# Electronic Transport Through Double Quantum Dot Coupled to Majorana Bound States and Ferromagnetic Leads

- Department of Materials Engineering, Xiangtan University, Xiangtan, China

We have studied theoretically the properties of electrical current and tunnel magnetoresistance (TMR) through a serially connected double quantum dot (DQD) sandwiched between two ferromagnetic leads by using the nonequilibrium Green’s function technique. We consider that each of the DQD couples to one mode of the Majorana bound states (MBSs) formed at the ends of a topological superconductor nanowire with spin-dependent coupling strength. By adjusting the sign of the spin polarization of dot–MBS coupling strength and the arrangement of magnetic moments of the two leads, the currents’ magnitude can be effectively enhanced or suppressed. Under some conditions, a negative TMR emerges which is useful in detection of the MBSs, a research subject currently under extensive investigations. Moreover, the amplitude of the TMR can be adjusted in a large regime by variation of several system parameters, such as direct hybridization strength between the MBSs or the dots and the positions of the dots’ energy levels. Such tunable currents and TMR may also find use in high-efficiency spintronic devices or information processes.

## 1 Introduction

Electronic transport through structures composing of quantum dots (QDs) hybridized with a topological superconductor nanowire (TSNW) hosting Majorana bound states (MBSs) [1–3] has aroused much interest in recent years. The zero-energy MBSs are exotic self-conjugate quasiparticles that have been successfully realized in encouraging experiments [4–6] during the last decade. One of the most attractive platforms [7, 8] to host and detect MBSs is a superconductor proximitized semiconductor nanowire having the spin–orbit interaction and strong Zeeman splitting. Previous theoretical work predicted that the combined effects of the spin–orbit interaction and the large enough Zeeman splitting will convert an ordinary

Some previous theoretical and experimental work has shown that the MBSs will affect the electronic transport processes through QD-based devices in a significant way [19, 20]. For example, Baranger and his co-author proved that the value of the conductance in a single QD, which is coupled to the left and right leads, will remain at half of its quantum value

It is known that the zero-energy MBSs exert remarkable effects on the electrical conductance and current around the zero-bias regimes. But under such a condition, the amplitude and changes of the above quantities are usually small and hard to be detected. In fact, the quantity of tunnel magnetoresistance (TMR) [28–30], which measures the relative change of the currents’ amplitude, is frequently used as detection means in electronic transport. It is also a key quantity in spintronic devices. The TMR is defined as

**FIGURE 1**. Schematic plot of the DQD coupled to ferromagnetic leads with coupling strength

## 2 Model and Methods

The Hamiltonian of the studied system shown in Figure 1, which is composed of the DQD each of which connected to the left and right ferromagnetic leads and to one mode of the MBSs, can be written as the following form [21, 24],

where the creation (annihilation) operator

in which

In terms of the Hamiltonians given by Eqs 1 and 3, spin-dependent electrical currents

in which the Fermi distribution function for the left and right leads are given by

in which the diagonal matrix is

with

in which

The spin-dependent transmission is then calculated by [24]

## 3 Results and Discussion

In this section, we present our numerical results for the spin-dependent currents and TMR. We choose the leads’ bandwidth

**FIGURE 2**. Electronic current

With the results in Figure 2, we now study the currents in Figure 3 for different values of

**FIGURE 3**. Spin-dependent and total currents for parallel configuration in A–C, and antiparallel one in D–F with

As is seen from Figures 2, 3, the variation of the currents around the zero bias, where the MBSs play an important role, is quite nonobvious. We then present the TMR in Figure 4 varying with respect to the bias voltage for both positive and negative

**FIGURE 4**. TMR for positive

Figure 5 shows the influences of the ferromagnetism of the leads on the currents and the TMR for

**FIGURE 5**. Total currents in parallel configuration **,**3.

Figure 6 presents the impacts of

**FIGURE 6**. TMR for different values of **,**3.

## 4 Summary

In summary, we have studied the spin-polarized currents and TMR in a DQD coupled to both ferromagnetic leads and MBSs formed at the ends of a topological superconductor nanowire. Our calculation results show that the currents through the system can be effectively adjusted in terms of the spin polarization of either ferromagnetic leads or coupling strength between the dots and the MBSs. When the two spin polarizations are the same in sign, the currents’ amplitude in the antiparallel configuration can be larger than that in the parallel one, which results in an obvious negative TMR that can be used for detecting the existence of the MBSs. If the two spin polarizations are different in sign, however, then the TMR is positive and can be further enhanced by adjusting system’s parameters. Such a result is useful in designing high-efficiency spintronic devices. The negative or positive TMR is robust against variations of the overlapping between the MBSs, the tunnel coupling between the two dots, or even the difference between the dots’ energy levels.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

## Author Contributions

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.

## Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772287 and 11572277).

## Conflict of Interest

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.

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Keywords: double quantum dots, Majorana bound states, spin-dependent coupling strength, tunnel magnetoresistance, ferromagnetic leads

Citation: Tang L-W and Mao W-G (2021) Electronic Transport Through Double Quantum Dot Coupled to Majorana Bound States and Ferromagnetic Leads. *Front. Phys.* **8**:616107. doi: 10.3389/fphy.2020.616107

Received: 11 October 2020; Accepted: 30 October 2020;

Published: 18 January 2021.

Edited by:

Qiang Xu, Nanyang Technological University, SingaporeCopyright © 2021 Tang and Mao. 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.

*Correspondence: Li-Wen Tang, qdtang2020@163.com