Topological isoconductance signatures in Majorana nanowires

We consider transport properties of a hybrid device composed by a quantum dot placed between normal and superconducting reservoirs, and coupled to a Majorana nanowire: a topological superconducting segment hosting Majorana bound states (MBSs) at the opposite ends. It is demonstrated that if highly nonlocal and nonoverlapping MBSs are formed in the system, the zero-bias Andreev conductance through the dot exhibits characteristic isoconductance profiles with the shape depending on the spin asymmetry of the coupling between the dot and the topological superconductor. Otherwise, for overlapping MBSs with less degree of nonlocality, the conductance is insensitive to the spin polarization and the isoconductance signatures disappear. This allows to propose an alternative experimental protocol for probing the nonlocality of the MBSs in Majorana nanowires.

www.nature.com/scientificreports/ nanowire using a quantum dot (QD) as a local probe, which was then experimentally performed in Ref 13 . Theoretically, this nonlocality is estimated by computing the ratio between the couplings of the QD with both the MBSs hosted in the nanowire 38 , which also defines the so-called topological quality factor 37,57 .
In the current work, by analyzing the Andreev current through a quantum dot (QD) placed between metallic (N) and superconducting (S) reservoirs and coupled to a TSC hosting MBSs (Majorana nanowire), see Fig. 1a [58][59][60][61] , we theoretically propose an additional protocol to differentiate between the corresponding foregoing cases of: (A) Highly nonlocal and nonoverlapping MBSs: corresponds to the case of long nanowires, L ≫ ξ M , wherein L is the TSC section length and ξ M is the Majorana coherence length 28 . In such a situation, the wave functions of the MBSs are well localized at the TSC nanowire ends, leading to both a zero overlap between them and a zero coupling of QD state with the faraway (right) MBS. (B) Nonlocal and overlapping MBSs: describes the opposite case of shorter nanowires ( L ξ M ), wherein the wave functions of MBSs can overlap with each other 28,51,53,62 and the QD also can couple with the outer (right) Majorana state 13,34,37,38 .
For the ideal situation of nonoverlapping and highly nonlocal MBSs (A), the Andreev conductance profiles reveal strong dependence on the parameter which characterizes the spin asymmetry of the coupling between the QD and the TSC. More specifically, the zero-bias Andreev conductance as a function of both the gate-voltage defining the position of the energy level of the QD and the strength of the hybridization between the QD and superconducting lead exhibits isoconductance lines with maximum value of e 2 /h . Their shape strongly depends on the spin asymmetry of the system. However, for the opposite case of overlapping MBSs with lower degrees of nonlocality (B), the sub-gap Andreev conductance becomes spin-independent, and the aforementioned isoconductance profiles with its characteristic e 2 /h value disappear. Instead, the Andreev conductance at zero-bias shows either a non-quantized peak or a dip, depending on the relative values of parameters which characterizes the direct overlap between the MBSs and the coupling between the QD and the outer (right) MBS. Thus, our findings contribute to the endeavor of characterizing the nonlocality of MBSs by means of sub-gap Andreev conductance measurements using a QD as a local probe.

Results and discussion
In what follows, we analyze the sub-gap Andreev conductance at zero-temperature limit using the value of Ŵ N as energy unit, with has the order of µeV 18  It consists of a QD working as a local probe coupled to normal (N) and superconducting (S) leads and a segment of a semiconductor nanowire covered by an s-wave superconductor layer. In the presence of an external magnetic field parallel to the wire, the latter is driven into a topological superconducting state, with Majoranas bound states (MBSs) formed at its opposite ends. (b) The scheme illustrating spin-dependent transport channels in the system. Finite bias voltage eV is applied between superconducting (S) and normal (N) reservoirs. An incoming electron from the normal reservoir with a certain spin is injected into the QD and is reflected back as a hole. In the same time, a Cooper pair is formed either in the superconducting reservoir, where it has ordinary s-wave character, or in the TSC, where it has a p-wave symmetry. The interplay between the transport through S and TSC terminals defines the spin orientation of the reflected hole with respect to the spin of the incoming electron. Highly nonlocal and nonoverlapping MBSs. We start with the ideal situation of nonoverlapping ( ε M = 0 ) and highly nonlocal MBS ( Rσ = 0 ), with spin-independent QD-TSC coupling, putting p = 0.5 , Figure 2a shows the Andreev conductance as a function of both the bias-voltage eV and the gate-voltage eV g , which shifts the position of the energy levels of the QD, for Ŵ S = 3.0Ŵ N . One can clearly notice the presence of the pronounced four peak structure around eV = 0 corresponding to the well resolved Andreev levels, appearing due to the QD-TSC coupling and splitted by the external magnetic field. Moreover, there is a visible zero-bias structure present because of the leakage of an isolated MBSs into the QD 18,58,61,65 , whose amplitude G A (eV = 0) changes with eV g , and reaches the maximal value of e 2 /h for eV g = −1.0Ŵ N .
