Plasmonic interference nanolithography with a double-layer planar silver lens structure

Beibei Zeng; Xufeng Yang; Changtao Wang; Xiangang Luo

doi:10.1364/OE.17.016783

1. Introduction

Recently, developments in the microelectronics industry and nanotechnology have posed great challenges to traditional micro-fabrication techniques. Several methods, such as nanoimprint lithography [1], electron-beam lithography [2], and scanning probe lithography [3], have been proposed to achieve nanometer-scale features. Compared with these techniques, however, photolithography has been considered as a useful technology because of its easy repetition and suitability for large-area fabrication. However, in order to overcome the diffraction limit that restricts the conventional photolithography method, optical contact or near field lithography, such as evanescent near field lithography [4,5], has been demonstrated to produce sub-100nm features.

In addition, a new kind of surface plasmons assisted lithography method has been proposed to produce nano-scale patterns with simple conventional light source [6,7]. It employs surface plasmons (SPs) excited by the grating mask made of noble metal, which has an optical frequency but a much shorter wavelength than incident light, to yield deep subwavelength interference patterns. As we all know, surface plasmons will split into two modes, the symmetric mode and antisymmetric mode, in the thin metal film [8]. Due to its large wave-vector, the antisymmetric mode shows potential application in nanolithography with conventional light source. Therefore some methods, such as metal-layer lithography scheme [9] and metamaterial lithography [10], have been presented to realize surface plasmon interference lithography. Nevertheless, these methods suffer from some problems like low contrast of interference patterns and difficulty in fabrication, which restrict their practical applications in nanolithography.

In order to overcome these difficulties, we put forward a double-layer planar silver lens (DLSL) structure with the mismatched permittivities between the metal and surrounding dielectric medium to give high resolution interference patterns in this paper. A feature size far beyond the diffraction limit with sufficient intensity contrast could be obtained by affiliating the DLSL structure below the conventional chromium grating mask, which could be practical for lithography process. In addition, some factors influencing the interference fringe sizes such as the thickness of silver and dielectric film are also investigated. Based on theoretical analysis, we propose a resolution limit of about 1/12 wavelength to the surface plasmon interference lithography method. In the following, the lithography principle and characteristics will be described in detail.

2. Numerical analysis and simulation results

The proposed structure consists of chromium grating mask, a layer of PMMA, two layers of $A g - A l_{2} O_{3}$ stack, photoresist and silica substrate, as shown in Fig. 1 . All the components are treated as semi-infinite in the y direction. In our configurations, the thicknesses of $A g$ , $P M M A$ , and $A l_{2} O_{3}$ layers are all equal to 30nm. And the thickness of chromium grating mask is set to be 50nm. The permittivities of the $P M M A$ , $A l_{2} O_{3}$ , and photoresist are set as $ε_{P M M A} = 2.31$ , $ε_{A l_{2} O_{3}} = 3.22$ and $ε_{p r} = 2.89$ [11], respectively. Drude model $ε (ω) = ε_{0} - ω_{p}^{2} / [ω (ω + i V_{c})]$ is used to describe the permittivity of silver, where $ε_{0}$ = 4.9638, plasma frequency $ω_{p}$ = 1.4497e16 rad/s and collision frequency $V_{c}$ = 8.33689e13 rad/s. The calculated permittivity of silver is $ε_{A g} = - 6.5875 + 0.2258 i$ at the operating wavelength 442nm. This calculated permittivity agrees well with the experiment data [12]. All the materials are assumed to be nonmagnetic so that the magnetic permeability μ is equal to 1 and only the permittivity ε has been taken into account.

Fig. 1 Schematic drawing of the proposed double-layer planar silver lens (DLSL) structure. The thicknesses of the $P M M A$ , $A g$ and $A l_{2} O_{3}$ layers are all equal to 30nm.

