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Edited by: Zichuan Yi, University of Electronic Science and Technology of China, China

Reviewed by: Feng Chi, University of Electronic Science and Technology of China, China; Qiang Xu, Nanyang Technological University, Singapore; Zhen-Guo Fu, Institute of Applied Physics and Computational Mathematics (IAPCM), China

This article was submitted to Optics and Photonics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We study quantum transport and tunneling magnetoresistance (TMR) through an interacting quantum dot (QD) attached to ferromagnetic electrodes in the Coulomb blockade regime. The QD is also side-coupled to a superconductor nanowire hosting Majarana bound states (MBSs). It is found that when the electrodes' magnetic moments are arranged in antiparallel, the current's intensity will be enhanced to be larger than that of the parallel configuration by the hybridization between the QD and the MBSs. This change in the current induces anomalous negative TMR unique to the existence of MBSs, providing an efficient detection way of the MBSs. The negative TMR is weakened by the overlap between the two bound states but obviously enhanced by the left-right asymmetry between the QD and the electrodes. We also find that the TMR value changes non-monotonously with the spin polarization of the electrodes. Our results may find real use in energy saving spintronic devices and quantum information processing.

Tunneling of spin-polarized electrons in nano- and micro-scale devices are at the core of quantum information data processings [_{P} − _{AP})/(_{P} + _{AP}), where _{P/AP} is the electronic current in parallel (antiparallel) configuration. Usually, the TMR value is positive as _{P} > _{AP} [_{B}

Recently, there is much interest in preparation and detection of Majorana fermions, which is a kind of elementary particle being of its own antiparticle, due to its promising applications in quantum computation free from decoherence [

In recent experiments, semiconductor spacers of InAs QD has been inserted in between nickel or cobalt leads [

Schematic diagram for a QD coupled to MBSs and the left and right ferromagnetic leads with coupling strength Γ_{L/R}. Due to the chiral nature of the Majorana fermions, only spin-up is assumed to couple to the MBSs, formed at opposite ends of a nanowire. The two MBSs are denoted by η_{1} and η_{2} and coupled to the QD with strength of λ_{1} and λ_{2}, respectively. If the dot level ε_{d} is occupied by an spin-up electron, then the spin-down electron will by pushed up to the level of ε_{d} +

The Hamiltonian of the QD coupled to MBSs and ferromagnetic electrodes takes the following form [

where _{kβσ}) creates (annihilates) an electron of momentum _{kβσ} and spin σ = ↑, ↓ in the ferromagnetic electrode β = _{σ}) is the creation (annihilation) operator of an electron having energy level ε_{d}, spin-σ and intradot Coulomb interaction _{d} and _{kβσ}, which is spin-dependent due to the ferromagnetism on the electrodes. The last term _{MBSs} in Equation (1) stands for the zero-energy MBSs located on the opposite ends of the semiconducting nanowire and their coupling to the QD [

in which δ_{M} is the overlap strength between the two MBSs with operator satisfying both _{i}, η_{j}} = δ_{i,j}. The hopping amplitude between MBSs and spin-↑ electrons in the QD is accounted by λ_{j}. Following previous work [_{j} in terms of the regular fermionic operators _{MBSs} becomes

Within the standard Keldysh Green's function technique, the spin-dependent electric current is obtained as [

where _{L/R}, temperature _{B}. The transmission coefficient _{σ}(ε) can be expressed with the help of the retarded Green's function

where _{β} as _{L} = _{R}) and antiparallel (_{L} = −_{R}) configurations of the two electrodes. In both of the two cases, we have

By applying the equation of motion method, the retarded Green's function in Equation (5) is obtained as (we have truncated the higher-order Green's functions by following reference [

where the self-energies considering the MBSs are given by

and

To our knowledge, there are three kinds of schemes that are mainly used for studying the transport phenomena [

In this section, we present our numerical results for the spin-dependent current, TMR and differential conductance varying as functions of the bias voltage. The intradot Coulomb interaction _{L} = −μ_{R} = _{↑} − _{↓})/(_{↑} + _{↓}) for parallel and antiparallel configurations, respectively. The currents exhibit typical Coulomb blockade effect as shown in _{d} are within the transport window. The current then is increased. With an increase from _{d} + _{d} and ε_{d} +

