A Short wire case

Kondo and Majorana signatures near the singlet-doublet quantum phase transition

Abstract

We study the low energy quasiparticles driven by the Kondo effect and the proximity induced Majorana state in a setup, comprising the correlated quantum dot coupled between the conducting and superconducting reservoirs and additionally hybridized with the topological superconducting nanowire. We investigate mutual interplay of the Kondo and Majorana states appearing near the quantum phase transition/crossover from the (spinless) BCS-type singlet to the (spinful) doublet ground states. We find novel properties, steming from the subgap Kondo effect combined with the Majorana quasiparticle and discuss their signatures in the tunneling conductance, originating from the particle-to-hole (Andreev scattering) mechanism.

I Introduction

Recently there has been enormous activity devoted to quasiparticles, resembling the Majorana fermions Alicea (2012); Leijnse and Flensberg (2012); Stanescu and Tewari (2013); Beenakker (2013); Elliott and Franz (2015) that are identical with their own antiparticles. Such exotic objects have been predicted at topological defects Volovik (1999) or at boundaries of the symmetry broken systems Read and Green (2000); Kitaev (2001) and their non-Abelian character could be useful for the quantum computing and/or novel spintronic devices Liu et al. (2016). Although such Majorana quasiparticles has been predicted in various systems Tewari et al. (2007); Fu and Kane (2008); Nilsson et al. (2008); Sato and Fujimoto (2009); Wimmer et al. (2010); Sau et al. (2010); Oreg et al. (2010); Lutchyn et al. (2010); Choy et al. (2011); Jiang et al. (2011); San-Jose et al. (2012), their experimental realization has been so far observed in the ballistic tunneling Mourik et al. (2012); et al. (2016) and in STM measurements Nadj-Perge et al. (2014); Pawlak et al. (2016); Ruby et al. (2015); Jeon et al. (2017) via nanowires proximity-coupled to the bulk -wave superconductors.

Inspired by the work by M.T. Deng et al. Deng et al., 2016, who provided experimental evidence for the Majorana modes leaking onto the quantum dot side-attached to InAs nanowire in its topologically nontrivial superconducting state, we consider the setup (displayed in Fig. 1) studying interplay between the Kondo effect and Majorana quasiparticle (both manifested near the zero energy). Andreev states coalescing into the zero-energy Majorana state have been discussed at length by several authors Liu et al. (2017); Hoffman et al. (2017); Ptok et al. (2017), but influence of the correlations has been addressed on the Hartree-Fock level Prada et al. (2017), the equation of motion approach Barański et al. (2017), or by means of NRG Chirla and Moca (2016) for very weak coupling to the conducting lead. Our present study focuses on the subgap Kondo effect, originating in the proximitized and correlated quantum dot solely due to this coupling. Such situation could be realized in STM-type geometry, similar to what has been used by the Princeton Nadj-Perge et al. (2014) and Basel Pawlak et al. (2016) groups. We thus propose to use a nanoscopic chain of magnetic atoms (e.g. Fe) with one (or a few) side-coupled nonmagnetic atoms (for instance Ag or Au) deposited on the superconducting substrate (like Pb) and probed either by the normal or ferromagnetic STM tip. The latter possibility has been recently achieved by A. Yazdani and coworkers Jeon et al., 2017.

Figure 1: Schematic illustration of the quantum dot (QD) laterally coupled to the metallic (N) and superconducting (S) electrodes and additionally hybridized with the Rashba nanowire, hosting the Majorana quasiparticles and at its edges.

We thus analyze the quantum dot (QD) placed between a superconducting (S) and metallic (N) lead and side-coupled to the Rashba nonowire, hosting the Majorana quasiparticles at its ends (Fig. 1). This N-QD-S setup hybridized with the Majorana quasiparticle has a geometry of T-shape junction. We shall investigate here the Majorana quasiparticle near the quantum phase transition/crossover from the (spinless) BCS-type to the (spinfull) singly occupied ground state Žitko et al. (2015), studying its interplay with the Kondo effect and electron pairing.

