**2. Theory on monomer and dimer systems**

#### **2.1 Single metal sphere: monomer systems**

We refer to monomer systems when considering a single plasmonic nanostructure. Practically, for each system, in which metal nanoparticles are sparsely distributed in a dielectric environment, the interaction between the individual nanostructures can often be neglected so that each nanostructure can be considered as a monomer. Here, we address the equations for a single metal sphere regarding the light-matter interaction in detail [38–40], while the solution for further arbitrarily shaped single elements, which may appear more frequently in reality, is not derived, but related theoretical work can be found in refs. [41–46].

In the quasistatic approximation, light scattering by a spherical particle, the radius of which is *a a*ð Þ ≪ *λ* , in a uniformly distributed electric field of *E* ¼ *E*0*r* cos *θ*

*The Influence of Geometry on Plasmonic Resonances in Surface- and Tip-Enhanced Raman… DOI: http://dx.doi.org/10.5772/intechopen.108182*

**Figure 1.** *Schematic sketch of a metal nanosphere in an electric field [38].*

(as shown in **Figure 1** [40]) is described by the Laplace equations for the scalar electric potential [38].

$$E\_{in} = -\nabla \Phi\_{in} \tag{1}$$

$$E\_{out} = -\nabla \Phi\_{out} \tag{2}$$

$$\nabla^2 \phi\_{\text{int}} = \mathbf{0} \ (r < a) \tag{3}$$

$$\nabla^2 \phi\_{out} = \mathbf{0} \ (r > a) \tag{4}$$

with the continuous boundary conditions

$$
\phi\_{in} = \phi\_{out}, \ \varepsilon\_m \frac{\partial \phi\_{in}}{\partial r} = \varepsilon\_d \frac{\partial \phi\_{out}}{\partial r} \ (r = a) \tag{5}
$$

*Ein* and *Eout* indicate the electric fields inside and outside the metal particle with their electrical potential written as *ϕⅈ<sup>n</sup>*ð Þ *r*, *θ* and *ϕout*ð Þ *r*, *θ* . *ε<sup>m</sup>* and *ε<sup>d</sup>* are the dielectric functions of the metal sphere and the dielectric environment, respectively. If we consider that the electric field at infinite distance is not disturbed by the metal sphere, the solution of Eqs. (1)–(4) can be written as [38].

$$\phi\_{in} = \frac{-\mathfrak{z}\varepsilon\_d}{\varepsilon\_m + 2\varepsilon\_d} E\_0 r \cos \theta \tag{6}$$

$$\phi\_{out} = -E\_0 r \cos \theta + a^3 E\_0 \frac{\varepsilon\_m - \varepsilon\_d}{\varepsilon\_m + 2\varepsilon\_d} \frac{\cos \theta}{r^2} \tag{7}$$

Eq. (7) indicates that the potential outside the sphere can be considered as an addition of the incident field �*E*0*rcos*θ and a dipole with its dipole moment defined according to Eq. (8) [38],

$$p = 4\pi a^3 \varepsilon\_0 \varepsilon\_m \frac{\varepsilon\_m - \varepsilon\_d}{\varepsilon\_m + 2\varepsilon\_d} E\_0 \tag{8}$$

with its polarizability α of [38]:

$$\mathfrak{a} = 4\pi \mathfrak{a}^3 \frac{\mathfrak{e}\_m - \mathfrak{e}\_d}{\mathfrak{e}\_m + 2\mathfrak{e}\_d} \tag{9}$$

This is to say that we can consider a metal sphere, the dimension of which is much smaller than the wavelength of the incident light, as a simple dipole. Its polarizability is a function of the dielectric constant and size of the metal sphere.

