Introduction

Multi-terminal systems enable measurements of both the local and the nonlocal voltages/currents between selected electrode pairs1. The non-local transport of charge2,3,4,5,6,7, heat8 and spin9 via hybrid devices consisting of the normal and superconducting reservoirs are currently of interest for the basic research and innovative applications. Electrons traversing metal-superconductor interface are glued into the Cooper pairs and conversely, the Cooper pairs are split into the individual electrons10. In both processes there emerge the entangled carriers, leading to nonlocal correlations. These effects can be amplified by inserting the quantum dots between the reservoirs11. In this regard, the three-terminal structures are especially useful, because they allow for efficient splitting of the Cooper pairs12,13,14, give rise to spin filtering15, generate the correlated spin currents16, separate the charge from heat currents17, enable realization of the exotic Weyl or Majorana-type quasi-particles18, etc.

Very spectacular non-local effects are provided by the crossed Andreev reflections (CAR), operating in a subgap regime. The ‘driving’ current applied to one side of the multi-terminal junction can yield either positive or negative nonlocal voltage response at the other interface, depending on a competition between the ballistic electron transfer (ET) and the CAR processes. Such changeover has been observed in three-terminal planar junctions2,3,4,5, using a piece of superconducting sample sandwiched between two conducting (normal or magnetic) electrodes. The induced non-local conductance, however, was much weaker from the local one in agreement with theoretical predictions19,20,21.

Here we propose a different configuration, where the quantum dot is built into the three-terminal hybrid as sketched in Fig. 1. Proximity effect converts the quantum dot into, a kind of, superconducting grain and its subgap spectrum develops the, so called, Andreev or Shiba bound states22,23,24,25,26, which substantially enhance the non-local transport. We show that effective non-local conductance can be comparable to the local one and can change sign from the positive to negative values by increasing the coupling to superconducting electrode or by appropriate tuning of the gate potential. The gate potential is also controlling symmetry of this effect. Experimental tests of such effects should be feasible using the three-terminal architecture with such quantum dots as the carbon nano-tubes7,27, semiconducting nano-wires28,29 or self-assembled InAs islands30,31.

Figure 1
figure 1

Scheme of the three-terminal device consisting of two conducting leads (L and R), superconducting reservoir (S) and the quantum dot (QD).

The ‘driving’ current in the L − QD − S loop induces the non-local voltage ‘response’ of the floating R electrode.

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It has to be stressed that similar structure has been analyzed previously6,20. Futterer et al.6 have considered the single level quantum dot with a strong Coulomb interaction and found a regime of large voltages when the CAR processes dominate and the non-local resistance is negative. Even though the analysis was limited to extremally non-linear transport with voltages as large as they have not considered charge redistribution in the electrodes and the screening effects due to the long range Coulomb interactions. Golubev and Zaikin20 have studied the many level chaotic and non-interacting quantum dot in a similar setup. They allowed for the proximity induced superconducting order parameter on the dot. They neglected interactions and the effects related to non-linear transport32,33,34,35. Contrary to those earlier studies we consider a quantum dot with on dot level modified by the long range Coulomb potential which is important to account for the non-linear transport.

Results

Microscopic model

Some aspects of the local and non-local transport properties for this three-terminal device could be inferred by extending the Landauer-Büttiker approach36,37,38,39,40,41,42 (see the 1-st subsection of Methods). On a microscopic level, we describe this system in the tunneling approximation43 by the Hamiltonian

with standard notation for the annihilation (creation) operators of the itinerant and localized dot electrons. The first term describes the left (α = L) and the right (α = R) conducting leads. The subsequent term refers to the quantum dot (QD) with its energy level ε0 shifted by the long-range potential U(r). Hybridization between the QD and itinerant electrons is characterized by the matrix elements tα. The last two terms in (1) correspond to the BCS-type superconducting reservoir with an isotropic energy gap Δ. Addressing here the subgap (low-energy) transport we assume the constant tunneling rates , where is the (normal state) density of states of α lead. In what follows, we assume the superconducting gap Δ to be the largest energy scale in the problem.

