Terahertz (THz) scattering-type scanning near-field optical microscopy (s-SNOM) is an important means of studying and revealing material properties at the nanoscale. The nanotip is one of the core components of THz s-SNOM, which has a decisive impact on the resolution of the system. In this paper, we focus on the theory and design of the nanotip and conduct comprehensive research on it through simulation. The theoretical model is based on full-wave numerical simulation and dipole moment analysis, which can describe the overall nanotip electromagnetic response under the incident field. A comprehensive design model of nanotip geometry, sample materials, and incident field is established to significantly improve the near-field coupling efficiency and spatial resolution to achieve optimal performance.

- Chinese Optics Letters
- Vol. 22, Issue 9, 090002 (2024)
Abstract
1. Introduction
Terahertz (THz) radiation is located at an important position in the electromagnetic spectrum. THz spectroscopy has been widely applied to resonantly probe collective charge, spin, and lattice oscillations in solids, the rotations of small polar molecules, and the structural vibrations of large biomolecules[1-3]. THz imaging has become the distinctive detection technique in biological tissue samples and new materials[4-6]. However, the spatial resolution with conventional optical methods is constrained by the diffraction limit, which obstructs the application and development of THz imaging. Scattering scanning near-field microscopy is a promising scanning probe technique, which conveys near-field dielectric information by nanotip scattering and is detected by conventional far-field detectors[7,8]. The surface charge density of the tip apex is polarized by the incident field, and the surface charge is concentrated to the nanoscale by the tip apex[9,10]. When the nanotip-sample distance reduces to the near-field region, the tip scattering field is influenced by the local tip-sample system’s dielectric properties. The nanotip-sample system scattering field can be transferred and recorded at the far-field. The dielectric information and surface topography can be recorded simultaneously[11-14].
In the THz scattering-type scanning near-field optical microscopy (s-SNOM) system, the spatial resolution and near-field coupling efficiency are predominantly determined by the tip geometry, as the nanoscale tips provide the basic near-field environment[15,16]. Nanotips with different performance characteristics have different fabrication processes, including photolithography, electron beam lithography (EBL), and reactive ion etching (RIE). Besides, the dielectric response of the sample surface and incident field intensity and polarization also affect the near-field coupling efficiency. In this paper, we comprehensively characterize the performance improvement for near-field spatial resolution and near-field coupling efficiency induced by the nanotip geometry, nanotip material, sample material, and incident field. The theoretical model is a numerical full-wave simulation based on solving Maxwell’s equations in the frequency domain[17]. The spatial resolution is dominated by the tip apex radius, tip cone angle, and tip-sample distance. The near-field coupling efficiency is dominated by the tip geometry and incident field polarization.
Current THz s-SNOM systems are typically constructed with continuous THz sources or photoconductive antennas, which yield THz pulses with low peak power and are unfavorable for enhancing near-field coupling efficiency. Common methods for generating intense THz pulses include optical rectification and femtosecond laser filamentation. The spectral energy distribution of these intense THz sources is primarily centered around 0.5 THz[18,19]. Consequently, we concentrate on the design of THz s-SNOM probes within the 0.5 THz range. These intense THz sources typically necessitate the utilization of femtosecond laser amplifiers as their driving light sources, leading to lower repetition rates for these THz sources (the disadvantage of low repetition rate is being overcome)[20]. This simulation method extends the application of such high peak power THz sources in THz s-SNOM.
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2. Theoretical Model
Full-wave numerical simulations offer a distinct advantage in comprehensively characterizing the electromagnetic response of a nanotip illuminated by the THz field. In full-wave numerical simulations, the electric dipole moment of charge distribution is defined by the integral expression[21]:
In Eq. (1), and are the positions where the charge density is evaluated, which describes the electromagnetic response of the nanotip scattering and sample surface scattering. determines the charge density of the nanotip apex surface. Charge conservation relates the charge density to the current density by the continuity equation[21,22]:
As shown in Eq. (3), the electric dipole moment is proportional to the current density. The tip-scattered electric field is proportional to the complex-valued dipole moment , calculated numerically according to[21,22]
In this paper, we focus on designing a nanotip design method to improve the THz s-SNOM spatial resolution and near-field coupling efficiency by solving Maxwell’s equations in the frequency (full-wave numerical simulations). The research process of the nanotip aims to achieve high spatial resolution and high near-field coupling efficiency. Figure 1 demonstrates the research process details of the nanotips used in THz s-SNOM. An excellent nanotip geometry design can reach an extreme spatial resolution while maintaining the near-field coupling efficiency. The optimized nanotip needs an ideal incident field condition to achieve maximum performance. The Au nanotips and the Au surface (or surface) are described by the Drude model with plasma frequency and collision frequency [23]. The nanotip length is 17 µm. The nanotip apex radius range is 0.5–500 nm. The cone angle range is 10°–40°.
Figure 1.The research process of the nanotip of THz s-SNOM. The research process involves two aspects: the influence of the nanotip geometry and the incident field on the near-field scattering signal.
