Novel non-local effects in three-terminal hybrid devices with quantum dot

We predict non-local effect in the three-terminal hybrid device consisting of the quantum dot (QD) tunnel coupled to two normal and one superconducting reservoirs. It manifests itself as the negative non-local resistance and results from the competition between the ballistic electron transfer (ET) and the crossed Andreev scattering (CAR). The effect is robust both in the linear and non-linear regimes. In the latter case the screening of charges and the long-range interactions play significant role. We show that sign change of the non-local conductance depends on the subgap Shiba/Andreev states, and it takes place even in absence of the Coulomb interactions. The effect is large and can be experimentally verified using the four probe setup. Since the induced non-local voltage changes sign and magnitude upon varying the gate potential and/or coupling of the quantum dot to the superconducting lead, such measurement could hence provide a controlled and precise method to determine the positions of the Shiba/Andreev states. Our predictions ought to be contrasted with non-local effects observed hitherto in the three-terminal planar junctions where the residual negative non-local conductance has been observed at large voltages, related to the Thouless energy of quasiparticles tunneling through the superconducting slab.

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 Γ S 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-tubes 7,27 , semiconducting nano-wires 28,29 or self-assembled InAs islands 30,31 .
It has to be stressed that similar structure has been analyzed previously 6,20 . 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 ≈ Γ 1000 L they have not considered charge redistribution in the electrodes and the screening effects due to the long range Coulomb interactions. Golubev and Zaikin 20 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 transport 32   -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 approach [36][37][38][39][40][41][42] (see the 1-st subsection of Methods). On a microscopic level, we describe this system in the tunneling approximation 43  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 ε 0 shifted by the long-range potential U(r). Hybridization between the QD and itinerant electrons is characterized by the matrix elements t α . 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 ρ α 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 J α flowing from an arbitrary lead α = , , L R S { } can be evaluated using the Heisenberg equation 44 . In particular, the current J L(R) from the normal L (R) electrode is given by 44 11 11 where G r 11 and < G 11 denote the matrix elements (in the Nambu representation) of the retarded and lesser QD Green functions, respectively. From now onwards we consider the current J L 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 as 45 Transmission of the anomalous CAR channel, on the other hand, depends on the off-diagonal part of the matrix Green function . It also has maxima around the same Shiba states ± E A but with a different amplitude, sensitive to the induced pairing 〈 〉 ↓ ↑ d d . This is a reason why T CAR (E) quickly diminishes whenever Γ S is decreased or the QD level ε 0 departs from μ S = 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 Γ S (to superconducting electrode) is sufficiently strong and the QD level ε 0 is close to the chemical potential μ S . Otherwise, the non-local effects are dominated by the single electron tunneling (ET). The related changeover can be detected by measuring the voltage V R in the floating R electrode, in response to the current in the L − QD − S branch. Such voltage V R 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 scheme [36][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 β L ij for β = ET, DAR or CAR can be determined from the equations (3)-(5) and they read Treating the potential V S as a reference level we analyze the induced voltage V R in response to the 'driving' current The left panel shows that , R RS LS has a negative sign (signifying the dominant CAR processes) only for sufficiently strong coupling Γ > Γ S N . 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 ε 0 . In the linear regime the negative nonlocal resistance occurs when ε 0 ~ μ S for sufficiently strong coupling Γ > Γ S N . Since Γ S and ε 0 can be experimentally varied in the realizations of the superconducting-metallic devices with the quantum dots 7,27-31 , such qualitative changes should be observable.
Beyond the linear response limit. To confront these findings with the non-local effects observed so far in the 'planar' junctions 2-5 we now go beyond the linear response framework. For arbitrary value of the 'driving' voltage V L we computed self-consistently V R , guaranteeing the net current J R 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 dot 47 . It affects the chemical potentials and the injectivities of the leads and contributes to the screening effect [32][33][34][35] . The potential U(r) has to be properly adjusted, depending on specific polarization of the system 33 (for details see the 2-nd subsection of Methods). Figure 4 shows the induced non-local voltage V R and its derivative with respect to V L for several couplings Γ S and temperatures, obtained for U(r) = 0. At low voltage V L the induced potential V R is proportional to V L , as we discussed in the linear response regime (Fig. 3). Upon increasing the 'driving' voltage V L the Shiba states ± E A (indicated by vertical lines in Fig. 4) are gradually activated, amplifying the non-local processes. For Γ > Γ S N we hence observe local minima (maxima) of V R at the quasipar-  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 eV L exceeding the characteristic Thouless energy [19][20][21] .
Feedback effect of the long-range potential 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 way [32][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 V R and dV R /dV L . The right panel of Fig. 5 shows dV R /dV L with respect to V L outside the particle-hole symmetry point, i.e. for ε = ± Γ L 0 . 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 Γ S and tuning the QD level ε 0 .
Some of these effects have been previously addressed theoretically using the perturbative real-time diagrammatic calculations 6 . The authors of the paper 6 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 paper 6 focuses on the extremely strong interaction limit /Γ = U 1000 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 authors 6 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 ↑ ↓ U n n in the Hamiltonian Eq. (1) treated within Hubbard I approximation reproduces the results of paper 6 . In the related work 20 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 reported 20 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 ± E A . 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 Γ > Γ + Γ S L R . 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 ε 0 (tunable by the gate potential) can additionally control asymmetry of the non-linear transport properties, affecting the CAR transmission ε (± ) ∝ + ( /Γ ) 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) 2-5 . Russo et al. 2 reported evolution from the positive to negative non-local voltage V R induced in response to the 'driving' bias V L . At low V L the ET processes dominated, whereas for higher V L the CAR took over. The sign change of V R occurred at voltage V L 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 configurations 4,5 . In the planar junctions the non-local conductance was roughly 2 orders of the magnitude weaker than the local one 4 .

