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Abstract
Tip-enhanced Raman spectroscopy (TERS) can provide correlated topographic and chemical information at the nanoscale, with great sensitivity and spatial resolution depending on the configuration of the TERS probe. The sensitivity of the TERS probe is largely determined by two effects: the lightning-rod effect and local surface plasmon resonance (LSPR). While 3D numerical simulations have traditionally been used to optimize the TERS probe structure by sweeping two or more parameters, this method is extremely resource-intensive, with computation times growing exponentially as the number of parameters increases. In this work, we propose an alternative rapid theoretical method that reduces computational loading while still achieving effective TERS probe optimization through the inverse design method. By applying this method to optimize a TERS probe with four free-structural parameters, we observed a nearly 1 order of magnitude improvement in enhancement factor (|E/E0|2), in contrast to a parameter sweeping 3D simulation that would take ∼7000 hours of computation. Our method, therefore, shows great promise as a useful tool for designing not only TERS probes but also other near-field optical probes and optical antennas.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tip-enhanced Raman spectroscopy (TERS) is an important division under the scanning near-field optical microscopy (SNOM) branch. By using a metallic or metal-coated scanning probe, the excitation light confines the strongly enhanced electromagnetic (EM) field in the vicinity of the tip apex due to the lightning-rod effect and localized surface plasmon resonance (LSPR). The strongly enhanced EM field near the tip enhances the Raman signal of molecules with an enhancement factor of up to 106 [1]. When the TERS probe scans raster over the surface, the technique correlates the vibrational spectra with the topography. As invented in the 2000s [2–4], TERS has shown great advantages in catalysis, electrochemistry, semiconductor, biology, and other spectra analytical regions [5–7].
Two main factors affect the sensitivity of the spectrum in TERS including the lightning-rod effect and LSPR, which are closely dependent on the structure of TERS probes. So the metallic structure of the TERS probe plays an essential role in the performance of TERS, which has been reported by many works [7–9]. Methods for designing and fabricating TERS probes have always been a hot topic, focusing on promoting lifetime and enhancement factors (EF). A well-designed TERS tip would improve the EF and facilitate obtaining Raman spectra with a high signal-to-noise ratio (SNR). Therefore, it’s important to figure out what factors may influence the performance of TERS probes in certain experimental conditions. These critical factors, as reported, include excitation laser profiles (polarization [10,11], and incident angle [12]), tip configuration (materials [13–15], tip radius [14], and coating thickness [8]), nanophotonic structures on the tip (grating [16], nanoparticles [17], photonic crystal [18] et al.) and sample properties [19,20]. To improve TERS probes’ performance, many researchers have proposed effective approaches. Xu improved the TERS enhancement by radially polarized incident light rather than linear polarized light [21]. Kawata utilized the silicon oxide tip modified by metal spheres or film to induce a strongly localized electromagnetic field [17]. Ren group fabricated TERS tips by pulsed electrodeposition and verified that the electrodeposited tips achieved maximum TERS enhancement when the tip radius was distributed between 60 and 75 nm [22]. Meanwhile, some special tip structures were also proposed such as the cone coupled with photonic crystal [18], grating coupled TERS tip [16], campanile-shaped tip [23] and silver nanowire coupled with AFM tip [24], or optic fiber [25]. These tip-related factors (summarized in Fig. 1(a)) may correlatively impact the TERS performance.
Fig. 1. (a) Tip-related issues that could affect TERS performance. (b) Four factors optimized in this paper. The EF of the model is evaluated in the region where the white double-headed arrow exists. The double-headed arrow is 2 nm in length, in the middle of the nanogap.