3. Results and Discussion
Research on THz s-SNOM simulation primarily relies on point dipole models and finite-dipole models, in which the most prominent feature is that the nanotip apex is simplified into a sphere[24-27]. We further extend this model by simplifying the nanotip into a cone.
In Fig. 2(a), the traditional dipole moment model simplifies the nanotip into a sphere. In Fig. 2(b), we extend this model by simplifying the nanotip into a cone (the illustration shows the cross-section of a cone). is the cone angle of the nanotip, and is the circumscribed circle radius. The cone cross-section is divided into three isosceles triangles to simplify calculations, thereby . The effective polarizability of near-field coupling can be expressed as[28]
Figure 2.Two approximation methods of near-field nanotips. (a) The near-field nanotip is simplified as a sphere. (b) The near-field nanotip is simplified as a cone. θ is the cone angle of the nanotip, r0 is the circumscribed circle radius, and ξ = θ.
The polarizability of a cone can be written as
Equations (7) and (8) indicate that the near-field effective polarizability comprises the nanotip polarization and mirror dipole polarization. The intensity of the mirror dipole polarization is closely related to the sample material dielectric constant. The near-field enhancement effect of the Au–Au system is greater than that of the system; the reason is that Au has a larger dielectric constant. Furthermore, the localized field spatial distribution depends on the near-field polarization at the nanotip apex. As the cone angle increases, the near-field polarization increases, resulting in a larger focusing spot diameter at the apex of the nanotip.
Figure 3 describes two near-field systems, which are the Au nanotip–Au surface system and the Au nanotip– system, respectively. In the Au nanotip–Au surface system, as the cone angle of the probe tip varies from 10° to 40°, there is a 1.6-fold increase in localized field enhancement at the probe’s apex and a corresponding 1.2-fold increase in the diameter of the focusing spot at this apex. In the Au nanotip– surface system, as the cone angle of the probe tip varies from 10° to 40°, there is a 1.1-fold increase in localized field enhancement at the probe’s apex and a corresponding 1.2-fold increase in the diameter of the focusing spot at this apex. Although the spatial resolution and the local field enhancement in different nanotip-sample systems are different, the overall trend is the same.
Figure 3.The evolution trend of the localized field enhancement at the nanotip apex and the FWHM of the focus diameter are influenced by the nanotip cone angle. (a) Field enhancement at the nanotip apex on two material surfaces. (b) Focusing spot diameter at the nanotip apex on two material surfaces.
Figure 4 shows the dipole moment and focus point diameter of the focus points for several nanotips that are statically placed at a distance above the sample. The nanotip parameters are as follows: nanotip apex radius 5, 100, and 250 nm; cone angle 20°; length 17 µm; and nanotips-to-sample distance . The nanotip-sample dielectric system’s dipole moment is according to[28,29]
Figure 4.The evolution trend of the near-field induced dipole moment of the overall nanotip is influenced by nanotip apex-sample surface distance.
We systematically studied the influence of the nanotip apex radius on the near-field coupling efficiency, and the spatial resolution is shown in Fig. 5, in which the nanotip geometry parameters are set as follows: the cone angle is 20° and the nanotip length is 17 µm. In Fig. 5(a), the increase of the near-field enhancement at the nanotip apex can be achieved by reducing the nanotip radius. This simulation result is consistent with Eqs. (10) and (11), and a smaller nanotip apex radius leads to a larger near-field enhancement at the nanotip apex.
Figure 5.The evolution trend of the near-field localized field enhancement at the nanotip apex and the full width at half-maximum (FWHM) of the focus diameter are influenced by the nanotip apex radius. (a) Field enhancement distribution at the nanotip apex on two material surfaces. (b) Dipole moment distribution of the overall nanotip on two material surfaces.
However, in Fig. 5(b), the increased field enhancement at a smaller nanotip apex does not result in an increased induced dipole moment of the overall nanotip. The specific analysis is as follows. The dipole moment of the overall nanotip reflects the current density integration of each cross-section along the -direction of the nanotip, thereby comprehensively representing the scattered field generated by the nanotip. The scattering field and the induced dipole moment are shown as[30]
Apart from the scattering nanotip geometry, the incident field polarization is also influencing the near-field coupling efficiency. The relationship between the scattering field I and the incident field polarization can be simplified as
Figure 6.The evolution trend of the near-field localized field enhancement at the nanotip apex and the FWHM of the focus diameter are influenced by the incident field polarization status. (a) Field enhancement distribution at the nanotip apex on two material surfaces. (b) The evolution trend of the focusing spot diameter at the nanotip apex with the incident field polarization.
For reasons similar to those mentioned in Eqs. (7) and (8), the variations in near-field coupling efficiency and the near-field focusing spot among different nanotip-sample systems in Fig. 6 are attributed to the differences in the dielectric constants of the sample materials. The near-field enhancement effect of the Au–Au system is greater than that of the system; the reason is that Au has a larger dielectric constant. Furthermore, the localized field spatial distribution depends on the near-field polarization at the nanotip apex. As the incident field angle changes, the near-field polarization increases, resulting in a larger focusing spot diameter at the apex of the nanotip.