Similar configurations with two metallic Liu and Baranger (2011); López et al. (2014); Lee et al. (2013); Cao et al. (2012); Vernek et al. (2014); Gong et al. (2014); Liu et al. (2015); Stefański (2015); Li et al. (2015) or ferromagnetic Weymann (2017); Weymann and Wójcik (2017) electrodes have been studied by other groups. It was found that the Majorana quasiparticle ,,leaking” into the QD region strongly affects the Kondo resonance. Its influence could be observed in the linear conductance. It was established that for infinitely long chain (where overlap between Majorana end-modes is absent) the linear conductance reaches , whereas in the case of short chain (with nonzero overlap between Majoranas) the conductance peak reaches López et al. (2014); Lee et al. (2013); Weymann (2017); Weymann and Wójcik (2017). Further analysis of the thermoelectric properties of such N-QD-N setup revealed that for small overlap the thermopower reverses its sign López et al. (2014); Weymann (2017); Weymann and Wójcik (2017).

Such T-shape junctions with one normal and one superconducting electrode have been much less explored Chirla and Moca (2016); Barański et al. (2017); Gong et al. (2014); Wang et al. (2016). Owing to the proximity effect electrons of the QD are bound in pairs Balatsky et al. (2006); Domański (2010), so physical processes engaging electrons of a given spin would simultaneously affect its opposite spin partner Golub (2015). In particular, the side-attached Majorana quasiparticle hybridized with, let’s say, electron affects the energy spectrum of electrons. Consequently, both spin channels become interdependent. This effect is particularly important when considering the Andreev processes, in which both of the spin components equally participate in the subgap transport process.

The paper is organized as follows. In Sec. II we formulate the microscopic model. Next, in Sec. III we study effects appearing in a subgap spectrum and in tunneling conductance of the uncorrelated QD. Sections IV and V discuss the correlation effects in the superconducting atomic limit and the Kondo regime, respectively. In Sec. VI we summarize the main results additional data for the short Rashba wire are presented in Appendix A.

Ii Low energy model

Empirical realizations of the topological superconductivity in the semiconducting wires Mourik et al. (2012); et al. (2016) or magnetic atoms’ chain Nadj-Perge et al. (2014); Pawlak et al. (2016); Ruby et al. (2015); Jeon et al. (2017) rely on the -wave pairing (of identical spins) between the nearest neighbor sites, reminiscent of the Kitaev scenario Kitaev (2001). In this paper we assume that such paring is induced for electrons, so only this particular spin component of QD is directly coupled to the Majorana quasiparticle Vernek et al. (2014); Ruiz-Tijerina et al. (2015). Due to the proximity induced on-dot pairing, however, the other () spin is indirectly influenced by the Majorana quasiparticle as well. For this reason, any process engaging electrons would simultaneously (although with different efficiency) affect the opposite spin Golub (2015). This would be very important for the Andreev (particle to hole conversion) scattering, which is the only subgap transport channel at low temperatures.

Our setup (Fig. 1) can be described by the following Anderson-type Hamiltonian

(1)

where describes the metallic electrode, refers to -wave superconducting substrate and electron energies are measured with respect to the chemical potentials . The correlated QD is described by , where denotes the energy level and stands for the repulsive interaction between opposite spin electrons. The QD is coupled to both external reservoirs via , where denote the matrix elements. In a wide bandwidth limit, it is convenient to introduce the auxiliary couplings , which we assume to be constant. For energies it has been shown Bauer et al. (2007); Yamada et al. (2011); Martín-Rodero and Levy Yeyati (2011); Barański and Domański (2013) that the superconducting electrode induces the static pairing . We make use of this low energy model, whose extension on the realistic has been widely discussed in the literature Liu et al., 2017.

The zero-energy end modes of the topological nanowire can be modeled by the following term Liu et al. (2015)

(2)

with the Majorana (selfhermitian) operators , where corresponds to their overlap. We recast these Majorana by the standard fermionic operators Elliott and Franz (2015) and so that the term (2) can be expressed as

(3)

where .

Iii Majorana vs electron pairing

We first consider the case of uncorrelated QD (). Let us calculate the Green’s function in the following matrix notation

(4)

where . For (i.e. without any overlap between the Majorana fermions) the Greens function can be expressed explicitly by

(5)
(6)
(7)

where and . These Green’s functions (5)-(7) are composed of the part, representing solution for the quantum dot coupled only to N and S electrodes () (first terms of r.h.s Eqs. (5)-(7)) and additional term dependent on coupling to the Majorana fermions. In the SC atomic limit, , such Green’s function is characterized by five poles: two of them corresponding to the Andreev bound states () and three additional states resulting from the Majorana fermions (0, ). The correlation effects shall be discussed in sections IV and V.

iii.1 Free quasiparticle spectrum

Figure 2: Spectral function of the uncorrelated dot obtained for , and various couplings .

