Quantitative Modeling of Near-Field Interactions in Terahertz Near-Field Microscopy

Abstract

Terahertz scattering-scanning near-field optical microscopy (THz s-SNOM), combining the best features of terahertz technology and s-SNOM technology, has shown unique advantages in various applications. Consequently, building a model to characterize near-field interactions and investigate practical issues has become a popular topic in THz s-SNOM research. In this study, a finite element model (FEM) is proposed to quantify the near-field interactions, and to investigate the edge effect and antenna effect in THz s-SNOM. Our results indicate that the proposed model can give us a better understanding of the near-field interactions and direct the parameter design of the probe for THz s-SNOM.

1. Introduction

It has been demonstrated that scattering-scanning near-field optical microscopy (s-SNOM) is a powerful instrument for determining nanoscale dielectric characteristics of the material [1]. The s-SNOM provides a new method for nanoimaging or nanomaterial identification by combining with an infrared spectral source or terahertz source technology, while the traditional imaging system has limited spatial resolution which is roughly half of the incident wavelength. In the past few years, many researchers have investigated some nanomaterial properties, such as electromagnetic field localization [2,3,4,5], nanoscale free-carrier profiling of semiconductors [6,7], and biomolecular identification [8,9,10]. In s-SNOM, the incident light can create a “hotspot” around the atomic force microscope (AFM) tip by exciting a modulated and enhanced electric field. The tip-scattered light intensity is determined by the tip radius, the tip length, the sample characteristics, and the tip-to-sample distance. Thus, it is proven that the tip vibration with the incident light would introduce an amplitude modulation signal by considering the tip-sample model as a dipole system [11]. The background scattered signal can be eliminated by acquiring the signal demodulated at higher harmonics of the tip-vibration frequency, and leaving the pure near-field information. However, the dipole model does not account for the influence of the tip geometry or surface properties of the sample, and that cannot accurately represent the experimental findings [12]. Thus, it is necessary to develop an appropriate model considering the tip geometry to explain the experimental phenomenon.

A. Cvitkovic developed the finite dipole model (FDM) based on the dipole model [12]. In the finite dipole model, the tip is represented as an ellipse, which may more accurately reflect the form of an expanded tip. In the terahertz band, the finite dipole model cannot accurately predict the experimental result while the AFM tip length is much greater than the tip radius [13]. Alexander. S. Mcleod presented a lightning-rod model for predicting and interpreting the experimental observables of near-field spectroscopy and the probe length influence was given in the article [14]. X. Chen et al. set different distances between the tip and the sample in their model to simulate the tapping process in an s-SNOM experiment and to obtain the higher harmonics tip-sample scattered signal [15]. As an alternate approach for describing the coupled effect between the tip and the sample, the finite element model (FEM) is not constrained by the complicated geometry of the tip or the heterogeneity of the sample. Althoughthe FEM consumes more computer power than the FDM, simplifying the real model could solve the problem. F. Mooshammer et al. developed a model which provided a scattering process into the far field at a given demodulation order, with the near-field evolution as the tip-sample distance [16]. G. Conrad et al., in their model, demonstrated that FEM simulations were an appropriate choice for the strong tip-sample coupling and the model could be tailored to specific nanostructured systems and geometries of interest [17]. Moreover, X. Chen et al. proposed an FEM model to simulate the tip-sample interaction and tip scattering in s-SNOM experiments with the tip geometry on demand [18]. All of these models concentrated on the strong coupling between the tip and sample in the infrared band, but did not account for the antenna effect in the terahertz band. If the length of the tip was consistent, matching with the incident wavelength, there would be a significant resonance effect. It has been shown that a 200 μm-long tip described in Dai’s terahertz s-SNOM (THz s-SNOM) is superior to a commercial tip [13]. In order to match the experimental parameters of the terahertz s-SNOM system, a new model should be proposed.

