Abstract
Nonlocal entanglement is a key ingredient to quantum information processing. For photons, entanglement has been demonstrated^{1}, but it is more difficult to observe for electrons. One approach is to use a superconductor, where electrons form spinentangled Cooper pairs, which is a natural source for entangled electrons. For a threeterminal device consisting of a superconductor sandwiched between two normal metals, it has been predicted that Cooper pairs can split into spinentangled electrons flowing in the two spatially separated normal metals^{2,3,4,5}, resulting in a negative nonlocal resistance and a positive currentâ€“current correlation^{6,7}. The former prosperity has been observed^{8,9}, but not the latter. Here we show that both characteristics can be observed, consistent with Cooperpair splitting. Moreover, the splitting efficiency can be tuned by independently controlling the energy of the electrons passing the two superconductor/normalmetal interfaces, which may lead to better understanding and control of nonlocal entanglement.
Main
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 twoparticle interferometer was observed using current crosscorrelation measurements^{10,11}, but further investigation is still required to verify the entangled states^{12}. Spinentanglement has been predicted to exist at the superconductor/normalmetal (SN) interface^{2,3} and can be understood in the context of Andreev reflection^{13}, in which a lowenergy 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 (Î¾_{S}), 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 nonlocal analogue of Andreev reflection called crossed Andreev reflection^{6,7} (CAR). As a Cooper pair splits into two coupled electrons with opposite spin orientation that are then injected into the two normalmetal leads, instantaneous currents of the same sign are generated across the two SN interfaces, giving rise to a negative nonlocal resistance as well as a positive currentâ€“current correlation between the SN junctions. Previous experimental attempts focused on nonlocal resistance measurements^{8,9,14,15}, but the observation of CAR is complicated by another nonlocal process called elastic cotunnelling, in which electrons in the normalmetal leads tunnel across the superconductor with the help of Cooper pairs, resulting in a positive nonlocal resistance and a negative currentâ€“current correlation. The electron keeps its spin orientation during the tunnelling. In terms of the conductance, theoretical studies have found that these two nonlocal processes tend to cancel each other exactly in the lowest order approximation in the tunnelling limit^{16}, but experimental results and further theoretical investigation show that exact cancellation does not occur if the normalmetal leads are ferromagnetic^{8}, if the NS interfaces are highly transparent^{14,17} or if there is strong Coulomb interaction^{9,18}. For tunnelling junctions it was shown that Coulomb interaction may lead to a transition from elastic cotunnelling to CAR with increasing voltage bias^{9,18}.
The noise crosscorrelation measurement is an especially powerful tool to probe the correlation between charge carriers in mesoscopic systems^{19}; for example, recently it has been used for probing orbital entanglement of electrons from independent sources^{11}. For a threeterminal NSN device, it was also predicted that currentâ€“current correlation can be used to directly probe CAR and elastic cotunnelling without the drawbacks of the resistance measurement^{20,21}. However, despite a considerable amount of theoretical work, no experimental observation has been reported. Below we describe both resistance and noise crosscorrelation 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 splitting^{22,23}, but the observation of both a negative nonlocal resistance and a positive crosscorrelation for our devices provides compelling evidence of CAR, which involves spinentanglement 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 normalmetal/insulator/superconductor (NIS) junctions are formed between Al and Cu leads, with roomtemperature resistances in the range of 10â€“20 kÎ©. 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 voltage^{14,17}, whereas for tunnelling junctions CAR can dominate elastic cotunnelling at high bias^{9,15,18}. Third, the Coulomb interactions between the electrons in the normal metals may lead to a â€˜dynamicalâ€™ Coulomb blockade effect (DCB), which prevents both electrons from being injected into the same normalmetal lead and thus facilitates the separation of the two spinentangled electrons^{3}. 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 currentâ€“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 wellseparated 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 socalled shot noise reflects the particle nature of the charge carriers^{19}. Here the voltage noise power S_{V} is measured and it is related to the current noise power S_{I} by , with
where dV/dI is the differential resistance at bias voltage V, R is the resistance, k_{B} 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 noise^{19,24}. At e V â‰«k_{B}T, the hyperbolic cotangent term approaches 1 and S_{I} 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 interfaces^{25}. However, for typical NIS junctions such as ours, the probability of Andreev reflection is very small^{26,27}; 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 junctions^{28}. As for CAR, although in principle its presence can also be inferred from shot noise (autocorrelation) measurements^{20}, it is even harder because CAR is expected to be dominated by the Andreev reflection process^{14,20}. Another practical issue for autocorrelation measurements is that the extrinsic current noise from the measurement setup (back action) can be much larger than the intrinsic shot noise.