Fig. 2 shows the spin-resolved spectral function of the uncorrelated QD obtained at half-filling () for various couplings . As a reference shape we also present the spectrum in absence of the Majorana quasiparticles (), revealing the Andreev quasiparticle peaks at whose broadening is . For the spin-resolved spectra are no longer identical due to the direct (indirect) coupling of () QD electrons with the side-attached Majorana state. The most significant differences show up near . In particular, direct hybridization of electrons depletes their spectrum near the Majorana state. Exactly at their spectral function is reduced by half, , similarly to what has been reported for the same geometry with both nonsuperconducting leads Lee et al. (2013); Ruiz-Tijerina et al. (2015); Weymann (2017). Contrary to this behavior, the spin electrons (indirectly coupled to the Majorana state via on-dot pairing) clearly gain the electronic states. Again, at the spectral function does not depend on (unless vanishes). This constructive feedback of the side-attached Majorana state on electrons has no analogy to any normal systems Lee et al. (2013); Ruiz-Tijerina et al. (2015); Weymann (2017).

Upon increasing the coupling we observe a gradual splitting of the Andreev quasiparticles, leading to emergence of the effective molecular structure. We can notice some differences appearing in the spectrum of and electrons, especially in the low energy region.

iii.2 Quasiparticle features in tunneling spectroscopy

Figure 3: The differential Andreev conductance obtained at zero temperature for the same model parameters as in Fig. 2.

Low energy quasiparticles of the QD, which is side-attached to the Majorana mode, can be probed in our setup (Fig. 1) only indirectly, via the tunneling current. When voltage applied between the normal tip and superconducting substrate is smaller than the energy gap the charge transport can be provided solely by the Andreev reflections Andreev (1964). Such transport channel can be described quantitatively by the Landauer expression

(8)

where is the Fermi distribution. The energy-dependent transmittance

(9)

describes probability that an electron (from STM tip) with spin is converted into a hole (reflected back to the STM tip) with an opposite spin , injecting one Cooper pair into the superconducting substrate. It has been previously emphasized Barański and Domański (2013), that the differential conductance can detect the subgap quasiparticle states, at expense of mixing the particle and with hole degrees of freedom. In particular, at zero temperature the differential conductance simplifies to .

Fig. 3 shows the differential Andreev conductance obtained at zero temperature for different values of , assuming . We observe, that for all finite couplings the linear conductance drops to the value . This result is qualitatively different from what have been obtained for N-QD-N junctions, where (see Ref. Lee et al., 2013). Upon increasing the coupling the nonlinear conductance develops four local maxima, two of them at and additional pair at . These local maxima are no longer equal to the perfect Andreev conductance . They originate from the bonding/antibonding Andreev states combined with the Majorana quasiparticle (see Fig. 2).

Iv Majorana state near the singlet-doublet transition

To gain some insight into the correlation effects let us first consider the strongly asymmetric limit , assuming . Influence of the Coulomb repulsion on the subgap Andreev states in the large gap limit with a vanishing coupling to the normal lead has been first addressed by E. Vecino et al Vecino et al. (2003). The effective bound states of this superconducting atomic limit have been discussed in the review paper Martín-Rodero and Levy Yeyati, 2012. In the present context we consider the proximitized QD coupled to the Majorana modes, i.e.

(10)

From now onwards we introduce the shorthand notations for the induced pairing gap and for the shifted QD level , respectively.

Figure 4: The spectral function of the correlated and proximitized QD obtained for , , using (top panel), (middle panel), and (bottom panel). Calculations are done by exact numerical diagonalization of the Hamiltonian (10) and imposing a small line broadening.

In absence of the hybridization the true eigenstates of QD are represented by the doublet configurations and (corresponding to QD spin ) and the BCS-like singlet states ()

(11)
(12)

with the eigenenergies and the coefficients , where . In the present case, however, we have to extend the Hilbert space by the additional fermion state (which can be either empty or occupied). We have numerically diagonalized the Hamiltonian matrix in the occupancy representation , determining the eigenvalues and eigenvectors. Using the spectral (Lehmann) representation we have computed the spectral functions and calculated the Andreev transmittance.

