This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MAGNETICS 1 Dyadic Green’s Function for the Tensor Surface Conductivity Boundary Condition Shiva Hayati Raad 1 , Zahra Atlasbaf 1 , Mahmoud Shahabadi 2 , and Jalil Rashed-Mohassel 2 1 Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Tehran 1411713116, Iran 2 School of Electrical and Computer Engineering, Center of Excellence on Applied Electromagnetic Systems, College of Engineering, University of Tehran, Tehran 1417614418, Iran Dyadic Green’s function (DGF) for the tensor surface conductivity boundary condition (TSCBC) is formulated in this paper. Electrically biased spatially dispersive graphene sheet, densely packed graphene strips under electric bias, and magnetically biased graphene nano-patch array can be treated as the special cases of the formulation by assigning proper values to the components of the surface conductivity. In the proposed problem, due to the anisotropic nature of surface conductivity, TE and TM modes are coupled. Based on this fact, scattering Green’s functions in the upper and lower regions of the interface are expanded in terms of an appropriate linear combination of vector wave functions. The unknowns are obtained after applying the necessary boundary conditions on the tangential components of the electric and magnetic Green’s functions. The validity of the technique is verified by calculating the propagation constant of surface waves, and reflectance and transmittance of a plane-wave by graphene-based structure for the electric and magnetic biases along with the spatially dispersive sheets. It is important to note that the numerical simulation of graphene interface with tensor surface conductivity using commercial software packages is challenging due to the lack of efficiently developed models in their libraries. Index Terms— Dyadic Green’s function (DGF), electric bias, graphene, hyperbolic metamaterial, magnetic bias, magneto-plasmonic, nano-patch, spatial dispersion, spontaneous emission. I. I NTRODUCTION M AGNETICALLY biased graphene sheet has anisotropic surface conductivity and shows strong Faraday rota- tion [1]. This feature makes it a promising candidate for the realization of novel non-reciprocal devices such as iso- lators, special couplers, and phase shifters [2]. Moreover, a tunable polarization rotation is attainable by patterning the magnetically biased graphene sheet [3]. Furthermore, spatial dispersion in an unbiased graphene sheet is due to the finite size of 2-D material at the micro-scale and can also be ana- lyzed through a tensor surface conductivity [4]. The influence of the spatial dispersion on the propagation constant of the surface plasmons has been studied using transverse equivalent network [5]. It is observed that nonlocal spatial effects degrade the optical performance by up-shifting the frequency response and limiting the maximum tunable operating range, and they should be considered in the design of practical terahertz devices [6]. Two approaches are usually used to model anisotropic graphene sheet for the purpose of numerical analysis: 1) assigning a surface conductivity tensor to graphene sheet or 2) replacing it by a thin dielectric layer with an equivalent complex permittivity tensor [7]. Usually, the former model in which the problems of the huge contrast between graphene thickness and other dimensions and fine 3-D vol- umetric meshing in the numerical analysis are avoided is preferred [8]. For instance, the periodic method of moment Manuscript received October 21, 2018; revised March 7, 2019; accepted June 26, 2019. Corresponding author: Z. Atlasbaf (e-mail: atlasbaf@ modares.ac.ir). Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2019.2926363 (PMoM) has been implemented to analyze the reflectance and transmittance of graphene meta-surface by imposing the surface conductivity boundary condition [9]. Furthermore, tunable transmission characteristics of a periodic arrangement of the magnetically biased split-ring resonators (SRRs) are investigated with the finite-difference-time-domain (FDTD) method by converting the surface conductivity to a rational polynomial [10]. Recently, a recursive convolution formu- lation is incorporated in the FDTD algorithm to analyze a magnetically biased graphene sheet [11]. In another method, the similarity of graphene with lossy ferrite has been proposed through the Duality theorem for the numerical simulation of anisotropic graphene when gyrotropic materials are not included in the software library [12]. Dyadic Green’s functions (DGFs) for the magnetically biased graphene, spatially dispersive graphene, and graphene ribbons using scattering superposition method have not been reported yet, and it is the subject of this paper. Employing the tensor surface conductivity boundary condition (TSCBC), all cases are treated simultaneously. The most crucial feature of TSCBC is that it supports hybrid TE/TM surface waves. By considering the tensor impedance as the inverse of tensor conductivity, TSCBC can be considered as a generalization of scalar surface impedance (SIBC) and perfect electromagnetic conductor (PEMC) boundary conditions [13]. It is worth noting that Green’s function has been extracted for an electrically biased graphene thin sheet utilizing plane- wave expansion and for a magnetically biased graphene sheet using the Hertzian potentials by considering the particular case of a sheet residing between two identical dielectrics [14], [15]. Unlike the Hertzian potential technique which requires differ- ent potentials for vertical and horizontal current sources, our formulation is applicable for arbitrary currents [16]. 0018-9464 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 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