In Fig. 2b we demonstrate how Andreev conductance amplitude at zero-bias also changes as a function of both eV g and QD-S hybridization strength Ŵ S for the same case of p = 0.5 . Experimentally, one can change Ŵ S continuously while tuning the QD level eV g by employing a dual-gate device geometry 66 . The maximal value of the conductance e 2 /h is reached along the white vertical dotted line, which we call isoconductance line. For this particular spin-independent situation, the position of this line is defined by the condition of particle-hole symmetry, reached when eV g = −1.0Ŵ N . This condition is broken in spin asymmetric case, when ↑ � = ↓ 60 , which leads to the distortion of the isoconductance line in the (eV g , Ŵ S ) space, as we shall see. Note also that along the isoconductance line, the zero bias conductance does not depend on the value of Ŵ S , so the QD becomes effectively decoupled from the S lead and the transport through it is uniquely defined by its pairing to the TSC.
The opposite case of fully spin polarized transport, corresponding to p = 0 , ↑ = and ↓ = 0 is illustrated by Fig. 2c,d. The profile of the conductance as a function of the bias and gate-voltages becomes asymmetric, as it can be clearly seen in Fig. 2c. Zero-bias conductance peak still appears, but the isoconductance line defined by the condition G A (eV = 0) = e 2 /h is not a straight vertical line, but has a more complicated shape shown in Fig. 2d. Note that differently from the case shown in Fig. 2b, the isoconductance line has a minimum, which means that maximal value of the zero-bias conductance e 2 /h can not be reached below certain critical value of the coupling between the QD and the S lead. The intermediate case of p = 0.3 is illustrated by Fig. 2e,f. www.nature.com/scientificreports/ The comparison between the three sets of panels of Fig. 2 allows us to conclude that the presence of an isoconductance plateau corresponding to the vertical isoconductance line in eV g , Ŵ S coordinates can be considered as a hallmark of spin symmetric coupling between the QD and the TSC.
It is worth mentioning that e 2 /h quantized Andreev conductance amplitude characteristic for highly nonlocal and nonoverlapping MBSs ( ε M = Rσ = 0) 60 at T = 0 is distinct from 2e 2 /h value typical for the normal conductance through a N-QD-N geometry 67 without the presence of the TSC section ( Lσ = 0 ). In such case, away from the Kondo regime, for e.g. spin-polarized coupling ( L↑ � = 0 and L↓ = 0 ), the density of states corresponding to the spin ↑ drops to e 2 /2h owing to the coupling with an isolated MBS, while the same quantity for the spin ↓ remains unaffected ( e 2 /h ), giving rise to a ZBP height of 3e 2 /2h 67,68 . However, if a N-QD-S geometry is considered, the pairing induced into the QD by the S lead mixes the spins [Eq. (4)] and hence both spin channels are affected by the presence of the MBSs, even in the fully spin-polarized situation with p = 0 60 . This interference process between the spin channels mediated by the S-lead reduces the Andreev conductance maximum amplitude from 4e 2 /h to e 2 /h (see e.g. Fig. 5).
Although finite temperature effects can flatten the isoconductance plateaus of Fig. 2 64 , quantized conductance values near e 2 /h may still be obtained in a realistic experimental situation when the QD-TSC coupling Lσ is dominant over both the temperature k B T and the overlap ε M between the MBSs 41,69 . Previously some of us have studied the interplay between the thermal broadening k B T and the overlap strength ε M of the MBSs via NRG analysis 69 . It was shown that overlapping MBSs can become decoupled from each other when the system is driven into a specific fixed point due to finite temperature effects, fully recovering the ZBP signature.