Subgap charge transport

The charge current Jα flowing from an arbitrary lead can be evaluated using the Heisenberg equation 44. In particular, the current JL(R) from the normal L (R) electrode is given by44

where and denote the matrix elements (in the Nambu representation) of the retarded and lesser QD Green functions, respectively. From now onwards we consider the current JL focusing on the subgap voltage, smaller than the energy gap . In such regime there are possible: the ballistic electron transfer (ET) from L to R electrode, the direct Andreev reflection (DAR) when electron from L lead is converted into the Cooper pair in S reservoir and hole is scattered to L electrode and the crossed Andreev reflection (CAR) which is similar to DAR except that hole is scattered to R electrode. They can be expressed as45

where and are the Fermi-Dirac distribution functions for electrons and holes, respectively. Let us remark, that only the ET (3) and CAR (5) contributions lead to the non-local effects, because they depend on the chemical potentials of both conducting (L and R) electrodes. Since these ET and CAR processes deliver different types of the charge carriers to the right electrode, the induced voltage VR would be a probe of the dominant transport mechanism.

Relationship between the ET and CAR processes can be inspected by studying their transmissions, defined as and (see Fig. 2). Deep in a subgap regime (i.e. for ) the Green function simplifies to the familiar BCS structure46. Its diagonal part is given by with the quasi-particle energy , where . Subgap spectrum consists thus of two Shiba states at ±EA whose spectral weights are and with the quasiparticle broadening . The single electron transmission EET(E) is a quantitative measure of this subgap spectrum. The left panel in Fig. 2 illustrates evolution of the Shiba states upon increasing the coupling while the right panel shows a transfer of the corresponding spectral weights upon changing the QD level ε0 by an applied gate voltage.

Figure 2
figure 2

Transmissions of the ET (dashed lines) and CAR (solid lines) transport channels obtained at zero temperature for ΓR = ΓL.

The left panel refers to ε0 = 0 and the right one to .

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Transmission of the anomalous CAR channel, on the other hand, depends on the off-diagonal part of the matrix Green function , where . It also has maxima around the same Shiba states ±EA but with a different amplitude, sensitive to the induced pairing . This is a reason why TCAR(E) quickly diminishes whenever is decreased or the QD level ε0 departs from μS = 0 (solid lines in Fig. 2).

Confronting both these transmissions reveals that the non-local transport predominantly comes from the CAR process when the coupling (to superconducting electrode) is sufficiently strong and the QD level ε0 is close to the chemical potential μS. Otherwise, the non-local effects are dominated by the single electron tunneling (ET). The related changeover can be detected by measuring the voltage VR in the floating R electrode, in response to the current in the L − QD − S branch. Such voltage VR can vary between the positive and negative values and the non-local resistance can be tuned by the gate potential lifting/lowering the Shiba energies.

Linear response

Practical realizations of the setup (Fig. 1) would allow to measure the local and the non-local resistances/conductances within the four-probe scheme36,37,38,39,40, where the potentials and currents are treated on equal footing (see the Method). In a weak perturbation limit the response would be linear

The coefficients for β = ET, DAR or CAR can be determined from the equations (3)–(5), and they read

At zero temperature , hence depend on the transmissions .

Treating the potential VS as a reference level we analyze the induced voltage VR in response to the ‘driving’ current . The local resistance is due to the DAR processes whereas the non-local one results from the single electron tunneling (ET) competing with the anomalous crossed Andreev reflection (CAR) processes. Fig. 3 shows this non-local resistance RRS,LS normalized with respect to . The left panel shows that has a negative sign (signifying the dominant CAR processes) only for sufficiently strong coupling . This is a straightforward consequence of the (zero-energy) ET and CAR transmissions (Fig. 2). The right panel of Fig. 3 displays the non-local resistance versus the QD level ε0. In the linear regime the negative nonlocal resistance occurs when ε0 ~ μS for sufficiently strong coupling . Since and ε0 can be experimentally varied in the realizations of the superconducting-metallic devices with the quantum dots7,27,28,29,30,31, such qualitative changes should be observable.

Figure 3
figure 3

The non-local resistance 2RRS,LS/DR as a function of ΓS/ΓL ratio (left panel) and the QD dot level ε0 (right panel) obtained in the linear limit for three representative temperatures.

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Beyond the linear response limit

To confront these findings with the non-local effects observed so far in the ‘planar’ junctions2,3,4,5 we now go beyond the linear response framework. For arbitrary value of the ‘driving’ voltage VL we computed self-consistently VR, guaranteeing the net current JR to vanish. Under such non-equilibrium conditions the long-range potential U(r) plays an important role in the transport when the charges pile up in the electrodes and the quantum dot47. It affects the chemical potentials and the injectivities of the leads and contributes to the screening effect32,33,34,35. The potential U(r) has to be properly adjusted, depending on specific polarization of the system33 (for details see the 2-nd subsection of Methods).