Fortunately, with the nonlocal configuration shown in Fig. 1, the crosscorrelation between V_{A} and V_{B} is only determined by nonlocal 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 nonlocal voltageâ€“voltage correlation signal, which greatly helps the observation of the small nonlocal currentâ€“current correlation. However, the large differential resistance here also limits the bandwidth used for the crosscorrelation measurement because of the capacitance of the electrical wiring (see the Methods section), which prevents us from making a reliable measurement below about 0.2â€‰K.
Figure 3 shows the voltage noise power measured by crosscorrelation, S_{V}, for our device at three different temperatures: 0.4â€‰K, 0.3â€‰K and 0.25â€‰K. For NIS junctions in the absence of the Coulomb interaction in the leads, it was shown that when eV , k_{B}Tâ‰ªÎ”, where Î” is the superconducting gap, the currentâ€“current correlation between junction A and junction B is^{20}
where G_{Q}=2e^{2}/h; A^{CAR}(A^{EC}) 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, A^{CAR}=A^{EC} is predicted^{16}. We note that S^{CAR} and S^{EC} have a similar form to the shot noise in equation (1). As the precise value of A^{CAR} is not known, a normalized S_{AB} is plotted in Fig. 3 to compare with experimental results. It is easy to see that by setting V_{A}=V_{B}, the elastic cotunnelling term in the brackets of equation (2) reduces to a biasindependent term 2k_{B}T A^{EC}, and the CAR term is maximized. As current sources instead of voltage sources were used in the experiment, currentâ€“voltage characteristics from the semiconductor model (see Fig. 2) were used to convert S_{I}(V_{A},V_{B}) to S_{V}(I_{A},I_{B}) 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 S_{V}, whereas in the experiment, S_{V} changes from slightly positive to slightly negative as the temperature decreases, suggesting a reduced A^{EC}, which indicates that the assumption A^{CAR}=A^{EC} is oversimplified. As V_{A},V_{B}â†’0, although S_{AB} is predicted to be 2k_{B}T(A^{CAR}âˆ’A^{EC}), the experimental data show a positive correlation at 0.4â€‰K that evolves to a sharp negative dip at 0.25â€‰K, which is again not consistent with the assumption A^{CAR}=A^{EC}.
To understand this discrepancy, the Coulomb interactions in the leads need to be considered^{3,18,21}. We note that the charging effect of solitary junctions has a strong temperature and bias dependence^{29}. As the Coulomb energy associated with charging the superconducting lead and charging the normalmetal leads changes, A^{CAR} and A^{EC} also change and this may lead to a nonmonotonic temperature and bias dependence of S_{AB}. Note that here the junction capacitance is much smaller than that of the planar NISIN devices studied before^{9}. As the charging energy is E_{C}=e^{2}/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 approach^{21}, where the presence of Coulomb interactions in the normalmetal 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 normalmetal leads that enhances the DCB effect may facilitate comparison with theory^{3,21}. We also note that in most theoretical work only pair tunnelling (Andreev reflection, CAR and higherorder processes) is considered in the zerotemperature limit, whereas in the experiment, quasiparticle tunnelling dominates at finite temperatures because a reasonable differential resistance is required for measurement.
