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Applications of LSPR nano-particles in various areas of solar cells, LSPR biosensors, and SERS biosensors, based on interaction of light with noble metal nano-particles is increasing. Therefore, design and nano-fabrication of the LSPR devices is a key step in developing such applications. Design of nano-structures with desirable spectral properties using numerical techniques such as finite difference time domain (FDTD) is the first step in this work. A new structure called nano-sinusoid, satisfying the some desirable LSPR characteristics, is designed and simulated using the FDTD method. In the next stage, analytical method of electro static eigen mode method is used to validate the simulation results. The, nano-fabrications method of electron beam lithography (EBL) is implemented to fabricate the proposed profile with high precision. Finally, atomic force microscopy (AFM) is used to investigate the shape of the fabricated nano-particles, and the dark field microscopy is employed to demonstrate the particular spectral characteristics of the proposed nano-sinusoids.
© 2014 Optical Society of America
1. Introduction
When a light beam travels from a high refractive index medium to a low refractive index medium, an evanescent electric field wave, which is called surface plasmon, is generated on the interface surface of the two media, and mostly penetrated into the low refractive index medium. Coating the interface with a layer of a noble metal such as gold or silver highly enhances surface plasmon polariton (SPP), which propagates on the surface of the noble metal in contact with the low refractive index medium. Using noble metal nano-particles (NPs) instead of the metal film localizes the surface plasmon around the NPs surface, which is called localized surface plasmon resonance (LSPR). Compared to the SPP, the LSPR devices have advantages such as real time response, parallel monitoring of multiple species to decrease measurement time and sample volume, and increasing detection accuracy [1–4].
The LSPR behaviour is characterized using parameters such as extinction/scattering cross section, resonance wavelength, and spectrum sharpness, which is interpreted by the full width at half maximum (FWHM) parameter. The electromagnetic Maxwell’s equations are solved to determine the LSPR behaviour. In this work, the equations are numerically solved by means of the finite-difference time-domain (FDTD) [5]. The analytical method of electrostatic eigenmode method [6, 7] formulates the plasmon resonance wavelength of a single NP by:
where is the LSPR resonance wavelength, is the permittivity of the metal at the resonance wavelength, is the dielectric constant of the background medium, and is the eigenvalue of the single NP. For example, Eq. (1) gives the dipolar resonance condition of for nano-spheres in dipolar mode and within a limited range of size.For a set of particles, the interaction between the NPs results in a shift in the LSPR wavelength from to due to the coupling among particles. For interacting NPs, a coupling matrix is defined as [8]:
where is the coupling constant resulted from the interaction of NP on NP i, and are excitation amplitudes for NP i with and without interaction of other NPs, respectively, and is eigenvalue of the NPs ensemble.Usually the interactions between NPs are considered as symmetric (corresponding to the case of symmetric medium) and only between adjacent NPs, i.e.
In this case, the coupling constant c is calculated as follows:
According to Eq. (5), there are many parameters affecting the LSPR characteristics including shape, size, structure of the NPs, and the light wave polarisation [9, 10]. In addition, the effect of substrate and media on the plasmonic characteristics have been previously investigated by the authors [11, 12]. The authors have recently presented a new nano-particle shape, which is called nano-sinusoids [13] (Fig. 1), and formulated by:where is the amplitude of the sinusoid at the position along the dipole main axis, is the modulation constant, i.e. the in-plane width, and w is the overall in-plane width of the dipole.The cutoff frequency of this NP can be controlled by the modulation factor , where decreasing results in a lower cutoff frequency.
In this paper, the results of modelling and fabrication of the nano-sinusoids are compared with two existing sharp-tip nano-particles: nano-triangles [14] and nano-diamonds [15] (see Fig. 2).
Fig. 2 (a) AFM image of nano-triangles fabricated using Nanosphere lithography (NSL) method [14], (b) SEM image of nano-diamonds fabricated using self-assembly monolayer process [15].
