**2.3 Simulation on SERS and TERS**

In the technical application of plasmonics, commonly used experimental configurations for SERS and TERS are shown in **Figure 4**. It is not only convenient but also reasonable to simplify the experimental configurations to simple spheres thus reducing a lot the computational cost in simulations. These schematic sketches of typical configurations include SERS in colloidal solution, SERS with a solid substrate, gapmode SERS, TERS [73] and shell-isolated nanoparticle-enhanced Raman spectroscopy (SHINERS) [74–77].

To understand the impact of the geometry of plasmonic structures, simulations have an unbeatable advantage of freedom when designing the geometries. Besides straightforward experimental research, numerical simulations are gradually changing their role from a supporting approach to interpret the experimental results to a convenient and solid tool to investigate the mechanisms of plasmonic structures.

Various methods such as T-matrix [78–81], discrete dipole approximation (DDA) [82, 83], finite element method (FEM) [84, 85], and finite difference time domain (FDTD) [63, 64, 86] are used to address plasmonic systems. We can directly get the electric field distribution and use it for qualitative and even quantitative comparison with experiments. Classical theory based on solving Maxwell equations builds the backbone of many commonly used simulation tools while *ab initio* calculations may produce understanding beyond the knowledge obtained from classical theory. For a long time, we have tried to understand the mechanisms of light-matter interaction on metal nanoparticles. With the help of the fast development in the field of electronic and computer science, many computational methods were implemented to solve and

#### **Figure 4.**

*Schematic presentations of different configurations of plasmon enhanced Raman spectroscopy. a) a plasmonic colloidal solution as a substrate for SERS. b) A plasmonic solid substrate for SERS, comprising a glass or silicon support and plasmonic metal nanoparticles. c) Gap-mode SERS. d) Shell-isolated nanoparticle-enhanced Raman spectroscopy (SHINERS) uses nanoparticles coated with a layer of a dielectric material. e) TERS, the nanoparticles are replaced by a single metallic scanning probe microscope (SPM) tip [73].*

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

visualize this problem. As solving the Maxwell equations is the core mission in this field, solvers for a particular design have been developed in the form of either integral or differential equations [87, 88].

**Table 1** [89] provides a comparison of the most used simulation tools for plasmonic structures.


#### **Table 1.**

*Comparison on computation time, advantages, and disadvantages of different computational techniques [89].*

Among above-mentioned simulation methods, the finite element method (FEM) and the finite difference time domain (FDTD) method are the most commonly and widely used methods commercially available nowadays. In brief, FEM reduces the complex partial differential equations to simple algebraic equations. This method gives the approximate results at each discrete number of points over the domain. To solve the problem, it divides the whole problem into various numbers of discrete units generally termed as mesh elements. FEM can be applied to various physical problems such as structural analysis, fluid flow, electromagnetic potential, and mass transport [90]. FDTD, on the other hand, is usually suitable to solve transient change processes of a field under external excitation. If a pulsed excitation source is used, a single solution can yield a response over a wide-frequency band. Time domain methods have reliable accuracy and faster computational speed, and can truly reflect the nature of electromagnetic phenomena, especially in research areas requiring time domain measurements [91]. FDTD is more useful for nonlinear materials with offering a large range of wavelength-dependent dielectric constants and broadband simulations especially for the transient studies, while FEM benefits from unstructured gridding and is therefore more promising for higher-order curved elements with the advanced FEM codes [89, 92].

One of the most used simulation tools based on the FEM method is COMSOL Multiphysics [93]. This software includes various working packages for a variety of applications, among which the Wave Optics module is the one specifically used for plasmonic studies, because it enables to handle objects, the dimensions of which are comparable or smaller than the probing wavelength [94]. All modeling formulations are based on Maxwell's equations together with material laws for propagation in various media. The modeling capabilities are accessed *via* predefined physics interfaces, which allow the user to set up and solve the electromagnetic models in two- and three-dimensional spaces. The modeling of electromagnetic fields and waves can be performed in the frequency domain, time domain, eigenfrequency, and mode analysis. The modeling typically follows the sequence: definition of the geometry, selection of materials, selection of a suitable Wave Optics interface, definition of boundaries and initial conditions, definition of the finite element mesh, selection of a solver, and visualization of the results [94]. Most of the simulations presented in both SERS and TERS sections were performed with this tool.