Strong non-local properties of the nano-device (shown in
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 dots 7,27-31 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 correlations 45 , 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 method 36,37 is well established technique for measuring the resistance in a ballistic regime. Voltage V kl measured between k and l electrodes in response to the current J ij between i and j electrodes defines the local (ij = kl) or non-local ( ≠ ) ij kl resistance via l 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 μ S 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 L LR ET 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 L LL DAR 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 L LR CAR 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 J R . By symmetry reasons we have  (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 J L = 0 and, from the  We notice some analogy between the resistances (14)- (16) and the previous expressions (10)- (12). The significant difference appears between the non-local resistances R RS,LS (11) and R LR,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 β L ij . 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 Γ ≠ Γ R L the measurements can unambiguously determine the partial conductances Some inconvenience is related to the fact the tunneling rates Γ L , Γ R 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 V L (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 < ∆ e V (practically we assume ∆ Γ ) 100 L . Nevertheless, even at such small voltage (of the order of a few Γ ) L 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 Khmelnitskii 47 , Büttiker with coworkers 32,33 and others 34 . It has been also explored in the metal-superconductor (two-terminal) junctions 35 . Here we follow 35 , assuming that the long range interactions modify the on-dot energy ε 0 changing it to ε 0 − eU(r). In equilibrium the potential U(r) has a constant value, which we denote by U eq . In the presence of the applied voltages V α (where α = L,R,S) the deviations δ = ( ) − U U U r eq , in the lowest order, would be a linear function where (…) 0 denotes the derivative with all voltages set to zero and the gauge invariance implies that . 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)  depend on the screening potential U(r). During the flow of carriers the deviations of δU from the equilibrium value U eq 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 − n eq , where n eq denotes the equilibrium (i.e. calculated for all voltages set to zero) value of the charge 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 u L and u R . As in the earlier work 35 we assume C = 0 in the following. The inspection of the formula for n reveals that for the symmetric coupling Γ = Γ L R the functions of both electrodes take on the same value u L = u R . 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 > Γ V L L . Let us note that ( ) is obtained from matrix elements G r 11 and G r 12 of the the Green functions as they depend on the potential U. The calculation of the characteristic potentials u L/R require the derivatives of n with respect to voltages V L/R , which enter the distribution functions. The characteristic functions define in turn the potential = + U u V u V L L