Superlens for lithography

Yuan Zhang1, Daohua Zhang1, and Michael A. Fiddy*2 1School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore

2Center for Optoelectronics and Optical Communications, University of North Carolina at Charlotte, Charlotte NC, USA mafiddy@uncc.edu

Abstract—in this paper, we have examined the possibilityof using a planar negative index lens in lithography applications.We numerically studied the field patterns generated by a PEC and Cr mask propagating through a negative index slab to form a super-resolved image of the mask at the image plane. We discuss the advantages and disadvantages of this for next generation lithography.

superlens; negative refraction; lithography;

I. INTRODUCTION Negative index materials and metamaterials have

attracted much attention in recent years. An interesting and potentially high impact application of these artificial materials is to permit the development of an imaging system that provides a higher resolution image of an object beyond the diffract limit. Achieving superresolution has been the goal for much research over the last 60 years, but appropriately designed negative index materials provide one of the first real opportunities to achieve this physically, rather than purely via numerical methods. Following the pioneering work of Veselago [1], the paper by Pendry [2] fuelled enormous interest in validating and trying to fabricate a negative index metamaterial, especially one that would provide these properties at optical frequencies (e.g. [3]). It is well understood that fabrication issues remain a challenge and that further innovations are necessary in order to realize a practically useful negative index metamaterial with minimal losses [4]. Losses may be diminished through the use of gain but the ability to engineer a metamaterial for a superlens [2], with an index exactly equal to -1, remains highly controversial [5]. Nevertheless, many well known experiments have made progress in trying to realize super-lens and some encouraging results have been reported [6, 7].

Transferring high resolution images (or patterns) is also important for lithography which is a key and

expensive technology in the semiconductor industry.Progress is being made toward replicating patterns with linewidths of the order of tens of nm using extreme uv illumination. As circuit features reduce in size, new physical properties also reveal themselves, providing further opportunities for increasingly low power, highly functional electronic and optical chips. Moreover, improved subwavelength fabrication of abstracted circuit elements is also the key to advancing the entire field of optical metamaterials. Early contact lithography was cheap and straightforward for replication at high resolution but the (expensive) mask would degrade after some limited number of uses. Modern projection lithography, even with sophisticated phase mask techniques to improve resolution, is increasingly complex and thus expensive. If one could realize a low loss planar negative index lens; some of these difficulties may be eliminated. We study this possibility here, assuming a good planar super-lens is available and examine its role in lithography.

II. SUPER-LENS FOR LITHOGRAPHY

A. Numerical experiment setup The simulation setup is illustrated in Fig. 1. A

mask (black area, with thickness t) is positioned 100 nm before a super-lens which has a fixed thickness of 200 nm and refractive index n = -1 (which means ε = µ = -1). Thus, considering the focusing rule for such a planar negative index lens, the image of the field exiting the mask plane will be locate at x = 100nm (green line in Fig. 1). The wavelength of illumination we used is λ = 365 nm (i-line).The air gaps in the mask are 60 nm in width (~ λ/6), and the distance between these gaps are d1 = 120 nm (~ λ/3) and d2 = 180 nm (~ λ/2).

Figure 1. Schematic diagram of our simulation setup. Athin mask (blackstrips) with sub-wavelentg gaps (60 nm) is located 100 nm in front of a 200nm thick super-lens.

The scattering of the incident wave from the sub-wavelength features of the mask will generate evanescent waves. The spatial frequency ky of these waves is larger than the free space wave vector k0, and their field strength isexponentially decaying along the x direction. For the special case of ε = µ = -1, the negative index lens has the ability to amplify these evanescent waves [2]. Once any albeit exponentially decaying evanescent field reaches the left interface of the super-lens, its amplitude will increase exponentially when passing through the super-lens, and then exponentially decay again when it leaves the lens. As the index of the super-lens is well matched with the surrounding media (we assume air and n = 1), the field strength will have the same amplitude in the image plane and in the object plane and thus contribute the correct amplitude for that high spatial frequency, toward forming a perfect image.The exponentially decaying-increasing-decaying field has a coefficient of exp (±kyx) where ky is real and positive, and the positive sign corresponds to within the lens while the negative sign to air. This process is illustrated by the red curves in Fig. 1. For a lithography application, there should be a reasonable operating distance between the recording resist and the super-lens which means that in practice, the lens thickness should be as large as possible, e.g. 1 micron. However, the exponentially increasing evanescent field property is ill-conditioned, and a vanishingly small input field on the left of super-lens may be amplified in an unstable fashion if the thickness of the lens or the transverse wave vector ky is too large, since kyx is exponentiated. Over amplification (in a numerical experiment) will destroy the superresolution we seek and require the equivalent