Therefore, a general method is in demand to optimize the design of TERS probes with multi-parameters. In previous works, researchers proposed a TERS probe structure with a few tuning structural parameters and verified the performance by directly solving Maxwell’s equations in three dimensions, via the Finite Element Method (FEM) or other numerical methods [14,26,27]. However, the 3D numerical simulations are usually time-consuming and the computation cost exponentially increases when expanding the tuning range or adding new structural parameters. Therefore, the conventional theoretical design method is usually restricted to a limited tuning range of 2-3 parameters. Due to the limited parametric space, the optimization usually ends up with trivial performance. In this regard, two existing main challenges block the way of rapidly designing TERS probes with high performance: 1) the huge volume of 3D numerical simulation prevents probe designing from rapid multi-parametric optimization due to the vast consumption of computation resources, and 2) the parameter sweeping method transverses through the whole parameter space, leading to low efficiency in optimization. Regarding the first challenge, we apply the 2.5D method to simplify the physical model by the rotational symmetry [28–30], given that most TERS probes are highly axial symmetric. Considering the second challenge, we introduce the inverse design method to the optimization. Inverse design is an object-oriented design method in which the initial design would iterate many times until the design meets the target performance [31]. The inverse design method is a powerful tool for optimizing the design in robotics, chemical synthesis, and nano-optics [32–34]. Here we apply a clustering-based genetic algorithm called Polygamy Genetic Algorithm (PGA) to search for an optimal solution for our design [32].
In this work, we apply the 2.5D method in the tip-substrate-coupled model to simplify the complexity of 3D numerical simulation [28–30]. The optimization algorithm, PGA, runs in MATLAB 2018b and calls COMSOL Multiphysics interface for EM field solution via FEM. The PGA algorithm generates the initial population, encodes parameters, and performs crossover and mutation processes. Finally, an optimal parameter combination is searched and the new TERS probe enhances the local EM field |E/E0|2 as 5 order of magnitude.
2. Theoretical methods
2.1 Simulation of the TERS probe model
The classical physical model of AFM-TERS consists of a cone-shaped Si probe coated with metal and a semi-infinite flat substrate which is situated 1∼2 nm under the tip. When the p-polarized EM plane wave is illuminating, the local EM field would be strongly enhanced in the nanogap between the substrate and the apex of the tip. We propose the physical model in Fig. 1(b) scheme and consider four main factors in the model including the incident angle, the half-cone angle of the tip, the Si tip radius, and the coating thickness. We choose these factors for demonstration because experimentalists have reported these factors for improving the TERS probes’ performance and these factors are easy to control in setting up the TERS apparatus or milling TERS probes [8,12,14].
As illustrated in Fig. 2(b), the physical model is highly symmetric in the rotational axis. Since the incident plane wave breaks the axial symmetry, we expanded the plane wave into a series of cylindrical waves. More precisely, we solve the EM field excited by cylindrical waves of different harmonic number m and compose the solution collection at different m (from m = 1 to m = mmax) into the final EM distribution. The analytical derivation of 2.5D can be found in Supplement 1 (Section 1: 2.5D Method Verification). As equations S2-S4 show, the maximum harmonic number mmax determines the accuracy of planewave expansion at the cylindrical coordinates system. Thus, the final EM field can be solved more precisely while mmax is increasing (shown in Fig. S1). However, a larger mmax means more time to solve Maxwell’s equations. In this work, mmax = 5 is preset for a sufficient accuracy in the model.
Fig. 2. Method of optimizing the TERS probe by inverse design. (a) Scheme of design method; (b) simplified physical model by the 2.5D method.
By the 2.5D method, we reduce the degree of freedom (DoF) to ∼1/5 of that in the 3D model. As we increase the harmonic number m, the computation cost increases linearly, in contrast to the exponential growth of DoF in the 3D case. As a result, the total computation time accumulates slightly (see Fig. S2f) but is still less than 5 minutes, far away from the 3D method (∼60 minutes). Regarding the solution accuracy, we demonstrated the sphere scattering model via the 2.5D method compared to that in Mie theory. The result shows 2.5D method keeps a high solution accuracy and we can improve the harmonic order m to reduce the error compared to the analytical solution. We also present a detailed comparison (Fig. S1-S2) in Supplement 1 (Section 1: 2.5D Method Verification).
