1. Introduction

The rapid growth of nanotechnology is stimulated by the advancement of nanolithography, and the fabrication technologies on the nanometer scale are becoming increasingly important from the viewpoint of application such as the manufacture of quantum devices. Conventional optical lithography has been a major fabrication method in semiconductor industry over the past several decades because of its easy repetition and capability for large-area fabrication. However, it is limited by the physics limitation and its optical resolution is limited to the wavelength of the illumination light. To achieve nanoscale resolution, either the working wavelength is reduced, or the higher numerical aperture has to be adopted. The primary approach of reducing the working wavelength is to directly use light sources with higher photon energy, such as deep-ultraviolet light [1], extreme-ultraviolet light [2], soft X-rays [3], electron beams [4], and ion beams [5]. However, the cost for instrumentation and processing drastically increases with this reduction of the exposing wavelength. Furthermore, the industrial mass fabrication needs cannot be fulfilled. Another approach, such as immersion photolithography [6,7], which makes use of higher numerical aperture is constrained by the highest refractive index of the available nature materials. By collecting the evanescent waves at the near field, several near-field optical lithography methods [8, 9] have been developed with the resolution beyond the diffraction limit, however, the low optical throughput limits their practical application.

Surface plasmons are electro-magnetic excitations that propagate along a metal-dielectric surface with an exponentially decaying field in both neighboring media [10]. The potential of using SPs to manipulate light in the subwavelength space opens up a multitude of new possibilities for subwavelength lithography. Recently a new scheme to achieve nanoscale patterns has been proposed based on the unique properties of SPs. Surface plasmon polaritons (SPPs) which take place of photons as an exposure source have been applied in nanolithography field, providing a new method to produce fine patterns with a resolution of subwavelength scale [11–13]. A photolithography technique based on the interference of SPs can go far beyond the free-space diffraction limit of the light, and one-dimensional and two-dimensional periodical structures with 40–100nm features can be patterned [14].

As a recent explosive progress in nano-optics, achieving localization or focusing of SPs to truly nanoscale dimensions [22,23] induces many enhanced nonlinear-optical phenomena and has various prospective applications. As is well known, in addition to SPs at a planar dielectric-metal interface, localized surface plasmons (LSPs) can exist on other geometries [15,16], such as metallic spheres or long metallic cylinders, especially on subwavelength roughness of metal surface, where a very strong electromagnetic field enhancement can be observed. The magnitude of the electromagnetic field depends significantly on the shape and size of the above individual particles. Generally speaking, the shape of these particles where LSPs are confined is irregular or even particular, so it is difficult to control and modulate the magnitude of the localized energy. In our previous paper [17], a LSP nanolithography technique has been demonstrated numerically to produce patterns with the line width below 20nm. However, we haven’t described in detail that the localized energy can be modulated via interfering the SPs. In this paper, we propose a technique to control the localized energy at an invariable structure (the taper shape is fixed) where the LSPs are confined. Constructive or destructive interference, which causes enhancement or decrease of the localized energy, can be obtained just by tuning the distance of the grating and the taper. It is demonstrated that nano-scale exposure patterns with different depths and line widths can be patterned by properly controlling the interference of SPs by our FDTD simulation results.

2. Modulation of localized energy

A schematic view of the special nanolithography process is shown in Fig. 1 for the simplest case. The combined structure is composed of a taper waveguide integrated with grating structure at both sides. Here, two approaches are introduced to realize the excitation of SPs, one by the grating with reasonable period, and the other by the nanotaper. Generally, there would be a cut off at some waveguide radius beyond which the propagation is not possible [18] in the conventional optical fiber supporting guided photonic modes. However, in a plasmonic taper waveguide, the guided photonic modes could be coupled to plasmonic modes at the cut-off point [19], thus enabling corresponding SPs to propagate toward the taper tip. As is well known, the grating structure can excite the SPs with high efficiency by choosing a reasonable period, and also, a part of SPs can propagate toward the taper tip. It is noted that interference will be produced between SPs at the sidewalls of the taper structure, the resultant SPs then still propagate toward the tip. Subsequently the local field is sharply concentrated at the tip [15, 17], and finally the SPs are adiabatically transformed into LSPs, leading to nanofocusing [20].

 

Fig. 1. Schematic view of the composite structure with controllable localization energy.