of a regularization step, as used in numerical superresolution or other inverse problems. This can be achieved by applying a proper cutoff to ky (or possibly introducing a very low level of loss) and limiting the thickness of the lens. In our calculations below, we fix the thickness of our super-lens at 200 nm for this reason while arguing that this is still a practical thickness to use. B. Results and discussion

First, we use a perfect electrical conductor(PEC) as the mask layer with thicknesst = 0, and TE waves as the illuminating source.The electric field intensity pattern from the mask forthe case of incoherent illumination is shown in Fig. 2 (a), and here we suppose no lens is inserted after the mask. We can see that three bright intensity spots are well resolved and can be used for contact lithography. Next, we introduce the planar lens with n = -1 into the simulations. It is impossible to simulate a truly perfect situation since thiswould require that we include all of the ky components generated by the mask, i.e. -∞ < ky <+∞. Also, a real negative index metamaterial will have intrinsic properties that can only support a limited range of ky increasing exponentially. We defined this range as (-kc <ky <+kc), where kc could be determined by the finite size of the unit cell of metamaterials, or unavoidable material loss [4, 6]. Using sub-wavelength features in the mask, then we need an appropriate range of ky in order to achieve a satisfactory transfer of the mask’s features to the image plane. A larger cut-offkcmeans a higher resolution along they direction. Fig. 2 (b), (c) and (d) show the intensity distribution patterns for kc = 2k0 ,kc = 5k0and kc = 10k0 , respectively. As there is an exponential amplification of these high k fields in the lens, the field intensity for each ky ∈ [kc, kc] will be very high at the output interface (x = 0 nm) and then decay to their correct original value to be the same in the image plane as at the object plane (x = -300 nm). For easier observation these overly amplified fields are shown in dark red. From Fig. 2 (b) - (d) we can see that as kc increases, the image is increasingly more like the field pattern at the exit of the mask (Fig. 2(a)); the more components of ky that contribute to the image plane the better one canrepresent the sub-wavelength features.

Figure 3. Distributions of patterns of electrical intensity |Ez|2 for lossless case: (a) PEC mask without super-lens; (b) kc = 2k0 ; (c) kc = 5k0 ; (d) kc = 10k0 ; (e) kc = 5k0 with mask contacting the super-lens. Figure (f) showsthelossy case when kc = 5k0 and δ = 0.001, and the transmission coefficient of thesuperlens is shown in (g) for δ = 0.001 (blue circle curve) and δ = 0 (red curve, lossless case). The xcoordinates correspond to that shown in Fig. 1.

A photoresist layer in the image plane records the transfer of the field originating from the mask pattern. For better operational freedom, the larger the distance between the surface of the lens and the surface of the resist, the better. To achieve this we can move the mask closer to the lens or even place it in contact with, or fabricate it combined with, the negative index lens. This then provides an operating distance equal to the total thickness of the negative slab. Here, we show such a case in Fig. 2 (e), when the mask is placed at x= -200 nm (i.e. in contactwith the lens). We can see that the recovered image of

the mask appears at x = 200 nm without any other differences from (c).