In the demonstration above, the various geometric parameters of the nanotip are comprehensively discussed. We conclude that when only considering the influence of the nanotip geometry on the spatial resolution of THz near-field scanning, the nanotip apex radius is a primary factor. Therefore, we designed a conical tip of 17 µm length, and apex radius was scanned above a sample modeled by Au on the left side and on the right side to verify the THz-SNOM system’s achievable ultimate spatial resolution[32]. The material boundary at . The polarization of the input wave is parallel along the nanotips at 0.5 THz relative to the tip axis. The blue curve in Fig. 7(a) shows the result of field enhancement evolutionary trends at the nanotip apex. The scanning spatial resolution depends on the nanotip geometry and is described by the differential curve of the field enhancement evolutionary curve. By scanning the boundaries of two materials, the evolution of the field enhancement shows that the spatial resolution is 0.5 nm. Figure 7(b) shows the electric field near-field distribution in (the nanotip is above the Au), which shows that the focusing spot diameter is 0.5 nm. The near-field enhancement electric distribution of the nanotip apex position conforms to a Gaussian distribution, exhibiting good symmetry. This is attributed to the optimization of the nanotip parameters (nanotip cone angle, nanotip apex radius) and the near-field conditions (the nanotip apex-sample surface distance).
Figure 7.Numerical simulation of THz near-field one-dimensional scanning. A tip with apex radius of r = 0.5 nm and length of 17 µm is placed above a sample consisting of Au on the left side (x < 0 nm) and Al2O3 on the right (x > 0 nm) side. (a) Near-field localized field enhancement evolution curve and corresponding derived curve. (b) Electric near-field distribution in x = −0.5 nm. The red curve represents the Gaussian fitting curve of the near-field enhancement. (c) Electric field near-field distribution below the tip apex for different tip positions.
We calculated the scattering coefficient at the apex of the nanotip apex to characterize the localized field enhancement at the nanotip apex. The interactions between the nanotip and the mirror dipoles define the total current density located at the nanotip in the presence of a sample and, therefore, the scattering coefficient[33]:
The nanotip apex and the mirror dipole create a situation akin to an optical resonator, enabling an increase in localized field enhancement at the nanotip apex through near-field interaction.
Figure 7(c) shows the transverse scanning process continuously. The electric field near-field distribution shows significant changes when nanotips cross two materials’ boundaries. Similar to Eq. (8), the difference in the different scattering boundaries can be attributed to the variation of the local-dielectric function of the sample based on the method of image charges.
The near-field coupling efficiency of the nanotips depends on the field enhancement and dipole moment distribution in the nanotip apex, which can be described as effective polarizability. The scattering signal is influenced by the nanotip geometry, nanotip-sample distance, local dielectric properties of materials, and polarization of the incidence wave. The near-field scattering signal characterizes dielectric properties of the local materials by scanning the structure of the material and controlling the other factors unchanged. We improve the resolution to 0.5 nm with 0.5 THz () by optimizing the nanotip geometry, nanotip apex to sample surface distance, and incident field polarization. The optimization of the nanotip geometry can noticeably improve the spatial resolution, especially for specific incident wavelengths. This method can provide the optimized nanotip parameters and assist the search for the balance parameters between the near-field coupling efficiency and spatial resolution. The nanotip design process and method mentioned in this paper provide a new method for improving the spatial resolution of THz s-SNOM.
Furthermore, we provide a brief overview of the nanotip fabrication process, which varies depending on the desired apex radius of the nanotips. The high-resolution nanotip fabrication can be categorized into two methods. The first method focuses on refining the apex of the nanotip to achieve a radius of approximately 3 nm or less. The secondary method involves the growth of an additional nanotip, composed of high-density carbon, at the summit of the original nanotip pyramid. Consequently, the extra-nanotip extensions can be crafted with an apex radius as fine as 1 nm or even less, further enhancing the precision and resolution of the nanotips[34-36]. The fabrication process of the conventional nanotip (apex radius 10–500 nm) is based on photolithography and chemical etching. The nanotip geometry is defined by the photolithography, and the protective layer is removed by chemical etching[37].
4. Conclusion
In conclusion, ultrahigh-resolution THz near-field scattering nanotips are investigated. The nanotip geometry (cone angle, and apex radius), incident field, and local material dielectric constant are comprehensively considered to obtain the optimal parameters. From the perspective of the nanotip geometry, the spatial resolution and the near-field coupling efficiency mainly depend on the nanotip apex radius and the distance between the apex and the sample surface. Other parameters such as the polarization of the incident field and the sample surface materials mainly influence the near-field coupling efficiency, but they do not change the spatial resolution significantly. Finally, a feasible model is presented to design optimal parameters of the THz near-field scattering nanotip with 0.5 nm () spatial resolution, while maintaining an excellent near-field coupling efficiency (). This method describes the nanotip electromagnetic response in the near-field and provides a feasible method for designing the nanotip of the THz s-SNOM system.
References
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[22] J. D. Jackson. Classical Electrodynamics(1999).
[35] S. Sheng, D. M. Czajkowsky, Z. Shao. AFM tips: how sharp are they?. J. Microsc., 196, 1(1999).

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