1. Introduction

In recent years, developments in the microelectronics industry and nanotechnology have posed great challenges to traditional fabrication techniques. To achieve nanometer-scale features, several methods, such as nanoimprinting lithography (NIL) [1–2], electron-beam lithography (EBL) [3], and scanning probe lithography (SPL) [4], have been proposed. Compared to these techniques, however, photolithography is still highly attractive due to its low cost and high throughput for mass fabrication. Although conventional photolithography is restricted by the diffraction limit, about half of the operating wavelength, many improvements, e.g., phase shift mask lithography (PSML) [5], evanescent near field optical lithography (ENFOL) [6–7] and surface plasmon interference lithography (SPIN) [8–9], have been proposed to overcome this restriction. Among these methods, SPIN is particularly interesting. It employs a lithographical mask made of a noble metal to excite surface plasmons, which possess an optical frequency but a much shorter wavelength, to deliver sub-wavelength interference patterns. Nevertheless, this technique is usually suitable for fabricating small-area nanostructures. In addition, the dependence of resolution on materials other than the mask pitch is still a restriction.

Since Pendry proposed that using a slab of negative refractive index medium could realize super-resolution imaging [10], there has been a growing interest in researching left-handed materials. However, due to the absence of left-handed materials in nature, metamaterials are chosen as a substitute [11–12]. These engineered materials exhibit exceptional properties in their interaction with the electromagnetism at both microwave and optical frequencies, like left-handed material. These particular properties depend mainly on synthetic structures rather than those inherent to the materials they are composed of. This provides us with an avenue to design various structures for the metamaterial to realize specific functions. In the present work, we theoretically propose a sub-wavelength interference photolithography technique with sandwich anisotropic metamaterial structure (SAMS). A feature size far beyond the diffraction limit can be obtained by affiliating a specially designed SAMS below the conventional chromium mask. Moreover, in contrast to SPIN, the resolution in this method mainly depends on the mask pitch, and therefore it possesses better application flexibility. In the following, the lithography principle and characteristics will be described in detail.

2. Principle

The proposed structure consists of a chromium grating mask affiliated with a SAMS, a layer of photoresist with refractive index 1.7 and a silica substrate, as shown in Fig. 1. All the components are treated as semi-infinite in the y direction. The SAMS is constructed from the periodical unit cell which is composed of two thin slices of silver and fused silica with thicknesses of t₁ and t₂. The permittivities for silver and fused silica at the operating wavelength 442nm are ε_Ag=-5.77+0.225i and ε_fs=2.13, respectively [13].

 

Fig. 1. Schematic drawing of the proposed structure. All the components are treated as semi-infinite in the y direction.

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According to the effective medium theory [14], if the thickness of the unit cell (t₁+t₂) is far smaller than the operating wavelength, the SAMS can be regarded as an anisotropic dielectric with a permittivity tensor

where ε_x=(t₁ε_Ag+ t₂ε_fs)/(t₁+t₂) and ε_z=(t₁+t₂)ε_fsε_Ag/(t₁ε_fs+t₂ε_Ag). Here we note that since the signs of the permittivities for silver and fused silica are opposite, choosing the appropriate thicknesses for the two slices could also yield opposite signs for ε_x and ε_z. For a plane wave with k² _x/ε_z+k² _z/ε_x=k² ₀ in such a dielectric, this condition indicates that the dispersion relation represents a hyperbolic shape and therefore the dielectric only supports the propagation of field components with high spatial frequencies. Figure 2(a) gives the dispersion relation of the SAMS with ε_x>0 and ε_z<0. The proportion of thicknesses for silver and fused silica slices is set to be 2:3. From this figure, it is clearly shown that only the components with wavevectors larger than 2k₀ can propagate inside the SAMS. Here the function of silver slices in the SAMS is twofold. First, the silver slices completely restrain the transmission of propagating waves. Second, each silver slice acts as a superlens [15], which could enhance the evanescent waves by surface plasmon excitation at the interface of silver and fused silica.

