## 2 Phonons and Plasmons in Nanoshells  
In order to describe the interface and surface phonons, and their coupling to the surface plasmons, the problem of finding the electrostatic potential in a planar multilayer structure needs to be addressed. This potential has been derived for an arbitrary number of layers in [48] (see also [49; 50]). The method of [48] was later used to determine the electrostatic potential in multilayered spherical systems [51]. The phonon frequencies were obtained from the secular equation for the dynamical matrix [48; 51]. There exist four types of phonon modes in spherical nanoshells: \"sphere-like\", \"shell-like\", \"bulk-like\", and \"interface\" modes. The electrostatic potentials of sphere-like, shell-like, and bulk-like polar modes are confined to the core, shell, and outer regions, respectively, and do not penetrate the neighboring media. Due to their localization at interfaces, only the interface phonons strongly interact with the interface plasmons. The boundary conditions for the electrostatic potential of the  
Figure 2: Light absorption cross section versus the photon energy for the d-InAs/u-GaAs/Hexane nanostructure. The numbers used as peak labels correspond to the interface phonon-plasmon modes in Figure 1b. polarization field follow from the continuity of the normal components of the electric displacement at the uncharged interfaces between the core and the shell, and between the shell and the outer region. The resulting boundary conditions,  
\[ \begin{array}{l}b_{11}V_{l}(R_{1})+b_{12}V_{l}(R_{2})=0,\\ b_{21}V_{l}(R_{1})+b_{22}V_{l}(R_{2})=0\end{array}  \tag{1}\]  
determine the frequencies and the amplitudes of the interface phonons (cf. [51]). In Equation (1), \(R_{k}\) is the radius of \(k\)th spherical layer (\(k\) = 1,2) and \(V_{l}(R_{k})\equiv V(R_{k},l,m;\,\omega)\) is the amplitude of the polarization-induced electrostatic potential expanded in spherical harmonics with angular momentum quantum numbers \(l\),\(m\) and frequency \(\omega\). In the remainder of this paper, the last two quantum numbers and the frequency are omitted. The coefficients \(b_{11}\), \(b_{12}\), \(b_{21}\), and \(b_{22}\) in Equation (1) are  
\[ \begin{array}{l}b_{11}=lR_{1}\epsilon_{1}(\omega)+\frac{(R_{1}R_{2})^{l+1}}{ R_{2}^{R_{2}^{l+1}-R_{1}^{l}R_{1}^{l+1}}}[(l+1)(\frac{R_{1}}{R_{1}})^{l}+l( \frac{R_{1}}{R_{2}})^{l+1}]\epsilon_{2}(\omega),\\ b_{12}=b_{21}=(2l+1)R_{1}^{l+1}R_{2}^{l+1}/(R_{2}^{2l+1}-R_{1}^{2l+1}) \epsilon_{2}(\omega),\\ b_{22}(\omega)=\frac{(R_{1}R_{2})^{l+1}}{R_{2}^{l+1}R_{2}^{l+1}-R_{1}^{l}R_{1} ^{l+1}}[(l+1)(\frac{R_{1}}{R_{2}})^{l}+l(\frac{R_{2}}{R_{1}})^{l+1}]\epsilon_ {2}(\omega)+(l+1)R_{2}\epsilon_{3}(\omega),\end{array}  \tag{2}\]  
with the dielectric functions  
\[\varepsilon_{i}(\omega)=\left\{ \begin{array}{l}\varepsilon_{\omega,i}\frac{ \omega^{2}-\omega_{2\Omega,i}^{2}}{\omega^{2}-\omega_{\Omega,i}^{2}}-\frac{ \omega_{\Omega,i}^{2}}{\omega^{2}}),\text{ if the $i$th medium is doped}\\ \varepsilon_{\omega,i}\frac{\omega^{2}-\omega_{\Omega,i}^{2}}{\omega^{2}- \omega_{\Omega,i}^{2}}),\text{ if the $i$th medium is undoped}\end{array} \right. \tag{3}\]  
and  
\[\varepsilon_{3}(\omega)=\varepsilon_{\infty,3}. \tag{4}\]  
Equations (1) and (2) for the core/shell/outer medium nanostructures are a particular case of Equations (6) and (7) of [51], corresponding to the number of spherical layers \(K\) = 3 and the surface charge density \(\sigma^{\prime}=0\). The efficient control over the yield and lifetime of a charge-separated state, by optimizing the shell thickness, allows for engineering core/shell nanostructures suitable for charge-transfer applications [52]. The appearance of interface charges, for instance, due to interactions with a biomolecule, leads to a detectable shift of the surface plasmon resonance frequency of the nanoshell [53].