Figure 4 shows the induced non-local voltage VR and its derivative with respect to VL for several couplings and temperatures, obtained for U(r) = 0. At low voltage the induced potential VR is proportional to VL, as we discussed in the linear response regime (Fig. 3). Upon increasing the ‘driving’ voltage the Shiba states ±EA (indicated by vertical lines in Fig. 4) are gradually activated, amplifying the non-local processes. For we hence observe local minima (maxima) of VR at the quasiparticle energies EA (−EA). Further increase of leads to revival of the dominant ET channel. The derivative dVR/dVL, which is related to the ratio of the local and non-local differential resistances , can be measured by the standard lock-in method. The distinct features observed in the dependence of VR vs. VL in Figs 4 and 5 allow for simple determination of the positions of the Andreev bound states in the system. This is crucial microscopic parameter of the system under study.

Figure 4
figure 4

The non-local voltage VR (left panel) and its derivative dVR/dVL (right panel) induced in the floating R lead in response to the ‘driving’ voltage VL.

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Figure 5
figure 5

The non-local voltage VR and its derivative with respect to VL obtained at low temperature for ε0 = 0 (left panel) taking into account the screening effects U(r).

The right panel shows dVR/dVL for .

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Our results differ qualitatively from the properties of the planar junctions (where the ET and CAR dominated regions are completely interchanged)2,3,4,5 where the non-local transport occurs through the Andreev states, that are localized at two normal-superconductor interfaces separated by a distance d comparable to the coherence length of superconductor. In consequence, the anomalous CAR transport is possible only for eVL exceeding the characteristic Thouless energy19,20,21.

Feedback effect of the long-range potential (where Ueq denotes the equilibrium value incorporated into ε0) is illustrated in Fig. 5. The quantitative changes are observed for all voltages, however, the qualitative behavior is similar to that found in the linear regime (Fig. 4). The screening effects and injectivities are calculated here in the self-consistent way32,33,34,35,47 (discussed in the 2-nd subsection of Methods). This selfconsistent treatment of U(r) partly suppresses both the non-local voltage VR and dVR/dVL. The right panel of Fig. 5 shows dVR/dVL with respect to VL outside the particle-hole symmetry point, i.e. for . These asymmetric curves can be practically obtained by applying the gate potential to the quantum dot.

Discussion

We proposed the three-terminal hybrid device, where the quantum dot is tunnel-coupled to two normal and another superconducting electrode, for implementation of the efficient non-local transport properties. We investigated such effects in the linear and non-linear regimes. We found that in the both cases the non-local resistance/conductance can change from the positive (dominated by the usual electron transfer) to negative values (dominated by the crossed Andreev reflections) upon varying the coupling to superconducting electrode and tuning the QD level ε0.

Some of these effects have been previously addressed theoretically using the perturbative real-time diagrammatic calculations6. The authors of the paper6 argue that: (1) “the negative nonlocal conductance is not due to CAR” and (2) “can only be probed because of a large charging energy that prohibits direct transport between the normal leads”. To understand the seeming discrepancy with our results let’s note that the paper6 focuses on the extremely strong interaction limit , and , where the usual electron tunneling between normal electrodes is suppressed. Nevertheless Futterer et al.6 have found the region of negative non-local conductance/resistance for the bias voltages far from equilibrium. However, as mentioned above, in order to see the effect authors6 need small coupling to the normal electrodes that prohibits direct transport between normal leads, so the subgap transport is dominated by Andreev processes at the interface between quantum dot and the superconducting lead. On the contrary our careful analysis shows that there exists a region of much lower voltages for which the crossed Andreev processes dominate as is visible in Fig. 4 for non-interacting case and Fig. 5 for long range interactions. We have also checked that the Coulomb correlation term in the Hamiltonian Eq. (1) treated within Hubbard I approximation reproduces the results of paper6. In the related work20 dealing with non-interacting chaotic quantum dot the voltages are limited to the values of the order of superconducting gap. It has been demonstrated that the CAR and ET contributions ‘do not cancel each other beyond weak tunneling limit’. The authors find the diminishing of the non-local conductance with increase of the coupling between the dot and the superconducting electrode. However, they have not reported20 the situation with negative (differential) resistance.

