Tunable massive and acoustic plasmons in two-dimensional plasmonic crystals

TL;DR

The paper addresses tunable dispersion of plasmons in two-dimensional plasmonic crystals formed by grating gates on a 2DES. It employs a Kronig–Penney–type model to derive the Bloch dispersion and to distinguish bright and dark plasmon modes, showing quadratic edges at Brillouin-zone boundaries and an acoustically linear low-frequency branch. It reveals that the effective plasmon masses $m_{\rm b}$ and $m_{\rm d}$ and the acoustic velocity $s_{\rm ac}$ are highly tunable via lattice filling factor $f$ and gate voltages, with degeneracy points where $m_{\rm b}=m_{\rm d}=0$. It also demonstrates controllable spatial localization of plasmons within the unit cell through gate bias, frequency, and filling factor, enabling electrical band-structure engineering for THz applications.

Abstract

We theoretically investigate dispersion of plasma waves propagating in a lateral plasmonic crystal based on a two-dimensional electron system with grating gates. Two specific configurations are analyzed: a system with single grating gate having ungated gaps and a double-grating-gate system. We calculate the dispersion relations for the fundamental and several higher-order plasma modes, classifying them as either ${\it bright}$ or ${\it dark}$ excitations. At the boundaries of the Brillouin zones, the dispersion of both types of excitations is shown to be quadratic, justifying introduction of effective bright and dark plasmon masses. In the low-frequency limit, the plasmonic crystal spectrum exhibits an acoustic plasma mode characterized by a certain velocity. We demonstrate that the effective plasmon mass and acoustic velocity are highly sensitive to both the crystal geometry (specifically the lattice filling factor) and the gate voltages, enabling wide-range tunability.

Figures (9)

• Figure 1: Schematic depiction of gated-gated (top) and gated-ungated (middle) structures, and the model dependence of electron concentration $N$ on $x$ (bottom) within an elementary cell of the plasmonic crystal (consisting of regions $L_1$ and $L_2$). The 2D electron gas (2DEG in the figure) occupies the $(x, y)$-plane, is homogeneous in the $y$-direction, and is periodically modulated in the $x$-direction.
• Figure 2: Band diagrams of a gated-ungated plasmonic crystal, plotted using experimental parameters from Ref. Khisameeva2025 for different filling factors. Left: $f=0.15$; center: $f=f^*=0.28$; right: $f=0.5$. Circles denote the dark and bright modes at $q = 0$ solid circle – bright mode, dashed circle – dark mode. The filling factor at which band crossing occurs, $f=f^*$ (central panel) corresponds to the condition $q_{\rm g} L_{\rm g} = q_{\rm u} L_{\rm u} = \pi$ according to Eq. \ref{['Eq_case2']} for $N = M =1$. Note that as the filling factor increases above $f^*$, the dark state "jumps" from the lower branch to the upper one (a similar transition of the dark mode from one band to another can be observed by varying the parameter $\eta$, as shown in Fig. 10 of Ref. Gorbenko2024). Consistent with Fig. \ref{['Fig_3']}, the effective mass of the bright plasmon is positive for $f<f^*$ and negative for $f>f^*$. For $f=f^*$ there are three linearly dispersing modes at small $q$: a low-frequency acoustic mode and two modes arising from the intersection of the bright and dark excitations.
• Figure 3: Dependence of the frequencies of the several lowest plasma excitations at $q=0$ on the filling factor for the gated-gated case: top, $s_1 = 0.5 s_2$; bottom, $s_1 = 2 s_2$.
• Figure 4: Dependence of the plasmonic crystal eigen-frequencies at $q=0$ on the filling factor for the gated-ungated configuration ($h = L/25, \varepsilon^s = \varepsilon$): top, $N_{\rm u}/N_g =16$; bottom, $N_g = N_{\rm u}$. In the top panel, the horizontal dotted line divides the figure into two regions: below the line, $\eta <1$; above it, $\eta>1$ (in the gated-ungated case, $\eta$ depends on frequency according to Eq. \ref{['Eq_eta_gu']}). In the bottom panel, region $\eta>1$ corresponds to very high frequencies: $\omega \gtrsim 2\omega_g$.
• Figure 5: Dependence of plasmon effective masses, normalized on $m_0$ - the effective electron mass in GaAs, on the filling factor for the gated-gated case: top, $s_1 = 0.5 s_2$; bottom, $s_1 = 2 s_2$.