In Fig. 3, we analyze the behavior of the isoconductance profiles with the Coulomb correlation U for the ideal case of nonoverlapping ( ε M = 0 ) and highly nonlocal MBSs ( Rσ = 0 ), for the same values of spin anisotropy parameter p as in Fig. 2. For the spin-independent case (Fig. 3a-c), the isoconductance lines depicted by the white-dashed lines have their positions at the eV g -axis shifted as U changes, being pinned at eV g = −U/2 . The corresponding fully-spin polarized and partially spin-polarized cases are illustrated by the middle [(d-f)] and lower panels [(g-i)] of Fig. 3, respectively. They also reveal similar shift with the change of U, demonstrating the robustness of isoconductance with respect to electron-electron interactions.
It should be noted that introduction of the additional transport channels (multiple subbands) in the TSC nanowire of a considerable thickness can lead to the vanishing of isoconductance signatures of highly isolated MBSs 28,70,71 , as the nonzero occupancy of multiple subbands plays role of an effective disorder 71 , thus leading to the formation of trivial low-energy states which may emulate the signatures of MBSs. One thus needs to use In the upper panels, Andreev conductance as a function of the bias and gate-voltages for the fixed value of Ŵ S = 3.0Ŵ N is shown. Direct comparison with upper panels of Fig. 2 shows that conductance profiles are qualitatively the same for the cases of topological nonoverlapping MBSs. However, if one turns to zero-bias conductance as a function of the gate voltage eV g and QD-S lead coupling Ŵ S , the results are totally different. It was already stated that for the case of highly nonlocal and nonoverlapping MBS ( ε M = Rσ = 0 ), the maximal value G A (eV = 0) = e 2 /h is reached along certain open isoconductance lines (Fig. 2b,d,f). The situation for the case of overlapping MBSs is qualitatively different. Indeed, it can be clearly seen from the lower panels of Fig. 4 that the condition G A (eV = 0) = e 2 /h at zero-temperature is reached along the closed lines, which now can not be considered as isoconductance lines, as inside them the value of the conductance exceeds e 2 /h . This remarkable difference suggests a further experimental criterion for distinguishing between the topological cases 28,38 of overlapping and nonoverlapping MBSs localized at the edges of the TSC nanowire.
To study in more detail the corresponding crossover, we analyzed the zero-bias Andreev conductance as a function of eV g and Ŵ S for several values of the parameter ε M , characterizing the overlap between the different MBSs well-localized at the edges. The results are shown in  www.nature.com/scientificreports/ quasi-plateaus transform into non-monotonous curves corresponding to the onset of strongly overlapped MBSs, characterizing a nonlocal fermionic state with finite energy.
In Fig. 6 we study the case in which the wave function of right MBSs overlaps with the QD state, leading to a finite coupling between them ( R ≫ ε M ) 13,37,38 . Within this case, we also consider the situations of spinindependent ( p = 0.5 ), fully spin-polarized ( p = 0 ) and intermediary ( p = 0.3 ) QD-TSC couplings. In the upper panels, Andreev conductance profiles as a function of bias and gate-voltage for Ŵ S = 3.0Ŵ N , L = 2.0Ŵ N and R = 0.5Ŵ N are shown. One can clearly see, that for all the values of p the zero-bias Andreev conductance almost drops to zero, which is quite distinct from the cases of highly nonlocal and nonoverlapping MBSs and overlapping MBSs with R = 0 , see Figs. 2 and 4 . This pronounced drop is also seen in the lower panels showing G A (eV = 0) as a function of the QD gate-voltage eV g and QD-S lead coupling Ŵ S . Isoconductance signatures are completely absent for all values of the parameter p. Figure 7a-c shows Andreev conductance profiles as a function of eV g and Ŵ S for increasing values of R , allowing to investigate the crossover from highly nonlocal MBSs ( R = ε M = 0 ) to MBSs with lesser degree of nonlocality ( R ≫ ε M ). In panel (a), one can easily spot the isoconductance line with G A (eV = 0) = e 2 /h for the case of highly nonlocal Majoranas. However, as its nonlocal feature is suppressed with increase of R , the isoconductance profile disappears and G A approaches to zero, see panels (b) and (c), which is quite distinct from the previous situation of overlapping MBSs well-localized at the TSC section ends ( ε M = 0 , R = 0 ) (Fig. 5), where the zero-bias conductance almost reaches its maximal value of 4e 2 /h . The zero-bias Andreev conductance behavior as a function of Ŵ S for eV g = −1.0Ŵ N is shown in panel (d), where the plateau of e 2 /h appears only for the topologically protected case of highly nonlocal MBSs, corresponding to the flat red dotted line.
