1. Introduction

Advances in super-resolution optical imaging are being aggressively pursued for applications in imaging, data storage, and lithography. In biological imaging, for example, recent advances have seen reports of molecular-scale resolution using a range of nonlinear and stochastic processes [1,2], and commercial instruments are now available. For other applications—such as lithography imaging that is the subject of this paper—analogous techniques are not readily applied and alternative means of achieving super-resolved imaging must be developed.

The semiconductor industry has historically reduced the feature sizes used in integrated circuits by a factor of 2 every 4 years or so. The most important feature to reduce is the pitch of a repeating pattern, since this determines the packing density of transistors in the circuit. The resolution (R), defined as the minimum half-pitch of a pattern than can be imaged in a single exposure in a particular optical lithography system, is limited by the ratio of the wavelength (λ) to the numerical aperture of the imaging tool (NA):

To further improve resolution, one must lower the wavelength or increase NA. Today, a wavelength of 193 nm is common for semiconductor lithography and lowering it has proven difficult in manufacturing; for example, extreme ultraviolet (EUV) lithography, using 13.5 nm wavelength, is on the very near horizon for manufacturing but has taken considerable time and cost to develop. Increasing NA is also difficult, being limited by the lowest refractive index in the optical path on the wafer-side of the imaging tool. Today, water is used as the immersion fluid, giving a maximum practical NA of ~1.35. Higher index materials are employed in what is known as a solid-immersion system [5–10]. The limiting refractive index is then that of the photoresist, constraining NA to ~1.6. Higher NAs are extremely attractive and feasible but put the system into the evanescent regime. Unfortunately, evanescent images decay exponentially into the resist, leading to images that are practically useful only for extremely thin resists, making pattern transfer difficult.

Here we report new principles, designs, and experimental demonstration for a method whereby coupling of the evanescent component to a resonant surface state on an underlying medium can be used to significantly enhance the depth of focus of evanescent images. We analyze and experimentally verify our new techniques using a simple interference lithography (IL) arrangement, but the principles are adaptable to more-conventional lithography or other ultra-high resolution imaging techniques.

The rest of the paper is structured as follows. In the next section we review conventional IL techniques and introduce a solid-immersion Lloyd’s mirror interference lithography (SILMIL) technique. We have already developed a SILMIL testbed for performing evanescent interferometric lithography (EIL) and have reported its basic performance characteristics elsewhere [11]. Following this we analyze the use of surface states on an underlying interface to enhance the usable image depth of EIL, showing that there are three distinct regimes worthy of exploration. Idealized and practical examples are then presented, followed by experimental results demonstrating the technique for pattering 55-nm half-pitch lines with 100 nm image depth using 405 nm wavelength light.

2. Interference Lithography in the Propagating and Evanescent Regimes

Interference lithography is a straightforward way to achieve the lowest possible value for the resolution factor k₁ (k₁ = 0.25), with the simple principle to interfere two plane waves travelling at the opposite ends of the available NA in a given optical system [3]. The aim of research in IL is to be able to fabricate high resolution structures with high aspect ratios for large area patterning; this is a well-developed field in which many groups have made their contributions to fulfilling either one or more of these requirements in a variety of configurations [12–15].

Figure 1(a) illustrates a simple scheme for two-beam IL carried out in air upon photoresist in which case the NA of the interference pattern is less than one; the resultant periodic pattern has a period p = λ/2sinθ = λ/2NA (not super-resolved) and infinite image depth provided that the resist is lossless. This illustration is for transverse electric (TE) polarization, as defined in Fig. 1(a), which results in deep nulls in the interference pattern, but the use of transverse magnetic (TM) polarization is also possible [16]. In either case the image depth will be determined by the Lambert-law absorption decay length of the interference pattern in the photoresist, which may be many microns or tens of microns.

 

Fig. 1 Schematic representations of (a) basic interference lithography carried out with air as the ambient medium, (b) inside a prism at a high numerical aperture, and (c) in a prism at an ultra-high numerical aperture (NA), with θ beyond the critical angle at the prism-resist interface. These illustrations are for transverse electric (TE) polarisations, with the electric and magnetic field directions defined in part (a). Electric and magnetic field orientations are exchanged in the transverse magnetic (TM) configuration.