2. Fabrication and characterization
The proposed nano-sinusoid NPs are fabricated using nano-fabrication devices, and the results are utilized to verify the simulation results published by the authors in [13, 16].
The first step in any experimental development is to design the device. Generally, the device geometry and dimensions need to be defined precisely. Thus, the pattern to be exposed is designed by CAD software such as AutoCAD [17]. The nano-fabrication process is shown in Fig. 3.
First stage: In the first stage, the glass wafer is fully washed in aceton and PA2 solutions to remove any dusts or oily substances.
Second stage: A spin coater is used to coat the substrate (e.g. glass or silicon) with a photoresist material. The proper photoresist concentration is set according to the thickness of the resist that needs to be coated on the surface of the glass wafer. This thickness should be about four times the thickness of the noble metal that is going to be deposited on the surface of the wafer. For example, if the photoresist thickness needs to be 120 nm, the PMMA (poly-methyl methacrylate) type of A2 is selected based on the material graph.
Third and Fourth stages: In the electron beam lithography (EBL) process, a highly focused electron beam exposes the resist to print any two-dimensional nano-structures. Inelastic collisions of electrons with the photoresist material result in photoresist ionization. When a typical positive long chain polymer photoresist, like PMMA, is exposed to the electron beams, it is broken into smaller and more soluble fragments. In this experiment, the designed pattern is printed on the glass wafer using the EBL system which is EBPG5000 from Vistec Company. The writing, main field, and subfield resolutions are taken as 1 nm, 1 nm, and 0.5 nm, respectively. The beam current is 200 pa, aperture size is 200 µm, and electron dosage is from 800 to 1550 µc/m2. Electric charges are accumulated on top of the PMMA layer, when they are exposed with electrons during the EBL exposure, which deflects the beam and distorts the pattern. To resolve this problem, a thin layer of light metal (e.g. 5 nm) such as Chromium is deposited either on top of the PMMA layer or between the PMMA layer and the substrate to have electrical grounding and electron reflection features.
Fifth stage: The Cr layer is etched using Chromium etching solution. The wafer is put in a dish of Chromium etchants and shaken for around 15 minutes to dissolve the Chromium layer. Mixtures of percloric acid and ceric ammonium that are diluted in water are used as Chromium etchants.
Sixth stage: Wafer development means washing out the part of the PMMA layer on top of the wafer which has been affected through EBL. Through this process, the binding of the part of the PMMA layer exposed by electrons is weakened, and thus can be easily removed. This development is done using a developer like MIBK2 for 1 min. IPA2 is used to rinse the wafer, and it is then dried by nitrogen.
Seventh stage: Due to poor adhesion of the gold layer on the glass substrate, a Chromium adhesion layer (2 nm thick) is deposited on the substrate prior to the gold evaporation by using an E-beam evaporator instrument.
Eighth stage: The E-beam evaporator is used again to deposit gold on the surface of the developed wafer with the required thickness (20 nm in this experiment).
Ninth stage: Lift-off is performed in the last stage of the fabrication process. The patterned PMMA is removed from the surface to have nano-structures on the surface of the wafer as designed. The lift-off is done with IPA2 solution.
An AFM system is used to acquire images from our fabricated NPs in ambient conditions. The AFM is used in tapping mode. The tip shape is conical and it has resonance frequencies in between 280 and 320 kHz. The images are then analysed by using the AFM analysis software, to draw the thickness of the particles for each point. The AFM used is from Veeco Dimension Icon.
A Nikon Eclipse TE-2000 dark field microscopy is also utilized to collect the scattering spectra. It is fabricated by a dark-field condenser and a Nikon Plan Fluor ELWD objective, which is focused onto a MicroSpec 2150i imaging spectrometer and coupled with a TE-cooled PIXIS 1024 ACTON Princeton Instruments CCD camera.
3. Statistical model prediction
Response surface method (RSM) is a statistical method used to relate input and output data set of experimental or modelling results by a mathematical formula. The formula can be interpreted by means of the regression model, as where is the input data set, is the output data set, and is the experimental error term resulting from the estimation of the output . In a well-estimated model, the error term must be a normal and independent variable with the mean value of zero. The analysis of variance (ANOVA) method is employed to estimate the best model fitted to the input/output data sets [18].