In this study, a simplified two-dimensional (2D) model is proposed to simulate the interactions between the tip and the sample in THz s-SNOM. Using this model, we first simulate the practical tip vibration by changing the tip-sample distance, and then a near-field electric field image is obtained by extracting the signal demodulated at higher harmonics of the tip-vibration frequency. It shows a good consistence with the three-dimensional (3D) model finding obtained by F. Mooshammer et al. in 2020 [16], which proves the accuracy of our proposed 2D model. Moreover, compared with the calculation in the 3D model, our proposed 2D model requires less time consumption. The basic principles and methods used in our study are presented in Section 2. The results of our proposed 2D model are presented in Section 3 including the edge effect and antenna effect. The edge effect result explains why the resolution steadily improves with the order of the harmonic signal increasing. Moreover, the antenna effect result indicates that a superior near-field signal could be obtained by selecting the optimal tip-geometry parameters [19]. The paper closes with the conclusions in Section 4.

2. Methods

The probe’s operational schematic in s-SNOM is shown in Figure 1a. $E_i$ represents the incident electric field, $E_r$ represents the sample reflected electric field, and $E_s\left(\Omega \right)$ is the electric field containing the near-field interactions information of the sample and tip ($\Omega$ is the frequency of tip vibration). All of the FEM simulation software, such as ANSYS, ABAQUS, and COMSOL, are appropriate for the numerical calculation models.

To construct the model, we need to extract the longitudinal profile of the probe axis, as G. Conrad’s study has shown that the decrease in dimensions resulted in less information loss [17]. To cut down the calculation consumption, the x-direction range should not be excessively broad, and the y-direction range should completely cover the length of the modified probe so as to provide adequate space for tip. This reduces the amount of space for detailed calculations. The near-field scattering simulation model is seen in Figure 1b.

2.1. Scattering Model

The interactions between the tip and the sample can be described as two electric dipoles. The relationship between the scattering electric field and the incident electric field can be expressed in Equation (1).

$$E_s={\alpha }_{eff}{E}_i$$

(1)

$${\alpha }_{eff}=\frac{\alpha \left(1+\beta \right)}{1-\frac{\alpha \beta }{16\pi r^3}}$$

(2)

The ${\alpha }_{eff}$ is the effective polarizability of the coupled tip-sample system, and $\alpha$ and $\beta$ are the tip and sample parameters, respectively (further details can be found in Zenhausern 1995, [20] and Hillenbrand & Keilmann, 2000 [21]). In Equation (2), $r$ denotes the distance between the dipoles.

The Equation (2) shows that the scattering signal becomes progressively smaller with the r increase. As a consequence, the information of the near-field interactions will be included in the scattered signal while the tip is vibrating on the sample. By demodulating this part of the received signal, we can theoretically remove the influence of noise and recover the near-field interactions signal. The r in the Equation (2) can be expressed by introducing the information of tip vibration amplitude A, frequency f, and curvature radius a.

$$r=A\cos (2\pi ft)+a$$

(3)

By varying the tip sample distance d, we replicate the vibration of the tip in the model. Some experimental results indicate that satisfactory experimental results can be achieved when the vibration amplitude is nearly equal or over the radius of curvature of the tip [7,22]. Therefore, in Equations (4) and (5), we choose N points between 0 and 2a to mimic the electric field intensity at various positions of tip vibration, and recover the electric field intensity $S\left(r_l\right)$ at the corresponding points. Thus, the harmonic signals demodulating of tip-vibration frequency are obtained by computing the Fourier demodulation transformation [15].

$$t_l=\frac{1}{2\pi f}\arccos (\frac{A-r_l}{A}),l\in [1,N]$$

(4)

$$S_n\propto {\displaystyle \sum _{N}S\left(r_l\right)e^{-in2\pi f{t}_l}}\left(t_{l+1}-t_l\right)$$

(5)

In this model, the electric field of each point in the space range when the tip vibrates is obtained by acquiring values at equal intervals for fast Fourier transform analysis, and the signal demodulated at higher harmonics of tip-vibration frequency of each point can be extracted to obtain $S_n$ signal imaging in the small area around the tip. Through the theoretical demonstration and experiments, the relationship between near-field electric field strength and far-field reception strength is demonstrated in 2020 study [16]. By comparing the spatially dispersed electric field of each space point to the incident electric field, we are able to obtain the electric field enhancement data of each point and reconstruct the image of the harmonic signal, and explore some factors which affect the near-field electric field in s-SNOM. Different from the previous methods, this method is more intuitive for showing the effect of near-field signals. By changing some parameters, we can see its impact on near-field signals and explore the near-field interaction. However, the calculation amount of this method is greatly increased, so we also focus on simplifying the model to reduce the calculation amount in the subsequent model building.