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We are interested in exploring the ultimate limits of resolution at a particular wavelength, for which using liquid- or solid-immersion schemes as depicted in Figs. 1(b) and 1(c) are advantageous. Figure 1(b) demonstrates what we call the high-NA regime in which the numerical aperture exceeds that of air (NA > 1), but is less than the refractive index of the photoresist (n_r)—in this case imaging is through propagating waves and image depth is also unlimited as in the ideal case. The practical disadvantage is the need to use a prism with a liquid immersion layer in intimate contact with the resist, or to use careful gap control (in the nanometer range) in a solid-immersion system [5]. For the ultra-high NA scheme (NA greater than the resist refractive index, but less than n_p, the prism index) of Fig. 1(c) resolution is further improved, however now the fields in the resist are evanescent and image depth becomes a severe practical constraint [5, 7–11].

Evanescent Interference Lithography (EIL)

The maximum-achievable resolution for the system will be achieved at ultra-high NAs, in the EIL regime [Fig. 1(c)]. There has been a report of deep-UV imaging of 26 nm half-pitch features at 193 nm [5], corresponding to NA = 1.85, the highest reported for immersion EIL. We have also reported imaging sub-100 nm features at 405 nm using a low-cost solid-state laser source [11], making the nanolithography process accessible to a wider research community. However in both reports to date image fidelity and image depth has been poor due to the evanescent nature of the interference pattern.

Imaging at NAs larger than the index of the resist is only possible within the resist due to frustration of the evanescent field back into its propagating state [9]. This phenomenon has been rigorously analyzed by Zhou et al. [9] to show that the effective index and extinction coefficient of the resist are in fact a function of the NA when operating in the evanescent regime; this is indeed a very interesting result and confirms why it is possible to create an image within the resist at greater than the critical angle.

To recover and enhance an evanescent image deeper into an imaging layer we propose and analyze here the use of sub-resist reflectors. While the principles and derivations are presented for both the transverse magnetic (TM) and transverse electric (TE) polarizations of light, the feasible designs and experimental demonstrations are presented only for the TE polarization of light as this is the polarization that is desired by the lithographic industry. Achieving fine control of an individual component of the evanescent spectrum as such has tremendous potential and would make this technique considerably useful and attractive to a larger audience in the lithographic community.

The novelty is that such underlying media can be implemented using readily available materials and standard thin-film deposition techniques; the solution is a feasible one which has not been considered before. We provide design examples at λ = 193 nm and experimentally demonstrate the effect at a wavelength of 405 nm, which has been used to create 55-nm half-pitch structures with a considerably higher aspect ratio than is otherwise possible using evanescent interference lithography.

3. Surface states to improve image depth in EIL

Consider Fig. 2 that depicts a schematic of a solid-immersion IL system constituted of three separate materials, namely the prism, the resist, and an underlying substrate. It is by appropriate choice of the underlying substrate, either as a single material or as an appropriately-designed multilayer stack, that surface states can be induced on the lower resist-substrate boundary to provide significant image improvements. The principle is outlined in Fig. 3, which depicts how an evanescent field permeates the photoresist film (blue long-dashed line) and how this field may then induce a ‘reflected’ image (green dashed line) using a carefully chosen substrate, so that the resultant image field (red solid line) has higher intensity and less variation with depth compared to the original evanescently decaying image. The advantage in enhancing the otherwise decaying field in such a way is the ability to expose the photoresist to a greater depth and with a higher contrast and hence create taller or higher aspect ratio structures, which are needed to allow pattern transfer in lithographic applications.

 

Fig. 2 Schematic of solid immersion two-beam interference lithography.

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Fig. 3 Schematic diagram showing field formation within resist and motivation to consider evanescent field interaction with an underlying substrate.

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The evanescent enhancement may be accomplished by making use of a surface state to couple onto the decaying field. Upon careful choice of the underlayer, boundary conditions are modified and such coupling allows for the extraction of energy from the incident beam that would otherwise have reflected away at the prism-resist interface. This energy is then redistributed within the resist cavity in a uniform manner to allow a high depth of focus image to form. One can see from Fig. 3 how this results in a higher effective dosage due to energy extraction and redistribution. Finally a super-resolution pattern may be created that has a sufficiently high aspect ratio to be transferrable and be of practical use.