The characteristic drop in the Andreev conductance at eV = 0 shown in Figs. 6 and 7 for R ≫ ε M comes from interference phenomena between distinct transport channels due to the leakage of the left and right MBSs with different strengths ( L > R ). In other words, there is a formation of a nonlocal fermion through the QD coming from the unbalanced combination of left and right MBSs, leading to the above mentionated interference process. This underlying mechanism is quite distinct from that one for the opposite case of ε M ≫ R (Figs. 4 and  5), where the left and right MBSs localized at opposite ends of the TSC section overlap with each other directly. Hence, the QD perceives the TSC section as a nonlocal fermionic state with energy ε M , giving rise to a peak at eV = 0 in the Andreev conductance.
In Fig. 8, we investigate the crossover between the opposite cases of ε M ≫ R and ε M ≪ R for p = 0.5 , considering two distinct sets of parameters corresponding to left (a-g) and right panels (h-n). In the left panels of Fig. 8, the parameters are the same adopted in Fig. 7, but for the specific situation of finite overlap ε M = 0.05Ŵ N www.nature.com/scientificreports/ between the MBSs. Panel (a) exhibits the situation where the right MBS does not overlap with the QD ( Rσ = 0 ), depicting an Andreev conductance peak at zero-bias with its amplitude higher than the corresponding isoconductance plateau of e 2 /h . This peak at eV = 0 indicates the formation of a nonlocal fermionic state with energy ε M coming from the combination between the Majorana components at the opposite ends of the TSC section, as discussed earlier (Fig. 5). However, as the wave function of the right MBS overlaps with QD, the coupling Rσ acquire finite values and the Andreev conductance peak at eV = 0 is suppressed (Fig. 8b-d). Such a conductance drop gets more pronounced when the regime of R > ε M is reached (Fig. 8e), leading to the formation of a dip in which the Andreev conductance is strongly suppressed at zero-bias for the situation of R ≫ ε M (Fig. 8f,g). Similar behavior for this peak-dip transition is also found for other parameters adopted, as seen in the right panels of Fig. 8. The Andreev conductance profiles shown in Fig. 8 reveal that the peak-dip crossover mechanism at zero-bias is ruled by the relative values of two energy scales: the overlap ε M between the MBSs and the QD-right MBS hybridization R . It was previously shown that these quantities also govern the emergence of distinct profiles for the QD-MBSs energy spectrum in absence of the S-lead 13,37,38,57 .
In Fig. 9, we summarize the main differences between (A) highly nonlocal and nonoverlapping MBSs and (B) nonlocal and overlapping MBSs in the Andreev conductance spectra as a function of bias-voltage for the spin symmetric case ( p = 0.5 ), with L = 2.0Ŵ N , Ŵ S = 3.0Ŵ N and eV g = −1.0Ŵ N . For the ideal situation of highly nonlocal and nonoverlapping MBSs ( ε M = R = 0 , orange solid line), corresponding to Fig. 2a, a ZBP with quantized amplitude of e 2 /h and satellite peaks describing the Andreev levels formed in the QD due to the coupling with S-lead are observed. For well-localized, but overlapping MBSs ( ε M ≫ R , teal dotted line), corresponding to Fig. 4a, the ZBP is not quantized anymore and its height depends on the parameters of the system, as e.g. Ŵ S (see Fig. 5). A non-quantized ZBP also characterizes the situation where the overlap between the MBSs is comparable with the QD-right MBS coupling ( ε M = R , purple dash-dotted line). However, for the case where the right MBS wavefunction strongly overlaps with the QD ( ε M ≪ R , magenta dotted line) corresponding to Fig. 6a, the ZBP is replaced by a zero-bias dip, which reaches zero (Fig. 7). This peak-dip transition suggests an additional protocol for distinguish between overlapping but well-localized MBSs and less nonlocal MBSs.