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The basic principle of the method employed in this work is based on the interference modulation of SPs. The interaction of light and SP is described by the SP dispersion relation [10] in Eq. (1):

where k _sp is SP wave vector, k ₀ is the wave vector of light in vacuum, and ε _m and ε _d are dielectric constants of a metal and a surrounding dielectric material, respectively. According to the SP dispersion, the wavelength of the excited SPs can become shorter compared with the wavelength of the excitation light. Generally, light cannot excite SPs directly due to the momentum mismatch between the light wave and SP. But, the conversion between light and SPs can be achieved with the existence of momentum matching element, such as a grating structure. By selecting a proper periodicity of grating, the lights can be reasonably coupled to SPs, and the relationship can be described in Eq. (2):

where θ is the angle-of-incidence, n _s is refractive index of the substrate, and ∧ is the period of grating. When the incident light is normal to the substrate (θ=0), by choosing m=1 in our simulation, Eq. (2) can be simplified to

Numerical simulations were performed using the commercial software opti-FDTD. The numerical values of refractive indices for the quartz and the photoresist are 1.52 and 1.7, respectively. The dielectric constant of the aluminum is taken from the experimental measurement [21]. The incident light, with the wavelength of 436nm polarized along the x direction, is normal to the substrate. The grating periodicity of 280nm was chosen through solving Eq. (1) and Eq. (3) together by using the above-mentioned simulation parameters. The ridge number n was 4, and the depth of the grating d was 40nm, in this case, SPs could be highly excited. The detailed parameters are labeled in the Fig. 1. Here we take the values of w ₁=600nm, w ₂=160nm, h ₁=60nm, and h ₂=240nm.

The electric field distribution pattern in the case without grating is shown in Fig. 2(a), the SPs are only excited by the taper at the cut-off point, and finally the localized SPs are accumulated at the taper tip as mentioned above. By adding the gratings structure to the both sides of the taper bottom, the localized electric field and relevant localization energy can be modulated. We fix the taper structure, only changing the distance between the taper and the first ridge of the grating (labeled as l in Fig. 1). Figures 2(b)–2(d) show the electric field distribution in the cases of l=0nm, l=70nm and l=140nm, respectively, in which additional excitation of SPs by the grating is clearly revealed. In Fig. 2(b), the amplitude of the localized electric field intensity is obviously enhanced compared with that in the case without grating, indicating a constructive interference of SPs which eventually leads to the more accumulation of localized energy at the tip. Also, when we choose l=70nm as shown in Fig. 2(c), the localized electric field intensity is still enhanced, but this increase has weakened. Lastly, when l=140nm, a destructive SPs interference has taken place, the localized electric field is attenuated by 13 times, and corresponding localized energy is reduced by 2 orders of magnitude compared with that in the case without grating.

The calculated near-field intensity profiles at Δh=30nm (30nm below the interface of mask and photoresist) are displayed in Fig. 4(e) to describe the interference modulation to the localized field. We notice that all the fields have one maximum centered on the symmetry axis of the taper (or X=0nm). When l=0nm, the line width defined as full-width at 0.7 maximum is only 100nm which is smaller than the tip width w ₂=160nm, and this curve is the sharpest. It is expected that an acceptable pattern with high resolution and high contrast can be approached with this profile. When l=140nm, the magnitude of electric field is dramatically reduced due to the destructive interference of SPs, and yet, the line width is 154nm which is still smaller than the diffraction limit 218nm. It is suggested that this technique can break through diffraction limitation. For instance, the pattern with line width below 20nm has been numerically demonstrated [17].

Further investigation focuses on the relationship of the maximum electric field intensity versus the vertical (along Z direction) distance as shown in Fig. 2(f). The localized fields decay exponentially into the photoresist, and the expression for the peak of electric field E _m (h) as a function of vertical distance h can be given n by E _m(h)∝exp(-h/h0). Here, the decay length h0 which influences the exposures depth in photoresist is defined as the vertical attenuation distance from the mask-photoresist interface to the location where the electric field decreases to e ^-1 of the interface value. Through simulation data, we have h0 labeled by arrow lines with different colors in Fig. 2(f) being approximately 86nm, 83nm, 84nm, and 108nm for the cases without grating, l=0nm, l=70nm, and l=140nm with grating, respectively.

 

Fig. 2. (a). Electric field intensity distribution pattern in the case without grating. Electric field intensity distribution pattern in the case of l=0nm (b), l=70nm (c) and l=140nm (d) with grating. (e) Electric field intensity profiles at Δh=30nm (30nm below the interface of mask and photoresist). (f) Magnitude of the electric field intensity at constant X=0nm versus the vertical distance Z.