This nano-device would enable realization of the strong non-local conductance (comparable to the local one) by activating the Shiba states formed at sub-gap energies ±EA. They substantially enhance all the transport channels, in particular promoting the CAR mechanism (manifested by the negative non-local conductance/resistance) when the coupling to superconducting electrode is strong . We predict the negative non-local conductance/resistance both, in the linear regime and beyond it. For the latter case such behavior would be observable exclusively in the low bias voltage regime capturing the Shiba states. The quantum dot level ε0 (tunable by the gate potential) can additionally control asymmetry of the non-linear transport properties, affecting the CAR transmission .

Strong non-local properties of the nano-device (shown in Fig. 1) can be contrasted with the previous experimental measurements for the three-terminal planar junctions (consisting of two N − S interfaces separated by a superconducting mesoscopic island)2,3,4,5. Russo et al.2 reported evolution from the positive to negative non-local voltage VR induced in response to the ‘driving’ bias VL. At low VL the ET processes dominated, whereas for higher VL the CAR took over. The sign change of VR occurred at voltage VL related to the Thouless energy (such changeover completely disappeared when a width of the tunneling region via the superconducting sample exceeded the coherence length). Similar weak negative non-local resistance/conductance has been observed in the spin valve configurations4,5. In the planar junctions the non-local conductance was roughly 2 orders of the magnitude weaker than the local one4.

Summarizing, we proposed the nanoscopic three-terminal device for the tunable (controllable) and very efficient non-local conductance/resistance ranging between the positive to negative values. Our theoretical predictions can be verified experimentally (in the linear response regime and beyond it) using any quantum dots7,27,28,29,30,31 attached between one superconducting and two metallic reservoirs. It is well known that the interactions of electrons on the dot lead to various many-body phenomena as the Coulomb blockade and the Kondo correlations45, which modify charge transport in the system. These modifications should also be captured in the future experiments using the four probe setup. We provide all necessary details for a realization of this challenging but makable experimental project.

Methods

Landauer-Büttiker formalism

The four-point method36,37 is well established technique for measuring the resistance in a ballistic regime. Voltage Vkl measured between k and l electrodes in response to the current Jij between i and j electrodes defines the local (ij = kl) or non-local resistance via

where is a difference between the chemical potentials of k and l electrodes. The formalism has been later extended by Lambert et al.38,39 to systems, where electron tunneling occurs between one or more superconductors. The current from i-th lead depends on the chemical potential μS of superconducting reservoir, because the scattering region acts as a source or sink of quasi-particle charge due to the Andreev reflection (see e.g. ref. 40).

Adopting this approach, we analyze here the local and non-local transport properties of the three-terminal hybrid system consisting of two normal (L and R) leads coupled through the quantum dot with another superconducting (S) electrode. We consider the charge transport driven by small (subgap) voltages , when the single electron transfer to the superconductor is prohibited. In this limit the net current flowing from the normal L electrode consists of the following three contributions

The linear coefficient refers to the processes transferring single electrons between metallic L and R leads. We call this process as the electron transfer (ET). The other term with corresponds to the direct Andreev reflection, when electron from the normal L lead is converted into the Cooper pair (in S electrode) reflecting a hole back to the same lead L. The last coefficient describes the non-local crossed Andreev reflection, involving all three electrodes when a hole is reflected to the second R lead. In the subgap regime the competing ET and CAR channels are responsible for the non-local transport properties.

In the same way as (9) one can express the current JR. By symmetry reasons we have and , whereas the charge conservation (Kirchoff’s law) implies . From these linear response expressions one can estimate the relevant local and non-local resistances (8), assuming arbitrary configurations of the applied currents and induced voltages. Experimental measurements of such resistances (8) can be done, treating one of the electrodes as a voltage probe. In our three-terminal device with the quantum dot we can assume either the metallic or superconducting electrode to be floating. We now briefly discuss both such options.

Floating metallic electrode

We assume that the superconducting lead S is grounded and treat the metallic electrode (say L) as a voltage probe. This means that the net current vanishes JL = 0 and, from the charge conservation, one finds . In the linear response regime (9) implies the following potential differences

with a common denominator

According to the definition (8) and using (10)–(12) we obtain the local (RRS,RS) and non-local (RRS,RL, RRS,LS) resistances for the floating L lead. Let us notice, that a sign of the non-local resistance RRS,LS depends on a competition between the normal electron transfer (ET) and the crossed Andreev reflections (CAR). The local resistance RRS,RS is in turn a sum of the non-local resistances RRS,RL and RRS,LS. For the configuration, where the other (R) metallic lead is floating we obtain the equations similar to (10)–(12) with the exchanged indices .