### *2.3.1 SERS*

In the typical SERS configuration shown in **Figure 4a**, computer simulations showed that spherical Au and Ag NPs as monomers cannot generate a strong localized electric field on their surface [95], and nevertheless, they carry on being most widely used options in SERS and TERS experiments, where high-quality signals can be obtained from different analytes, due to their easy and fast synthesis. Since a main task in SERS and TERS research is to increase the sensitivity of the plasmonic systems, other alternative geometric structures have been investigated, in which the aspect ratio of the spheroid structures is investigated: particles with prolate or oblate spheroid geometries [96]. A practical approximation in ref. [97] shows the possibility to use a Taylor expansion to numerically predict the extinction spectra of metallic spheroidal particles for a wide range of the geometric aspect ratios.

For configurations as shown in **Figure 4b** and **c**, classical electrodynamics provides a good description down to gap distances of the order of >1 nm, after which quantum and non-local theory approaches have to be used [98–100]. The EM enhancement

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

continues until the distance between two metal surfaces becomes so small that electron spill-out and non-local effects become important, eventually leading to electronic tunneling and electrical shortcut [101]. Such phenomena are observed by simply bringing two or more spherical NPs close enough experimentally [102–104].

It is worth to mention that besides the spherical nanostructures in **Figure 4** other nanostructures, such as mesoporous gold particles [105], nanostars/flowers [106–113] or spiky structures/superstructures [114, 115], nanoshells [116], nanocubes [117–122], and hollow-structured particles [109, 112], have been considered. For those as SERS substrates, their edges work as hotspots concentrating the electromagnetic field of the probing light into small volumes. This enhances the local electromagnetic (EM) field near the edges of these metal nanostructures. The "hotspot" areas utilize the field enhancement properties of the metal nanostructures to amplify the usually weak Raman scattering signals.

Another way to boost the hotspots is to bring two or more particles in close vicinity. Therefore, many agglomerated structures are practically used to increase the SERS enhancement employing the interaction among the single monomers to fulfill the "dimer" condition, such as clusters [123], trimers [67, 124–128], tetramers [125, 129, 130], chains [131, 132], and arrays [133–137]. Detailed studies to understand such agglomerates were performed by several groups. Sergiienko et al. investigated the influence of NP agglomeration on the SERS signal [127]. The study was carried out on monomers, dimers, and trimers. In comparison with a single NP, the plasmonic absorption for dimers exhibits a new band at longer wavelength (red shift) due to the interparticle plasmonic coupling. Theoretically, the interparticle plasmonic coupling leads to more enhancement and red shifts the plasmonic absorption band with increasing degree of aggregation. When the nanoparticles in a chain are brought closer to each other (gaps decreasing from 2.5 to 0.5 nm), the maximum field enhancement at the gap becomes nearly 10 times larger and aggregation causes a large red shift of more than 200 nm. Overall, the SERS enhancement factor (EF) increases by 43% in average upon dimerization and 96% upon trimerization for both AuNPs and AgNPs. However, the maximum ratio of EFs for some dimers to the mean EF of monomers can be as high as 5.5 for AgNPs on gold substrates. For dimerization and trimerization of gold and silver NPs on silicon, the mean EF increases by 1–2 orders of magnitude relative to the mean EF of single NPs. Therefore, the hotspots in the interparticle gap between gold nanoparticles rather than hotspots between Au nanoparticles and substrate dominate the SERS enhancement for dimers and trimers on a silicon substrate. Raman-labeled noble metal nanoparticles on plasmonic metal films generate on average SERS enhancement of the same order of magnitude for both types of hotspot zones (i.e., NP/NP and NP/metal film). A summary of these results is presented in **Figure 5**. More details about this work can be found in ref. [124]

A SERS substrate can be composed of both monomer and dimer metal structures placed on metal/non-metal substrates. Arrays, as one of the important configurations, have been utilized in many fields [66, 138–140]. The fabrication of plasmonic arrays is also versatile including both top-down and bottom-up methods as described, for example, in ref. [138] and for instance, nanogap arrays using photolithography for nanogap arrays with swelling-induced nano-cracking [141], superimposition metal sputtering [135], and direct writing [142].

