1. Introduction

More attention has been paid on developing nanolithography to fabricate nanoscale devices for nanotechnology applications. Photolithography has remained as a useful technology because of easy repetition and suitability for large-area fabrication [1]. In order to overcome the diffraction limit [2], optical contact or near-field lithography, such as contact printing through a transmission mask or a binary phase-shift mask [3, 4], evanescent near-field lithography[5, 6], plasmon printing [7], and the use of light-coupling masks [8], have been successfully demonstrated to produce the sub-100-nm features. Recently, a new scheme of surface-plasmon-assisted nanolithography (SPAN) has been proposed [10, 11, 12, 13]. The interference field of SPs can double the spatial frequency of the lithography pattern, which effectively increases the resolution of lithography pattern [10]. Moreover, the near-field enhancement of intensity and large wave-vector characteristics of surface plasmons (SPs) [9] can effectively decrease the exposure time and the linewidth of lithography pattern. Furthermore, researchers have found that SPs on some metallic films can regenerate the evanescent field and focus the near-field image with a resolution far beyond diffraction limit [14, 15]. Melville et al. have investigated and confirmed experimentally the submicron imaging with a planar silver lens citeMelville.04, Melville.05, which motivated the combination of the SPs effects on both perforated metal film and planar metal film for nanolithography. They also have simulated to show that SPs on an underlying silver layer can be used to improve process latitude and depth of field [18]. Recently, Shao and Chen have experimental demonstrated the similar structure, where a polarized laser beam of 355nm wavelength was used as light source to photoinitiate an 80nm thick photoresist on a silicon substrate coated with titanium of 80nm thickness [19].

The main focus of SPAN is utilizing the larger wave vector of SPs compared with the light at the same frequency. It is known that SPs will split into two modes, the high-frequency coupled SPs (HFSPs) and the low-frequency coupled SPs (LFSPs), in the thin metal layer structure [20]. It should be pointed out that the LFSPs has larger wave-vector than the one of SPs on the semi-infinite metal surface, which indicates that it is a better candidate for the nanolithography. A novel resonant structure named metal-based grating waveguide structure (GWS) has been used to couple the incident light for the high-frequency coupled mode [22], which is also effective for the low-frequency one.

From metallic waveguide theory, the coupled SPs can be solved as symmetric (high-frequency) and antisymmetric (low-frequency) modes for waves guided by a metallic film [21]. In this letter, we use the metallic waveguide theory to design the subwavelength metallic grating waveguide structure (MGWS) that will excite appropriate coupled modes to give the high-resolution, spatial-frequency doubled lithography pattern. By comparing with the conventional SPAN, two potential lithography schemes will be carried out to improve the performance of the lithography patterns.

2. Simulation method and model

Two-dimensional Finite Difference Time Domain (FDTD) method [23] has been used in this work, because of its computational advantages of reducing memory requirement and ease in treating complex materials and shapes etc. Both the spatial area and the time interval are discretized. The Maxwell’s equations in differential form can be solved directly and the iterative expressions have been gained. The material can be represented in the mesh of the square cells. In our simulation, the cell size about 1nm is used. Time steps of less than 0.002 f s are used in order to satisfy the FDTD stability criteria. The perfectly matched layer(PML) with the width of 500nm has been used in the boundaries perpendicular to the propagating direction of the light [24]. The periodic boundary conditions have been used in the other boundaries. The Drude dispersion model is used to simulate the metallic film, whose expression is shown as follows,

$\epsilon \left(\omega \right)={\epsilon }_{\infty }+\frac{{\omega }_p^2}{2i\omega \nu -{\omega }^2}$

where ε _∞ is the relative permittivity at infinite frequency, ω_p the plasma frequency, and n the collision frequency. Here we set ε_∞=1. Other parameters can be automatically calculated in the simulation by the dispersion expression after specifying the permittivity of metal.

 

Fig. 1. Schematic diagram illustrating the MGWS and illumination conditions used for the simulations.