Concerning the zero-bias conductance profiles as a function of both the QD-S hybridization Ŵ S strength and the QD gate-voltage eV g , we notice that isoconductance lines appear only for the ideal case of highly nonlocal zero-energy MBSs ( Rσ = 0 ) and zero overlap ε M between each other (Fig. 2, lower panels). For the situation of almost zero-energy MBSs characterized by finite but small ε M , the plateau which originates the isoconductance www.nature.com/scientificreports/ lines is slightly distorted (Fig. 5f). Thus, within our effective model we can infer that the robustness of the isoconductance signatures arise from both the zero-energy and nonlocal nature of the MBSs, since for both overlapping MBSs with more ( ε M ≫ Rσ ) or less ( ε M ≪ Rσ ) nonlocal feature, the MBSs cannot be characterized as true zero-energy and highly nonlocal states anymore. However, since fine-tuned ABSs, quasi-MBSs or disordered-induced zero-energy modes can also induce ZBPs 19,28,30 , it should be emphasized that a study using more detailed models 27,39 is required in order to investigate if the isoconductance lines can also appear for these topologically trivial subgap states.

Conclusions
We have studied the sub-gap Andreev conductance G A through a quantum dot (QD) connected to metallic and superconducting leads and additionally coupled to a hybrid topological semiconducting nanowire (TSC) hosting Majorana bound-states (MBSs) at the opposite ends. For nonoverlapping and highly nonlocal MBSs, corresponding to the ideal case of long and pristine Majorana nanowires, the profiles of G A as functions of both quantum dot gate-voltage and hybridization between the dot and the superconducting reservoir reveal pronounced isoconductance signatures with maximum amplitude of e 2 /h , sensitive to spin anisotropy of the coupling between the QD and the TSC. However, in the situation of shorter Majorana nanowires, the MBSs remain nonlocal but overlap with each other or lose its nonlocal feature. Hence, such isoconductance signatures disappear, giving rise to a nonquantized zero-bias peak for the former situation and a zero-bias dip for the latter. This suggests that the analysis of the sub-gap Andreev conductance profiles by means of a local probe can be employed as an additional tool to distinguish MBSs with distinct degrees of overlap and nonlocality.

Methods
Theoretical model. To describe transport properties of the system sketched in Fig. 1, we use the following Anderson-type Hamiltonian 58,60,72 : w h e r e H N = kσ ε N k c † Nkσ c Nkσ a n d H S = kσ ε S k c † Skσ c Skσ − k (�c † Sk↑ c † S−k↓ + h.c.) r e p r esent the N and S reservoirs, respectively, with electron energies ε α k , spin σ =↑, ↓ and superconducting energy gap . www.nature.com/scientificreports/ reservoir and the QD, characterized by the coupling strength V αkσ . The QD is described by the Hamiltonian H QD = σ ε dσ d † σ d σ + Un d↑ n d↓ , corresponding to a pair of nondegenerate energy levels with the energies ε dσ = eV g − σ V Z , that can be tuned by a tunnel gate eV g in presence of an external magnetic field inducing the Zeeman splitting V Z , and U corresponds to the Coulomb repulsion between electrons with opposite spins.
The TSC section can be modeled by the following low-energy effective Hamiltonian 38,73 : where the Hermitian operators γ i = γ † i (i = L, R) describe the MBSs localized at the left (L) and right (R) of the TSC segment [marked in purple in Fig. 1a] 2,3 . The parameter ε M describes the overlap between the opposite MBSs, while i,σ (i = L, R) characterizes the coupling between the QD and the left/right MBSs, with spin σ =↑, ↓ . The overlap ε M decays exponentially with the TSC length and oscillates around zero with some system parameters, as the TSC length, chemical potential and applied magnetic field 38,[51][52][53]74 . Hence, ε M can reach zero at specific values of parameters space (oscillation parity crossings) for shorter TSC sections.
In the highly nonlocal and nonoverlapping case (A), the MBSs are well-localized at the ends of the TSC section ( R = 0 ) with an exponentially suppressed overlap between them ( ε M = 0 ). However, for the situation of nonlocal and overlapping MBSs (B), the overlap ε M can be either finite or zero owing to its oscillatory behavior.
(2) www.nature.com/scientificreports/ When considered, R also can oscillate, but remains finite 38 . For all situations R < L , once the left MBSs couples with the QD more strongly. The effective Hamiltonian of Eq. (2) was derived from its corresponding tight-binding model in Ref. 38 . It should be emphasized that the effective model of Eq. (2) trustingly describes the low-energy spectrum of the QD-TSC system 37,38,40 , being able to reproduce qualitatively the experimental results 12,13 . Eq. (2) can be rewritten in the regular spinless fermionic basis by using the transformation γ L = 1 75 , with f † (f ) being nonlocal fermions with ordinary Fermi-Dirac statistics.