Once the best model is found by the RSM, a nano-linear programming (NLP) optimizer is utilized to find the best set of morphological parameters for the NPs which presents desirable plasmonic features such as resonance wavelength and FWHM values, while offering the maximum scattering cross section as the objective function. Therefore, to the scalar objective function is maximized subject to constraints on the allowable :
subject to: where x is an element of the input date set, and are vectors of lower and upper bound values of the input data x, and and are the nonlinear constrains vectors on the input data vector x, respectively [19]. In this work, the NLP method is used based on the solution of the Karush-Kuhn-Tucker (KKT) equations [20].The input data set is defined as:
and the optimium LSPR is attaiend when:subject to the input data bounds: and the nonlinear constraints:In this work, the constraints are considered as follows:
As shown in Fig. 4, the proposed procedure for modelling and optimization of the LSPR biosensors, shown in Fig. 2, includes the following steps:
- 1. Design of Experiments/Simulations: To find a mathematical model that fits the LSPR behaviour with the least residual error, first we design the best input data set using the RSM. This step specifies the input values of the required simulations (or experiments) as stated in the first four columns of Table 1. The total number of data sets and the values of the data set elements are calculated by the RSM to have an efficient model. The optimum number of data sets suggested by the RSM is.
- 2. Electromagnetic Modelling: The FDTD method simulates Maxwell’s equations on the LSPR structure using the input data set in Table 1, generated in Step 1, row by row. The outputs of this electromagnetic modelling including extinction cross section, resonance wavelength, and FWHM values are inserted in the second three columns of Table 1.
- 3. Mathematical Modelling: The RSM builds statistical models on the input/output data sets denoted in Table 1, generated from the last two steps. This model gives three mathematical equations for each output data set as a quadratic polynomial function of all input data set.
- 4. Optimization: The extinction cross section formula attained from Step 3 is given to the NLP optimizer as the objective function, while the other two formulas (resonance wavelength and FWHM) are given to the optimizer as nonlinear constraints. In addition, the lower and the upper bounds of the input data set are defined for the optimizer. The optimizer is asked to find the best geometrical parameters for the LSPR biosensor with a specified range of resonance wavelength and FWHM values.
- 5. Mathematical Model Matching: The optimized parameters are evaluated using the RSM. Furthermore, the desirability of the attained input data is calculated by the RSM, which specifies how fit the data is to the mathematical model calculated by the RSM.
- 6. Experimental Evaluation: Finally, the optimized input/output data sets are evaluated using the experimental results.
Table 1. Input/output data sets.
4. Discussion and results
4.1. Simulation results
The interaction of a source light with a silver NP over glass substrate () is investigated using the finite-difference time-domain (FDTD) method [5], in air (). The NPs permittivity is matched with a fourth order Drude-Lorentz model. Since the silver NP is subject to changes when exposed to water, acids, halides, and air, the NP should be covered by a protecting layer over a reasonably long period of time [11, 21].
Simulation results demonstrate an enhanced plasmon spectrum for nano-sinusoids than the other sharp tip nano-structures like nano-triangles and nano-diamonds. Cross sections of silver nano-triangles, nano-diamonds, and nano-sinusoids with in-plane width versus wavelength, are compared in Fig. 5.Due to less multipoles excitation in nano-sinusoids compared to nano-triangles and nano-diamonds, smoother LSPR spectrum of nano-sinusoids is observe in this figure. As can be seen in the figures, the scattering and the extinction cross sections increases with increasing the size of the NPs, and at the same time, the plasmon wavelength red-shifts. These results conform findings reported in [22, 23].
Fig. 5 Extinction (blue), scattering (red), and absorption (green) cross sections for (a) silver nano-triangles of in-plane width 80 nm, (b) silver nano-diamonds of in-plane width 80 nm, and (c) silver nano-sinusoids of in-plane width 80 nm versus wavelength.