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We theoretically analyzed the physical mechanism behind the phenomenon of the modulation of the localized energy. From the viewpoint illustrated in the preceding paragraph, the SPs excited by the grating and the nanotaper can interfere at the sidewalls of the taper structure. There exists a phase difference φ arising from a combined path length and initial phaseangle difference of the SPs excited by above two different mechanisms. The phase difference can be changed by adjusting the distance l. Utilizing the model Δφ=ksp*Δl, we can find that the distance variation Δl is 280nm when Δφ=2π, which corresponds to a periodical oscillation of the localized energy. To validate theoretical calculation, the localized energy at the tip versus the distance l is simulated with FDTD, and corresponding results are shown in Fig. 3. As expected, the localized energy exhibits pronounced oscillation, and the oscillation periodicity is 280nm which is in well accordance with the preceding theoretical analysis results. The maximum energy are obtained when l=280nm, 560nm and 840nm where total constructive interference occurs. Whereas, the maxima decrease appreciably as the distance l increases, and we believe this phenomenon is related with the loss of SPs excited by the grating. The minimum energy are nearly approach to zero in the case of total destructive interference, when l=420nm, 700nm and 980nm. The ratio of maximum to minimum I _max/I _min is 958.18, which implies that the modulation range of localized energy is extremely wide.

 

Fig. 3. Relationship of localized energy versus the distance l. the red dash line with the value of 1 represents the energy value of a single taper structure without grating. Geometrical parameters are also w ₁=600nm, w ₂=160nm, h ₁=60nm, h ₂=240nm, p=280nm, n=4 and d=40nm.

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3. Controllable nanolithography

In what follows, a LSPs nanolithography technique is proposed and demonstrated to produce patterns with different depths and feature widths in one chip based on the interference modulation of SPs. To prove it in a specific instance, a structure including four tapers with the same shape is simulated. The distances between the tapers and adjacent gratings are chosen as 0nm, 50nm, 100nm, and 150nm respectively along the positive direction of X axis, other geometrical parameters are the same as the preceding section. Figure 4(a) represents the electric field distribution, in which the localized energy of the four tapers is gradually weakened along positive X direction as expected.

 

Fig. 4. (a). Electric field intensity distribution with four tapers. (b) Near-field intensity profiles at Δh=30nm. (c) Peak of the electric field intensity versus the vertical distance from the interface of the mask and the photoresist.

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As the photoresist is mostly sensitive to the electric field intensity, the feature size and the contrast of exposed patterns can be reflected by the electric field intensity profile, and the exposure depth is relevant to the attenuation of electric field. Next, we analyze the near-field intensity profiles of the four taper tips at Δh=30nm as shown in Fig. 4(b). The photoresist only records regions where the local electromagnetic field intensity is above the exposure threshold. It is therefore imaginable that the localized field of the four tapers will induce distinct patterns. Suppose the exposure threshold is 4 indicated as the purple solid line, only the intensities of taper 1 and taper 2 are high enough to react with the photoresist, and the corresponding feature widths are 84nm, 72nm, 0nm and 0nm, respectively. Furthermore, the pattern depths for exposure are 51nm, 41nm, 23nm and 0nm respectively, as revealed in Fig. 4(c) which represents the peak of electric field intensity versus vertical distance. Note that the electric field drops off gradually with increasing vertical distance from the mask-photoresist interface. When the threshold is 1 labeled as the purple dash line in Fig. 4(b), the line widths are 385nm, 356nm, 280nm and 94nm respectively, and the exposure depths will accordingly increase.

We only testify a mask with four tapers from the simulation results above, actually this method can be extended to other structures with multiple tapers. Our type of lithography, which can control the localized energy just by designing and controlling the interference of SPs, is different from previous contact lithography. It is expected that some acceptable nanolithography patterns with different depths and line widths can be approached with this technique. It should be noted that the effect of resist absorption on the localized SPs has not been taken into account in these simulations. By considering a highly absorbing resist layer, the exposure depth will be correspondingly increased,

4. Conclusions

In conclusion, we have proposed a type of nanolithography technique which can be used to produce patterns with different depths and feature widths in one chip. Physical analysis and FDTD simulation calculation have demonstrated that the localized exposure energy could be modulated by controlling the interference of SPs originate from two distinct excitation sources. As an example, sub-wavelength patterns with different line widths and depths have been obtained in the simulation with 436nm exposing light by using one mask with four tapers. This technique provides an alternative method for nanodevices fabrication.