Among such methods, the so-called nanosphere lithography (NSL) using selfassembled nanospheres as a shadow mask for metal deposition is a typical costefficient and fast technique [143, 144]. NSL details are reviewed elsewhere [145]. Various metals can be used, such as silver, gold, copper, and aluminum. SEM images

#### **Figure 5.**

*Raman and SERS spectra of analytes adsorbed on 60 nm Au NPs on an Au film: (A) 4-aminothiophenol, 1—Raman, 2—SERS, (B) 4-nitrobenzenethiol, 1—Raman, 2—SERS, (C) 2-methoxythiophenol, 1—Raman, 2—SERS, (D) SERS spectra of 2-methoxythiophenol (monomer, dimer, and trimer) [127].*

of such typical structures are shown in **Figure 6**. Tuning the plasmon resonance frequency of such structure can be performed in the range from the near infrared to the blue spectral using different metals and by annealing them at different temperatures [146]. This tuning is simply based on the change of the shape of each metal nanotriangle (NT) from triangular to roundish for the case of Au and Ag. For Cu and

#### **Figure 6.**

*SEM images of the nanostructures prepared by nanosphere lithography. Top row shows the structures as deposited and the bottom row after annealing at 500°C. The scale bar is 500 nm in all images [146].*

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

Al, the change in shape is not so dramatic as they form a dense oxide layer *via* annealing thus preventing further shape changes (see **Figure 6**).

FEM simulations were performed using COMSOL 5.6 Wave Optic module [94]. The results are shown in **Figure 7**. There clearly is a blue shift of the resonant wavelength with increasing annealing temperature, that is, change of the shape for Ag, Au, and Cu arrays (**Figure 7a**–**c**), which is the main reason for the variation of the plasmonic resonance in this scenario. Additionally, the simulation for the Cu arrays is performed by adding a copper monoxide layer with the different thicknesses shown in **Figure 7c**. The results on the LSPR position support the hypothesis derived from the experimental results that annealing above 400°C produces thicker layers of copper oxide [146]. The electric field distribution (**Figure 7e**) reveals the position of the highest local enhancement for different copper oxide layer thicknesses.

The optical behavior of the metal NTs as a function of different annealing temperatures is a straightforward example of the flexibility in tuning the LSPR. Nanosphere lithography also allows other array structures to be fabricated, such as nanovoids [147] and "hedgehog-like" nanosphere arrays [148].

The metal nanotriangle structures have widely been used to study 2D materials, such as indium selenide (InSe) as shown in **Figure 8** [149]. One up to seven layers of InSe were deposited on arrays of plasmonic NTs composed of different metals. To

#### **Figure 7.**

*Simulated transmission spectra and the electric field enhancement distribution at the LSPR position using geometries corresponding to as prepared and annealed Ag NPs (a) and Au NPs (b). Simulated LSPR positions vs. geometry change in a Cu array by changing the radius of the edge of the Cu triangles shown in the inset as a sketch mimicking the change of the geometry at the lower annealing temperature shown in (c) and the electric field enhancement distribution for Cu arrays at LSPR conditions shown in (d); (e) simulated LSPR for further increase of the oxide thickness mimicking the situation as annealing temperature further increases [146].*

**Figure 8.**

*FEM simulation of plasmonic coupling between InSe and metal nanotriangles (MNTs). (a) Sketch of the model used in the FEM simulation of InSe/metal NTs. (b) Simulated electric field intensity M distribution at two different wavelengths for Ag and Al NTs. (c) Absolute value of calculated maximum M at three different excitations. A similar plasmonic behavior is expected for Ag and Au NTs on InSe for three selected wavelengths, while InSe with Al NTs shows a maximum at 1.94-eV excitation. (d) Raman spectra of 7 L InSe with Al NTs acquired under three different wavelengths [149].*

study the enhancement behavior, simulations were performed using the same conditions as in the experiment. The enhancement factor, *M*, is defined as the square of the local electric field strength enhancement. We can see dramatic enhancement for gold and silver nanotriangles (NTs) with excitation energy of 1.58 eV and a relatively large enhancement at 1.94 eV.