According to the simulation results, the spectra of nano-sinusoids is compared with that of nano-diamond and nano-triangles in terms of the following aspects:
- a. The LSPR spectrum of nano-sinusoids are smoother than that of nano-triangles and nano-diamonds, because less multipoles are excited in nano-sinusoids than others. Full width at half maximum (FWHM) factor is a measure to compare the sharpness of the spectra together. A comparison of the three nano-structures (see Fig. 6(a)) indicates an extremely sharper plasmon spectrum for nano-sinuoids than nano-triangles in the visible wavelength range less than around 615 nm; however, for higher wavelengths, nano-triangles present a sharper spectrum [16].
- b. Figure 6(b) demonstrates more extinction and scattering cross sections in nano-sinusoids than nano-triangles and nano-diamonds.
- c. As shown in Fig. 6(c), nano-sinusoids present more linearity of the cross sections versus wavelength than the other two nano-structures, which make them an excellent LSPR resonator in a wider range of size and frequency.
- d. Figure 6(d) demonstrates stronger coupling between NPs in the case of the nano-sinusoids than the other two sharp tip NPs [16, 24].
Fig. 6 Comparison of nano-sinusoids (purple colour) with nano-triangles (red-colour) and nano-diamonds (green colour) in terms of: (a) FWHM, (b) extinction cross section, (c) optimum in-plane width versus plasmon wavelengths, and (d) coupling constant for two NPs of 80 nm in-plane size versus spacing between NPs [16].
The above-mentioned simulation results will be validated through characterizing the fabricated nano-structure as described in the next section.
4.2. Fabrication results
The FDTD method [5] has been used to model the nano-sinusoid NPs. In the simulations, the incident light is taken as a total field scattered field (TFSF) source [5] on z-direction with polarization along y-axis, which is along the NPs chain axis, to guarantee the maximum enhancement [25] over the optical wavelength ranging from 300 to 900 nm. To verify the modeling results, gold nano-structures with varible physical dimensions were also fabricated on a glass wafer using the EBL, and the E-beam evaporator, as explained in the previous section.
Since a glass wafer that is not conductive was used in this research, capturing SEM images was not possible. Therefore, to investigate the shape of the fabricated nano-particles, AFM imaging was used. The shape of the nano-particles imaged using the AFM is shown in Fig. 7(a) for a single nano-sinosoid NP, and in Fig. 7(b) for a dimer of nano-sinosoid NPs with spacing of 35 nm. The in-plane width of the Nps is 200 nm and their height is around 20 nm.
Fig. 7 AFM images of (a) single nano-sinusoid NP, (b) dimensions of the single nano-sinusoid, NP (c) double nano-sinusoid NPs, (d) dimensions of the double nano-sinusoid NPs.
Figures 7(c) and 7(d) show the in-plane width and height of the single and double NPs shown in Figs. 7(a) and 7(b), respectively.
Rayleigh scattering spectrum of a single nano-sinusoid NP with variable in-plane width (from 140 nm to 200 nm) and height of 20 nm was chracterized using dark field microscopy, as shown in Fig. 8(a).Figures 8(b) and 8(c) compare the scattering spectra of a single NP with double and triple NPs with in-plane width of 140 nm and height of 20 nm, for various spacings of 35 nm and 50 nm, respectively. Figure 8(d) shows the effect of NPs in a chain on the LSPR spectrum for multiple NPs with in-plane width of 140 nm and height of 20 nm.
Fig. 8 Scattering spectrum of (a) a single nano-sinusoid NP with various width of 140, 160, 180and 200 nm; (b) double nano-sinusoid NPS with width of 140 nm and spacing of 35 and 50 nm; (c) triple nano-sinusoid NPs with width of 140 nm and spacing of 35 and 50 nm; (d) multiple nano-sinusoid NPs with width of 140 nm and spacing of 50 nm; caught by dark fild microscopy.