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From the waveguide theory, as the thicknesses of $A l_{2} O_{3}$ and silver layers decrease, SPs modes at these metal-dielectric interfaces couple and split into an antisymmetric/symmetric pair [8]. For example, for TM-polarized waves in a metal-dielectric-metal structure, the transverse wave vector $k_{x}$ can be defined by the dispersion relations:

ε_{1} k_{z 2} + ε_{2} k_{z 1} \tanh (\frac{- i k_{z 1} d}{2}) = 0, for antisymmetric mode

ε_{1} k_{z 2} + ε_{2} k_{z 1} \coth (\frac{- i k_{z 1} d}{2}) = 0, for symmetric mode

and

k_{z}

defined by the momentum conservation:

k_{z 1, 2}^{2} = ε_{1, 2} {(\frac{ω}{c})}^{2} - k_{x}^{2}

where the subscripts 1 and 2 denote the case of

A l_{2} O_{3}

and silver layer, respectively, and d is the thickness of the

A l_{2} O_{3}

layer. The thicknesses of

A l_{2} O_{3}

and silver layer determine the wave vector of the coupled SPs mode, and they should be properly chosen to ensure that SPs modes at these metal-dielectric interfaces interact with each other to generate the coupled modes. We will discuss about this later. Furthermore, the coupled modes interfere with each other and the electric field intensity distribution performs the profile of standing wave with doubled spatial frequency.

Here the chromium mask with a subwavelength periodicity Λ plays the role of grating coupler in this structure rather than as an object in the superlens imaging structure [13]. According to the grating law:

k_{x} = k_{0} \sin θ + 2 m π / Λ

incident light transmits through the mask in several orders of diffraction, where

k_{x}

means the transmitted transverse wavevector;

k_{0}

and θ are incident wavevector and angle, respectively; m is the diffraction order. When the incident plane wave impinges normally on the mask, the grating law could be simplified to

k_{x} = 2 m π / Λ

. In order to obtain uniform interference fringes, it is necessary to ensure that only one pair of identical diffraction orders (

\pm 1, \pm 2...

) of the incident light pass through the DLSL structure to interfere with each other. However, although the transmitted components from ±2 order (

m = \pm 2

) could generate interference fringes as the ±1 order (

m = \pm 1

) components do, the intensity distribution of the fringes would be irregular. The main reason is that if we want to make ±2 order components transmit through the DLSL structure and generate interference patterns, ±1 order components with high amplitude will also transmit through the DLSL structure and mix with the ±2 order components to generate irregular patterns. This is clearly shown in the optical transfer function curves of the DLSL structure and will be discussed in the next section. Therefore, only ±1 order components would be used to produce interference fringes in our analysis.

In Fig. 2 (a) and Fig. 2 (b), total electric field intensity distribution for the proposed structure, which corresponds to the coupling of antisymmetric mode on the silver layer, has been performed by the two-dimensional rigorous coupled wave analysis (RCWA) method [14]. The simulated structure is the same as that shown in Fig. 1. From Fig. 2 (a), we can see that antisymmetric mode SPs has been excited on the silver layers and the total electric field intensity distribution performs the profiles of standing wave with doubled spatial frequency. A regular interference electric field distribution with periodicity of 82 nm in the photoresist layer could be obtained, as shown in Fig. 2 (b). If half peak (hp) width is taken as the feature size, fringes of 41nm about 1/11 of incident wavelength can be achieved.

Fig. 2 (a)(b) Calculated distribution of total electrical field intensity for the proposed DSDL structure. The geometrical parameters are the same as those mentioned above.

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For the TM-polarized incident wave, Fig. 3 (a) shows the calculated H-field transmission factor in the photoresist layer as a function of wavelength and transverse wave number through the DLSL structure. Parameters of the structure are the same as those mentioned before. From this figure, two main modes can be clearly distinguished and it is clearly shown that the antisymmetric mode has a larger wave vector than the symmetric one at the same wavelength, such as 442nm [9]. In addition, it is also seen in Fig. 3 (a) that when the wavelength is close to 365nm (i-line), the two split modes begin to converge due to the resonant excitation of SPPs at the condition of $Re (ε_{m}) \approx - ε_{d}$ , which means that the metal and the surrounding dielectric have a matched permittivity at the working wavelength. It indicates that components with wide band of large wave vectors can be enhanced and transmitted through this structure and has been used in the sub-diffraction imaging structure [13,15] and double-layer planar lens lithography [16]. However, this is unexpected in interference lithography because the interference of the evanescent components with different wave vectors would result in an irregular pattern. Therefore, we choose the working wavelength as 442nm (HeCd laser) where the gap between two split modes is relatively wide and only the components with some particular wave vectors can be enhanced and transmitted through the DLSL structure. As shown in Fig. 3(a), the transverse wave number corresponding to the antisymmetric mode is about 2.7 $k_{0}$ at the wavelength of 442nm, which corresponds to the grating periodicity 164nm due to the grating law mentioned above ( $k_{x} = 2 m π / Λ$ ). The duty cycle of the grating is chosen to be 0.25 to decrease the influence of grating slits.