### *2.3.2 TERS*

TERS is another important experimental technique based on plasmonic enhancement. In a typical TERS configuration, there is a metal tip that is used for scanning a substrate usually decorated with the analytes, for example, molecules or nanostructures. If the substrate is a metal, then such a configuration is called gapmode TERS. One of the critical targets in this technique is to maximize the TERS signal enhancement and achieve very good spatial resolution in the nanometer range, well below the diffraction limit of light. Here, gold and silver are the two preferred materials for the plasmonic tips. For both materials, their LSPRs locate in the near infrared to visible range, where laser wavelengths are available to match the LSPR. Therefore, most of the experimental and simulation studies are performed using these two materials. Additionally, Au is normally the first choice because it is more chemically stable when exposed to air, enabling its use for longer periods of time, while Ag tends to form sulfides when exposed to air deteriorating the TERS performance [150].

### **2.4 Tip effects**

A "sharp" metallic tip promises good spatial resolution. Therefore, as the most critical component in TERS, the tip is considered as a sharp "corner" of a metal rod. In the macroscopic world, a spark would form at the end of a long metal rod due to lightning bolt during a thunderstorm, and similarly in the nano-world, this effect also plays a vital role and contributes to the TERS signal enhancement. This was first explained in 1980 by Gersten and Nitzan [151] and then in 1982 by Wokaun [152] using the formulation of depolarization factors.

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

Considering a metallic ellipsoid with the major axis *a* and minor axis *b* with *a*,*b* ≪ *λ*, so that the electrostatic approximation can be utilized. A uniform electric field *EL* is applied along the major axis, and this leads to a uniform polarization density within such an ellipsoid [152].

$$P = \frac{1}{4\pi} \frac{\varepsilon\_{ellip} - 1}{1 + (\varepsilon\_{ellip} - 1)A\_a} E\_L \tag{15}$$

Then, the field at the tip of the ellipsoid can be written as:

$$E\_{tip} = \frac{(\mathbf{1} - A\_d) \left(\varepsilon\_{ellip} - \mathbf{1}\right)}{\mathbf{1} + \left(\varepsilon\_{ellip} - \mathbf{1}\right) A\_d} E\_L + E\_L \tag{16}$$

where *εellip* is the dielectric constant of the ellipsoid material and the *Aa* is the depolarization factor defined as:

$$A\_d = \frac{ab^2}{2} \int\_0^\infty \frac{ds}{(s+a^2)R} \ (a=a,b) \tag{17}$$

with *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> *<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>2</sup> ð Þ *<sup>s</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>2</sup> � �<sup>2</sup> . For a sphere, *Aa* <sup>¼</sup> <sup>1</sup> <sup>3</sup> and for a prolate ellipsoid with ratio *a* : *b* ¼ 3 : 1, *Aa* ¼ 0*:*1087. The prominent effect that the depolarization factor gives is the shift of the plasmon resonance frequency, that is, when the denominators in Eq. (15) and Eq. (16) approach zero at a specific wavelength [152].

Now, we consider the nanoparticle dipole moment *μ* obtained by integrating Eq. (15) over the whole volume of the nanoparticle. This leads to *<sup>μ</sup>* <sup>¼</sup> <sup>4</sup>*πab*<sup>2</sup> *P=*3 [152].

The field of the nanoparticle is determined by the simple dipolar field of *μ* when at large distance. However, the situation changes when we look at the tip of the ellipsoid particle. A factor *γ* must be considered since the specific shape concentrates the field on the narrower parts of the structure. This phenomenon is called the lightning rod effect. We can then rewrite the Eq. (16) in a form of dipolar field *Edipolar* <sup>¼</sup> <sup>2</sup>*μ=a*<sup>3</sup> and the new *Etip* is written as [152].

$$E\_{\rm tip} = \chi E\_{\rm dipolar} + E\_L \text{ where } \chi = \frac{3}{2} \left(\frac{a}{b}\right)^2 (1 - A\_a) \tag{18}$$

We can see that for a sphere, *<sup>γ</sup>* <sup>¼</sup> 1 with *<sup>a</sup>* <sup>¼</sup> *<sup>b</sup>* and therefore, *Aa* <sup>¼</sup> <sup>1</sup> 3 . For a prolate where *a* : *b* ¼ 3 : 1, *γ* ¼ 12. In the more extreme situation with a needle-like ellipsoid, we have *Aa*≈0 and *γ*≈ <sup>3</sup> 2 *a b* � �<sup>2</sup> .