2.2 Inverse design of TERS probe
Inverse design is an object-oriented design method in which the initial design would iterate many times until the design meets the target performance [31]. In the iterations, the design model is optimized by algorithms like Gradient Descend (GD) method, Particle Swarm Optimization (PSO), Genetic Algorithm (GA), or even Neutral Network (NN). The inverse design method has been a powerful tool for optimizing the design in robotics, chemical synthesis, and nano-optics [32–34]. In this work, we choose the clustering-based GA, named Polygamy Genetic Algorithm (PGA) as the optimization strategy. GA has many advantages such as ease to use, good robustness, and the ability to support multi-objective optimization in various areas like data processing, optimization and planning [31]. PGA includes the advantages of GA and features a faster convergence within a few iterations and the compatible ability for real-time applications [32].
PGA divides the population into several clusters based on the Euclidean distance of the parameter vectors and the clustering rule is k-means. That means parameter vectors close to each other would more probably gather into the same cluster and produce their offspring.
In every cluster, the best individual matches not only one but several individuals simultaneously for offspring succeeding the best gene of the cluster. Meanwhile, mutation and crossover happen in multiplying new genes so that the population can search the rest part of the parameter space (crossover and mutation scheme can be found in Fig. 3). The newborn individuals go through natural selection by calculating their fitness and become parents of the next generation. Every generation repeats the clustering, mutation, crossover, and natural section until the iteration meets some convergence criterion. Here, we use 4 as number of clusters while performing K-means clustering. The discussion on number of clusters can be found in Supplement 1 (Section 2: Number of Clusters in Polygamy Genetic Algorithm).
Fig. 3. Scheme of crossover and mutation in Polygamy Genetic Algorithm. The yellow star indicates the best individual of one cluster in generation N and it would mate with other individuals to produce generation N + 1.
In this work, we formulate the optimization problem as Equations (1),
The illumination laser in TERS experiments is monochrome and numerical simulation is demonstrated with excitation at a single wavelength. For broadband optimization, the objective function needs to be reconstructed as Eq. (2)
By defining a central wavelength λ, the objective function can be written as the sum of EFs of 2n wavelengths. Δ is the step when varying wavelength.
3. Results and discussions
We run three rounds of the 40-generation optimization via PGA. The results of three rounds can be compared to each other, to see how the molecular layer and tip-substrate distance (dgap) affect the performance of TERS experiments. The first round is set with no molecular layer under the tip and 2 nm as dgap (black curve in Fig. 4(a)). The second round is added the 0.5 nm molecular layer (approximation in Fig. S4) and dgap is set as 2.5 nm (red curve in Fig. 4(a)). The third round keeps dgap as 2 nm and adds a 0.5 nm molecular layer in the nanogap (green curve in Fig. 4(a)). The refractive index of the molecular layer is 1.49 (2.22 as the relative permittivity, the same with pyridine) and its scale is marked in Fig. 4(b). The substrate and tip coating of the models are set as gold (Detailed information for numerical simulation can be found in Supplement 1 (Section 3: Numerical Simulation Details). The optimizations of all rounds converge within 10∼15 iterations as Fig. 4(a) shows and it would take about ∼60 hours to finish the whole 40-generation optimization in PGA (64GB RAM, Xeon Silver 4114 CPU). Whether with or without the molecular layer, the EF (|E/E0|2) of TERS probes can be improved over 0.5 orders of magnitude.
Fig. 4. (a) Optimization process under different conditions. PGA optimization process converges within 10-15 generations; (b) electric fields of optimal results by PGA in (a). The molecular layer is 0.5 nm thick with a diameter of 20 nm.
The electric fields of all three optimal structures are shown in Fig. 4(b). The molecules under the tip play an essential role in the performance of TERS experiments. With the same tip-substrate distance, existence of molecules improves the performance of the TERS tip (see the electric field in black and green box Fig. 4(b)). As we widen the nanogap between the tip and substrate, the optimal EF of TERS probes decreases drastically due to weak plasmonic coupling (shown in red and green box in Fig. 4(b)). Table 1 compares the results of the optimized parameters under different conditions. The comparison shows that the optimal conditions for different experiments may change quite differently, illustrating the complexity of designing TERS probes.
