## 4 Effect of Nanoparticle Shape: Contact Angle between Ga NPs and Graphene  
A critical issue when considering the near-field enhancement produced in metal nanoparticle-graphene-metal substrate systems is the contact angle, \(\theta_{c}\), between the NP and the graphene (see Figure 5a). When metal NPs are evaporated on graphene, as in Refs. [1; 2; 3], their contact angle with the graphene layer is related to their interfacial free energy. Common plasmonic noble metals such as Au, Ag, and Cu present values of the contact angle with graphene of 113\({}^{\circ}\)[22], 114\({}^{\circ}\)[23], and 122\({}^{\circ}\)[24], respectively. It is also worth mentioning that while for Ag the contact angle remains stable in time [23], for copper the contact angle progressively reduces to a value of 60\({}^{\circ}\) in 30 min [24]. In the case of Ga, the contact angle with smooth graphene layers, like the ones in our experiments, is close to a hemispherical NP shape, \(\sim\)80\({}^{\circ}\). However, for rough and rippled graphene surfaces, the value of the contact angle can go up to 150\({}^{\circ}\). The contact angle that defines the shape of the NP is a key parameter to be considered when modeling the electromagnetic interactions in this type of system [25]. Figure 4: Ga NPs on G/Ni support. (**a**) Experimental extinction coefficient spectra \(\langle k\rangle\) recorded in real time during Ga NP deposition on the G/Ni substrate. Each spectrum is acquired every 1 sec; the black spectrum refers to the initial G/Ni sample; the final yellow spectrum refers to Ga NP/G/Ni; the arrow indicates Ga NP deposition. For comparison, the \(\langle k\rangle\) spectra for equivalent Ga NPs on G/glass is also shown. (**b**) Raman spectra of the graphene layer before and after Ga NP deposition. (**c**) XPS spectra of the photoelectron core level Ga3\(d\) for Ga/G/Ni. The inset shows the AFM image of the Ga NP morphology. In light of this, Figure 5b shows the absorption cross-section \(C_{abs}\) of spherical Ga NPs of \(R=55\) nm illuminated at normal incidence (see sketch in Figure 5a) forming different contact angles with a G/Cu substrate: 90\({}^{\circ}\) (hemisphere), 120\({}^{\circ}\), 150\({}^{\circ}\) and 180\({}^{\circ}\) (sphere). The \(C_{abs}\) spectra for the sphere (\(\theta_{c}=180^{\circ}\)) show two clear resonant peaks: a low-energy peak associated with a dipolar mode (related to the LO mode in the hemisphere), and a high-energy one associated with a higher-order mode (related to the TO mode in the hemisphere). The dipolar mode (LO) progressively quenches and slightly blue-shifts with the decrease of \(\theta_{c}\), while the higher-order plasmon mode (TO) is severely blue-shifted and damped. The near-field distribution in Figure 5c shows high enhancement at the NP-substrate boundary for the spherical nanoparticle, which decreases with the decrease in \(\theta_{c}\) (going from the spherical to the hemispherical geometry). One of the factors affecting the near-field enhancement is the contrast between the optical properties of the NP and its surrounding medium (electromagnetic boundary conditions). Dipolar modes are produced by oscillations of the electrons at the equatorial plane of the spherical NP. For high values of \(\theta_{c}\), the effective surrounding medium around the equatorial plane of the NP is air (high contrast, metal-insulator) leading to high near-field enhancements. This effect is additionally enhanced by the coupling of the mode to the cavity formed in the NP-air-substrate region. Moving to the hemispherical shape (decreasing values of \(\theta_{c}\)), the contrast between the NP and the surrounding medium decreases due to the proximity of the conducting substrate, and the near-field enhancement becomes lower. Moreover, the cavity region becomes smaller, hampering the coupling of the mode until its full disappearance at \(\theta_{c}=90^{\circ}\). It is worth mentioning that this mode is relevant for SERS applications [1; 2].