To calculate the total Raman enhancement, that is, an analyte molecule, for instance, is beneath the TERS tip, we consider the large Raman molecular moment that is induced by the intense local field at the ellipsoid and then the polarization of the ellipsoid induced by the molecular field. For simplicity, we can treat both the molecule and the ellipsoid as point dipoles. The molecular dipole field in turn polarizes the ellipsoid, giving an ellipsoid dipole at the Stokes frequency *ωs*. This added-up molecular dipole is larger than the usual Raman molecular dipole by a factor of [152].

$$f = \frac{4}{9} \frac{\varepsilon(o\_\varepsilon) - \mathbf{1}}{[\varepsilon(o\_\varepsilon) - \mathbf{1}]A\_d + \mathbf{1}} \times \frac{\varepsilon(o) - \mathbf{1}}{\mathbf{1} + [\varepsilon(o) - \mathbf{1}]A\_d} \left(\frac{b^2}{a^2}\right)^2 \tag{19}$$

The Raman intensity enhancement is then given by j j *<sup>f</sup>* <sup>2</sup> . Note that the depolarization factor *Aa* can be approximately written as follows when *a=b* is very large [152].

$$A\_d \sim \left(\frac{b}{a}\right)^2 \ln\left(\frac{a}{b}\right) \tag{20}$$

The lightning rod effect can be critical when simulating the electric field enhancement using geometrical simplifications. Therefore, for a tip geometry that is considered as a single sphere, the contribution to the signal enhancement due to the lightning rod effect is neglected.

The two most important aspects in TERS are the TERS enhancement factor (EF) and the spatial resolution. The TERS EF scales with *Eloc Einc* <sup>4</sup> , where *Eloc* is the local electric field and *Einc* is the incident electric field. The following classical theory provides a straightforward understanding [153]. To define the spatial resolution, the authors built a straightforward model for the TERS tip as shown in **Figure 9**. Here, the geometry of the tip can be approximated as a metal sphere, for which the solution was introduced in the previous section. The full-width at half maximum (FWHM) of the field distribution (mind that the real TERS spatial resolution is actually derived from the fourth power of the local electric field distribution) along the horizontal direction under the tip apex at a specific distance *d* from the tip apex [153]:

$$\text{FWHM} = \mathbf{1.346}(R+d) \tag{21}$$

The derived term indicates a supreme confined region of the local field, which is limited by the radius of the metal sphere or the curvature radius of a tip [154].

Simulations for an Ag tip with varying geometrical parameters were performed using COMSOL in ref. [155]. In this systematic simulation work, the simulated tip length, the tip radius, and the conical tip angle are varied [155]. As can be seen in **Figure 10**, there is a dramatic difference of the field enhancements between short and long tips, while a short, truncated tip can produce a better enhancement than a long one because of the excitation of the localized plasmon resonance. Regarding the tip radius, there is a significant improvement of the field enhancement observed when r decreases from 20 to 10 nm. Finally, the setting of the cone angle indeed influences the results but not as dramatic as the other two factors. Unfortunately, a direct proof of the spatial resolution derived in Eq. (20) cannot be found in the literature to our best knowledge.

**Figure 9.**

*Sketch of the geometric structure and local field distribution of a metal tip and its approximation as a single sphere [153].*

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

**Figure 10.**

*a) Local electric field enhancement spectra for conical tips with different tip lengths l with the tip radius r = 20 nm and the tip cone angle α = 15°; b) local electric field enhancement spectra at the tip apex for different tip radii, and the tip length is kept as infinite and tip angle = 15°; c) local electric field enhancement spectra at the tip apex for different tip angles. The tip length is infinite and the tip radius r = 20 nm. Inset indicates the simulated tip geometry [155].*

Due to the significance of the TERS tips, their fabrication became an important branch in the field of plasmonic [156, 157]. Different methods have been developed to produce metal TERS tips, such as electrochemical etching [158–160], electrodeposition [161, 162], and tip tailoring [163, 164]. Recently, Zhang developed an approach by concentrating the light *via* a waveguide and thus producing a low background hotspot at the tip apex [165].