Table 1. Optimization results of parameters under different conditions
By our method, the optimization search about ∼3000 combinations of parameters in the four-dimensional parameter space, while ∼10000 (or 104) combinations of parameters are required to be simulated for sweeping 10 steps for each of the four parameters. As we can see in the optimal parameters of the TERS tip in Table 1, the tip radius is about 66 nm, which is the sum of RCoat (coating thickness) and RSi (Si tip radius). Previous work has reported that the tip radius should be 60-75 nm to achieve the best TERS performance, which is consistent with our results [35]. By reasonably limiting the optimizing range of the parameters, the method can obtain the optimal results feasible for fabrication.
In contrast to the initial structure, the optimal tip structure shows a stronger enhancement of the local field We extract the magnitude of the local field in optimal structure (Fig. 5(a) black curve). and the initial one respectively (Fig. 5(b)). By comparing the full width half maximum (FWHM) under each tip, the FWHM of the optimal structure shows 3 nm less than that of the initial one with a larger magnitude of EF. After iterations of the inverse design, the optimal structure evolves with higher performance and better confinement of the electric field than the initial one, which improves the spatial resolution of TERS more effectively.
Fig. 5. (a) The EM field of optimal result without molecular layer and the initial one is shown in the x-z plane (i and ii) and the x-y plane (iii and iv). The initial result is the best individual in the first generation of iterations. (b) The comparison of electric field distribution shows the optimal results with better FWHM. The scale bar in (a) is 50 nm.
Moreover, our method shows the ability to tune the resonance wavelength of the input structure. As Fig. 6 shows, the optimal structure is resonant at 610 nm, close to the incident wavelength of 633 nm. And the initial structure shows the strongest EF at the wavelength of 600 nm, illustrating that our method slightly modifies the resonance wavelength closer to the incident wavelength.
Fig. 6. The EF of the final tip structure without molecular layer excited by the wavelength between 500-800 nm.
Principle component analysis (PCA) is a method to simplify high-dimensional data into two or three dimensions, which are called principal components (PC) [36]. Here the PCA method is applied to analyze the data features in the optimization process. We choose 1000 combinations of parameters before the convergence and convert them from 4-dimensional vectors into 2-dimensional vectors, which are colored points on the plane consisting of two PCs. These points are colored by the normalized EF for recognizing the correlation between their EF and positions.
We apply the PCA method for the first and the last 200 points respectively before the convergence and color the two groups of points by normalized EF. The result shows that at the beginning of optimization, the first 200 points (Fig. 7(a)) locate separately and the points with high EF are embedded in points with low EF. Here, the points disperse with no distinct pattern since the initial population of PGA is randomly generated. However, the last 200 points (Fig. 7(b)) before convergence with high EF assemble at the origin of the plane and show a high density, which indicates that our method selects out the points of high EF. Then, we could see all the 1000 colored points (Fig. 7(c)) before the convergence. Generally, the points with high EF (red ones) gather at the origin of the plane, which are consistent with the features of the last 200 points. The PCA analysis illustrates that the population of PGA evolved and gradually converged to the final optimal results.
Fig. 7. The first 200 (a), last 200 (b), and first 1000 (c) parameter vectors are analyzed by PCA. The points are colored by the normalized EF. (d) Contribution rates of optimized parameters. The data is extracted from the optimization results without the molecular layer (the red line in Fig. 4(a)).
When applying the PCA method, we could also analyze which factor dominates the performance of the TERS probes. Figure 7(d) shows that the incident angle is the most important factor with Au coating thickness followed. These two parameters claim nearly 60 percent of the contribution to the objective function. Additional attention should be paid that the contribution rates are relative and also depend on the optimizing range. If one of the parameters could be only optimized in a narrow range, it may lead to a less contribution rate and have a less weigh on the objective function.
4. Conclusions
In conclusion, our method offers a new perspective on optimizing the performance of TERS probes by alleviating the computational burden brought by the 2.5D method and optimizing the structure parameters via the PGA algorithm. The results show that our method has great advantages in terms of time saving and optimization of performance. Further, the PCA analysis confirms the effectiveness of our method and summarizes the data features, to help researchers identify the key parameters in the probe design. Moreover, our method has great potential in designing both TERS and near-field optical probes, which may be a useful tool for solving design problems in terms of time-saving and performance optimization.
Funding
National Key Research and Development Program of China (2021YFA1201502); National Natural Science Foundation of China (21727807, 22272140).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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