Fano resonance and zero-bias anomaly in parallel double quantum dots coupled to Luttinger liquid leads

doi:10.1016/j.physleta.2020.127095

Physics Letters A

Volume 389, 15 February 2021, 127095

https://doi.org/10.1016/j.physleta.2020.127095 Get rights and content

Highlights

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Nonequilibrium transport through parallel-coupled dot connected to Luttinger liquid.
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A current formula is derived using the nonequilibrium Green's function method.
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Effects of interdot-tunneling, Coulomb interaction, asymmetric-coupling on transport.
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The coexistence of zero-bias dip and Fano resonance for asymmetric coupling.
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Scaling analysis supports two-channel-Kondo theory of zero-bias dip.

Abstract

We study the nonequilibrium transport through a parallel-coupled double quantum dot coupled to Luttinger liquid leads using the nonequilibrium Green's function method. A current formula is derived, where the inter-dot tunneling, the intralead interaction, and asymmetric dot-lead coupling are taken into account. The interplay of these interactions on the differential conductance is investigated. The differential conductance of the double-dot system exhibits the coexistence of zero-bias dip and Fano resonance for asymmetric coupling. The zero-bias dip of differential conductance scales onto a single universal curve for different temperature with the same scaling exponent as a system of Kondo dot. As temperature increases, the zero-bias minimum is gradually lifted.

Introduction

Double quantum dot (DQD) systems have become a good paradigm of studying variation of electron transport upon Coulomb interaction and quantum phase coherence. The DQD is more easily manipulatable and controllable than a single quantum dot (QD). Due to the ultra small size of a DQD, the Coulomb interaction is strengthened and quantum phase coherence is preserved. The preservation of coherence in electron transport process can make DQD systems as promising candidates for quantum computers [1], [2]. The most convenient method to test coherence is quantum interference between different conduction channels [3]. Quantum interference plays an important role in electron transport through a DQD system. The interference between resonant and nonresonant tunneling processes across the parallel DQD leads to the well-known Fano resonance [4], manifesting itself as an asymmetric conductance line shape [5]. The quantum interference is also influenced by the Coulomb interactions. The interplay of interference and interaction on transport has been observed in experiments [6], [7], [8], [9], [10] and theoretically investigated by means of various methods [11], [12], [13], [14], [15], [16], [17]. These works have also been devoted to the interesting Fano resonance. In a one-dimensional channel, Johnson et al. have studied the modulation of Fano line by Coulomb interaction in QDs [9]. Sato et al. also observed the competition and coexistence of Fano effect and Kondo state [10].

In addition to the above traditional QD systems, several groups have reported the appearance of Fano resonance in various of carbon nanotube systems [18], [19], [20], [21]. For example, Kim et al. first reported Fano interference of conductance in crossed multi-wall carbon nanotubes [18], while Yi et al. observed the coexistence of Coulomb blockade and Fano effect in multi-wall carbon nanotube bundles [19]. Furthermore, Fano resonances have been measured both in individual multi-wall carbon nanotubes [20] and in single-wall carbon nanotubes (SWNT) [21]. In spite of the observed striking Fano resonance behavior, its physical origination has not been explained consistently. The origination was ascribed to the scattering from the contact of two crossed tubes [18], either an additional carbon nanotube [19] or defects in the nanotube [20]. It was speculated [21] that the interference occurred between two transport channels in a single SWNT or different nanotubes in a SWNT bundle.

Besides the Fano resonance, a zero-bias dip in differential conductance was observed [19]. The zero-bias minima in differential conductance has been also observed in other experiments such as in Cu point contacts [22], disordered metal point contacts [23], quasi-one-dimensional crystals [24], quantum Hall line junctions [25], strongly coupled nanodot [26], Luttinger liquid (LL) [27], multiwall carbon nanotubes [28]. Although the experimental observation of zero-bias minima is interesting, its physical origin has remained controversial. To date, the interpretation about the origin for zero-bias anomaly includes two-channel Kondo (2CK) scattering arising from two-level systems [22], the electron-electron interaction in disorder metals [29], magnetic impurities [30], [31], Kondo scattering from spontaneous electron spin polarization [32], the environmental Coulomb blockade theory [28], LL theory [27], as well as the interplay of Kondo effect and intralead interaction [33].

By means of combination of renormalization group technique and scaling theory [34], the electron-assisted tunneling between two sites of a two-level system could be modeled as a 2CK system. The two-level system plays the role of a pseudospin. One interpretation based on scaling analysis method [22] was that the zero-bias minima was 2CK physics. However, this interpretation remained ambiguous [29], [35], owing to the lack of intuitive and straightforward treatment method. Few studies have been done on the transport properties of DQDs coupled to LL. Recently Durganandini [36] reported a theoretical transport study through a DQD in series weakly coupled to LL leads, using an master equation method assuming incoherent sequential tunneling. While Kawaguchi [37] applied the Green's functions method to investigate the properties of the shot noise in parallel-structure at the special Thoulouse point. Relatively little attention has been paid to the parallel configuration case with any intralead interaction strength.