Scanning electron microscope (SEM) images of commonly used tip geometries, namely of a) a sharp Au AFM tip, b) Au-coated spherical AFM tip and c) electrochemically deposited Au nanoparticle on a Pt AFM tip, are shown in **Figure 11** [166]. The related simulation work (shown in **Figure 12**) explained the experimental results.

**Figures 11** and **12** show an example from tip fabrication to characterization and finally simulation. Here, the importance of the simulation is emphasized, and it not only helps to understand the optical response of the tip but can also help to design tips for specific resonance requirement. Thus, the simulation nowadays allows us from finding suitable experimental analytes to fulfilling instrument requirements and designing suitable experimental instruments for specific purposes.

#### **2.5 Tip-substrate systems**

Previously, it was demonstrated that a single metal tip alone can already enhance the electric field intensity in the vicinity of the tip apex due to its plasmonic resonance and the geometrical lightning rod effect. In a real TERS experiment, a metal substrate

#### **Figure 11.**

*SEM images of a) a sharp gold AFM tip; b) gold-coated spherical AFM tip, and c) electrochemically deposited gold nanoparticles on platinum AFM tip [166].*

**Figure 12.**

*a) Numerically simulated near-field spectra of spherical Au and AuNP-on-Pt tips with (b), (c) near-field maps of the main resonance as highlighted by circles in (a). Simulated tips have 300 nm spherical radii, 120 nm neck widths, 20° opening angles, and 1.88 μm lengths to best match the typical experimental tip geometries and avoid truncation artifacts. Tips are illuminated by plane waves oriented along the tip axis. (d) Interpolated field enhancement map with superimposed resonant wavelengths, as the neck width varies from a spherical to a sharp tip. Tips have a 250 nm apex diameter, 1.88 μm length, and 10° opening angle [166].*

is usually present to further boost the signal. This type of configuration is called gap-mode TERS.

As shown in **Figure 13**, a gap-mode tip-substrate system can be considered as a dimer system composed of a metal sphere and its image dipole that is created in the metal film substrate [167]. Xu et al. gave a simple geometrical argument to estimate the local electric field in the gap of such dimer systems taking into consideration the drop in potential for the incident field *Eloc* [168].

The drop in potential between the two spheres shown in **Figure 13** (dashed circles) can be expressed as ΔV ¼ j j Eloc *d*, while the potential difference between these two sites in the absence of two metal spheres can be expressed as [153]:

$$
\Delta \mathbf{V} = |E\_{inc}| (2\mathbf{R} + d) \tag{22}
$$

Since the two spheres can be considered as equipotential bodies, we can write

$$
\Delta V = |E\_{inc}|(2R + d) = |E\_{loc}|d\tag{23}
$$

In a specific geometry, where the radius of the sphere *R* and distance right beneath the tip apex *d* are fixed, the lateral offset of electric field from the center can be written as

*Schematic presentation of metal tip-substrate structure and its approximation as a metal-sphere dimer [153].*

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

$$|E\_{loc}(\mathbf{x})| = \frac{\Delta V}{2R + d - 2\sqrt{R^2 - \mathbf{x}^2}} \tag{24}$$

Therefore, the FWHM of the local field is given by *FWHMEloc* <sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffi *Rd* <sup>p</sup> [153]. Considering the TERS intensity, which is proportional to the fourth power of the local electric field [169], we get

$$|E\_{\rm loc}(\mathbf{x})|^4 = \frac{\Delta V^4}{\left(2R + d - 2\sqrt{R^2 - \mathbf{x}^2}\right)^4} \tag{25}$$

For a very small *d,* we can obtain the FWHM of the TERS intensity distribution as [170]:

$$\text{FWHM}\_{\text{TERS}} = 2\sqrt{\left(\left(\sqrt[4]{2} - 1\right)Rd\right)} \approx 0.87\sqrt{Rd} \tag{26}$$

