Plasmon spectrum of single-layer antimonene

The collective excitation spectrum of two-dimensional (2D) antimonene is calculated beyond the low-energy continuum approximation. The dynamical polarizability is computed using a six-orbital tight-binding model that properly accounts for the band structure of antimonene in a broad energy range. Electron-electron interaction is considered within the random phase approximation. The obtained spectrum is rich, containing the standard intraband 2D plasmon and a set of single interband modes. We find that spin-orbit interaction plays a fundamental role in the reconstruction of the excitation spectrum, with the emergence of novel interband branches in the continuum that interact with the plasmon.

Figure 1

(a) The buckled atomic structure and (c) Brillouin zone of SL-Sb. (b) The band structure of SL-Sb as obtained in Ref. [20] using their TB Hamiltonian with (solid red lines) and without (dashed black lines) spin-orbit coupling.

Figure 2

Imaginary part of the polarization function $\mathrm{\Pi }(\mathbf{q},\omega )$ for different values of wave-vector $\mathbf{q}$ and chemical potential $\mu$. In the top images, $\mathbf{q}$ lies on the $\mathrm{\Gamma }$-M line in the BZ, while in the bottom images $\mathbf{q}$ lies on the $\mathrm{\Gamma }$-K line.

Figure 3

Real part of the dielectric function $\epsilon (\mathbf{q},\omega )$ for different wave-vector $\mathbf{q}$ and chemical potential $\mu$. In the top images, $\mathbf{q}$ lies along the $\mathrm{\Gamma }$-M direction of the BZ, while in the bottom images $\mathbf{q}$ lies along $\mathrm{\Gamma }$-K. For a given $\mathbf{q}$ and $\mu$, the energy of the collective plasmon mode is given by the zeros of the dielectric function.

Figure 4

Real part of the dielectric function $\epsilon (0,\omega )$ for different chemical potential $\mu$, calculated using Eq. (4).

Figure 5

The loss function $-\mathrm{Im}(1/\epsilon )$ in the low-energy and wave-vector region of the spectrum. Panels in the first row correspond to $-\mathrm{Im}(1/\epsilon )$ without SOC, while the second row is with SOC. For each image, the loss function is calculated for 50 $q$ points, with fitted points in between. Due to this, some of the figures have a chopped appearance.

Figure 6

Comparison of the band structure and density of states of single-layer antimonene with (solid red) and without (dashed black) spin-orbit coupling. Green dotted lines show the values of the chemical potential $\mu$ used in the calculations of the loss function in Fig. 5.

Figure 7

The loss function $-\mathrm{Im}(1/\epsilon )$ in the low-energy and wave-vector region of the spectrum for electron doping. For each image, the loss function is calculated for 50 $q$ points, with fitted points in between. Due to this, some of the figures have a chopped appearance.

Figure 8

Electron-hole excitation spectrum of single-layer antimonene. Imaginary part of the polarization function $\mathrm{Im}{\mathrm{\Pi }}^0(\mathbf{q},\omega )$ for different values of the chemical potential $\mu$, with $\mathbf{q}$ along the $\mathrm{\Gamma }$-K line in the BZ. The dotted black lines represent the $\mathbf{q}$ values depicted in the bottom images of Fig. 2.

Figure 9

Real part of the dielectric function $\epsilon (\mathbf{q},\omega )$ (top panels) and the loss function $-\mathrm{Im}(1/\epsilon (\mathbf{q},\omega \left)\right)$ (bottom panels) for different values of the chemical potential $\mu$, with $\mathbf{q}$ along $\mathrm{\Gamma }$-K of the Brillouin zone. The black dotted lines indicate the values of $\mathbf{q}$ used in Fig. 3.