2.2. Simulation Settings

To match the tip parameters in THz s-SNOM, the tip is set as a gold-cone longitudinal section with main length L = 100 μm and tip radius α = 100 nm. The physical domain is set as a 30 μm × 200 μm rectangular range surrounded on all sides by a 20 μm perfectly matched layer, and the sample thickness is 30 µm. In order to improve the calculation efficiency, the grid size needs to be distinguished. The maximum mesh size at the small range of the tip is set to 1 nm to facilitate subsequent simulation of the vibration fluctuation of the tip. By setting a transition range, the general grid size of other areas can be transitioned to nano-size to ensure the accuracy of the simulation. Theoretically, the smaller the grid size near the tip, the more accurate the calculation results will be, but the amount of calculation will be greatly increased. For THz s-SNOM, the radius of curvature of the tip is generally above 20 nm, and this model can meet the research accuracy requirements for such systems. The specific meshing setting is shown in Figure 2a.

In most cases, to acquire the scattering field of the tip sample, the solution is solved in two steps. In the first step, the scattering boundary conditions are established in our 2D model. The top boundary of the model is set as a port, and launched a plane wave towards the surface at an angle of 60° relative to the vertical. The left and right bounds are specified as periodic boundary conditions to simulate parallel light incident on infinite plane. The perfectly matched layer set does not need to be enabled in the first step. In the second step, the electric field calculated in the first part should be used as the input electric field in the second part when there is a tip-sample scatterer. In this way, the perfectly matched layer can absorb the scattered wave radiated outward and eliminate the reflection caused by setting the boundary. In Figure 2, the scattering simulation model is displayed. The different near-field electric field values are generated by adjusting the distance value of several tip samples to replicate the actual tip shaking. We choose 20 simulating values (the tip-sample distances) between 2–200 nm and obtain the electric field information of each point in our 2D model. Using these data, the Fast Fourier Transform (FFT) is employed to extract the various order signals of tip vibration in order to imitate the near-field signal extracted from the harmonic frequency of tip vibration during the experiment.

3. Results and Discussion

3.1. Electric Enhanced Image

To validate the accuracy of the simplified model, we present simulation results for two common materials, silicon and gold. As an exceptional scattering metal, gold is a better scattering material than silicon, which is represented in the simulation results. In many s-SNOM imaging investigations, silicon substrates are etched with metal strips to test the imaging effect [23]. The frequency of the incident wave is 4.2 THz, and the other simulation settings were described in Section 2.2. Figure 3 depicts the electric field enhancement factor as a function of the tip-sample distance with the gold or silicon sample. The electric field value of each point at different tip-sample distances is simulated, and the maximum value of the electric field is selected to represent the electric field enhancement effect. There are 100 values for the tip-sample distances (d from 2–200 nm). For the purpose of better data comparison, we only extracted the data where the d is less than 100 nm in Figure 3, and did not show the data indicating that the maximum electric field intensity does not change significantly after the d exceeds 100 nm. The results show that the enhancement effect of the electric field decreases sharply with the increase of the tip-sample distance. Thus, the stronger electric field enhancement proves that gold is a better scattering material than silicon.

Through the data processing of the simulation, the signal of each order is extracted, and the near-field electric field intensity image near the tip is reconstructed. Figure 4 depicts the near-field electric field distribution image for the two materials at the extracted harmonic order signal (S₁, S₂, and S₃), which corresponds to the vertical section electric field division diagram produced by others in the 3D simulation [16]. The incident electric field is confined to a tiny region close to the tip, and a “hotspot” is created by the effect. Figure 5 depicts the width of the field distribution with the gold sample. For the increasing order n = 1−3, the lateral full width (2 nm above the sample) at half-maximum (fwhm) 2Γ of the field profiles decreases continuously and the value could be smaller than the tip radius (100 nm). All the results of our 2D model are in good agreement with those of others’ 3D models [16,18], which proves that our 2D model can reduce the amount of calculation and ensure its correctness. The results depicted in Figure 3 indicate that the electric field enhancement can penetrate non-metallic materials such as silicon, and that the scattering electric field of gold is stronger than that of such materials, so it can be used to detect metal structures covered by layers of penetrable materials. This is also an application direction for s-SNOM technology, and significant development has been carried out in this field [24]. It can image the nm antenna covered by layers of penetrable materials and detect whether the antenna is damaged without damaging the surface. Thus, with the development of nm technology, there will be more scenarios that need to use this technology to solve more nm imaging problems.