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The system geometry depicted in Fig. 1 shows that a MGWS suspended in a uniform dielectric medium. This structure is comprised of a subwavelength silver grating and a silver layer. The grating period and thickness are denoted by p and t₁, respectively. The width of slit is denoted by w. The thickness of the silver layer is denoted by t₂. The spacing between the grating and the layer is denoted by s. The TM-polarized plane wave is normally incident on the structure from the top down. The situation shown in Fig. 1 represents the case in which a photoresist layer in the near field of the structure has index matched to the material surrounding theMGWS, which can be fulfilled by choosing experimental conditions. In our simulation, the wavelength of incident light is 436nm. The permittivity of surrounding medium is ε=2.56 corresponding to typical photoresist materials at blue and UV wavelengths. The frequency-dependent permittivity of silver is ε_s=-6.1063+i0.2699 at this wavelength, which is incorporated by use of a second-order polynomial fitted to tabulated data over the 300–700nm range[25].

Two lithography schemes withMGWS will be classified by the different regions in the structure where the waveguide mode is excited, which can be named metal-layer and metal-cladding schemes, respectively. The characteristics of lithography pattern involving visibility, resolution and the influence of deviated grating period will be numerically analyzed in the following text.

3. Metal-layer lithography scheme with MGWS

In this scheme, the silver layer is treated as a planar waveguide, while the silver grating serves as a coupling structure to excite the appropriate mode on the silver layer. So it can be named metal-layer lithography scheme. From the waveguide theory, the mode of metal layer can be expressed by the coupling equations shown as follows,

where k ₀ is the value of the wave vector of illumination light, k the value of the wave vector of coupled mode. ε ₁, ε ₂ and ε ₃ denote the values of permittivity in the region above, inside and below the silver layer, respectively (here ε ₁=ε ₃=2.56). The thickness of silver layer t ₂ determines the wave vector of coupled mode, and it should be small enough to ensure the SPs modes on the either surface can interact each other, which is the foundation of generating the coupled modes. Considering the absorption of silver, we set t ₂=40nm. From Eq. (1), two values of k can be solved as k ₁=1.8451k ₀ and k ₂=2.5003k ₀, which correspond to the HFSPs (symmetric mode) and LFSPs (antisymmetic mode) solutions, respectively. According to the wave-vector matching condition, 2π/p=k, the period of the silver grating can be solved as p ₁=236nm and p ₂=174nm. The width of slit is set w=30nm, which is small enough to weaken the influence of slits on the SPs’ dispersion relation of smooth metal film.

Firstly, the typical SPAN structures, namely the single metallic gratings, supporting HFSPs (p ₁=236nm) and LFSPs (p ₂=174nm) have been simulated by FDTD method, respectively. The average distributions of E-field intensity of two modes are shown in Figs. 2(a) and (b). One can see that the distribution of (a) is the typical pattern of SPs interference on the grating surface, which is used to generate the spatial-frequency doubled lithography field in previous work [10]. The image of Fig. 2(b) shows the same distribution near the output surface, but the pattern disappears rapidly with increasing output distance (about 10nm), which is almost impossible to be used in industry. That is why the interference pattern of LFSPs has not been used for nanolithography yet. We think that the interference pattern is submerged rapidly by the background intensity because of the high attenuation characteristics of output pattern. So the performance of LFSPs’ interference pattern may be improved through decreasing the background signal.

The metal-layer scheme with MGWS has been designed to confirm our thought. The FDTD method has also been performed to get the average distributions of E-field intensity of this scheme with two periods p ₁=236nm and p ₂=174nm corresponding to the coupling of HFSPs and LFSPs modes on the silver layer, which are shown in Figs. 3(a) and (b), respectively. In the simulation, the thickness of grating is equal to the metal layer, i.e., t ₁=t ₂=40nm, and the spacing between the grating and layer s is chosen as 60nm. One can see that two different coupled SPs modes have been excited on the silver layers, while there is no coupled mode existing in the region between the grating and the layer. The distributions on the silver layer both perform the profiles of standing wave with doubled spatial frequency.