Further derivation shows the cross sections for scattering and absorption are obtained from the scattered field radiated by this induced dipole interacting with the incident plane wave. They can be written as [38]:

$$\mathbf{C}\_{\text{scattering}} = \frac{k^4}{6\pi} |a|^2 = \frac{8}{3} k^4 \pi a^6 |\frac{\mathbf{e}\_m - \mathbf{e}\_d}{\mathbf{e}\_m + 2\varepsilon\_d}|^2 \tag{10}$$

$$C\_{absorption} = kIm\{a\} = 4k\pi a^3 Im\left\{\frac{\varepsilon\_m - \varepsilon\_d}{\varepsilon\_m + 2\varepsilon\_d}\right\} \tag{11}$$

where *k* is the wave vector of the incident light.

For a specific metal in a specific environment where ε*<sup>d</sup>* and ε*<sup>m</sup>* are defined and fixed, the absorption coefficient is proportional to the third power of the radius of the particle, while the scattering cross section is proportional to the sixth power of this radius. The efficiency of absorption dominates over the scattering efficiency when the particle size decreases.

Additionally, one can also notice a resonant enhancement for scattering and absorption when the condition *Re*ð Þ¼ *ε<sup>m</sup>* þ 2*ε<sup>d</sup>* 0 is satisfied, which is called Fröhlich condition [30]. This resonance is due to resonant excitation of the dipole surface plasmon. With the Drude model of the dielectric function, the frequency of the dipole surface plasmon can be written as ω*sp*≈ω*p=* ffiffiffi <sup>3</sup> <sup>p</sup> with <sup>ω</sup>*<sup>p</sup>* corresponding to the plasma frequency of the bulk metal.

The theory mentioned above can only be applied to particles that are much smaller than the excitation wavelength so that we can consider the electromagnetic field uniformly distributed across the entire metal particle. For particles with dimensions comparable to the excitation wavelength, in which the electrical field can no longer be considered uniform across the particle, a modified long wavelength approximation (MLWA) based on perturbative corrections has to be used [47–49].

The localized surface plasmon resonances (LSPRs) of noble metal particles with sizes of >10 nm were characterized well experimentally [50–53]. However, the characterization and understanding for sizes smaller than 10 nm is still poor and challenging from both experimental and theoretical points of view [54, 55]. This is mainly due to the fact that both quantum effects and surface interactions become important as the electrons interact more strongly with the surface including the spill-over of conduction electrons at the particle surface, which complicates geometrical analysis [56]; these effects cannot be described directly and solely by electrodynamics, and they require detailed calculations of the electronic structure for the actual atomic arrangements in the nanostructure of interest. In what concerns the experiments, the optical detection in the far field becomes difficult for small particles due to the size-dependent reduction in scattering intensity. In what concerns theory, time-dependent density functional theory-based methods are in general limited at present to particles with the sizes below 1–2 nm [57–59]. This mismatch between what can be achieved experimentally and what can be addressed theoretically make it difficult to benchmark both approaches.

#### **2.2 Coupled elements: dimer systems**

When we bring two or more single elements together, a new system is formed due to the interaction among those single elements and their light-matter interaction can

*The Influence of Geometry on Plasmonic Resonances in Surface- and Tip-Enhanced Raman… DOI: http://dx.doi.org/10.5772/intechopen.108182*

consequently be quite different. Here, we address a dimer system, which is composed of two metal spheres with a sufficiently small gap distance in the range of a few nanometers. These spheres are normally but not necessarily identical with respect to their size, geometry, and material.

A widely accepted and discussed theory stems from Nordlander [60], who gave an intuitive explanation to define the extinction cross section of dimer systems based on the gap distance between the nanoparticles. The dimer plasmons can be considered as a combination of bonding and antibonding states derived from the individual nanosphere plasmons. In this theory [61], the conduction electrons are considered as a charged and incompressible liquid sitting on top of rigid, positively charged ion cores. Ion cores are treated within the jellium approximation and the positive charge *n*<sup>0</sup> is uniformly distributed within the particle boundaries [62]. The plasmon modes are considered as self-sustained deformations of the electron liquid. Only the surface charges are responsible for such deformation since the liquid is incompressible. Therefore, the surface charge for a single solid metal sphere can be written as [61]:

$$
\sigma(\mathfrak{Q}, t) = n\_0 \varepsilon \sum\_{l,m} \sqrt{\frac{l}{R^3}} \mathfrak{S}\_{lm}(t) Y\_{lm}(\mathfrak{Q}) \tag{12}
$$

where *Ylm*ð Þ Ω indicates the spherical harmonic of the solid angle Ω, *R* is the radius of sphere, *Slm* represent the new degrees of freedom, and *l* is the angular momentum of a nanosphere. When the polar axis is chosen along the dimer axis, for a real representation that is adopted for the spherical harmonics, the interaction is diagonal in azimuthal quantum number *m*.

Therefore, the dynamics of the deformation is described by [61]:

$$L\_s = \frac{n\_0 m\_e}{2} \sum \left[ \dot{\mathbf{S}}\_{lm}^2 - a \rho\_{\mathbf{S},l}^2 \mathbf{S}\_{lm}^2 \right] \tag{13}$$

where ω*<sup>S</sup>*,*<sup>l</sup>* ¼ ω*<sup>B</sup>* ffiffiffiffiffiffiffi *l* 2*l*þ1 q represents the solid sphere plasmon resonance and ω*<sup>B</sup>* ¼ ffiffiffiffiffiffiffi *e*2*n*<sup>0</sup> *meε*<sup>0</sup> q , represents the bulk plasmon frequency, \_ *Slm* represents the time derivative of the term *Slm*. For the dimer system, when the distance between the centres of the two spheres is smaller than λ*B=*4, retardation effects can be neglected and the dynamics of the plasmons is defined by the instantaneous Coulomb interaction between the surface charges as [61]:

$$V(D) = \int R\_1^2 d\Omega\_1 \int R\_2^2 d\Omega\_2 \frac{\sigma^1(\Omega\_1)\sigma^2(\Omega\_2)}{\left|\overrightarrow{r\_1} - \overrightarrow{r\_2}\right|}\tag{14}$$

where *D* is the separation between the centres of the two spheres in a dimer system.

The left panel in **Figure 2a** shows the dimer plasmon energies as a function of dimer separation for plasmon polarizations along the dimer axis (*m* ¼ 0). At large separation, the interaction of plasmons on different nanoparticles is weak and the dimer plasmons are essentially bonding and antibonding combinations of plasmons of the same angular momentum *l* belonging to the nanoparticle.

When the separation is relatively large (�35 nm), the splitting of the bonding and antibonding dimer plasmons is symmetric. The splitting increases as their interaction increases. The bonding/antibonding configuration corresponds to the two dipole

#### **Figure 2.**

*Calculated plasmon energies of a nanosphere dimer with identical sphere radii of 10 nm as a function of interparticle separation (left); calculated plasmon energies of a heterodimer as a function of interparticle separation (right) with two spheres that have different radii of 10 nm and 5 nm. Panels (a) is for the azimuthal quantum number m* ¼ 0 *and panels (b) for m* ¼ 1*. The curves represent the bonding and antibonding dimer plasmons derived from the individual nanosphere plasmons with increasing angular momentum l. The arrows indicate the orientation of their dipole moments, see ref. [61].*

moments moving in phase/out of phase (positive/negative parity of dipole moments or symmetric/asymmetric fields). For identical spheres, the net dipole moment of the negative parity plasmon (asymmetric field) is zero and they can hardly be excited by light, and they are therefore considered as dark plasmons, while the positive parity (symmetric field) plasmons are referred to as bright or luminous plasmons.

As the separation decreases, the splitting of *l* ¼ 1 plasmon becomes asymmetric. Since the lower energy plasmon branch shifts faster than the higher energy plasmon branch, the overall non-dipole-like red shift effect is caused by the interaction of the *l* ¼ 1 nanosphere plasmons with the higher *l* plasmons of the other nanosphere.