Many studies considered the easiest case of a substrate composed of a flat surface of a bulk material. The geometric parameters of the metal tip and the metal substrate with specific excitation wavelength using side illumination were evaluated numerically in ref. [169]. The study was performed using three-dimensional finite-difference time domain simulation (FDTD) and the effect of the presence of a substrate is demonstrated in **Figure 14**. Without a substrate, the electric field is enhanced by a factor *M* ¼ 20 (see **Figure 14**, left), while it reaches *M* ¼ 189 (see **Figure 14**, right) when there is a metal substrate.

Our own FEM simulation results on tip-substrate systems with various tip radii using the wave optics module in COMSOL 5.4 [93] are shown in **Figure 15**. To save the computational time, a two-dimensional model was built with a non-uniform mesh that guaranteed a very fine mesh grid element (less than 1 nm) in the region of the gap between the tip apex and the substrate [170].

To study the spectral dependence of each geometrical setting of various tip apex radii, a spectral sweep is performed from 500 nm to 800 nm. The results shown in

#### **Figure 14.**

*FDTD simulations of the electric field distribution for a single Au tip (a), and a gold tip held at distance d = 2 nm from a gold substrate surface. The electric field E and wave vector k of the incoming light are displayed in the figures. M stands for the maximum enhancement [169].*

**Figure 15.**

*Simulation results of (a) the spectral dependence of the TERS EF as a function of tip diameter, (b) max. TERS EF, and (c) FWHM of the local TERS profile in the gap-mode TERS geometry as shown in the inset of Figure 15b. The simulated FWHM as a function of the tip diameter is compared with the FWHM profile derived from Eq. (26) (red asterisks in Figure 15c). The blue-shaded area presents the error bar of the FWHM [170].*

**Figure 15** demonstrate a strong relation between the tip diameter and TERS enhancement factor with high spatial resolution, which is represented by the FWHM of the fourth power of the local field distribution beneath the tip apex. As the tip diameter increases from 30 nm to 160 nm, the increasing scattering cross section and the increasing radiative damping [171–173] both affect the enhancement factor, which increases and finally shows a saturation tendency.

#### **Figure 16.**

*a) Scheme of the TERS experiment. b) A magnified SEM image of a TERS tip revealing the formation of Au nanoclusters at and around the tip. C) A representative TERS spectrum of a MoS2 monolayer on an Au nanocluster array in comparison with the spectrum excited by 785.3 nm light; d) Gaussian fit of an intensity profile obtained for a scan across the rim of a nanodisk. The spatial resolution of the TERS image is equal to the full width at half maximum (FWHM) of the fit (2.3 nm). Reproduced from ref. [174] with permission from the Royal Society of Chemistry.*

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

**Figure 17.**

*a) Schematic illustration of the STM-TERS experiments. b) STM topography of an isolated CNT on Ag (111) (1 V, 10 pA). Inset: Line profile of the CNT along the blue arrow line. c) Apparent height profile and TERS intensity profiles along the long end of a carbon nanotube [175].*

Utilizing gap-mode TERS, Rahaman et al. performed TERS studies of MoS2 layers on gold nanodisk arrays under the ambient conditions. They used a side illuminated AFM-TERS (**Figure 16a**) experimental configuration with a gold-coated Si AFM tip. The SEM image of the tip apex is shown in **Figure 16b**. The TERS enhancement factor was calculated using the A1g mode of MoS2 and it reached 5*:*<sup>6</sup> <sup>10</sup><sup>8</sup> (**Figure 16c**), while a spatial resolution of 2.3 nm was achieved (**Figure 16d**) [174].

STM-based TERS, on the other hand, can produce even better spatial resolution due to the sharpness of the STM tips and the controlled experimental environment. Liao used a STM-TERS system as sketched in **Figure 17a** on a carbon nanotube placed on an Ag (111) substrate. A STM topography image can be seen in **Figure 17b** [175]. They claimed a spatial resolution of 0.7 nm (**Figure 17c**) with a TERS EF of approximately 108.
