Transmission line theory of collective plasma excitations in periodic two-dimensional electron systems: Finite plasmonic crystals and Tamm states

Abstract

We present a comprehensive theory of the one-dimensional plasmonic crystal formed in the grating gated two-dimensional electron gas (2DEG) in semiconductor heterostructures. To describe collective plasma excitations in the 2DEG, we develop a generalized transmission line theoretical formalism consistent with the plasma hydrodynamic model. We then apply this formalism to analyze the plasmonic spectra of 2DEG systems with step-like periodic changes of electron density and/or gate screening. We show that in a periodically modulated 2DEG, a plasmonic crystal is formed and derive closed-form analytical expressions describing its energy band spectrum for both infinite and finite size crystals. Our results demonstrate a non-monotonic dependence of the plasmonic band gap width on the electron density modulation. At so-called transparency points where the plasmon propagates through the periodic 2DEG in a resonant manner, the plasmonic band gaps vanish. In semi-infinite plasmonic crystals, we demonstrate the formation of plasmonic Tamm states and analytically derive their energy dispersion and spatial localization. Finally, we present detailed numerical analysis of the plasmonic band structure of a finite four-period plasmonic crystal terminated either by an Ohmic contact or by an infinite barrier on each side. We trace the evolution of the plasmonic band spectrum, including the Tamm states, with changing electron density modulation and analyze the boundary conditions necessary for formation of the Tamm states. We also analyze interaction between the Tamm states formed at the opposite edges of the short length plasmonic crystal. The validity of our theoretical approach was confirmed in experimental studies of plasmonic crystals in short modulated plasmonic cavities (G. C. Dyer et. al., Phys. Rev. Lett. 109, 126803 (2012)) which demonstrated excellent quantitative agreement between theory and experiment. * gregory.aizin@kbcc.cuny.edu † gcdyer@sandia.gov 2

Transmission and Reflection of Terahertz Plasmons in Two-Dimensional Plasmonic Devices

“…(18)- (21), the dependence on the coordinate x can be eliminated by multiplying the equations by the fields of plasmons and bulk modes, and integrating over x. Mode orthogonality simplifies the expressions, so that the boundary condition for the magnetic field gives the following equations (27) and (29) and…”

“…In terms of the transmission-line model [13], [14], [29], the condition k znp ≈ k zgp means small impedance mismatch between the two waveguides. Fig.…”

“…Two-dimensional structures were also studied using the Fourier analysis [28], a transmission-line model [13], [14], [29], and finite-difference-time-domain simulations [30], [31]. Although mode-decomposition techniques have been overlooked for passive structures, they can offer several advantages.…”

“…In contrast to the transmission-line model of Refs. [13], [14], [29], mode-decomposition relies on rigorous solutions of Maxwell's equations, considers the plasmon fields in the dielectric as well as in the channel, and takes non-plasmonic modes into account. On the other hand, compared to fullwave numerical simulations, mode-decomposition techniques offer the advantage of fast calculations and even analytical approximations.…”

Modeling terahertz plasmons in coupled semiconductor resonators

“…Often, however, this approach can only provide a qualitative explanation but no quantitative agreement with experiment. More advanced models include a Fourier-integral approach [1], a transmission-line model [2], a mode-matching technique [3]- [5], and ubiquitous full-wave numerical solvers. There has, however, been little effort to compare these various approaches to each other.…”