Fig. 3 (a) H-field transmission factor shown as a function of wavelength and transverse wave number through the proposed DLSL structure. The dashed red line represents SPPs resonant wavelength 365nm, and the solid red line denotes the operating wavelength 442nm. (b) Intensity contrast (visibility) of interference patterns at different cross sections beneath the output surface of the DLSL structure, d denotes the distance between the output surface and the cross section. The inset depicts the different interference intensities distributions at different distances (d) beneath the output surface of the DLSL structure. The periodicity of the chromium grating mask is set to be 164nm. Other parameters are the same as those mentioned above.

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Since pattern contrast is a key parameter in the lithography process, we should pay attention to it. Theoretically, due to the existence of half wavelength shift of $E_{x}$ and $E_{z}$ components for TM-polarized waves, the contrast can be calculated from the equation (only considering ± 1 diffraction order components from the grating mask):

\begin{array}{c} V i s i b i l i t y = (I_{\max} - I_{\min}) / (I_{\max} + I_{\min}) = (E_{z}^{2} - E_{x}^{2}) / (E_{x}^{2} + E_{z}^{2}) \\ = ε_{p r} k_{0}^{2} / (2 k_{x}^{2} - ε_{p r} k_{0}^{2}) \end{array}

In comparison with the previous single-layer silver structure [9], the DLSL structure acts as a better filter to block the unwanted components in the near field. All of the wave vector noise and background field except one of the coupled SPs could be filtered more thoroughly than the previous single-layer silver structure. Therefore, better uniformity and pattern contrast could be obtained by using the DLSL structure mentioned above. In Fig. 3 (b), intensity contrast ( $V i s i b i l i t y = (I_{\max} - I_{\min}) / (I_{\max} + I_{\min})$ ) of interference patterns at different cross sections in the photoresist is described as a function of distance d beneath the output surface of the DLSL structure. Given the fact that typical minimum contrasts required for the common photoresist are about 0.2 [17], the contrast is sufficient as the distance ranges from 0 to 40nm for the lithography purpose. In contrast, in previous scheme the contrast is less than 0.2 near the output surface and it could be difficult to fulfill the requirement in lithography. In addition, as depicted in the inset of Fig. 3 (b), the pattern contrast reaches a peak at 15nm beneath the output surface. The total electric field intensity distribution becomes from uniform to a little nonuniform periodically, which is due to the phase of the direct current (DC) component changes during the propagating process. So the output pattern is regular and uniform at 15nm beneath the output surface and the pattern contrast has a maximum value. In theory, if we add more silver layers to the DLSL structure, the intensity contrast will be enhanced to a little more than that of the DLSL structure and the output pattern could be more uniform by reducing the influence of the DC component [10]. But it would be very difficult to fabricate such multilayer structure and it is easy to bring fabrication errors into such structure. Therefore, the DLSL structure is the optimum choice for antisymmetric mode plasmonic interference lithography.

3. Discussion

3.1 Influence of the thickness of silver layer

Next, let us consider the influence of the silver layer’s thickness on the antisymmetric mode by the conventional optical transfer function (OTF), which is defined as the ratio of image magnetic field to object magnetic field, $H_{i m g} / H_{o b j}$ , with a given lateral component of wave vector $k_{x}$ [18]. Figure 3(a) gives the OTF curves as a function of the thickness of silver layer and transverse wave number through the DLSL structure. Here the incident wavelength is fixed at 442nm and other parameters are the same as previous one. In this figure, as the thickness of silver layer decreases, the second peak of the OTF curves corresponding to the antisymmetric mode shifts to larger wave number. When the thickness of the silver layer decreases to 10nm, the antisymmetric mode disappears. Due to the competition between SPs resonance and intrinsic loss in the metal, in the structure mentioned above, the optimal thickness of silver layer is 30nm. Although the peak position shifts to larger wave number which will result in higher resolution if we choose thinner silver layer, the amplitude of the second peak would be smaller and even disappear due to the intrinsic loss in the metal, as shown by green and blue curves in Fig. 3(a). This is a factor that would limit the resolution of the interference patterns.