 

Fig. 2. Dispersion relation of the SAMS with ε_x>0 and ε_z<0. Here the proportion of thicknesses for silver and fused silica slices is set to be 2:3. (b) The optical transfer function for the designed SAMS. The structure is composed of 30 pairs of 20nm-thick silver and 30nm-thick fused silica slices. The operating wavelength is 442nm.

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Figure 2(b) shows the optical transfer function (OTF) of an example SAMS, which is composed of 30 pairs of 20nm-thick silver and 30nm-thick fused silica multilayers. For the p-polarized plane wave with a wavelength 442nm, only the evanescent components with wavevectors ranging from 2.2k₀ to 3.8k₀ can be enhanced and transmitted through the structure. On the other hand, due to the unsupported surface plasmon excitation for the s-polarization, the amplitude of field in this polarization is almost completely restrained by SAMS. This result reveals that unpolarized exposure light can be used directly in practical applications, and the SAMS will automatically allow propagation for p-polarized components.

A chromium grating with periodicity d is employed as the lithographic mask. Incident light is transmitted through the mask in several orders of diffraction, following the grating law:

where k’_x is the transmitted transverse wavevector; k₀ and θ are the incident wavevector and angle, respectively; and m is the diffraction order. First, let us consider the case of a normal plane wave impinging on the mask where Eq. (1) can be simplified to k’_x=2mπ/d. In order to obtain uniform evanescent wave interference fringes, the key point here is to ensure that only one pair of identical diffraction orders (±1 or ±2…) of the transmitted light pass the SAMS so that the coexistence of multiple wave vectors with different diffraction efficiencies can be avoided, which would result in an irregular interference intensity distribution. Referring to Fig. 2(b), this precondition indicates the grating reciprocal lattice k_G corresponding to the designed SAMS should be in the range from 2.2k₀ to 3.8k₀ for ±1 order diffraction or 1.3k₀ to 1.9k₀ for ±2 order diffraction. For the sake of achieving an interference pattern with periodicity Λ 80nm, the theoretical calculation k’_x=±π/Λ yields about ±2.76k₀ for the wavevectors of interference evanescent components, which can be delivered by a chromium grating with periodicity d=160nm (k_G=2.76k₀) for ±1 order diffraction or d=320nm (k_G=1.38k0) for ±2 order diffraction. For the case of d=160nm, transverse wavevectors from ±2 order diffraction are ±5.52k₀. For the case of d=320nm, transverse wavevectors from ±1 and ±3 order diffraction are ±1.38k₀ and ±4.14k₀, respectively. It can be seen from the OTF that the evanescent components with these wavevectors are all unsupported for transmission. Therefore, both of the two cases fulfill the requirement that only one pair of identical diffraction orders transmit through the structure.

3. Simulation and discussion

Figure 3 gives the distributions of total electrical intensity (E² _x+E² _z) for the above-mentioned two structures. Two chromium gratings with periodicities 160nm and 320nm (duty-cycle 0.5) are posited on the same SAMS, which consists of 30 pairs of silver-fused silica thin slices with thicknesses of 20nm and 30nm, respectively. A normal plane wave (at a wavelength of 442nm) in p-polarization is incident on the top side of the mask. Both the simulations are performed by the rigorous coupled wave analysis (RCWA) [16].

 

Fig. 3. Calculated distributions and cross-sections of total electrical field (E²_x+E²_z) for the two proposed configurations. Both of the two structures contain the same SAMS (30 pairs of 20nm-thick silver and 30nm-thick fused silica slices) under the mask. The p-polarized plane waves (at a wavelength 442nm) are vertically incident to the chromium masks. (a) (b) for the chromium mask with 160nm periodicity, (c) (d) for the chromium mask with 320nm periodicity.