By obtaining a higher harmonic signal, the action range of the strong scattering electric field at the tip is condensed. Although extracting higher-order signals will reduce the signal intensity, this part of the energy information more concentrated on the tip can be obtained to improve the resolution during imaging. Thus, through this model, we could further explore the influencing factors of the near-field electric field.

3.2. Edge Effect

In the definition of s-SNOM system resolution, the theoretical resolution of the system is the transition distance between the maximum signal and the minimum signal of the scattering signal between silicon and gold in the silicon–gold grating, which ranges from 10% intensity to 90% intensity, as shown in Figure 6.

According to the mirror dipole theory that the resolution is just related to the tip radius [25]. However, in our experimentation, we observed that the transition distance decreased when extracting a higher harmonic signal. The phenomenon is explained such that, as the order of the signal extraction increases, the influence of the scattering signal which is unrelated to the amplification of the tip sample diminishes. Thus, a better resolution can be obtained.

Our model can simulate the effect of electric field augmentation at any location within the radius of curvature of the tip. This feature enables us to describe the effect of the metal steps on the near-field scattering signals of each order. As shown in Figure 7, the following is a schematic representation of three typical states corresponding to the tip and edge in the scanning process. By simulating the change of electric field intensity of each order’s signal under these three conditions, we can further understand the influence of the step effect in the imaging process, so as to find ways to reduce the influence of the edge effect on s-SNOM imaging. The thickness of the gold strip is set to 100 nm. The frequency of the incident wave is 4.2 THz, and the other simulation settings were described in Section 2.2. The scattering signals of each order in three positions are shown in Figure 8.

As shown in Figure 8, there is an electric field enhancement with the tip’s side when the tip transitioned from the middle to right. From top to bottom, the retrieved signals (S₁, S₂, and S₃) are shown in ascending order. The absolute strength of the electric field enhancement between the gold strip and the gold tip is stronger than the electric field enhancement effect of the silicon sample and the gold tip. Even when the needle tip leaves the upper space of the gold sample and is above the silicon sample, the electric field intensity is still strong where the needle tip is close to the gold. For the received signal, an overall signal strength is obtained, which will lead to the conclusion that the needle tip is still in the gold sample during imaging. Due to this reason, the electric field enhancement effect of the interaction between the tip and the sample silicon is hidden. This phenomenon is most pronounced in S1. AS the signal order n increases, the edge effect is steadily reducing. In practical application, with the increase of the order of the demodulated signal, the theoretical resolution that can be achieved in imaging will be further improved as the edge effect of gold is further suppressed. To further comprehend the lowering of the step impact, we extract and process additional image data. The electric field strength values are extracted along the white dashed line, and we extract the electric field strength value and obtain the findings depicted in Figure 9.

To facilitate the comparison of the influence of the edge effect, we draw the extracted electric field intensities of the edge effect and sample effect together. The blue line represents the edge effect (white vertical line), and the red line represents the sample effect (white horizontal line). As the harmonic signal order n increases, the electric field becomes more confined underneath the tip, while the red line has narrower protrusions. The influence of the edge effect increasingly diminishes because the maximum electric field strength of the blue line is significantly reduced. In addition, the influence of the edge effect becomes very small when it comes to S₃. Therefore, for the THz s-SNOM system, the higher-order signal can effectively eliminate the influence of the edge effect and improve the resolution of image. For this system, with the increase of the order of the extracted signal, the edge effect will continue to decrease, but the increase of the order of the signal will also increase the difficulty of extracting the signal, and the decline of the signal-to-noise ratio will lead to the deterioration of imaging quality. Therefore, in practical applications, it is necessary to extract the appropriate signal order to obtain better imaging quality. A more intuitive understanding of the edge effect in the harmonic signal is represented in our proposed model.