Fig. 3 Optical transfer function (OTF) for the proposed DLSL structure as a function of the transverse wave number and (a) silver thickness when the thickness of the dielectric is fixed at 30nm; (b) dielectric thickness when the thickness of the silver is fixed at 30nm. Here the wavelength of the TM-polarized incident light is fixed at 442nm and other parameters are the same as those mentioned before.

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3.2 Influence of the thickness of dielectric layer

On the other hand, when the wavelength of the incident light and the thickness of the silver layer are fixed at 442nm and 30nm, respectively, the OTF curves as a function of the thickness of $A l_{2} O_{3}$ layer and transverse wave number are depicted in Fig. 3(b). It can be clearly seen that the second peak of the OTF curves corresponding to the antisymmetric mode shifts to larger wave vector as the thickness of the $A l_{2} O_{3}$ layer increases. It indicates that employing thicker $A l_{2} O_{3}$ film and appropriate grating mask periodicity can result in smaller interference patterns. However, if the thickness of the $A l_{2} O_{3}$ layer increases beyond 50nm, the antisymmetric mode will disappear due to rapid decay of SPs away from metal-dielectric interfaces.

3.3 Influence of the periodicity of grating mask

As depicted by the red curve in Fig. 3(a) or Fig. 3(b), the transverse wave number of the second peak is 2.7 $k_{0}$ , which corresponds to the transverse wave number of ±1 diffraction order components of the incident light according to the grating law mentioned above. If we change the periodicity of the grating to make the transverse wave number of ±2 diffraction order components correspond to that of the second peak, ±1 order components with transverse wave number about 1.35 $k_{0}$ will also transmit through the DLSL structure with high amplitude and mix with ±2 order components to generate irregular patterns. That is the reason why we choose only ±1 order components to produce interference fringes in our analysis.

From above analysis, if we optimize the structural parameters such as thicknesses of silver and dielectric films, it is believed that this method could be used to produce interference fringes with different pitch sizes. Furthermore, when the wavelength and the thicknesses of the silver and $A l_{2} O_{3}$ layers are fixed at 442nm, 30nm, and 30nm, respectively, it is clearly represented by red line in Fig. 3(a) or Fig. 3(b) that the evanescent components with wave vector changing from 2.2 $k_{0}$ to 3 $k_{0}$ , which varies in the vicinity of the second peak of the OTF curves, can be enhanced and transmitted through the DLSL structure. Therefore, as shown in Fig. 4 , changing the periodicity of grating mask when other parameters are fixed could produce interference fringes with different feature sizes.

Fig. 4 Feature size versus the mask periodicity and the intensity contrast (visibility). The inset depicts interference intensity distributions with different mask periodicities. The wavelength and the thickness of the silver and $A l_{2} O_{3}$ layer are fixed at 442nm, 30nm, and 30nm, respectively.

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In Fig. 4, the green line, which represents the relation between the feature size and theoretical intensity contrast, is depicted from Eq. (5). It is shown that as the feature size decreases to 37nm, the pattern contrast reduces to 0.2 and can be difficultly used in lithography. On the other hand, when the wavelength and the thickness of the silver and $A l_{2} O_{3}$ layer are fixed at 442nm, 30nm, and 30nm, respectively, the relation between the feature size and intensity contrast is calculated by two-dimensional RCWA method. The calculated relation based on electric field intensity distribution at the same cross section, which is described by red line, is a little different from the theoretical one due to the existence of wave vector noise and background field. Blue line shows the relation between the mask periodicity and feature size produced by the interference of evanescent components from ± 1 diffraction order. By considering the feasible intensity contrast (>0.2), employing masks with different periodicities when other parameters are maintained could yield periodical line patterns with feature sizes ranging from 40nm to 50nm.