 

Fig. 2. Near field intensity profiles of electric field of (a) HFSPs (symmetric mode, p ₁=236nm) and (b) LFSPs (antisymmetric mode, p ₂=174nm) in single silver grating illuminated at 436nm. The other parameters are t=40nm and w=30nm. The lateral scale is two periods for (a) and three periods for (b) for convenience of compare.

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Fig. 3. Near field intensity profiles of electric field for the MGWS of metal-layer scheme supporting (a) the HFSPs mode (p ₁=236nm) and (b) the LFSPs mode (p ₂=174nm) illuminated at 436nm. The other parameters are t ₁=t ₂=40nm, w=30nm and s=60nm. The lateral scale is two periods for (a) and three periods for (b) for convenience of compare.

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It should be emphasized that there is another mode supported by this structure. It is the non-coupled SPs on the bottom surface of silver layer. According to the dispersion of SPs on the semi-infinity metal surface [9], the value of SPs’ wave-vector is k_sp=2.0995k ₀, from which the grating period can be deduced p=208nm. The average distribution of E-field intensity in this situation is shown in Fig. 4. One can see that the SPs mode excited by the silver grating only exists on the bottom surface of silver film, there is no coupled mode excited on the silver layer.

 

Fig. 4. Near field intensity profile of electric field for the MGWS of metal-layer scheme supporting SPs mode on the output surface. The parameters are p=208nm, t=40nm and w=30nm. The lateral scale is two periods.

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Fig. 5. The average intensity profiles of electric field along the lateral direction y at different distance x beneath the output surface of (A) typical SPAN structure and (B) MGWS of metal-layer scheme supporting HFSPs. The lateral scale is two periods.

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The influence of silver layer can been represented clearly through comparing the average distributions of E-field intensity along the lateral direction y at different distance x beneath the output surface of (A) typical SAPN structure and (B)MGWS of metal-layer scheme supporting HFSPs, which is shown in Fig. 5. The output surface is the bottom surface of silver layer for the metal-layer scheme or of the silver grating for the SPAN structure. The output patterns of two structures present the same profiles with doubled spatial frequency. However the distinct phenomenon is performed on the evolution of the shallow valleys. The depth of the shallow valley of structure (A) decreases with increasing the distance x, while the one of structure (B) is almost constant. We deem the difference is due to the different formation mechanism of output pattern. In SPAN structure, it is formed by the multiple scattering of array of slits. So the standing field formed by the interference of LFSPs has not only the wave vector of LFSPs, but also small amount of other components in the near field, and then the attenuation speed of field on the output surface is different, which leads to the gradual combination of two peaks with increasing the output distance. In contrast, the output pattern of metal-layer scheme is formed by the coupling effect. Here the silver layer acts as a filter, which blocks all the wave-vector “noise” except the one of coupled SPs. So the pattern almost keeps uniformity at different distance beneath the output surface. The profile and evolution of output pattern generated by MGWS supporting LFSPs or SPs are similar to that of HFSPs because of the same formation mechanism.

Although the influence of slits has been blocked by the silver layer in this scheme, the approximation in our analysis can also lead to the disorder of the output pattern. The grating structure has been ignored in our calculation of resonant modes of metal-layer scheme. which is correct in the condition of weak coupling. However in the structure with subwavelength scales, the negligible deviation of calculated grating period from the resonant one induced by the approximation makes the obviously influence on the performance of output pattern. Furthermore, the background field directly passing through the thin silver film also affects the output pattern. In order to analyze the performance of output pattern, the image visibility V is defined as V=(I_max-I_min)/(I_max+I_min), where I_max is the maximum intensity, I_min is the intensity of dip of the shallow valley. The visibility profiles of output field at various distances x beneath the output surface of metal-layer scheme supporting different modes (HFSPs, LFSPs and SPs) and of SPAN are described in Fig. 6. The profile of SPAN (blue line) monotone decreases with increasing the distance due to the wave-vector noise generated by the slits. In contrast, the visibility profiles of metal-layer scheme supporting coupled SPs modes (HFSPs, black line