3.3. Antenna Effect

This model can also be used to investigate the antenna effect, a key contributor to the increase of the electric field in terahertz s-SNOM [26]. In the theoretical model, the interaction effect between the scattering probe and the sample corresponds to a model of a mirror dipole. As part of the transmission of the dipole antenna, there is an impedance matching condition with the scattering probe in the actual work process. When the tip length aligns with an excited electromagnetic wave, resonance will occur. In engineering applications, the typical antenna length is $\frac{1}{4}\lambda$ (λ is the incident wavelength) [27]. For the s-SNOM, the tip is not a circuit-matching emission. It is stimulated by the incident electric field to produce a new emission source. Thus, the engineering experience is no longer relevant. Nevertheless, using the FEM model, we can obtain the influence of the tip length on the electric field enhancement in the original signal. In the model, we extract the maximum electric field intensity as the evaluation standard of the electric field enhancement, and simulate the results under different needle tip lengths.

Shown in Figure 10 is the result of the influence of varying the tip length in the electric field at different frequencies. The electric field enhancement effect shows a positive correlation with the increase of tip length at the beginning, and when it reached a peak, its enhancement effect began to decline. Therefore, in practical applications, we expect to find the tip length corresponding to the point with the strongest electric field enhancement effect as the actual scattering probe to obtain a better imaging effect. With the condition that the tip radius does not change, there is a clear correlation between the maximum electric field intensity corresponding to the tip length and the incident wave frequency. The influence of the electric field enhancement is illustrated by matching the length of the tip to the incident wavelength, as shown in Figure 11.

This is performed to visualize the effect more clearly. The strongest enhancement effect is found when the length of the tip is equal to 0.3λ, where λ is the wavelength of the simulated electromagnetic wave that associated with it. Notably, this finding is inconsistent with the maximum enhancement effect of $\frac{1}{2}\lambda$ described in the study [14]. A possible reason is that the simulation frequency in our model is lower, resulting in a longer electromagnetic wave. Our model’s tip length was therefore increased to match the actual tip model. Concurrently, the probe model is defined as the longitudinal section of the cone, which differs from the ellipsoid probe in the B. Hauer model [28]. These factors may cause the different result. The design of the scattering tip has always been a focus of this kind of research. Although our model is simplified and designed to reduce the amount of calculation, we can explore some parameters of the scattering tip. Because it is an FEM model, the shape of the tip can also be different from some traditional shapes. We can change the shape of the tip according to our own needs to explore its impact on near-field signals. Therefore, it is possible to find a new scattering tip structure through this model to achieve a better imaging effect, which is no longer limited to the traditional commercial tip shape. In the subsequent experiment, we will validate the result which we discovered in our 2D model, and try to explain the reasons for the different results clearly to verify whether our conclusion is universal in terahertz systems.

4. Conclusions

In this study, we propose an FEM model to investigate the near-field interactions between the tip and sample in THz s-SNOM. By modeling the tip vibration, the electric field of each point in the near field can be determined. Moreover, the result of our proposed 2D model is consistent with the result of F. Mooshammer’s model which proves the model’s accuracy. After that, the edge effect in THz s-SNOM is discussed by our proposed 2D model. The third-order harmonic signal S₃ is proven to reduce the effect effectively. In addition, in the actual imaging process, extracting the S₃ signal for imaging can obtain a more ideal imaging effect; the reason can also be better explained by our 2D model. Finally, the proposed 2D model is used to examine the effects of different incident wave frequencies and tip lengths on the enhancement of the near-field electric field, by considering the antenna effect. Our results do not show a good consistence with A. S. McLeod’s study [14]. The strongest enhancement effect occurs when the length of the tip is equal to $0.3\lambda$ in our model, but $0.5\lambda$ in A. S. McLeod’s model. The inconsistencies can be explained by the fact that our simulation settings considered a lower incident wave frequency and longer tip length. Further study would verify the result in experiments. Moreover, our model can simulate the influence of any tip geometry on the electric field strength of each point in the near-field space range and can obtain images of signals of all orders of the harmonic signal. Our 2D model allows us to change the shape of the tip or the surface of the sample as needed to simulate the situation of a complex tip or samples in actual experiments. The simulation model and method can help us to better understand the phenomenon occurring in THz s-SNOM, such as the edge effect and antenna effect, and serve as a reference for the s-SNOM probe design.