Journal of Nanotechnology

Volume 2018, Article ID 3898524, 6 pages

https://doi.org/10.1155/2018/3898524

## Simple Modeling of the Ratio of Fields at a Tip and a Contacting Surface with External Illumination

Institute of Semiconductor Physics of NAS of Ukraine, Pr.Nauki 41, Kyiv 03028, Ukraine

Correspondence should be addressed to E. Bortchagovsky; moc.oohay@hctrob

Received 5 May 2018; Accepted 14 June 2018; Published 1 August 2018

Academic Editor: Paresh Chandra Ray

Copyright © 2018 E. Bortchagovsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The analysis of the relation of fields generated at a tip and a contacting surface is performed in the Rayleigh approximation of a simple dipole model for the standard configuration of tip-enhanced Raman scattering experiments with external excitation. A comparison of the present results with the previous ones obtained for the case of tip-source reveals the role of tip-surface configuration as the amplifier of the exciting field and the stronger influence of roughness on the field distribution at external illumination, as roughness is directly excited by the external field producing second source of field in addition to the tip.

#### 1. Introduction

A previous study [1] considered the ratio of fields at a tip and a contacting surface calculated in the dipole approximation in static limit (Rayleigh approximation) for the case of tip-source. The analysis revealed the influence of surface plasmons on the metallic surface, which makes the field at the plane surface slightly higher than the field at the tip with small tip-surface separation. It happens in spite of the fact, that it is the tip, which is the primary and the only source of the field in the considered system.

However, in spite of the fact that the use of a tip-source is favorable for tip-enhanced Raman scattering (TERS) [2], until now tetrahedral tip demonstrating excellent properties in TERS [2, 3] is the only literal tip-source without external illumination, which has simple construction. Another proposed construction [4] is much more complex in the preparation and practically exaggerates the list of tip-sources. One more possible variant [5, 6] was not yet used for TERS. Until now, the main TERS configuration is the external illumination of the tip apex by a focused light beam.

Thus, this article is devoted to the analysis of the standard case of the TERS configuration with external illumination. The aim is to reveal the angular dependency of fields and the change of the external field at different distances to the surface due to interference of incident and reflected light. As well as, in the previous article [1], the main attention is devoted to the behavior of the ratio of fields at the tip apex and at the surface , calculated on the basis of the same simple dipole model. As it was noticed previously [1], the used simple approach is not appropriate for modeling the whole field distribution and determining the absolute field values but is rather robust just to analysis of the ratio of fields. The latter is enough to compare enhancement conditions at the tip apex and at the contacting surface, which is necessary for the application of functionalized tips as Raman probe [7] and their use as an internal standard in TERS [8].

#### 2. Plane Surface

In contrast to the case of tip-source, which is the primary source of the field, external illumination excites the tip and contributes to the field at any point of the system. As in the previous study, the tip is modeled as a spherical particle, the polarizability of which in Rayleigh approximation is given by a standard expression [9]:which includes the radius of the particle and dielectric functions of the tip material and the ambient . In such an approach, the sphere is substituted by the dipole situated at the center of the sphere with given isotropic polarizability. It is well known that approaching a surface renormalizes polarizability of a dipole due to its interaction with its own image in the surface [9] towhat generates anisotropy with for the longitudinal polarizability along the surface and for the transverse polarizability perpendicular to the surface. is the distance from the center of the sphere to the surface, and is the static reflection coefficient:where is the dielectric function of the ambient and the dielectric function of the surface. It was underlined in [1] that this coefficient is bigger than the unit for metallic surfaces supporting surface plasmons, which makes the dipole image larger than the source one and creates the minimum close to the surface in the distance dependency of the ratio of fields at the tip apex and at the surface.

With the external illumination, the field exciting the tip is the sum of fields of the incident light and light reflected by the surface. Besides the angle of incidence and Fresnel reflection coefficients and for *p*- or *s*-polarization, respectively, this field is also defined by the distance to the surface *z*. If to define the phase of the field on the surface as zero, the fields are:where and are fields of the incident *p*- and *s*-polarized light and is the wave vector in vacuum, which is equal to , where is the wavelength. This field together with the additional field of polarized surface excites the tip, which in turn polarizes the surface producing noticed additional field, which in the present approximation is the field of the dipole image situated at the distance −*z* under the surface as a mirror image of the source. “+” corresponds to the transverse and “−” to longitudinal polarization as is defined by (3). Self-consistent account of this mutual polarization gives the value of renormalized polarizability (2), and the dipole generated on the tip is not but .

Finally, the total field at any point in the space is defined as the sum of external incident and reflected fields and fields generated by the dipole of the tip and its image in the surface. In the previous case [1], the tip dipole was constant; in this study, it depends on the illumination conditions and on the tip-surface separation.

The same parameters as in [1] were used for calculations, namely, silver tip mimicked by a sphere with radius 30 nm in front of gold surface, which is close to real experiment [8]. Now, the system is illuminated by the *p*- or *s*-polarized light of unit intensity. The angle of incidence is taken as 70°, which is close as to conditions of maximal exciting field what is proved later as to rather common experimental value. The ratio of fields at the tip and at the contacting surface, which depends on tip-surface separation, is shown in Figure 1, together with previous results for tip-source. As in the previous study, intensities in the case of *s*-polarization are very small, so results are shown for *p*-polarized light only.