4. Fresnel reflection analysis and fictitious-metal underlayer example

The reflective enhancement is a direct result of the coupling of the evanescent field to the substrate; this coupling can be analyzed using Fresnel reflection analysis. Under optimal conditions, such coupling is encouraged via the formation of a surface state, which again is naturally incorporated in a Fresnel reflection analysis in the evanescent regime. The most commonly considered surface state for super-resolution imaging is that of the surface plasmon polariton (SPP) [17–19], which occurs on a metal-dielectric interface, so that case is considered here first. It turns out that this is not the optimum surface state to consider for super-resolution EIL, as will be described later.

The Fresnel reflection equations (derived using Maxwell’s equations at the boundary of two media) may be used to discover what substrate materials encourage such coupling. For TM polarization of light, this is achieved using metals or lossy dielectrics as the substrate materials, which support SPP surface states. The case of a high-loss dielectric underlayer is also interesting to consider for TM polarization, as it supports the so-called surface exciton polaritons (SEP) surface state [20–22]. Although this surface state has not received the same attention as the SPP surface state for super-resolution imaging, we note this point here for future consideration—we have undertaken preliminary studies on SEP-enhanced EIL, but will report those elsewhere.

Evanescent-wave reflections from metal underlayers can provide strong field enhancements, particularly close to surface-plasmon resonances, and hence allow the evanescent decay of the original imaging field to be countered. An analytical case study is presented here using a fictitious metal to illustrate the point. It is to be noted that silver has been proposed as a reasonable candidate for such enhancement at 365 nm wavelength [11, 23]. In fact, the superlens suggested by John Pendry [24], and demonstrated experimentally using thin silver films [25–29], also utilizes this principle; enhancement is achieved via field redistribution by coupling to surface states on both sides of the superlens allowing energy extraction and redistribution from the source apertures to photoresist.

Two SPP-enhanced EIL scenarios are illustrated in Fig. 4, alongside the non-enhanced intensity decay of a transverse-magnetic (TM) polarized evanescent-wave image [Fig. 4(a)]. Finite-element method (FEM) numerical simulations, using a fictitious metal underlayer (permittivity ε = −29.8 + j0.0), is used to set the scene for the approach we present here. Considering Fig. 4, the evanescent decay in the simple index-matched system of Fig. 4(a) is seen to be overcome, or even overwhelmed, by resonant coupling to SPP states [Fig. 4(b) and 4(c)]. The difference between Figs. 4(b) and 4(c) is a small change in the NA of the imaging condition (1.79 compared with 1.85 respectively) illustrating how sensitive this resonant phenomenon is to NA, and how this tuning can be used effectively to tailor the final field profile in the imaging (resist) layer.

 

Fig. 4 Evanescent image (TM light) in 82.5-nm thick resist, λ = 193 nm with (a) resist underlayer, fictious metal (ε = –29.8) underlayer for (b) optimal off-resonant enhancement (NA = 1.85), and (c) non-optimal resonant enhancement (NA = 1.79).

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The underlying principle is energy extraction which would otherwise totally internally reflect back into the prism and its redistribution in the resist cavity. The principle is evident by reference to the attenuated total reflectance (ATR) spectra of Fig. 5, shown for the fictitious metal case of Fig. 4; setting the operating NA around the resonant dip in the ATR spectrum (NA = 1.79) allows fine control of the field profiles to produce a symmetrical intensity distribution [Fig. 4(b)] at NA = 1.85.

 

Fig. 5 Attenuated total reflectance (ATR) spectra (using analytical transfer-matrix calculations) for the fictitious metal reflector of Fig. 4(b) and 4(c).