and LFSPs, green line) both present a peak, which is due to the influence of the deviation of resonant wave-vector. The small deviation slowly detaches or combines the peaks shown in Fig. 5(b) with increasing the output distance x. Furthermore, the background field disorders the pattern obviously in the large distance, which induces the asymmetry of visibility curves. The visibility of structure supporting LFSPs reaches the peak more rapidly (~40nm) than the one supporting HFSPs (~80nm) because of the larger deviation in the structure supporting LFSPs with smaller parameters, and the higher attenuation of LFSPs with larger wave-vector along the direction x. The situation of structure supporting SPs (red line) is different from the others. The profile is almost flat with increasing the distance x, which indicates the excellent uniformity of its output pattern. In this situation, the deviation of wave-vector is the smallest, because the mode is the exact solution solved by the dispersion of non-coupled SPs on the metal surface. But the weak influence of background field still makes the small bend of profile. The optimal resolutions of this scheme supporting different modes are ~58nm for HFSPs, ~50nm for SPs and ~43nm for LFSPs, respectively. The output pattern of LFSPs with larger wave-vector has higher resolution but lower visibility, in contrast the one of HFSPs has better visibility but lower resolution. The output pattern of SPs has appropriate resolution and visibility. Furthermore, it has better tolerance of exposed distance (~100nm), which makes it more suitable for the practical nanolithography.

 

Fig. 6. Image visibility V as a function of distance x beneath the output surface of MGWS of metal-layer scheme supporting different modes (HFSPs p=236nm, SPs p=208nm and LFSPs p=174nm) and typical SPAN.

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In order to consider the range of grating period being usefully patterned in this scheme, the patterns in non-resonant conditions are numerically analyzed. The influence can be analyzed by considering the profiles of visibility V in non-resonant condition (p=240nm, p=222nm p=190nm and p=170nm) shown in Fig. 7. It should be noted that two different situations should be classified in the non-resonant conditions. One is the period between two resonant modes, such as p=222nm (between HFSPs and SPs modes, red line) and p=190nm (between SPs and LFSPs modes, green line). The output patterns in this situation still keep useful for the lithography compared with the resonant ones shown in Fig. 6. The other is the period deviated from the resonant mode, such as p=240nm and p=170nm. In this situation the performance of output pattern becomes worse due to the drop of the coupling efficiency, which should be avoided in practical lithography. So the effectively grating period of metal-layer scheme can be changed from 240nm to 170nm because there are three usefully resonant modes supporting in this scheme.

 

Fig. 7. Image visibility V as a function of distance x beneath the output surface of MGWS of metal-layer scheme in non-resonant condition. (p=240nm, p=222nm, p=190nm and p=170nm))

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Through properly designing of grating period, the metal-layer lithography scheme with MGWS can excite three different modes (HFSPs, SPs and LFSPs) on the metal layer to generate the spatial-frequency doubled lithography patterns with different performance. Especially the structure supporting SPs mode effectively improves the tolerance of exposed distance, as well as generates the lithography pattern with high resolution and visibility. The different modes can be used in the different practical conditions and for the different purposes.

4. Metal-cladding lithography scheme with MGWS

In this analysis, MGWS has been treated as a multi-layer planar waveguide. The LFSPs mode excited in the silver cladding region, i.e., the region between the grating and the layer, will be used in this scheme. So it can be named metal-cladding lithography scheme. The multi-layer coupled effect has been considered to calculate the coupled mode. The same width of slit as that of metal-layer scheme, w=30nm, is chosen. The thickness of silver grating t ₁ is supposed to be equal to the one of silver layer t ₂, so the LFSPs mode in the silver cladding region can be solved as the coupled mode of the symmetric five-layer waveguide structure according to the waveguide theory. The thin-film transfer matrix method will be used to calculate the modes in this equivalent model [26]. The coupling equations of LFSPs mode in this scheme can be deduced as follows:

where k is the value of the wave vector of coupled mode, i.e., LFSPs mode. ε ₁ denotes the value of permittivity of metal, ε ₂ and ε ₃ denote the values of permittivity in the region outside and inside of metal-cladding region, respectively. In this calculation, the thickness of grating and layer are set as t ₁=t ₂=40nm. The spacing between the grating and the layer s determines the number of the modes in the metal-cladding region. Only LFSPs mode need to be excited for keeping the uniformity of lithography pattern, so we set s=60nm which is smaller than the cutoff thickness of TM1 mode of metal-cladding structure s_c⋍86nm. ε ₁=-6.1063+i0.2699, ε ₂=ε ₃=2.56. After initialization of the necessary parameters, the value of k can be calculated as k=2.8416k ₀, from which the period of metallic grating can be deduced, p=154nm.