Spin-resolved current, non-linear differential conductance, and current' spin polarization as functions of the bias voltage when the magnetic moments of the leads are arranged in parallel _{1} = λ_{2} = λ. Other parameters are: δ_{M} = 0, ε_{d} = −0.1, Γ_{L} = Γ_{R} = _{B}_{L} = ±_{R} = 0.6 (± for parallel and antiparallel cases, respectively).

In the parallel configuration, the current intensity of _{↑} is obviously larger than that of _{↓} in the absence of QD-MBSs. The reason is that the spin-up electrons will enter and leave the QD faster than the spin-down ones because of the ferromagnetism on the electrons, i.e., _{1} = λ_{2} = 0), which is not shown in the figure. Turning on the coupling between the QD and MBSs, the intensity of spin-up currents in both parallel and antiparallel configuration are enhanced as shown in _{↑} in the antiparallel is more obvious than that of the parallel one, which results in negative TMR as will be shown in the following. For the differential conductance, an obvious zero bias anomaly (ZBA) emerges in addition to the increase in the intensity [^{2}/_{d} as compared to ε_{d} +

As demonstrated by previous work, the influence of the MBSs on the current is rather weak since a pair of Majarana fermions are charge neutral [_{↑} + _{↓} in _{d} and _{d} + _{d} < _{d} + _{d} and ε_{d} + _{d} (ε_{d} + _{d} is negative (see the dotted line), and then serves as a detection method for the existence of the Majorana fermions in the superconductor nanowire side-coupled to the QD. We emphasize that this phenomenon originates from the fact that the current intensity of the antiparallel configuration is more sensitive to the MBSs as compared to the parallel one.

Total current, non-linear differential conductance for the magnetic moments' configurations of parallel

In _{L} = _{R}) and overlap amplitude between the two MBSs δ_{M}. For _{d} for _{M} on the TMR for fixed spin polarization of the electrodes _{M} = 0), an obvious negative TMR emerges around the dot level ε_{d}. With increasing δ_{M}, we find that the strength of the TMR is enhanced, especially in the Coulomb blockade regime. For strong enough overlap between the two states δ_{M} = 0.05

TMR as a function of the bias voltage for different values of spin polarization of the leads _{M} in _{M}, respectively. Other parameters are as in

In _{R}/Γ_{L}. First of all, the currents' strength is monotonously enhanced by increasing Γ_{R}/Γ_{L}. This is because that for fixed ingoing tunneling rate Γ_{L}, the electrons will tunnel through the dot faster with increased outgoing tunneling rate Γ_{R}. Obviously, this holds true for both of the two spin component electrons regardless of the magnetic configuration. The TMR in _{R}/Γ_{L}. In the regimes of _{d} and _{d} + _{R}/Γ_{L}. This is the usual case in the absence of the QD-MBSs. In the Coulomb blockade regime ε_{d} < _{d} +

Total currents for parallel _{0} = 0.02

In summary, spin-dependent current and TMR in a QD sandwiched between two ferromagnetic leads and side-coupled to a pair of MBSs formed at the opposite ends of a superconductor nanowire is investigated within the non-equilibrium Green's technique. An unique negative TMR induced by the hybridization between the QD and the MBSs is found, which serves as a detection means of the Majorana fermions. This negative TMR is more likely to emerge in longer nanowire in which the two MBSs are well-separated from each other and the overlap between them is weak. By increasing the left-right asymmetry of the coupling strength between the QD and electrodes, the negative TMR becomes more obvious. It is also found that the intensity of the TMR depends on the spin-polarization of the electrodes in a non-monotonic way and is positive for large spin-polarization regardless of the existence of the MBSs.

All datasets presented in this study are included in the article/supplementary material.

W-GM and L-WT contributed the ideas equally and performed the numerical calculations. L-WT derived the formulae in the paper and wrote the original manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.