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5. TE polarization analysis and fictitious ‘gain material’ solution

The SPP- and SEP-enhanced evanescent imaging scenarios for TM polarisations are not the only ones that can be considered. A cursory analysis of the TE Fresnel equations [30] reveals no naturally occurring surface states for TE polarization, which is a disappointing initial finding, as the deep interference null possible with TE polarization makes it much preferred for lithographic imaging [3]. However, further analysis of the Fresnel equations reveals that if a material were to have a negative loss (i.e., a material that causes a wave to grow in intensity as it propagates) then such an enhancement would be achievable with TE polarization as well (see Appendix and [30]). Although passive gain media cannot be found naturally, it may still be possible to reproduce the desired reflection using effective medium theory (EMT) or metamaterial designs. EMT allows the realization of certain non-naturally occurring properties by creating a composite material, for example by layering several passive materials as we see in the next section. Such a finding has not been previously reported in literature to the best of our knowledge, and the concept is explored both theoretically and experimentally here.

To summarize, we have stated (with supplementary material in the Appendix) that metals, lossy dielectrics and hypothetical gain media would all support evanescent reflection enhancements (reflections greater than unity) within a low loss dielectric (such as photoresist) when using the TM polarization of light. Such an enhancement may only be achieved with hypothetical gain media, or a metamaterial with effective-gain properties, when using TE polarization. It is well known that both metals and lossy dielectrics are capable of confining and enhancing an E-field at their interfaces by making use of SPP and SEP resonances, and the conditions for achieving effective-gain surface states are developed here. The details of these various conditions are shown in Table 1.

Table 1. Substrate properties required for surface-state EIL enhancement, for TM and TE polarized light

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6. Effective gain medium (EGM) solution

Consider a high-index dielectric capable of supporting an ultra-high NA in a propagating form. Like a waveguide, the fields may be contained within it by satisfying relevant boundary conditions, i.e., if the high-index medium (n_hi > NA) is sandwiched between low-index media (n_lo < NA) that cannot support the ultra-high NAs. Control over thickness and index of the high index medium allows selection of the spatial frequency that is resonated. Figure 6 illustrates this arrangement with approximate field profiles within the media. The ATR spectrum would also show the characteristic dip at resonance, similar to that shown in Fig. 5. In the SPP case, electric fields are confined and enhanced at the interface while decaying into surrounding media [Fig. 6(a)]. Here, a similar purpose is fulfilled by a pseudo-interface [Fig. 6(b)].

 

Fig. 6 Evanescent wave enhancement (a) at a metal-dielectric interface through SPP resonance, and (b) at a pseudo-interface formed by sandwiching a high index dielectric between two low-index dielectrics.

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The high index on low index stack is in fact an EGM provided we concern ourselves with only reflection at the particular NA. The resonant spatial frequency here is the effective surface state. It should be noted that while the EGM presented here mimics the workings of a waveguide, other EGMs may also be created for the TM polarization of light using several layers of metals, lossy dielectrics and dielectrics. Such a setup would then support a composite surface state that would create an effective gain medium for the purposes of an enhanced reflection.

7. Design and implementation

Proposed design for industry standard λ = 193 nm

Figure 7(a) illustrates imaging of 26-nm half-pitch features where the image depth is only 20 nm into thick photoresist (n = 1.7 + i0.02). Figure 7(b) illustrates the significant enhancement when a tuned stack of 43 nm layer of Al₂O₃ (sapphire) with index n₃ = 2.08 [11] upon 50nm SiO₂ with index n₄ = 1.56 is used. The resulting image is 82.5 nm deep. We have also included photoresist loss in this design and with this the new reflection from the real-world stack is calculated as 6.77e^i0.212 instead of the optimum 7.39. This design, however, still allows ultra-high NA patterning at an aspect ratio of ~3.2. For the purposes of evanescent wave coupling at an ultra-high NA of 1.85, the stack behaves the same way as a hypothetical gain medium. The ATR spectrum of the system is shown in Fig. 8, together with the ATR spectrum for the fictitious-metal case of Fig. 5. As well as being a much sharper resonance, the ATR spectrum for the EGM case has a resonant dip at a spatial frequency corresponding to a NA of ~1.83 indicative of the off-resonance operation mentioned earlier.