 

Fig. 8. Near field intensity profile of electric field for the MGWS of metal-cladding scheme illuminated at 436nm. The parameters are t ₁=t ₂=40nm, p=154nm, w=30nm and s=60nm. The lateral scale is three periods.

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The FDTD method is also employed to calculate the average distribution of E-field intensity of this scheme with the lateral scale of three periods (462nm), which is depicted in Fig. 8. In contrast to the distributions of Fig. 3, this image presents obviously the coupled modes in the metal-cladding region, and the standing wave has been formed. Furthermore, the periodic field on the output surface is also built by the mutual coupling of SPs on the both sides of silver film. The visibility of intensity field in resonant (p=154nm) and non-resonant conditions (p=152nm and p=156nm) as the function of distance x beneath the bottom surface of silver grating is shown in Fig. 9 (a). The region of silver film is bounded by the dash-lines. In resonant condition, the curve presents that a peak arises in the point at x=23nm (the peak in the region of silver film can be ignored), and the visibility in the output region is small and decreases quickly, which is because of the high evanescent character of the output pattern with the high wave vector. Fig. 9(b) shows the average distribution of E-field intensity along the lateral direction y at the position of peak x=23nm. The resolution of the curve is ~34nm which is improved compared with the ones of the metal-layer scheme. However, in non-resonant condition, the intensity of lithography pattern decreases due to the drop of the coupling efficiency. Furthermore, the visibility shown in Fig. 9(a) also obviously decreases compared with the resonant one. The influence of deviation of grating period is larger than that of metal-layer scheme, which means that the metal-cladding scheme needs more rigorous control of grating period. The effective range of grating period is approximately <5nm.

 

Fig. 9. (a) Image visibility V as a function of distance x beneath the bottom surface of silver grating in the MGWS of metal-cladding scheme with different periods (p=154nm, p=152nm and p=156nm). The region of silver film is bounded by the dash-lines. (b) The average intensity profiles of electric field along the lateral direction y at the distance x=23nm beneath the bottom surface of silver grating in resonant condition (p=154nm). The lateral scale is three periods.

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Through properly designing of grating period, the metal-cladding lithography scheme with MGWS can generate the output pattern with higher resolution and approximately same visibility compared with the metal-layer one. Compared with the lithography scheme with a metallic layer proposed before [18, 19], the multi-layer coupled effect (here is five-layer structure) is considered in our scheme which increases the resolution effectively.

5. Conclusion

In summary, based on the metallic waveguide theory, MGWS has been designed to excite appropriate waveguide modes to give high-resolution, spatial-frequency doubled patterns. Two lithography schemes named metal-layer and metal-cladding schemes have been classified by the region supporting the modes. The numerical analysis of lithography pattern has been carried out for the fixed wavelength 436nm by the FDTD method. The metal-layer lithography scheme can excite three different modes (HFSPs, LFSPs and SPs) to generate the lithography field with different resolution, which can be used in the different practical conditions and for the different purposes. The metal-cladding lithography scheme has been proposed to use the LFSPs mode in the silver cladding region, which utilizes the multi-layer coupled effect to effectively increase the resolution of the lithography pattern (~34nm), and generates a field with approximately same visibility to the former scheme. The effectively deviated ranges of grating period in two schemes have also been numerically analyzed for practical application. The numerical simulation not only reveals the MGWS’s potential application in the fields of nearfield photolithography, but also confirms the validity of the analysis of some SPs related novel phenomena based on the metallic waveguide theory.