For the plasmons corresponding to *m* ¼ �1 (polarization-oriented perpendicular to the dimer axis shown in **Figure 2** left panel (b)), the overall phenomena are similar. Note that the assignment for the bright/dark plasmons is reversed in this case because the perpendicular polarization coupling has opposite signs.

The right panel of **Figure 2** depicts the dimer plasmon for a heterodimer as a function of dimer separation. The behavior with the separation is different compared with results shown in the left panel of **Figure 2**. As the parity of the dimer is broken, the lines representing the dimer plasmon energies exhibit avoided crossings in the figure. All dimer plasmons with j j *m* ≤ 1 are bright.

As the separation decreases, the interactions get particularly strong when antibonding plasmons approach the bonding dimer plasmons of higher *l* manifolds, meaning that the higher *l* dimer plasmon also carries a finite dipole moment and becomes dipole active. Therefore, multiple peaks in the absorption spectra or a broad absorption region in the case of overlapping resonances are expected.

*The Influence of Geometry on Plasmonic Resonances in Surface- and Tip-Enhanced Raman… DOI: http://dx.doi.org/10.5772/intechopen.108182*

The result from the plasmon hybridization method is further compared with the finite difference time domain method (FDTD). And the results agree well with each other [63]. **Figure 3** from ref. [64] shows the scattering cross section of a nanosphere dimer system with radii of 40 nm and a separation distance varying from 0.5 to 200 nm compared with that of a monomer with a diameter of 80 nm. When the separation of the dimer is large, for example, 200 nm, the dimer system behaves the same as the monomer system, while decreasing the gap distance induces a shift toward higher energy and creating additional modes when the gap reaches 4 nm or even smaller distances.

Besides the above-mentioned hybridization method, another practical approach, which is used to describe coupled plasmon resonances in the so-called capacitive and conductive coupling regimes using an equivalent circuit model, was put forward by Benz et al. [65]. They claimed that such model can be used to calculate analytically the resonance wavelengths for different gap sizes, nanoparticle sizes, refractive indices, and linker conductivities.

To understand the dimer system further, different dimer systems are developed for fundamental studies. For example, Jeong et al. used an approach to fabricate plasmonic dimers in a very large scale with precise control of size, nanogap, material, and orientation [66]. They found that the optical response of each dimer is found to be identical with a highly uniform gap maintained across the array over centimeter distances. The existence of the transverse dipole mode and/or the longitudinal coupled resonance mode is highly dependent on the polarization of the incident light with respect to the dimer axis. A red shift can be observed with increasing gold nanoparticle size. Arbuz et al. recently studied the influence of the interparticle gap in dimers of gold nanoparticles on gold (Au), aluminum (Al), silver (Ag) films, and silicon (Si) wafers as substrates [67]. They claimed that the influence of the substrate vanishes when the dimer gap becomes larger than 2 nm. Nevertheless, the relation between the gap and the SERS intensity and enhancement factor is still under debate [68–71]. Also, Song et al. designed an experiment using an electromechanical method to tune the distance in the nm range between two Au nanoclusters as a strongly coupled plasmonic dimer, right before detrimental quantum effects set in. Different plasmon modes followed different trends as the bonding dipole (BDP) mode, a small blue shift

#### **Figure 3.**

*FDTD simulation results for the scattering cross section of a nanosphere dimer system with radii of 40 nm and a separation distance varying from 0.5 to 200 nm compared with that of a monomer [64].*

of the anti-bonding dipole (ADP) mode, and a negligible shift of anti-bonding vertical quadrupole (AVQP) mode with decreasing gap of the nanodisk dimer [72].

Dimer systems, as a basic metal nanostructure coupled system, provides a simple but very practical approximation for many application situations, especially in the two techniques, that is, SERS and TERS, that are discussed in the following sections.
