e2/2C, where C is the capacitance of the tunnelling junction, a much stronger DCB effect is expected. The effect of DCB on noise correlation was recently considered with a renormalization group approach21, where the presence of Coulomb interactions in the normal-metal leads is treated as a reduction of interface transparency, resulting in a change of the sign and magnitude of the correlation. However, this approach does not seem to be applicable here as the Coulomb interaction in the superconducting lead was not taken into account. Further experiments with a modified local impedance near the normal-metal leads that enhances the DCB effect may facilitate comparison with theory3,21. We also note that in most theoretical work only pair tunnelling (Andreev reflection, CAR and higher-order processes) is considered in the zero-temperature limit, whereas in the experiment, quasiparticle tunnelling dominates at finite temperatures because a reasonable differential resistance is required for measurement.
The local and non-local differential resistance corroborate the observation from the noise cross-correlation measurements: they also demonstrate that CAR clearly dominates, as shown in Fig. 4. (In Supplementary Notes 2, more details are given to show that both the reduction of local resistance and the negative non-local resistance are consistent with the noise measurements.) Surprisingly, even for the local differential resistance, there is clearly a reduction of resistance when the d.c. bias VA≈VB, as indicated by the dashed line in Fig. 4. This strong reduction is not expected as the differential resistance is mainly determined by quasiparticle tunnelling, and even if there is a small probability of pair tunnelling, direct Andreev reflection is expected to dominate CAR and elastic cotunnelling16,20. However, this might suggest that DCB indeed enhances the splitting of a Cooper pair, especially when the electrons passing the two SN interfaces are of the same energy, in agreement with predictions3.
Figure 4b shows that at 0.4 K, the non-local resistance is negative in the subgap regime for both junctions, consistent with CAR. As the temperature decreases to 0.3 K, a non-local resistance peak evolves near zero bias. On further cooling to 0.25 K, the non-local resistance at the peak becomes positive, see Supplementary Information S1. Remarkably, if we take a one-dimensional cut at IB=0, the bias dependence of the non-local resistance is close to that previously reported where only one junction was biased9,15. This confirms that the non-local resistance measured in the current set-up may have the same origin; although a more complete picture is obtained here with the two-dimensional scan. Further experiments with spin-selective normal-metal leads and Bell-inequality-type measurements21,30 will provide a conclusive check of non-local spin entanglement.
The devices were fabricated using standard two-angle electron-beam lithography and electron-gun evaporation. A polymethyl methacrylate/polydimethyl glutarimide bilayer was spin-coated on Si substrates with a 300 nm SiO2 insulating layer for patterning the devices. For the particular device reported here, a 27 nm 99.999% pure Al film was deposited first in an electron-gun evaporator with a base pressure of 2.2×10−7 torr at a 40° angle and at a rate of 0.1 nm s−1. After deposition, 0.2 torr pure oxygen gas was allowed into the chamber for about 5 min to create a thin layer of oxide. Then a 50 nm layer of Cu film was deposited at a normal angle. Devices from three batches were measured in an Oxford KelvinOx 100 or an Oxford KelvinOx 300 dilution refrigerator. The results obtained on these devices were consistent with each other and here we concentrate on a single device with the most complete data. As shown in Fig. 1, the junction size is about 0.3 μm by 0.45 μm, and the distance between the two junctions is about 0.28 μm, comparable to the superconducting coherence length ξS of diffusive Al. The junction resistance is much larger than that of the normal-metal lead, which ensures a sensitive measurement of current across the junction, as voltage is the quantity actually measured here.
The current sources shown in Fig. 1 were realized by putting a large ballast resistor (1 GΩ) in series with a filtered voltage source. The a.c. voltage signals from the junctions were amplified using home-made battery-powered low-noise amplifiers. The voltage signals after amplification were sent to data acquisition cards and were analysed by a computer after digitization. To reduce noise coupled to devices at low temperature, π-filters on top of the cryostat were used. The measurement signals from the cryostat were amplified inside a μ-metal enclosure close to the cryostat. The bandwidth of measurement was limited by the roll-off effect because of the capacitance of the π-filter (about a few nanofarads).
The autocorrelation data shown in Fig. 2b at 4.2 K were measured in the frequency range from 500 to 1,000 Hz, whereas at 0.4 K different frequency ranges, from 200 to 600 Hz and from 120 to 130 Hz, were used and similar results were obtained. The voltage noise power SV is fitted to
where In is the extrinsic current noise and Vn is the extrinsic correlated voltage noise. A good fit with data was found by using In=30 fA (Hz)−1/2 and Vn=0.7 nV (Hz)−1/2. These values are consistent with standard junction field-effect transistor amplifiers. Hence, in the subgap regime, the extrinsic current noise dominates the intrinsic shot noise.
The cross-correlation data shown in Fig. 3 were obtained in the frequency range from 2 to 6 Hz to avoid the roll-off effect and the side lobe of the 60 Hz line frequency peak. The efficiency of the cross-correlation measurement at this frequency range was compromised by the 1/f noise of the amplifiers. Nevertheless, owing to the large differential resistance, the bias dependence of SAB is clearly visible.
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We thank M. Büttiker, W. Belzig and A. Levy Yeyati for comments on our preprint. This work was sponsored by the National Science Foundation through grant no. DMR-0604601.
The authors declare no competing financial interests.
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Wei, J., Chandrasekhar, V. Positive noise cross-correlation in hybrid superconducting and normal-metal three-terminal devices. Nature Phys 6, 494–498 (2010). https://doi.org/10.1038/nphys1669
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