3.4 Three factors that limit the minimum feature size

From the analysis above, conclusions could be made that the minimum pitch size of SP interference pattern here is mainly restricted by three factors. The first restriction comes from the intrinsic loss of metal. Theoretically, infinite small SP wavelength mode can be excited as we decrease silver film thickness. But the antisymmetric mode with large wave number suffers from the great loss in metal hence low transmission, which delivers decreased fringe visibility. The leakage transmission of the zero-order (DC) component through the DLSL structure is another restriction. As shown in the OTF curves in Fig. 3, transmitted light with zero-order component will not be completely filtered because of the finite layers of metal films. Due to the influence of these unwanted components, the interference fringes become less uniformity and the pattern contrast becomes lower, as shown in the inset of Fig. 4. Adding more layers of silver lens to the DLSL structure would reduce the unexpected zero-order component, but it is not practical considering the fabrication issues and accompanied influences.

The third restriction, which we think is dominant here, is in connection with the intrinsic TM polarization property of SPs. The $π / 2$ phase shift of $E_{x}$ and $E_{z}$ components results in the greatly decreased interference fringe visibility as the feature size of interference pattern goes far beyond the diffraction limit with larger $k_{x}$ . In fact, the minimum fringe feature size is mainly and physically determined by polarization property of SPs, if we assume metal material with low loss and optimize the structure from the viewpoint of filtering unwanted components and fabrication. In this ideal situation (material without loss and no DC component), due to the TM polarization property of SPs the intensity contrast of the interference pattern is determined by Eq. (5) ( $V i s i b i l i t y = ε_{p r} k_{0}^{2} / (2 k_{x}^{2} - ε_{p r} k_{0}^{2})$ ). From this equation, it is clearly seen that when the permittivity of the photoresist ( $ε_{p r}$ ) and incident wavelength ( $k_{0}^{} = 2 π / λ_{0}$ ) are fixed, intensity contrast decreases as the feature size of the interference pattern ( $k_{x}^{} = π / (2 Δ)$ , Δ represents the feature size of the interference pattern) reduces. If the minimum contrast 0.2 is required for feasible lithography, the theoretical resolution limit can be determined. Therefore, it is the polarization property of SPs that theoretically limits the minimum feature size of the interference pattern. For instance, considering the feasible intensity contrast ( $V i s i b i l i t y \geq 0.2$ ), when the incident wavelength is fixed at $442 n m$ the minimum feature size is about $Δ = π / (2 k_{x}^{}) = π / 2 \cdot {(3 ε_{p r})}^{1 / 2} k_{0}^{} \approx 37 n m$ ( $~ 1 / 12$ wavelength). So, it is justified to give a theoretical resolution limit of about 1/12 wavelength to the surface plasmon interference lithography methods presented here and those in Ref [6,7,9,10]. And it is believed that much higher resolution can be obtained by employing negative refractive index materials, in which metal loss would be the dominating factor influencing resolution.

4. Conclusion

In conclusion, we present a numerical analysis of plasmonic interference nanaolithography method with a double-layer planar silver lens (DLSL) structure. Distribution of SPs modes inside the DLSL structure are calculated by RCWA method with practical parameters. Structural parameters have been designed to excite appropriate coupled SPs modes to give high-resolution patterns. A uniform interference pattern with feature size 41nm, about 1/11 of incident wavelength, could be obtained. Three different factors that limit the resolution of the interference fringes have been discussed. It is pointed out that the polarization property of the SPs mainly determines the resolution of the interference fringes in the surface plasmon interference lithography methods. In contrast to previous work, the intensity contrast is sufficient near the output surface and this structure is feasible for fabrication. Therefore, this method could be practically used in lithography.

Acknowledgments

The work was supported by 973 Program of China (No.2006-CB302900) and the Chinese Nature Science Grant (No.60507014).

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Plasmonic interference nanolithography with a double-layer planar silver lens structure

Abstract

1. Introduction

2. Numerical analysis and simulation results

3. Discussion

3.1 Influence of the thickness of silver layer

3.2 Influence of the thickness of dielectric layer

3.3 Influence of the periodicity of grating mask

3.4 Three factors that limit the minimum feature size

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (5)

Optics Express