Fig. 4 shows the QD spectrum for three representative values of the Coulomb potential. In the weak interaction limit (top panel) the spectrum exhibits the non-crossing Andreev quasiparticle branches coexisting with the zero-energy mode leaking into the QD. For we observe a tendency towards the Andreev bound states crossing at (). However, due to hybridization with the Majorana mode, we rather observe emergence of the three-peak (molecular) structure. In the strong correlation case (bottom panel) there appear two crossing points aside from the half-filling Bauer et al. (2007), but we still can see the 3-peak spectrum. This behavior shows that quantum phase transition from the (spinful) doublet to the (spinless) BCS-type configurations is obscured by a leakage of the Majorana mode. Actually, the Majorana and Andreev modes give rise to the novel 3-peak structure. In section V we shall examine possible formation of the subgap Kondo effect.

V Majorana vs Kondo

Now, we analyze the correlated quantum dot taking into account the finite coupling to the normal STM tip. It has been recently shown Domański et al. (2016, 2017), that in absence of the Majorana quasiparticle the induced on-dot pairing cooperates with the Coulomb repulsion, enhancing the subgap Kondo effect. Its spectroscopic observation is possible only nearby the quantum phase transition, on the (spinful) doublet side.

Our main purpose here is to examine how this subgap Kondo effect (usually appearing at zero energy) gets along with the leaking Majorana mode. Some earlier studies of the correlated quantum dot coupled to both normal (conducting) electrodes in presence of the side-attached Rashba chain indicated a competition between the Kondo and Majorana physics Ruiz-Tijerina et al. (2015); Lee et al. (2013); Cheng et al. (2014); van Beek and Braunecker (2016); Weymann (2017); Weymann and Wójcik (2017). For sufficiently long wire () the Kondo effect is preserved only for the spin-channel (which is not coupled to the Majorana zero-energy mode), whereas for the other spin-channel there appears a dip in the spectral density at (reminiscent to what we observed in the upper panel of Fig. 2). In consequence, the total transmission is partly blocked suppressing the linear conductance from to the fractional value Ruiz-Tijerina et al. (2015); Lee et al. (2013); Weymann (2017); Weymann and Wójcik (2017); López et al. (2014). For the short Rashba wires () the Kondo physics survives in both of the spin channels, but a width of the Abrikosov-Suhl peak depends on . The initial Kondo features are fully recovered in both spin-channels only when . Results for the short length Rashba wires are shown in Appendix A.

When the correlated quantum dot is embedded between the metallic and superconducting leads (N-QD-S) the eventual subgap Kondo effect is controlled by ratio and , which decide whether the ground-state is either the (spinful) doublet or the (spinless) BCS-type configuration Domański et al. (2016); Žitko et al. (2015); Yamada et al. (2011); Tanaka et al. (2007); Barański and Domański (2013). In particular, for the half-filled QD () the BCS singlet exists in the weak correlation regime , whereas the doublet is preferred in the strongly correlated limit . In the latter case the Kondo temperature eventually occurs, and moreover the effective Kondo temperature can be amplified by the increasing coupling strength Domański et al. (2016, 2017).

In the present setup the Coulomb interactions (responsible for the Kondo effect) are additionally confronted with the Majorana physics. To study such effects qualitatively we compute the self-energy of the matrix Green’s function (4), treating the Coulomb interaction perturbatively. Formally we solve the Dyson equation , adopting the second-order perturbation theory (SOPT) Domański et al. (2016); Yamada et al. (2011); Vecino et al. (2003)

(13)
(14)
(15)

The imaginary parts of the second-order contributions are expressed by the following convolutions Domański et al. (2017)

(16)
(17)

where

(18)

and the auxiliary functions denote the occupancies obtained from the uncorrelated Green’s functions (4), but taking into account the effective dot level and the effective superconducting coupling strength Yamada et al. (2011).

Figure 5: The total spectral function of the half-filled qunatum dot () obtained at for and several values of the Coulomb potential (as indicated), assuming (a) , (b) , and (c) . All energies are expressed in units of .