Floating superconducting electrode

We encounter a bit different situation, assuming the superconducting S electrode to be floating (i.e. JS = 0). The charge conservation and Eq. (9) imply

We notice some analogy between the resistances (14)–(16) and the previous expressions (10)–(12). The significant difference appears between the non-local resistances RRS,LS (11) and RLR,SR (15). Because of a minus sign in (11) the former configuration seems to be more sensitive for probing the local versus non-local transport properties.

Remarks on the determination of partial conductances

Measurements of the local/non-local resistances provide information about the competition between various tunneling processes. Similar information can be also deduced about the linear coefficients . Let’s combine the results obtained for L (or R) and S floating electrodes. We have three independent equations, but we have to determine four coefficients

In general, we thus cannot obtain a complete information about all conductances from the separate measurements of the currents and voltages. This situation differs from the case when the quantum dot is coupled to all three normal electrodes, where electrical transport can be characterized only by three conductances.

Fortunately, for the case with asymmetric couplings the measurements can unambiguously determine the partial conductances

Some inconvenience is related to the fact the tunneling rates , must be measured as well.

Non-linear transport

The non-linear effects are of vital importance in the transport studies of nanostructures inter alia due to limited screening of charge and access to far from equilibrium states of the system. Non-equilibrium transport driven by the voltage VL (beyond the linear regime) in nanostructures is accompanied by substantial redistribution of the charges. This affects the occupancy of the quantum dot and leads to piling up of the charge in the electrodes. By long range Coulomb interactions the charge redistributions backreact on the transport properties. We shall address this effect in some detail.

Let’s note that we are considering here the charge transport driven by voltages safely below the superconducting gap (practically we assume . Nevertheless, even at such small voltage (of the order of a few the pile-up of electric charges in the electrodes and the dot affects the transport by shifting the chemical potentials and screening the charge on the dot. This is taken into account in the Hamiltonian (1) by the term eU(r).

The effect has been considered first in mesoscopic normal systems by Altshuler and Khmelnitskii47, Büttiker with coworkers32,33 and others34. It has been also explored in the metal-superconductor (two-terminal) junctions35. Here we follow35, assuming that the long range interactions modify the on-dot energy ε0 changing it to ε0 − eU(r). In equilibrium the potential U(r) has a constant value, which we denote by Ueq. In the presence of the applied voltages Vα (where α = L,R,S) the deviations , in the lowest order, would be a linear function

where denotes the derivative with all voltages set to zero and the gauge invariance implies that 32. Our treatment here relies on the mean field like approximation. In the three terminal device with the quantum dot the single electron transport occurs between the left and right normal electrodes, while the (direct and crossed) Andreev processes involve the normal and superconducting electrodes. The currents (3), (4), (5) and the quantum dot charge depend on the screening potential U(r). During the flow of carriers the deviations of δU from the equilibrium value Ueq can be related to the change of the charge carriers δn by the capacitance equation δn = CδU, where C is capacity of the system. The charge density as well as all currents depend on the voltages and δU. This allows to write the relation between δn = n − neq, where neq denotes the equilibrium (i.e. calculated for all voltages set to zero) value of the charge

where Π denotes the Lindhard function. Combining these equations we solve for known in the literature as the characteristic potentials and conveniently denoted by uα. They describe the response of the system to the applied voltages. One finds

For the analysis of voltages induced in the R electrode as a result of current flowing in the L − S branch of the system we need both uL and uR. As in the earlier work35 we assume C = 0 in the following. The inspection of the formula for n reveals that for the symmetric coupling the functions of both electrodes take on the same value uL = uR. The characteristic potentials enter the expression for the Green functions and as a result modify the relation shown in the Fig. 4. The modification is especially severe for .

Let us note that is obtained from matrix elements and of the the Green functions as they depend on the potential U. The calculation of the characteristic potentials uL/R require the derivatives of n with respect to voltages VL/R, which enter the distribution functions. The characteristic functions define in turn the potential , which has to be introduced into the Green functions entering the expressions (3), (4), (5) for the currents.

Additional Information

How to cite this article: Michałek, G. et al. Novel non-local effects in three-terminal hybrid devices with quantum dot. Sci. Rep. 5, 14572; doi: 10.1038/srep14572 (2015).