Despite efforts to introduce gain into metamaterials, we expect that some loss cannot be avoided. To model this for the superlens, we assume the medium’s permittivity and permeability have imaginary parts ε = -1 –δi, µ = -1 –δi. The imaginary parts of the permittivity and permeabilitybehave in a similar way to the clutter from the ky components shown above, for which a range of evanescent waveswill remain amplified while the higher components will not. We study the case of δ=0.001, and the transmission coefficient of the negative index slab is plotted in Fig. 2 (g) vs. ky (normalized by k0) as the x axis. We can see that for the lossless case the amplitude of the input field increases exponentially all the time ky is increasing, while for the lossy case there is initial exponential growth but then exponential decay when ky is beyond a certain threshold that is determined by the value of δ. The electric field intensity distribution is plotted in Fig. 2 (f), and we see that the resulting intensity patterns are better than those in (b) with kc = 2k0, because the threshold of cutoffky is larger than 2k0 (see Fig. 2 (g)). This confirms that: (1) loss in the negative index slab leads to a cutoff in the amplified ky components and (2) if we can control the loss of the “perfect” lens to keep it at a low level (such as using some gain medium in the metamaterial), the image qualityas well as the resolution, will be acceptable for lithography.

We also studied the case of using a Cr layer for the mask since this is a traditional mask choice. We set the Cr layer thickness to be t = 50 nm, and the permittivity of Cr at 365 nm is -8.6–9i [8]; the other geometric parameters are kept the same as in Fig. 1. The electric field intensity and magnetic field intensity are plotted in Fig. 3 (a) and (b), respectively. Here we set kc= 5k0 and the results show that the recovered image for the TE illumination case is better than that using TM illumination. The reason is that the well known surface plasmon polaritonsare excited by the diffracted TM wave. The additional surface wave at the surface of the mask will also be copied to the image plane. The surface wave on the mask surface is excited not by one but actually two surface modes when the Cr mask is thin, since there will be the so called long-range surface plasmon mode and short-

Figure 4. Calculated patterns of |Ez|2 ( |Hz|2 ) for TE (TM) illumination when using 50 nm thick Cr layer as the mask.

range surface plasmonsall associated with the many high-ky components generated from scattering. Consequently, these two special ky components, corresponding to the surface waves generated by the Cr layer, are over amplified and these can distort the image of the mask. We note that such a distortion is not too serious, the reason being that the imaginary part of the permittivity of Cr is quite large (even larger than the real part) and this forces the surface waves to damp very quickly thus diminishing the effects of the unwanted surface waves.

III. CONCLUSION To summarize, we investigated the feasibility of

achieving the transfer of subwavelength geometrical features encoded by a mask for the purposes of lithographic replication. While there are many practical difficulties still to be addressed, a low loss metamaterial with a negative index close to n = -1 can provide a one-step physical mechanism for this purpose. The thicker one can make a such a metamaterial with reasonably flat faces, then the better its performance. Since the metamaterial will itself likely be comprised of a homogenized 3D array of subwavelength abstracted circuit elements, such as split ring resonators, it makes sense to pattern the mask whose features are to be replicated,

at the same time, on one face of negative index slab. In this way the maximum operating distance between the face of the lens and the surface of the resist, equal to the thickness of the slab, will be obtained. In addition to developing metamaterials with optimized properties of the kind described, on-going studies are considering how to best manage the propagating and evanescent waves in order to improve the geometrical correspondence between a practical subwavelength-scaled mask and the scattered field immediately behind it. This scattered field is transferred to the surface of the photoresist and has to provide an exposure that results in an image of the mask. Of course, there is an identical need in the application of a negative index slab purely for the purposes of imaging. This also requires an improvement in the correspondance between the distribution of subwavelength index fluctuations in a thin penetrable 3D object of interest and the exiting near field distribution that gets transferred to the image plane. A greater challenge is to manage the illumination and subsequent evanescent wave amplification in order to achieve subwavelength resolution of index inhomogeneities in a thick 3D object. An interesting condition to be defined is how the subwavelength scale of the features in the metamaterial have to be reduced below that of increasingly smaller scale features within the object that we wish to image.

ACKNOWLEDGMENT The project is supported by the National Research Foundation, Singapore (NRF-G-CRP 2007-01).

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