 

Fig. 7 TE imaging of 26-nm (half-pitch) evanescent features into (a) semi-infinite lossy resist giving 20-nm image depth, and (b) 82.5 nm thick lossy resist on an effective gain medium made up of 43 nm of Al₂O₃ (Sapphire) on SiO₂, giving an image depth of 82.5 nm.

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Fig. 8 Attenuated total reflectance (ATR) spectra (using analytical transfer-matrix calculations) for the fictitious metal reflector of Fig. 4 (b) and 4(c) with TM light (solid line), and an artifical-gain-medium reflector of Fig. 7 with TE light (dashed line).

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Implementation at λ = 405 nm

An experimental demonstration of EIL image enhancement effect is presented in this section, using a 405 nm wavelength that will be more readily accessible in research laboratories. A solid immersion Lloyd’s mirror interference lithography (SILMIL) [11] system was used with a Yttrium Aluminium Garnet (YAG) prism at λ = 405 nm and a NA = 1.824. Since the index of the photoresist at this wavelength is n = 1.68 + j0.031, the NA is high enough to create evanescent images.

Two experiments are compared, with the imaging layers for each one shown in Fig. 9. The first experiment involves imaging into a single 450-nm thick layer of resist [Fig. 9(a)], which acts as a control. The thickness of the resist in the control sample was large enough to prevent reflections of the evanescent wave. The results are then compared with an optical stack that uses hafnium oxide (HfO) as the high index dielectric stacked upon silicon dioxide (SiO₂) as the low index dielectric [Fig. 9(b)]. The HfO upon SiO₂ stack is carefully designed to result in an enhanced reflectivity and hence serves as our EGM. Both imaging stacks were created on quartz substrates to ensure that the mechanical properties of the optical stacks were the same, as the stacks must deform similarly to the mechanical stress that is applied within the SILMIL system.

 

Fig. 9 Optical stacks for imaging at λ = 405 nm and NA = 1.824, including a water-soluble poly vinyl acetate (PVA) layer that is used as a barrier between the index matching liquid and the photoresist. (a) Without an EGM underlayer for conventional EIL, and (b) with an EGM underlayer (characterized by ATR earlier) for high aspect ratio imaging.

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Control experiment results – no underlying substrate

Exposure dose is an important parameter to control in these experiments, however the solid-immersion and evanescent nature of the imaging makes absolute dosage measurement difficult. To simplify dosage measurement, a pseudo-dosage (PD) parameter was used, whereby the intensity of the beam incident on the prism in mW/cm² was multiplied by the exposure time in seconds, resulting in a PD measured in mJ/cm². This PD does not take account reflection losses at prism-air and prims-resist interfaces, however it does allow images to be compared between control and reflection-enhanced scenarios (Fig. 9) for identical exposure conditions.

An example scanning electron microscope (SEM) scan of a control image exposed at NA = 1.824 and a PD = 257 mJ/cm2 is presented in Fig. 10, showing that deep sub-wavelength resolution can be achieved using the SILMIL solid-immersion scheme in the ultra-high NA regime. The 55-nm half-pitch spaces, compared to the 405-nm exposure wavelength, corresponds to better than λ/7 resolution.

 

Fig. 10 SEM plan views, at pseudo-dosage of 257 mJ/cm², NA = 1.824, λ = 405 nm, conventional EIL using SILMIL, resulting pitch ~111 nm (55.5 nm half-pitch)

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Representational cross-sections through this image are shown in Fig. 11 illustrating the shallow nature of such non-enhanced images, of 30-40 nm, as has previously been reported [11]. Control images were obtained over a wide range of exposure doses, with PD varied from 0 to 1540 mJ/cm² in 86 mJ/cm² steps, and the images presented here are the cleanest and deepest that were obtained. For lower PD the image depth was shallower than that of Fig. 11, and for higher doses image fidelity was lost (with no increase in image depth) due to unintended exposure in the imperfect nulls in the interference patterns. The limits that the evanescent nature of the imaging imposes on the maximum aspect ratio of the final resist lines (0.72 height-width ratio in this case) presents severe limitations for subsequent use of these lines for pattern transfer—aspect ratios of greater than unity are normally required for this.