In this work, we are going to propose a microscopic model of parallel DQD coupled to the LL leads without intra-dot and inter-dot Coulomb interactions. We only focus on how the Fano resonance and zero-bias dip develop and how they are affected by inter-dot tunneling and intralead Coulombic interaction in the coherent tunneling regime. For the parallel-coupled DQD system, it is convenient to investigate the Fano resonances, due to the interference effect between different paths across the two dots, and the influence of intralead Coulomb interactions on the interference. The current formula of the DQD coupled to LL is derived by using nonequilibrium Green's function method, which provides a good physical understanding the effects of inter-dot tunneling and intralead interaction on the Fano resonance and zero-bias dip. The results display that the appearance of zero-bias dip only depends on the inter-dot tunneling and intralead interaction. While the Fano resonance only relies on the asymmetric coupling. The coexistence of asymmetric Fano resonance and zero-bias dip resemble the results for Kondo dot coupled to LL leads subject to a magnetic field [38]. Very weak repulsive interaction drives the system to the 2CK physics in the presence of moderately strong inter-dot tunneling strength. This zero-bias dip does not appear in the Fermi liquid lead case even if the inter-dot tunneling is strong. The temperature dependence of the zero-bias dip is observed due to the inter-dot tunneling and intralead interaction. The differential conductance exhibits universal scaling behavior predicted by LL theory [39], with the same exponent as that in Ref. [40]. In addition, the zero-bias dip does not occur in ordinary three-dimensional Fermi liquid with magnetic impurities. These information should be useful for future numerical work on testing the physics properties of nonequilibrium Anderson mode, utilizing a adjustability and controllability of two-level systems.

The rest of the paper is organized as follows. In section 2 we shall give our model and method. The numerical results are presented in section 3. Finally, the conclusions are given in section 4.

Section snippets

Model Hamiltonian and formulation

Our model is a parallel DQD's coupled to two LL leads as shown in Fig. 1. Its Hamiltonian is $H = \sum_{α = L, R} H_{α} + H_{D Q D} + H_{T} .$ The first term $H_{α}$ describes the Hamiltonian of α LL lead. It is quadratic in terms of the bosonized form $H_{α} = ħ \int_{0}^{\infty} d k k v_{α} a_{α k}^{†} a_{α k}$ ( $α = L / R$ ) for spinless electrons, where $a_{α k}^{†} (a_{α k})$ is the boson creation (annihilation) operator of the α lead describing the charge density fluctuations with velocity $v_{α}$ . The parallel DQD Hamiltonian $H_{D Q D} = ε_{1} d_{1}^{†} d_{1} + ε_{2} d_{2}^{†} d_{2} - t_{c} (d_{1}^{†} d_{2} + d_{2}^{†} d_{1})$ , where $d_{n} (d_{n}^{†})$ represents

Numerical result and discussion

Here the $γ_{1}$ is taken as an energy unit. The energy of DQD is $ε_{1} = - ε_{2} = 0.25$ so that $\bar{ε} = 0$ . Since the energy levels of the two QD are different, there are always bonding and antibonding states. The intralead interaction parameters for the LL leads are the same, $g_{L} = g_{R} = g$ .

In Fig. 2 we study the effect of inter-dot tunneling on the differential conductance for $g = 1$ , i.e., the intralead interaction is absent. Let us first see Fig. 2(a). For $t_{c} = 0$ , the deep dip of differential conductance at $V = 0$ is

Summary

We have studied the effects of inter-dot tunneling coupling, intralead interaction, asymmetric dot-lead coupling strength, and temperature on quantum transport through two parallel-coupled QDs connected to LL leads. The nonequilibrium Green's function technique has been used to derive a basic current formula. The differential conductance shows the coexistence of Fano resonance and zero-bias anomaly at appropriate parameters of LL system. The typical characteristics are similar to that observed

CRediT authorship contribution statement

Kai-Hua Yang: Conceptualization, Formal analysis, Methodology, Writing – original draft. He-Yang Di: Investigation, Software, Validation. Huai-Yu Wang: Writing – review & editing. Xu Wang: Investigation, Software. Ai-ai Yang: Investigation.

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.

Acknowledgements

This work is supported by the National Key Research and Development Program of China under Grant No. 2018YFB0704304-3.

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Fano resonance and zero-bias anomaly in parallel double quantum dots coupled to Luttinger liquid leads

Highlights

Abstract

Introduction

Section snippets

Model Hamiltonian and formulation

Numerical result and discussion

Summary

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

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