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We present a formalism for calculating the image potential for a two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI) as well as for a 2D topological insulator (TI). The formalism is further generalized for including the Coulomb coupled multiple layers. Roles of broken inversion symmetry near the surface and the dielectric environment are investigated by using a surface-response function. The insignificant role of SOI in 2DEG is dramatically enhanced in TI by selecting a small relative permittivity $\epsilon_b$ for the dielectric environment. Manipulating $\epsilon_b$ is proven to provide an efficient way to drive electrons with opposite spins into two different integral quantum Hall states. The prediction made in this paper is expected to be experimentally observable for a 2DTI system, such as Bi$_2$Se$_3$, with a helical spin behavior and a dominant linear Rashba SOI-like term in the energy dispersion.

We perform a comprehensive analysis of the spectrum of graphene plasmons which arise when a pair of sheets are confined between conducting materials. The associated enhanced local fields may be employed in the manipulation of light on the nanoscale by adjusting the separation between the graphene layers, the energy band gap as well as the concentration of charge carriers in the conducting media surrounding the two-dimensional (2D) layers. We present a theoretical formalism, based on the calculation of the surface response function, for determining the plasmon spectrum of an encapsulated pair of 2D layers and apply it to graphene. We solve the coupled equations involving the continuity of the electric potential and discontinuity of the electric field at the interfaces separating the constituents of the hybrid structure. We have compared the plasmon modes for encapsulated gapped and gapless graphene. The associated nonlocal graphene plasmon spectrum coupled to the “sandwich” system show a linear acoustic plasmon mode as well as a low-frequency mode corresponding to in-phase oscillations of the adjacent 2D charge densities. These calculations are relevant to the study of energy transfer via plasmon excitations when graphene is confined by a pair of thick conducting materials.