Elsevier

Solid State Communications

Volume 318, September 2020, 113973
Solid State Communications

The modulation of Majorana bound states comb through quantum dots

https://doi.org/10.1016/j.ssc.2020.113973Get rights and content

Highlights

  • The current noise cross correlation and the differential conductance in the Majorana bound states (MBSs) comb are investigated.

  • The MBSs comb can be effectively modulated by the up-part quantum dots due to the effective resonance between the Majorana bound state and the dot.

  • For only one “QD-MBSs-QD” building block, N = 1, a zero bias conductance peak of 2e2/h can be found. For N = 2, the zero bias conductance peak is 4e2/h.

Abstract

Performing the scattering matrix approach, we calculate the current noise cross correlation and the differential conductance in the Majorana bound states (MBSs) comb. The teeth on the comb consist of topological super-conducting nanowire sandwiched between two quantum dots (QDs). We find that the nonlocal property of MBSs comb can be effectively modulated by the quantum dots. For only one “QD-MBSs-QD” building block, N = 1, the current noise cross correlation is negative at small bias and becomes positive at lager bias. A zero bias conductance peak of 2e2/h can be found, and it splits with ϵu increasing, where ϵu is the energy level of up-part QD. For N ≥ 2, the current noise cross correlation is zero at small bias and becomes positive at larger bias. Different from the case of N = 1, the zero bias conductance peak is 4e2/h due to the local Andreev reflection. Besides, we find that the differential conductance is robust against the influence of N.

Introduction

Majorana fermions, introduced into theoretical phy-sics by Ettore Majorana, have attracted great attention due to both their fundamental interests and potential applications for the decoherence-free quantum computation in condensed matter physics [[1], [2], [3], [4], [5], [6]]. Different from ordinary fermions, Majorana fermions are their own antiparticles, and two spatially separated Majorana bound states (MBSs) define a nonlocal fermionic state. Until now, a number of systems have been proposed to detect the MBSs, such as vortex core in a p-wave superconductor [7], superfluid [8], ultracold fermionic atoms with spin-orbit interaction [9], edge states of a topological insulator [10], optical lattices [11], quantum dot chains [12] and helical Shiba chains [[13], [14], [15]]. What is more, it was discovered that MBSs can also be realized at the ends of a one-dimensional semiconductor nanowire with strong spin-orbit interaction in the proximity of an s-wave superconductor [[16], [17], [18]]. This makes its applications more feasible, for Majorana fermions can be constructed in solid states.

The signature for a pair of spatially separated MBSs has been experimentally observed in terms of a zero bias conductance peak [6]. However, zero bias conductance peak can also be the signature of other phenomena such as Andreev bound states [19,20], weak antilocalization [21], disorder [22] or Kondo resonances [[23], [24], [25]]. Fortunately, we can utilize the nonlocal feature of MBSs to distinguish MBSs from other states. Thus, one can seek for characteristic of MBSs in current noise cross correlation [[26], [27], [28]]. In a previous work [29], it was observed that quantum dot (QD) can recover the Majorana fermion characteristic when the MBSs are sandwiched between two quantum dots. Apart from their discovery, more information beneath it can be found. For example, the nonlocal current noise correlation has not been investigated in the “QD-MBSs-QD” structure. In comparison with single quantum dot case, multi-quantum-dot system exhibits more interesting transport phenomenon due to more tunable structure parameters and abundant quantum interference mechanisms. Hence, we are going to investigate the MBSs comb (consists of many “QD-MBSs-QD” structure) with quantum dots by using scattering matrix method [[30], [31], [32]] in this paper.

In this paper, we mainly focus on the current noise cross correlation of the “QD-MBSs-QD” comb which is biased equally by the two leads as shown in Fig. 1. In this structure, these quantum dots are spinless, and are coupled to MBSs at two ends of a nanowire with spin-orbit coupling (SOC). We are going to present more details in later discussions. Different from the Ref. [29], not only the differential conductance but also the nonlocal current noise correlation had been investigated. First, we investigate the single “QD-MBSs-QD” building block by using the scattering matrix, the new result is that the nonlocal property of MBSs can be effectively modulated by both the two quantum dots. The current noise cross correlation is negative when the energy levels of the quantum dots (up-part ϵu and down-part ϵd) ally with the MBSs, and it increases to positive with the energy level of quantum dot increasing. There are two cases. One is that we let ϵd increase and keep ϵu equal to zero. In this case, the MBS wave function of down-part QD just move some weight to the up-part QD, and zero bias peak (ZBP) keeps. Another is vice versa. But in this case the MBS wave function of up-part QD “spill over” into the down-part QD and ZBP splits. The noise cross correlation can be remarkably different from former case at small bias. Next, we investigate the many-building-block structure. The result is different from the one-building-block case. It is observed that a zero bias conductance peak with height of 4e2/h appears. Correspondingly, the current noise cross correlation is zero at small bias and increases to positive value at larger bias. Finally, we find that differential conductance is robust against the influence of N.

Section snippets

Method

As is shown in Fig. 1, the model system under consideration is that a comb in which many “QD-MBSs-QD” building blocks are arranged in parallel is coupled to two leads. In this case, the semiconducting nanowire that hosts MBSs is in the proximity of an s-wave superconductor [33,34]. And two quantum dots are coupled to two ends of the nanowire, which makes the structure to be a “QD-MBSs-QD” building block. In the case of N building blocks, Hamiltonian can be written asH=Hdot+Hdotdot+Hlead+Hdotle

Results and discussion

In what follows, we are going to calculate the differential conductance ∂I/∂V and the zero frequency current noise cross correlation PLR at zero temperature. We consider ϵM as energy unit in this paper except for the case λ2 = 0. In our calculation, the current noise cross correlation is the research point. The quantum dot has only one energy level, and the electron-electron interaction is neglected.

Conclusions

In conclusion, we have calculated the differential conductance and the current noise cross correlation in the system of “QD-MBSs-QD” comb. In the beginning, we investigate the single “QD-MBSs-QD” building block by using the scattering matrix, we found that the nonlocal property of MBSs can be effectively modulated by both the two quantum dots. Next, we investigated the many-building-block case in this structure. We observed a zero bias conductance peak with height of 4e2/h which is ascribed to

Authors statement

Xiao-Feng Chen: Data curation, Writing - original draft. Long Liu: Data curation, Writing - original draft. Muhammad Aslam: Writing- Reviewing and Editing. Jing He: Writing- Reviewing and Editing. Juntao Song: Supervision. Yu-Xian Li: Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11874139, 11474085), the Natural Science Foundation of Hebei (Grant No. A2017205108, A2019205190), the youth talent support program of Hebei education department (Grant No. BJ2014038), the Outstanding Youth Foundation of HBTU (Grant No. L2016J01), and the youth talent support program of Hebei Province.

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