In what follows we discuss the total spectral function , because empirical detection (by the Andreev spectroscopy) would mix the quasiparticles of both spin orientations. Fig. 5 shows obtained at zero temperature for various Coulomb potentials, as indicated. The top panel refers to N-QD-S system without any coupling to the Majorana mode. In the weak interaction regime it reveals two Andreev peaks. Upon approaching these peaks merge, signaling the quantum phase transition (for it becomes a continuous crossover). In the strongly correlated limit (), the Kondo peak develops at and its broadening gradually shrinks upon increasing the ratio Domański et al. (2016); Yamada et al. (2011).

Coupling of the correlated QD to the Majorana mode (whose spectrum is displayed in panels b and c of Fig. 5) leads to appearance of additional structure near , even for . A width of this additional structure is controlled by . Upon increasing it becomes hardly distinguishable from Andreev peaks, forming a single broad peak. For the sufficiently strong Coulomb potential the spectral function at tends to . Similar result has been previously obtained from NRG calculations for the quantum dot - Majorana system coupled to both ferromagnetic electrodes Weymann and Wójcik (2017).

Figure 6: The differential subgap conductance as a function of the applied voltage obtained for the same set of parameters as in figure 5.

Practical observation of the above mentioned effects would be accessible by measuring the subgap current (8). Fig. 6 shows the Andreev conductance as a function of the Coulomb potential , ranging from the weak (a) to strong (b) hybridizations . In the limit these plots resemble the results shown in Fig. 3. Some qualitative changes appear at stronger interactions, especially on the doublet side .

In absence of the Majorana mode we have shown Domański et al. (2016), that differential conductance of the N-QD-S system is characterized by two peaks at the Andreev states and additional enhancement at zero-bias due to the Kondo effect, which operates only in the doublet region . From the generalized Schrieffer-Wolff transformation we have estimated that , which yields a broadening caused by the on-dot pairing Domański et al. (2016). For the Andreev scattering is quickly suppressed (because the off diagonal parts of Green’s function disappear). In the present case a leakage of the Majorana quasiparticle obscures the mentioned behavior. For we no longer observe any single peak structure, but rather a superposition of the induced Majorana feature (of a width dependent on ) and the initial Kondo peak (whose width is strictly controlled by ratio).

For larger voltages the differential conductance in the weak coupling is similar to N-QD-S system. For strong coupling we obtain the four-peak shape of . Internal peaks are related to the Andreev states, whereas the external ones are driven by the Majorana modes. With the increasing Coulomb potential the Andreev peaks merge into a single central peak and the side-peaks disappear.

Figure 7: The zero-bias Andreev conductance as a function of the Coulomb potential (in units of ) obtained for , , .

Fig. 7 shows the linear conductance as a function of the Coulomb interaction. The optimal conductance approaches the value , which is 4 times smaller than for the N-QD-S system without the side-attached Majorana mode (see Refs Tanaka et al. (2007); Górski (2016)). Maximum of occurs at , in a crossover between the BCS-type singlet and the doublet configurations Tanaka et al. (2007). In the present system (with ) the maximum of moves towards larger values , suggesting that the leaking Majorana mode competes with the Kondo state.

Vi Summary

We have analyzed the spin-resolved spectroscopic features of the correlated quantum dot side-coupled to the topologically superconducting nanowire, hosting the Majorana quasiparticles, focusing on STM-type geometry. Under such circumstances the subgap quasiparticles can be probed by the Andreev tunneling, involving simultaneously the particle and hole degrees of freedom.

In the uncorrelated case () a leakage of the Majorana quasiparticle induces either the zero-energy peak or dip in the QD spectral density, depending on the spin (Fig. 2). These effects originate from the constructive or destructive quantum interference, as can be inferred from the full counting statistics analysis Schuray et al. (2017). We predict, that the differential Andreev conductance would be predominantly influenced by a destructive interference, leading to the zero-bias dip feature preserved from the weak to strong hybridization regimes.