It should be specifically stressed that although the TSC section hosting MBSs is effectively spinless 4,37,53,76 , the coupling of the QD to the MBSs depends on the spin texture of the latter, by means of the canting angles θ L,R 38 of the left and right MBSs, with Lσ = L (sin θ L 2 , − cos θ L 2 ) and Rσ = −ı R (sin θ R 2 , cos θ R 2 ) , where iσ ≡ ( i↑ , i↓ ) . These canting angles depend on the applied magnetic field and spin-orbit coupling in the nanowire. These couplings also depend on the effective distance between the QD and the TSC segment 73 . A detailed analysis of the effects of canting angles in the QD-TSC spectrum is beyond our proposal. Thus, the spin-dependency in the QD-MBSs couplings is here accounted by the straightforward introduction of a generic polarization parameter p ∈ [0, 1] 60,77 only for ensuring the possibility of spin asymmetry in such couplings, so that i,↑ = i (1 − p) and i,↓ = i p , where i ≡ | i | stands for the maximum coupling amplitudes.
Since we are interested in sub-gap Andreev transport features through the QD and its relation with the MBSs, we restrict ourselves to the limiting case of large superconducting gap | | → ∞ 60,65,78 . It is well known that in this regime the S lead induces static s-wave pairing in the QD due to proximity effect. This allows to trace out the S lead from the Hamiltonian by using the substitution . Away from the Kondo regime 38,58,66,83 , the effects of the Coulomb blockade in the energy spectrum of the QD coupled to both S and N leads are well-described within the following self-consistent Hartree-Fock approximation (HFA) 38,[84][85][86] : are the average occupation and s-wave pairing amplitude in the QD, respectively. Both quantities should be numerically computed self-consistently. Thus, the system Hamiltonian given by Eq. (1) can be rewritten as: where ε dσ = ε dσ + U�n dσ � and Ŵ S = Ŵ S + U�d ↓ d ↑ �.
It should be noted that other methods can be employed to treat the effects of the Coulomb correlations in N-QD-S systems, as e.g., the slave-boson mean-field approximation (SBMFA) 81,87-89 in the strong correlated limit (3) Un d↓ n d↑ ≈ U(�n d↓ �n d↑ + n d↓ �n d↑ �−  www.nature.com/scientificreports/ U → ∞ , where the doubly-occupied state in the QD is traced out. Hence, the transformation introduced by the SBMFA reduces the problem into a Fermi liquid with renormalized parameters Ŵ N,S and ε d 81,89 . However, this approach is valid only in the deep Kondo regime T K ≫ , where T K is the Kondo temperature 89 , ruling out the possibility of forming a BCS-like singlet in the QD ground state. A faithful analysis of the transition between the Kondo spin-singlet and the BCS-like superconducting singlet is only possible via Numerical Renormalization Group (NRG) technique 66,78,81 , which also allows to study the interplay between the Kondo effect and the MBSs in N-QD-S junctions coupled to Majorana nanowires 60 . The analysis of this interplay, as well as the study of Kondo-BCS singlet transition, goes beyond the scope of the present work, in which we limit ourselves to the consideration of the case away from the Kondo regime 86 .
Sub-gap Andreev conductance. In a N-QD-S system, the total conductance through a QD is given by the sum of two channels, G N (V ) + G A (V ) , where the first term is the normal electron tunneling conductance and the second one is the Andreev conductance 65,80 . G N (V ) gives dominant contribution to the transport outside the gap ( |eV | ≥ ), while G A (V ) contributes mainly to the subgap electronic transport ( |eV | < � ). At very low temperatures, when the bias-voltage eV applied between the normal and superconducting reservoirs is far from the superconducting gap edges ( |eV | ≪ ), the electronic transport takes place exclusively due to the process of Andreev reflection 90 , see Fig. 1b. The corresponding differential Andreev conductance can be calculated as 58,60,91 : where eV ≡ µ N − µ S and is the sub-gap transmittance due to Andreev reflection processes, which depends on the anomalous Green's functions ��d † σ ; d † σ �� in the spectral domain ω , with Ŵ N = π k |V Nkσ | 2 δ(ω − ε α k ) being the effective broadening of the QD energy levels.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. www.nature.com/scientificreports/