Magnetoplasmons in $N$-layer structures

Abstract

We provide a systematic framework to investigate the magnetoplasmons of multilayer two-dimensional electron systems by using the Kac--Murdock--Szegő (KMS) Toeplitz matrix to consider interlayer Coulomb interactions. In the absence of interlayer tunneling, we show that the single-layer magnetoplasmon branch splits into $N$ collective modes -- one in-phase mode and $N-1$ out-of-phase modes -- and derive their asymptotic behaviors in the long-wavelength limit, as well as in the limit of large layer separation and strong magnetic fields. When interlayer tunneling is present, we clarify the magnetoplasmon dispersion, both qualitatively and quantitatively, by identifying the magnetoplasmon mode associated with each interband transition, as well as tunneling magnetoplasmons arising from interband transitions with the same Landau level index. Our study presents the hybridization between the modes governed by underlying symmetries, along with an enhanced tunneling magnetoplasmon gap exceeding the associated interband gap. The KMS-based analytic formalism thus provides a comprehensive physical understanding of magnetoplasmons in multilayer structures.

Figures (7)

• Figure 1: Schematic illustration of the dominant interband transitions for each mode and the corresponding mode dispersions in a trilayer system filled up to the lowest band: (a) the decoupled-layer limit and (b) the case with finite interlayer tunneling. The green dashed lines denote the Fermi energy, while the black (gray) horizontal lines represent occupied (unoccupied) Landau levels. The red (blue) arrows indicate the dominant transitions in the long-wavelength limit for the magnetoplasmon (tunneling magnetoplasmon) modes. The lower panels show the mode dispersions in the limits of $ql\to0$ and $ql\to\infty$. For a given mode, the same line style is used consistently across the figures. The insets in (a) show the corresponding charge oscillations in the long-wavelength limit. In (b), magnetoplasmon (tunneling magnetoplasmon) modes are depicted by black (blue) lines, and anticrossing behavior between modes is highlighted in yellow.
• Figure 2: Loss function plot $L(\mathbf{q},\omega)=-\text{Im}{\text{Tr}[\epsilon^{-1}(\mathbf{q},\omega)]}$ for a trilayer system in the decoupled limit, compared with the asymptotic forms in Eq. (\ref{['Eq:small_q_chg']}) and Eq. (\ref{['Eq:large_q_chg']}), which are drawn as black dashed lines. The inset displays Coulomb oscillations for the $\omega_\alpha^{(1)}$ ($\alpha=1,2,3$) modes; note that $\omega_\alpha^{(2)}$ exhibits an identical oscillation pattern. For the calculations, we used parameters typical for a GaAs quantum well: $m=0.067\,m_e$, $\kappa=10.9$, $d=100\,\text{\AA}$, and $\hbar\omega_c=15\,\text{meV}$, with a phenomenological broadening $\eta=10^{-3}\,\hbar\omega_c$.
• Figure 3: Schematic illustration of magnetoplasmons associated with each band transition and the loss function plot $L(\mathbf{q},\omega)=-\text{Im}{\text{Tr}[\epsilon^{-1}(\mathbf{q},\omega)]}$ for a coupled trilayer system filled up to the states (a) $\varepsilon_{01}$, (b) $\varepsilon_{02}$, and (c) $\varepsilon_{03}$. For band transitions indicated by red (blue) arrows, the corresponding magnetoplasmons (tunneling magnetoplasmons) are listed above the arrows. When multiple magnetoplasmons share the same gap, a splitting occurs unless the modes are related by subband-index-reversal symmetry. For example, in (b), $\omega_{1a}^{(1)}$ and $\omega_{1b}^{(1)}$ combine to form $\omega_{1,\pm}^{(1)}$, whereas $\omega_{2a}^{(1)}$ and $\omega_{2b}^{(1)}$ are degenerate and form only $\omega_{2}^{(1)}$. The red and blue stars indicate the magnetoplasmon and tunneling magnetoplasmon gap values evaluated using Eq. (\ref{['Eq:MP_gap']}) for the red stars in (a), Eq. (\ref{['Eq:MP_gap_modified']}) for the red stars in (b) and (c), and Eq. (\ref{['Eq:TMP_mode_gap']}) for the blue stars. The blue dashed lines represent Eq. (\ref{['Eq:TMP_mode_infty']}). The insets in each panel highlight the crossing and anticrossing behaviors between modes; the corresponding regions are indicated by black boxes. The calculations were performed using the same parameters as in Fig. (\ref{['Fig: schm. pic for N=3']}), except that $t=0.8\,\mathrm{meV}$.
• Figure 4: Schematic illustration of magnetoplasmons associated with each band transition in the weak Coulomb interaction limit, together with the loss function plot $L(\mathbf{q},\omega)=-\mathrm{Im}\,\mathrm{Tr}[\epsilon^{-1}(\mathbf{q},\omega)]$, for a coupled tetralayer system filled up to the state $\varepsilon_{01}$. The same parameters as in Fig. \ref{['Fig:numerical_results_N3_coupled']}(a) are used with $t=0.5\,\rm{meV}$. Unlike Fig. \ref{['Fig:numerical_results_N3_coupled']}(a), the blue stars obtained from Eq. (\ref{['Eq:TMP_mode_gap']}) deviate from the actual tunneling magnetoplasmon gaps, reflecting additional contributions from interband transitions other than the dominant one in the weak Coulomb interaction limit. By contrast, the red stars obtained from Eq. (\ref{['Eq:MP_gap']}) correctly reproduce the magnetoplasmon gaps.
• Figure 5: The top panel shows a schematic illustration of magnetoplasmons associated with each band transition for a trilayer system filled up to the state $\varepsilon_{01}$ in the limit $t \gg \hbar\omega_c$. The middle and bottom panels display the loss function plot $L(\mathbf{q},\omega)=-\mathrm{Im}\,\mathrm{Tr}[\epsilon^{-1}(\mathbf{q},\omega)]$ calculated for $t /(\hbar\omega_c)= 3$ and $t /(\hbar\omega_c)= 10$, respectively, with fixed $t = 0.5$ meV. As in Fig. \ref{['Fig:numerical_results_N3_coupled']}, the red and blue stars correspond to results from Eq. (\ref{['Eq:MP_gap']}) and Eq. (\ref{['Eq:TMP_mode_gap']}), respectively. The gray lines indicate the boundaries of the electron-hole continuum in a coupled trilayer 2DEG system at zero magnetic field, corresponding to total electron densities of $n_{\mathrm{tot}} = 2.33 \times 10^{9}\,\mathrm{cm}^{-2}$ (middle panel) and $n_{\mathrm{tot}} = 0.70 \times 10^{9}\,\mathrm{cm}^{-2}$ (bottom panel).