Exploring stable long-lifetime plasmon excitations in the Lieb lattice

Abstract

The subject of the present paper is a thorough numerical investigation of plasmon expectations, their dispersions and damping within a Lieb lattice. The Lieb lattice is known for its unique low-energy band structure which consists of a bandgap as well as a flat band intersecting the conduction band at its lowest point. In contrast to previously studied dice lattice, the location of the current flat band exhibits reduced and broken symmetries, which give rise to interesting electronic and optical properties of this new material. In this work, we have investigated the conditions for observing a well-defined and stable plasmon mode within a wide frequency range. Specifically, we have considered a free-standing layer with various doping levels, as well as different types of monolayers of the Lieb lattice interacting with a surface-plasmon mode localized on top of a semi-infinite conductor. In particular, we have observed and described fully long-living plasmon modes with unusual energy dispersions. Additionally, we have carried out a detailed investigation on the static screening associated with the Lieb lattice. Our study has further revealed that these predicted features seem to be quite different from those of pseudospin-1 materials but resemble those of graphene instead.

Figures (11)

• Figure 1: (Color online) The low-energy band structures (the energy dispersions) of a gapped dice lattice (left panel) and the Lieb lattice (right panel). These two dispersions demonstrate a lot of similarities, e.g. the presence of a flat band between the valence and conduction bands. However, the locations of these two flat bands are quite different. For a Lieb lattice, the flat band intersects with the conduction band at its lowest point, which is in contrast with a completely symmetric case for a dice lattice.
• Figure 2: (Color online) Numerically calculated dynamical polarization function $\Pi^{(0)}(q,\omega \, \vert \, \Delta_0)$ for a Lieb lattice. Here, each of upper panels $(a)$, $(b)$ and $(c)$ displays the imaginary part of $\Pi^{(0)}(q,\omega \, \vert \, \Delta_0)$ and the particle-hole modes (or the single-particle excitation regions), corresponding to a finite imaginary part of the polarization function. The three lower panels $(d)$, $(e)$ and $(f)$ represent the real part of $\Pi^{(0)}(q,\omega \, \vert \, \Delta_0)$ as functions of both wave vector $q$ and frequency $\omega$. Two left panels $(a)$ and $(d)$ connect to a zero-gap model of a Lieb lattice with $\Delta_0 = 0$, and the energy dispersions are equivalent to those of a zero-gap $\alpha$-$T_3$ model. The other four panels correspond to an actual Lieb lattice with $k_0 = 1.0\, k_F^{(0)}$ for different Fermi energies (doping levels) $E_F=1.0\,E_F^{(0)}$ and $E_F=2.0\,E_F^{(0)}$, as labeled.
• Figure 3: (Color online) Constant frequency cuts to the real part of the polarization function $\Pi^{(0)}(q,\omega \, \vert \, \Delta_0)$ for a Lieb lattice as a function of wave vector $q$. Panel $(a)$ and $(c)$ deal with a zero-gap model for a Lieb lattice, corresponding to $k_0 = 0$, and the energy dispersions become equivalent to those of a zero-gap $\alpha$-$T_3$ model. All the other plots $(b)$, $(c)$ and $(d)$ are associated with an actual Lieb lattice with $k_0 = 1.0\, k_F^{(0)}$ with different Fermi energies (doping levels) $E_F=0.9\,E_F^{(0)}$ and $E_F=2.0\,E_F^{(0)}$ and $E_F=1.0\,E_F^{(0)}$, respectively. Each curve connects to a specific fixed frequency $\omega = 0.5\,E_F^{(0)}/\hbar$, $1.0\,E_F^{(0)}/\hbar$, $1.5\,E_F^{(0)}/\hbar$, $2.0\,E_F^{(0)}/\hbar$ and $2.5\,E_F^{(0)}/\hbar$ for all panels, as labeled.
• Figure 4: (Color online) Plasmon dispersions for a monolayer Lieb lattice. The left panels $(a)$, $(c)$, $(e)$, $(g)$ and $(i)$, as well as the right panel $(f)$, represent density plots of the real part of inverse dielectric function $1/\epsilon(q,\omega \, \vert \, \Delta_0)$ whose peaks correspond to the plasmon modes. The broadened peaks of $1/\epsilon(q,\omega \, \vert \, \Delta_0)$, on the other hand, reveal damped plasmons. Here, the right-hand-side panels $(b)$, $(d)$, $(h)$ and $(j)$ display the exact numerical solutions for the plasmon branches. The Landau damped plasmons are presented in gray dashed curves while undamped ones are given by red solid curves. The first two upper rows (plots $(a)$ - $(c)$) connect to a zero-gap model with $k_0=0$ for a Lieb lattice, and then, the energy dispersions are equivalent to those of a zero-gap $\alpha$-$T_3$ model. The remaining panels $(d)$ - $(h)$ correspond to a regular Lieb lattice with the same $k_0 = 1.0 \,k_F^{(0)}$ but different Fermi energies (doping levels) $E_F=1.0\,E_F^{(0)}$ and $E_F=2.0\,E_F^{(0)}$. Meanwhile, we also present the plasmon branches for different values of the relative dielectric constant $\alpha_r=2.0$ and $7.0$, as labeled.
• Figure 5: (Color online) Plasmon dispersions for two Lieb lattice Coulomb-coupled monolayers. Here, three top panels $(a)$, $(b)$ and $(c)$ represent the density plots for the real part of an inverse effective dielectric function $1/\epsilon_{DL}^{(L)}(q,\omega \, \vert \, \Delta_0)$ for a Coulomb-coupled double layer whose peaks represent the plasmon modes. The broadened peaks of $1/\epsilon^{(L)}(q,\omega \, \vert \, \Delta_0)$, on the other hand, display damped plasmons. The lower panels $(b)$, $(d)$ and $(e)$ present the frequency dependence of the constant wave vector cuts of inverse effective dielectric function $1/\epsilon_{DL}^{(L)}(q,\omega \, \vert \, \Delta_0)$. The left panes $(a)$ and $(c)$ connect to the case of two identical Lieb lattices with $k_0 = 1.0 k_F^{(0)}$. The middle panels $(b)$ - $(e)$ are associated with two regular Lieb lattice with $k_0 = 1.0 \,k_F^{(0)}$ and different Fermi energies (doping levels) $E_F=1.0\,E_F^{(0)}$ and $E_F=2.0\,E_F^{(0)}$, and two equivalent Lieb lattices with $E_F=2.0\,E_F^{(0)}$ are presented in panels $(c)$ and $(f)$. All results for plasma branches correspond to the relative dielectric constant $\alpha_r=5.0$, as labeled.