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From Figs. 3(a) and 3(c), it is clearly shown that the evanescent components scattered from the grating are enhanced and transmitted through the SAMS. Both of the two structures deliver an interference pattern with 80nm periodicity in the vicinity of the interface between the SAMS and photoresist (Figs. 3(b) and 3(d)). If half peak is taken as the feature size, features about 40nm smaller than 1/11 of exposure wavelength can be achieved. Also, from these cross-sections, we can estimate the intensity contrast, which is another key parameter concerned in the lithography process. Theoretically, due to the existence of half a wavelength shift between the E_x and E_z components for p-polarization, the intensity contrast V can be calculated from the following equation:

where ε_pr is the permittivity of the photoresist. The simulation data show the contrast V is approximately 0.238 for both two structures. This agrees well with the value of 0.24 calculated from Eq. (3). Given the fact that typical minimum contrasts required for the common photoresist are about 0.2, this is sufficient for the lithography purposes [9]. Therefore, under the exposure of these spatially modulated electromagnetic fields, parallel lines with a half-pitch of 40nm can be developed in the photoresist. Moreover, since the propagation region of the SAMS ranges from 2.2k₀ to 3.8k₀, changing the periodicity of the grating mask while ensuring wavevectors with diffraction components inside the propagation region can produce other feature sizes. Figure 4 shows the relation between the mask periodicity and feature size delivered by the interference of evanescent components from ±1 order diffraction. Considering the feasible intensity contrast (>0.2), employing different masks associated with this SAMS can yield periodical patterns with feature sizes ranging from 38nm to 50nm.

 

Fig. 4. Feature size versus the mask periodicity and intensity contrast.

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Fig. 5. The relation between the field intensity under the SAMS and the duty cycle of the mask. The insets depict the interference intensities under the SAMS for case of a mask with same periodicity (160nm) but different duty cycles (0.3 and 0.65).

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Since the chromium mask upon the SAMS acts as a grating coupler, we consider an important grating parameter: duty cycle, which is defined as the ratio of the opening width to the grating periodicity. Although all of the above-mentioned masks are designed with a 0.5 duty cycle, the final structure may deviate from this value due to fabrication tolerances. Therefore, it is of great interest to better understand the sensitivity of interference pattern to the duty cycle. We take the first structure as an illustration, i.e., the periodicity of the grating is fixed as 160nm. Figure 5 gives the relation between the field intensity in the photoresist and a duty cycle varying from 0.2 to 0.8. It is shown that when the duty cycle is larger than 0.3, the intensity is basically inversely proportional to it. The reason for this is that with the continual shrinking of the opaque width, the ratio of the propagation components transmitted through the grating is increased. However, due to the filtrating of the SAMS, components with low spatial frequencies are completely suppressed, which leads to a low field intensity in the photoresist. Nevertheless, referring to Fig. 5, the intensity distribution is relatively stable for the duty cycles in the range from 0.2 to 0.5. On the other hand, since the interference periodicity and contrast are independent of the duty cycle of the mask in this technique, they are always invariable (insets of Fig. 5). Therefore, the tolerance of the interference pattern to the mask in this method is comparatively satisfying.

4. Conclusion

In conclusion, a sub-diffraction-limited interference photolithography technique with metamaterials has been proposed and numerically demonstrated by the RCWA method. The simulation with practical parameters shows that by positing a specially designed sandwich anisotropic metamaterial structure beneath a conventional chromium mask, a uniform interference pattern with feature size about 40nm, smaller than 1/11 of operating wavelength (442nm), can be obtained. In contrast to other near field interference photolithography techniques such as SPIN, this method is suitable for fabricating large-area periodical nanostructures and does not have rigorous requirements for the lithographic mask. Moreover, due to the dependence of resolution on mask periodicity rather than material, it affords better application flexibility. We believe this technique provides an alternative approach in semiconductor fabrication. Only grating like masks are considered in this paper for obtaining periodical interference patterns on the photo resist. The cases for complex mask structures are believed to have potential ability of generating irregular patterns, which is under investigation and would be published elsewhere.