Figure 9(a) demonstrates the linear variation of the dipolar resonance wavelength of the single nano-sinusoids against the size of the NPs. As expected from the simulation results, the resonance wavelength is linearly proportional to the size of the nano-sinusoids, while for other sharp NPs such a relationship is not observed. For example, the resonance wavelength variation of triangular nano-prisms synthesised from seed particles developed by Jin et al. [26], shown in Fig. 9(b), proves more linear relationship between the resonance wavelength and the size of the nano-sinusoids than the nano-triangles.
Fig. 9 Dipolar resonance wavelength of (a) single nano-sinusoids against their size for real data (blue) and simulated data(red), (b) synthesised nano-triangles by Jin et al. [26].
In addition, Fig. 10 shows red-shifting of the dipolar resonance wavelength with increasing length of the particle chains. Also, it demonstrates that by increasing the number of particles in a chain, the resonance wavelength gradually reaches a limit.
Fig. 10 Multi nano-sinusoids against chain length for in-plane width of 140 nm and spacing of 50 nm (blue), width of 140 nm and spacing of 35 nm (orange), width of 180 nm and spacing of 50 nm (green), and width of 180 nm and spacing of 35 nm (brown).
According to the presented results, it is possible to design our structure with any size and shape using EBL. The capability of EBL in fabrication of nano-sinusoids with different in-plane or modulation factor values has been demonstrated.
5. Optimization results
A 3D representation of the three stated plasmon factors versus each pair of the input parameters (thickness, in-plane width) and (spacing, chain_no) are shown in Figs. 11, 12, and 13.
Fig. 11 3D representation of the FWHM variations against (a) (thickness, in-plane width) for spacing of 33 nm and chain_no of 2, and (b) (spacing, chain_no) for thickness of 35 nm and in-plane width of 138 nm.
Fig. 12 3D representation of the resonance wavelength variations against (a) (thickness, in-plane width) for spacing of 29 nm and chain_no of 3, and (b) (spacing, chain_no) for thickness of 13 nm and in-plane width of 126 nm.
Fig. 13 3D representation of the extinction cross section variations against (a) (thickness, in-plane width) for spacing of 31 nm and chain_no of 3, and (b) (spacing, chain_no) for thickness of 20 nm and in-plane width of 111 nm. The value 1000 m2 shows that the vertical axes of the graphs have been down scaled by 1000. The ECS values actually range from around 50,000 to 300,000 m2.
Figure 11 demonstrates that the FWHM is reduced by decreasing the thickness or the in-plane width of the nano-sinusoids, increasing the NPs spacing, or shortening the length of the NPs chain. However, this reduction is more pronounced for thinner or smaller NPs, and is also less sensitive to the length of the longer chains (e.g. chain_no > = 4).
Figure 12 illustrates how the resonance wavelength is red-shifted by decreasing the thickness or increasing the in-plane width of the nano-sinusoids, decreasing the NPs spacing, or lengthening the NPs chain. However, this drop is more pronounced for thinner NPs or closer spacing.
Figure 13 shows how the extinction cross section almost linearly is reduced by increasing the thickness or decreasing the in-plane width of the nano-sinusoids, increasing the NPs spacing, or decreasing the length of the NPs chain.
The trends explained above obey the analytical electrostatic eigenmode method formula. This method explains how the interaction between multi-NPs grows while a longer chain of NPs is utilized, the NPs in the chain approach each other, or the in-plane width of the NPs increases. In these cases, the coupling constant in Eq. (5) increases which in turn causes more red-shift of the plasmon resonance and wider plasmon spectrum (larger FWHM) due to the multipoles interference. However, as observed from Eq. (5), as the chain_no parameter increases, the effect of the quantity on the plasmon spectrum fades.
These graphs show that the resonance wavelength and the FWHM responses have a curved surface shape, while the extinction cross section response has a smooth surface shape. This difference can be due to the formulation of the three output parameters as a function of the four independent factors. Therefore, the quadratic approximation of the extinction cross section response can be simplified using a first order polynomial approximation. Having a look at the extinction cross section formula, many insignificant terms related to its nonlinear formulation is observed.
