Research ArticleSurface plasmon polaritons in a waveguide composed of Weyl and Dirac semimetals
Introduction
The lattice model proposed by Haldane [1], in 1980, revealed that materials could have topologically nontrivial band structures even without an external magnetic field. A surge in research on topological materials started in 2005 [[2], [3], [4], [5], [6], [7]]. The topological properties originating from strong spin-orbit interactions have been proposed theoretically and observed experimentally in the materials so-called topological insulators (TIs) [8] and topological semimetals [9,10]. Three-dimensional topological insulators are materials that have a bandgap in their bulk energy spectrum, while their surface states are gapless. These surface states exhibit a Dirac cone dispersion with spin degree locked to the momentum direction. Breaking the time-reversal symmetry in these materials leads to opening a gap in their surface state spectrum which results in remarkable magnetoelectric effects [[11], [12], [13]]. In contrast, bulk Dirac semimetals (BDSs) and Weyl semimetals (WSMs) are three-dimensional materials with linear dispersion around nodes connecting conduction and valence bands [9] of their bulk energy spectrum. These nodes are called Dirac or Weyl nodes which appear in pairs and are protected by topology and symmetries against the small perturbations. This means that applying a small disorder to BDSs/WSMs can move the location of these nodes, but it can not create a bandgap. Two degenerate Dirac nodes of BDSs are split in momentum or energy space into pair of Weyl nodes due to breaking the time-reversal symmetry or inversion symmetry, respectively. In the case of breaking the time-reversal symmetry, Weyl nodes have opposite chiralities which act as magnetic monopoles for Bery curvature in the momentum space. The topological properties of these materials manifest in unusual surface states as Fermi arcs [[14], [15], [16], [17], [18]] and chiral anomaly [[19], [20], [21], [22]] that lead to several novel exotic effects such as negative magnetoresistance [23], anomalous Hall effect [24], and chiral magnetic effects [25]. Furthermore, the effect of the chiral anomaly causes optical anisotropy and unusual optical transitions. The exotic properties of WSMs have great potential for applications. WSMs can be employed in high-speed electronics and spintronics due to their high mobility and large magnetoresistance properties [26].
Electromagnetic waves that are confined to the interface of a conductor with a dielectric and can propagate along it [27,28] are called surface plasmon polaritons (SPPs). SPPs were proposed and observed by electron energy loss spectroscopy and optical methods of surface gratings or attenuated total reflection [[29], [30], [31], [32]]. They are utilized intensively in the fields of photonics, imaging, solar cells, biosensors, surface microscopy, biomolecular detection, or lithography [[33], [34], [35], [36], [37], [38], [39], [40], [41]] due to their localization at sizes smaller than the wavelength of light.
Theoretical [13,42] and experimental [43,44] investigations of SPPs formed on the surface of TIs have been performed recently. Properties of SPPs also have been studied on BDSs [45] and WSMs [46] surfaces and different structures such as thin films [45,47,48] or waveguides [49,50]. We have recently investigated SPPs in a slot waveguide comprised of two semi-infinite Weyl semimetals with different configurations [49] and symmetries [50]. We have presented a detailed study of symmetric slot waveguide in Voigt-Voigt and Faraday-Faraday configurations which result in bidirectional SPPs propagation. Moreover, we have introduced exotic asymmetric structures based on hybrid configurations of Voigt-Faraday, Voigt-Voigt configurations with different magnitudes or orientations of the vector connecting Weyl nodes [49], and two WSMs with different symmetries [50]. We have shown that SPPs have nonreciprocal dispersion and can propagate unidirectionally in these structures. Furthermore, We have demonstrated that these properties of the SPPs can be tuned by the topological properties and chemical potential of the WSMs without the need to application of an external magnetic field. To complement our studies on the features of SPPs propagation on the surface of WSMs, in the present work, we explore the dispersion of SPPs in a slot waveguide composed of WSM and BDS media. We investigate SPPs propagation at the interface of WSM in both Voigt and Faraday configurations. We show that hybridization of SPP modes localized at the interfaces of WSM and BDS leads to the nonreciprocal dispersion for them with bidirectional or even unidirectional propagation in Voigt configuration and to reciprocal SPP modes in Faraday configuration. SPPs dispersion can be controlled by the waveguide thickness and the physical parameters of WSM and BDS such as chemical potential. Our findings may be employed in unidirectional optical devices.
The remainder of the paper is organized as follows. In Section 2 we introduce our theoretical model and give the necessary equations for calculating dispersion relation in Voigt and Faraday configurations. Moreover, we present our main results and related discussions. Finally, we end by giving a conclusion in Section 3.
Section snippets
The theoretical model and equations
The topological properties of WSMs result in a modified displacement electric field [[51], [52], [53], [54]] that is given by the following relation in the case of the broken time-reversal symmetry [46],where is the unite vector along the vector connecting two Weyl nodes. The diagonal element of the dielectric tensor ϵ(ω) has only nonzero real part for Ω = ℏω/Ef < 2, with Ef being the Fermi energy, and is given by Ref. [45],where two
Conclusion
In conclusion, we have investigated features of SPP modes in an asymmetric slot waveguide composed of a WSM in two different Voigt and Faraday configurations and a BDS connected via a dielectric layer. The nonreciprocal dispersion of the SPP modes in the Voigt configuration emerges as unidirectional for some ranges of the frequencies depending on the choice of the materials. In particular, we demonstrated that the waveguide thickness and the physical parameters of the Weyl and Dirac semimetals
Funding
No funding was received for this work.
Intellectual property
We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property.
Research ethics
We further confirm that any aspect of the work covered in this manuscript that has involved human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript.
IRB approval was obtained (required for studies and series of 3 or more cases).
Written consent to publish potentially identifying information, such as details or the case and photographs, was obtained from the patient(s) or their legal guardian(s).
CRediT authorship contribution statement
S. Oskoui Abdol: Software, Validation, Data curation, Writing – original draft. B. Abdollahipour: Conceptualization, Methodology, Writing – review & editing, Visualization. A. Soltani Vala: Conceptualization, Methodology, 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.
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