Screened topological plasmons in graphene plasmonic crystals

Abstract

We study topological effects in an one-dimensional plasmonic crystal formed by the screened acoustic plasmons emerging in a periodically modulated graphene sheet, placed on top of a metallic substrate. To this end, we develop the theory of quantization of screened plasmons, as appropriate for lossless graphene described by a Drude conductivity. By analyzing the resulting band structure, we show that the crystal sustains nontrivial topological bands, with quantized geometric phase. We further show that in a finite, open system, edge states appear within the band gap, which undergo a topological phase transition and merge with bulk states as the modulation increases. Our work provides a robust theoretical framework for the study of band structure and topology of layered media, and extends the possibilities for engineering two-dimensional materials with external modulation.

Figures (6)

• Figure 1: Modulation of the Fermi energy level in space, across four unit cells of length $s$. The modulation widths $a$ in the lower step of the Fermi level, and $s-a$ in the upper step, are marked with red arrows.
• Figure 2: Spectrum of the Hamiltonian given in Eqs. \ref{['eq:FullHamiltonian']} for two values of graphene--metal separation $d^\prime=d/s$, with parameters $s=500$ nm, $a=250$ nm, $E_{\mathrm{F},0}=500$ meV and $\Delta E=E_{\mathrm{F},0}/5$. In black line, a large separation $d^\prime=0.8$ yields an approximately parabolic dispersion, and in red line a very small separation, $d^\prime=0.1$, produces a linear dispersion with acoustic plasmons.
• Figure 3: Profile of the particle-component (first entry in Eq. (\ref{['eq:WavefunctionSpinor']})) of the wavefunction spinor (real part in red, imaginary part in black) before (plots (a) and (b)) and after (plots (c) and (d)) a topological phase transition. The parities $\xi_n$, $n=2,3$, of the second and third bands are denoted in each panel, and are swapped in the two phases. These two bands become gapless at $k=0$ as the modulation width increases past a critical point $a_c \approx 0.43 \,s$ at fixed $\Delta E = E_{\mathrm{F},0}/5$. At the band edge, the bands are still gapped, so their parity eigenvalue remains fixed, such that the their Zak phases are also exchanged at this point.
• Figure 4: Phase diagram of the topological invariant $\nu$ (blue) as a function of the modulation width $a$. In red, the value of the gap just above the second (top) and third (bottom) gap. As the modulation width changes, the plasmon bands become gapless at some critical values of $a$, allowing for an exchange of Zak phase between the crossing bands. The second gap closes once.
• Figure 5: (a) Modulation profile of a single supercell. It has a region of plasmonic vacuum of length $Z$, where the Fermi energy is equal to zero, followed by a plasmonic crystal containing $N_s$ unit cells, equal to the ones used in the previous sections. (b) Sketch of the first BZ for the periodic lattice of the previous sections (in light blue) and the supercell (in light red). In the sketch, $T_m=5s$, but for large supercells the red interval becomes much smaller.