Interpreting the extinction cross section as a first order polynomial gives the following formula. This linear estimation decreases the F-value from 939.02 to 119.23 which is still significant.
The effectiveness of this method in finding the optimized physical dimension of the nano-sinusoids on 10 randomly selected target values of [Resonance wavelength, FWHM] is demonstrated in Fig. 14.The figure compares how close are the resonance wavelength and the FWHM of the RSM and fabricated NPs with the target values. In these calculations, the extinction cross section, which is the objective function subject to the bounds and constraints, is maximized by the NLP optimizer. Putting these data in the RSM gives the desirability value of 1 which shows 100% validity of the results on the predicted model. However, solving the optimization problem directly using the RSM is impossible because matching of the input/output data sets was not achieved for the particular data sets.
Fig. 14 (a) Resonance wavelength and (b) FWHM calculated by FDTD (green), RSM (blue) modelling, and fabricated NPs (red) for an optimized set of physical dimensions for given target wavelength and FWHM vales for single nano-particles. The reference line is shown in green colour.
6. Conclusion
An investigation on the spectral properties and the scattering characteristics of three different NPs in the visible frequency range was carried out. The numerical analysis demonstrates the advantages of nano-diamonds and nano-sinusoids with respect to nano-triangles, since the first two NPs present two hot spots at the two end vertices, a wider range of applicable plasmon wavelength, and higher electric field enhancement for the same resonant wavelengths. Furthermore, increasing the NPs size or aspect ratio incurs more dipolar and less multipolar extinction and scattering cross sections, besides shifting the dipolar resonance towards red light. Moreover, nano-triangles and with a higher extent nano-diamonds exhibit more non-linearity in their plasmon spectrum-size relationship because of their higher singularities due to many sharp edges, which makes the engineering of their plasmon wavelength uncontrollable. The linear relationship is very essential in solving the plasmon interactions between the NPs and tuning the plasmon wavelengths for applications such as SERS biosensing using the electrostatic eigenmode interaction method. The nano-sinusoids shape introduced in this paper possesses such characteristics, making it a valid alternative to other NP shapes. In addition, it was analytically demonstrated that the surface plasmon excitations have a lower bound cutoff at a finite frequency on the singular structures; in addition, the electric field diverges below a critical frequency even when metallic losses are considered.
Furthermore, it was shown that the nano-sinusoids exhibit higher coupling constants than the nano-triangles and the nano-diamonds at the same resonance frequencies. Thus, more enhanced surface electric fields are created at the hot spots of the nano-sinusoids than the other two NP shapes. Moreover, the closer the NPs are the higher coupling constant (i.e. the more enhanced surface electric fields at the hot spots) will be achieved. In addition, the shorter chain of NPs presents the higher coupling constants at the same resonance wavelength; while, less red-shift of the resonance wavelength and sharper LSPR spectrum is attained for the NPs which are close enough in a chain. However, if the chain length increases beyond almost more than five NPs per chain, the coupling constants and the resonance wavelengths will not be changed significantly for the NPs that are far enough from each other. The 2D simulation outputs also result in the generation of high multipoles over the whole NPs and dipolar only in hot spots of the outer layers.
Furthermore, the interaction of NPs in terms of the LSPR phenomenon based on the nano-sinusoids shape was modelled by means of the RSM. The extracted formulas can be employed to predict the plasmon response of a particular structure before any modelling or fabrication. A NLP procedure is applied on these formulas to optimally design a LSPR structure maximizing the extinction cross section function while maintaining a desirable constrain on the two other output data (i.e. resonance wavelength and FWHM) on all input data. The results of applying the NLP optimization on the model predicted by the RSM nearly matched with those of the FDTD simulation.
Acknowledgments
We acknowledge assistance of Professor Paul Mulvany and Dr Xing Zhan of Melbourne University by allowing us to access their dark field microscopy instrument. We also appreciate Melbourne Centre for Nanofabrication’s technical staff for helping us with the utilization of nanofabrication devices.
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