Comparison of Tensor Boundary Conditions With Generalized Sheet Transition Conditions

This paper compares the tensor boundary conditions (TBCs) with the surface-susceptibility-based generalized sheet transition conditions (GSTCs) for the modeling of metasurfaces and 2-D material allotropes. First, we recall the GSTCs, distinguishing the full-tensor (FT) GSTCs and the tangential-tensor (TT) GSTCs, which correspond to the most general and most reported GSTC forms, respectively. We show, by separating tangential and normal polarizations, that the FT-GSTCs involve 36 independent susceptibility parameters, associated with $3\times 3$ electric, magnetic, electric-to-magnetic, and magnetic-to-electric susceptibility tensors, despite the 2-D nature of the structure. Moreover, we find that suppressing the normal polarizations nontrivially reduces the number of FT-GSTC parameters to 24, which is greater than the 16 parameters of the TT-GSTCs. Then, the paper recalls the TBCs as originally reported in a previous study, called here scalar-parameter (SP) TBCs, and extends them to their tensorial-parameter (TP) counterparts, called the TP TBCs. In both formulations, we derive the equivalent susceptibilities in terms of the TBC parameters. We show that the SP-TBCs involve eight equivalent susceptibility parameters, among which only three are independent, while the TP-TBCs involve 16 independent susceptibility parameters. Next, we compare the two models, with their two respective formulations, in terms of both generality and physicality. We deduce from the number of independent susceptibility parameters the following ranking between the four formulations: 1) FT-GSTCs (36 independent parameters); 2) TT-GSTCs = TP-TBCs (16 independent parameters); 3) SP-TBCs (3 independent parameters), and illustrate with examples the property and functionality limitations of the TT-GSTCs, TP-TBCs, and SP-TBCs due to their parameter restrictions. Finally, we show that while the GSTCs appropriately describe the physics of the problem, the TBCs are discordant with it.