We have also inspected the correlated quantum dot () case, studying an interplay between the Kondo effect and the Majorana quasiparticle. Coulomb interaction along with the proximity induced on-dot pairing can cause a transition from the spinless to the spinful configurations, manifested by a crossing of the Andreev/Shiba quasiparticles Martín-Rodero and Levy Yeyati (2011). Finite coupling with the STM tip could drive the virtual exchange between the QD and conducting electrons Žitko et al. (2015); Domański et al. (2016), inducing the subgap Kondo effect Martín-Rodero and Levy Yeyati (2011). We have found, that the side-attached Majorana quasiparticle gives rise to unusual behavior near this quantum phase transition/crossover.

Interplay of the subgap Kondo effect and Majorana quasiparticle shows up in the low-energy QD spectrum and in the tunneling conductance. In the weak correlation regime the Majorana state prevails, whereas in the strong correlation limit the Kondo physics plays a major role. The most interesting intermediate regime reveals completely novel signatures Béri and Cooper (2012); Cheng et al. (2014); Galpin et al. (2014); van Beek and Braunecker (2016); Plugge et al. (2016); Béri (2017), without analogy to any known behavior. In the present context, we predict the linear conductance to be reduced to 25 % of its perfect value typical for N-QD-S junctions Martín-Rodero and Levy Yeyati (2011). This results is in stark contrast from what has been predicted for N-QD-N junctions, where the single particle conductance is reduced to only 75 % of the unitary value Lee et al. (2013). In both cases, however, underlying mechanism is related to a fractional character of the Majorana quasiparticle.

We hope that the present analysis would encourage experimental attempts to realize the STM setup proposed in Fig. 1, that could reveal unusual physics driven by: electron pairing, strong correlations, and exotic nature of the Majorana quasiparticles.

Acknowledgments

This work is supported by the National Science Centre in Poland via project DEC-2014/13/B/ST3/04451 (TD) and the Faculty of Mathematics and Natural Sciences of the University of Rzeszów through the Project No. WMP/GD-06/2017 (GG).

Appendix A Short wire case

Main part of this paper is devoted to the infinitely long nanowire, where the edge Majorana do not overlap with each other (). In such case the Kondo resonance and the Majorana quasiparticle are pinned to the Fermi level of the superconducting lead, leading to competition between these effects Lee et al. (2013); Ruiz-Tijerina et al. (2015).

Figure 8: The total spectral function of the correlated dot obtained for different values of , using , , , (a) and (b).

In the case of finite size nanowires the Majorana modes are characterized by a finite overlap (). Under such circumstances for and we obtain six quasiparticle states: two of them being the Andreev bound states () and the other four states being related to Majorana quasiparticles , where . In presence of correlations and for finite these quasiparticle states appear away from the Fermi energy, therefore their competition with the Kondo effect should be rather negligible. In Fig. 8 we present the total spectral function obtained for two values of in the Kondo regime. For small (Fig. 8 (a)) and in the weak interaction limit we clearly observe six subgap quasiparticle peaks. With increasing two of them merge into the Kondo resonance peak (for ) with a tiny superstructure at . Contrary to such behavior, for large overlap (Fig. 8 (b)) we obtain the spectrum reminiscent of the N-QD-S system (see Fig. 5 (a)) with only some redistribution of the spectral weight at higher energies.

Figure 9: The differential Andreev conductance as a function of the applied voltage for , , (a) and (b).

Finite overlap has also influence on the differential Andreev conductance. In Fig. 9 we present as a function of the applied voltage for the long () and short () nanowires, respectively. In the first case we observe well pronounced zero-bias peak which is characterized by a narrow width, controlled by . Optimal value of the zero-bias peak approaches for . For small we observe two Andreev and two Majorana-Andreev hybridized peaks showing up in differential conductance. With the increasing Coulomb interaction the Andreev peaks evolve into the central peak, appearing at . On the other hand, for the short nanowire () and at small voltages the differential conductance is similar as for N-QD-S system Domański et al. (2016).

Figure 10: The linear conductance as a function of the Coulomb potential for , and .

Fig. 10 shows the linear conductance as a function of the Coulomb potential obtained for and several values of the overlap . We can notice, that optimal value of is indeed , which occurs at the quantum phase transition (for the finite coupling it is changed to a smooth crossover). For the ground states evolves into the spinful doublet with a gradual suppression of the on-dot pairing, therefore the anomalous Andreev conductance monotonously vanishes. For larger overlap this changeover of the Andreev conductance slightly moves towards higher values of the Coulomb potential.

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