# Multipolar Plasmonic Resonances of Aluminum Nanoantenna Tuned by Graphene
## 2 Methods  
The schematic of the proposed hybrid graphene-Al nanoribbon (G-AlNR) structure on alumina (Al\({}_{2}\)O\({}_{3}\)) substrate is indicated in Figure 1a. The monolayer graphene is inserted into the Al\({}_{2}\)O\({}_{3}\) substrate to avoid the contact and carrier transport between AlNR and graphene. The period, width of the AlNR, height of the AlNR and the vertical distance between graphene and AlNR are denoted as \(P\), \(w\), \(t_{1}\) and \(t_{2}\), respectively. The length of AlNR is infinite along the \(y\) direction. The monolayer graphene is considered as a passive and lossy thin film with a thickness (\(t_{\text{g}}\)) of 0.34 nm. Unlike the previous study of graphene from infrared to terahertz, the surface conductivity (\(\sigma_{\text{g}}\)) of graphene in the UV range can be described by a Fano model according to the many-body effects [43; 47]:  
\[\sigma_{\text{g}}(\omega)=\frac{\sigma_{\text{CB}}(\omega)\cdot(q+E_{\text{n}} )^{2}}{1+E_{\text{n}}^{2}} \tag{1}\]\[E_{\rm n}=\frac{\hbar\omega-E_{\rm r}}{\Gamma/2} \tag{2}\]  
where \(\hbar,w\) and \(E_{\rm n}\) represent the Plank constant, the angular frequency of incident wave and the normalized energy by width \(\Gamma=0.78\) eV relative to the resonance energy \(E_{\rm r}=5.02\) eV of the perturbed exciton, respectively. \(\sigma_{\rm CB}(w)\) is the continuum background obtained by the calculation of a many-body system which denotes the response away from the singularity. The Fano parameter \(q\) determines the asymmetry of the conductivity line shape, which describes the excitonic transition strength to the unperturbed band transitions. Here, \(q\) is fixed at \(-1\). Then, the permittivity of graphene is expressed as \(\varepsilon_{\rm g}=1+i\
u_{\rm g}/(\varepsilon_{0}\omega t_{\rm g})\). \(\omega\) and \(\varepsilon_{0}\) are the angular frequency of incident wave and vacuum permittivity. The refractive index of Al\({}_{2}\)O\({}_{3}\) the substrate is set as 1.76. The dielectric permittivity of Al is taken from the literature [48]. The optical performance of the proposed G-AINR structure was investigated by the commercial software package \"EastFDTD, version 5.0\" based on the well-known finite difference time domain method. Due to the length of AINR along the \(y\) direction being infinite, the simulations of optical properties of GNRs were performed in a 2D module. Periodic boundary conditions are used along the \(x\)-axis, and two perfectly matched layers (PML) are set along the \(z\) direction to eliminate boundary scattering of the electromagnetic waves. A Gauss pulse is set as light source, and the transmission spectra is obtained by Fourier transform. The electric field distributions on a plane can be recorded directly. In the regions of graphene and AINR, the minimum mesh size is set to be 0.05 and 5 nm. For the other region, the homogeneous mesh size is set to be \(\Delta s=20\) nm, and the corresponding time step \(\Delta t=\Delta s/2c\), where \(c\) is light speed in vacuum. The proposed hybrid structure is able to be fabricated experimentally through the following fabrication processes: a large-area graphene monolayer was firstly transferred to the top surface of the Al\({}_{2}\)O\({}_{3}\) substrate. Then, an Al\({}_{2}\)O\({}_{3}\) spacer with thickness of several nanometers is deposited on the top surface of the graphene monolayer by the electronic beam evaporation method. Finally, through electron beam lithography method, the periodical arrays of AINR are fabricated directly on top of the Al\({}_{2}\)O\({}_{3}\) spacer with high quality.