The local and nonlocal differential resistance corroborate the observation from the noise crosscorrelation 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 nonlocal 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 V_{A}â‰ˆV_{B}, 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 cotunnelling^{16,20}. 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 predictions^{3}.
Figure 4b shows that at 0.4â€‰K, the nonlocal resistance is negative in the subgap regime for both junctions, consistent with CAR. As the temperature decreases to 0.3â€‰K, a nonlocal resistance peak evolves near zero bias. On further cooling to 0.25â€‰K, the nonlocal resistance at the peak becomes positive, see Supplementary Information S1. Remarkably, if we take a onedimensional cut at I_{B}=0, the bias dependence of the nonlocal resistance is close to that previously reported where only one junction was biased^{9,15}. This confirms that the nonlocal resistance measured in the current setup may have the same origin; although a more complete picture is obtained here with the twodimensional scan. Further experiments with spinselective normalmetal leads and Bellinequalitytype measurements^{21,30} will provide a conclusive check of nonlocal spin entanglement.
Methods
The devices were fabricated using standard twoangle electronbeam lithography and electrongun evaporation. A polymethyl methacrylate/polydimethyl glutarimide bilayer was spincoated on Si substrates with a 300â€‰nm SiO_{2} insulating layer for patterning the devices. For the particular device reported here, a 27â€‰nm 99.999% pure Al film was deposited first in an electrongun evaporator with a base pressure of 2.2Ă—10^{âˆ’7}â€‰torr at a 40Â° angle and at a rate of 0.1â€‰nmâ€‰s^{âˆ’1}. After deposition, 0.2â€‰torr pure oxygen gas was allowed into the chamber for about 5â€‰min to create a thin layer of oxide. Then a 50â€‰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 0.3â€‰Î¼m by 0.45â€‰Î¼m, and the distance between the two junctions is about 0.28â€‰Î¼m, comparable to the superconducting coherence length Î¾_{S} of diffusive Al. The junction resistance is much larger than that of the normalmetal 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 (1â€‰GÎ©) in series with a filtered voltage source. The a.c. voltage signals from the junctions were amplified using homemade batterypowered lownoise 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, Ï€filters on top of the cryostat were used. The measurement signals from the cryostat were amplified inside a Î¼metal enclosure close to the cryostat. The bandwidth of measurement was limited by the rolloff effect because of the capacitance of the Ï€filter (about a few nanofarads).
The autocorrelation data shown in Fig. 2b at 4.2â€‰K were measured in the frequency range from 500 to 1,000â€‰Hz, whereas at 0.4â€‰K different frequency ranges, from 200 to 600â€‰Hz and from 120 to 130â€‰Hz, were used and similar results were obtained. The voltage noise power S_{V} is fitted to
where I_{n} is the extrinsic current noise and V_{n} is the extrinsic correlated voltage noise. A good fit with data was found by using I_{n}=30â€‰fAâ€‰(Hz)^{âˆ’1/2} and V_{n}=0.7â€‰nVâ€‰(Hz)^{âˆ’1/2}. These values are consistent with standard junction fieldeffect transistor amplifiers. Hence, in the subgap regime, the extrinsic current noise dominates the intrinsic shot noise.
The crosscorrelation data shown in Fig. 3 were obtained in the frequency range from 2 to 6â€‰Hz to avoid the rolloff effect and the side lobe of the 60â€‰Hz line frequency peak. The efficiency of the crosscorrelation 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 S_{AB} is clearly visible.
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Acknowledgements
We thank M. BĂ¼ttiker, W. Belzig and A. Levy Yeyati for comments on our preprint. This work was sponsored by the National Science Foundation through grant no. DMR0604601.
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J.W. fabricated samples and carried out measurements and analysis. J.W. and V.C. prepared the manuscript.
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Wei, J., Chandrasekhar, V. Positive noise crosscorrelation in hybrid superconducting and normalmetal threeterminal devices. Nature Phys 6, 494â€“498 (2010). https://doi.org/10.1038/nphys1669
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DOI: https://doi.org/10.1038/nphys1669
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