 

Fig. 11 Cross-sectional views, at NA = 1.824, λ = 405 nm, conventional EIL using SILMIL, resulting pitch ~111 nm (55.5 nm half-pitch) at pseudo-dosage (PD) of 257 mJ/cm², giving 30-40 nm image depths. The (a) and (b) SEM scans are at different positions on the sample.

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EGM experiment – HfO on quartz underlayer

The schematic for the stack using the EGM is depicted in Fig. 9(b). Modeling results for this stack are presented in Fig. 12, compared to the control, showing that the underlying medium in the new optical stack has a considerable impact on the depth of focus of the evanescent interferometric pattern when compared side by side with the imaging stack with no underlying substrate.

 

Fig. 12 Finitie-element simulation of imaging with the stacks in Fig. 9 for NA = 1.824, λ = 405 nm, into (a) semi-infinite resist, and (b) 105 nm resist with an EGM underlayer.

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It is clear from Fig. 12(b) that the EGM is not perfect, as the image intensity is not symmetrically distributed vertically through the resist layer. However it suffices to demonstrate proof of concept as we see from the experimental AFM and SEM scans that are depicted below. Figure 13 depicts AFM scans of structures that have been created using the imaging stack in Fig. 9(b) at a pseudo-dosage of 214 mJ/cm². These scans demonstrate a considerably large image depth of 90 - 105 nm.

 

Fig. 13 – AFM scan demonstrating EIL with SILMIL using the imaging stack of Fig. 9 (b), high aspect ratio (~1.8) structure imaged at a NA of 1.824 at λ = 405 nm with pattern half-pitch ~55.5 nm. The average depth measured using AFM software was 100 nm. (a) a 2 µm by 1 µm AFM scan, and (b) a corresponding 2D-like perspective view to depict the tall standing structures.

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To compare with the control samples (Fig. 11) cross-sectional SEM images are presented in Fig. 14, again showing image depth of more than 90 nm compared with the weak, shallow images from the control samples. The underlying HfO layer can be seen in these SEM images (marked in Fig. 14), and it is noted that full clearance of the resist down to the underlying substrate has not been achieved for all the lines presented in this image. Difficulties with sample cleavage and charging on the non-crystalline, insulating substrates used for these studies make this cross-sectional SEM imaging challenging, but it is acknowledged that this imperfect development may be a result of imperfect coupling into the HfO resonant underlayer in this part of the sample. Further optimization of the imaging stack will be required to improve image quality and overcome this issue.

 

Fig. 14 SEM cross-sectional views showing tall standing structures, at PD of 214 mJ/cm², NA = 1.824, λ = 405 nm, EIL using SILMIL using the imaging stack of Fig. 9 (b), resulting half-pitch ~55.5 nm, with pattern depth ~96 nm .

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There is clear indication that the use of an EGM has made a significant difference. This is due to energy extraction and redistribution right down to the bottom of the resist which was facilitated through near-field interaction with the EGM, i.e., evanescent coupling of the image with the EGM.

Dose variation and resist collapse – proof of high aspect ratio imaging

As with the control images, EGM-enhanced image exposures were performed over a range of pseudo dosages (PD = 86-684 mJ/cm² in 86 mJ/cm² steps). Image depth, as measured from AFM scans, increased linearly for PD from 86 to 257 mJ/cm², at which point full development through the imaging resist layer was achieved. For higher doses the resist depth decreased due to background development in the imperfect interference nulls, accompanied by the appearance of a phenomenon previously unreported for evanescent-field imaging—resist pattern collapse. This is illustrated in Fig. 15, which shows AFM scans for samples exposed at PD = 342 mJ/cm², where there is clear indication of resist collapse in some areas of the sample.

 

Fig. 15 AFM scans demonstrating resist collapse using the imaging stack of Fig. 9(b), for an exposure at PD = 342 mJ/cm² at a NA of 1.824 and λ = 405 nm with pattern half-pitch ~55 nm. (a) A 5 µm by 5 µm AFM scan showing some resist collapse, and (b) a large area scan cropped to 5 µm by 2.5 µm demonstrating greater resist collapse, a result of over-dosage/exposure/development.

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Resist collapse is a side-effect of over-exposure/development, which is generally undesirable. Nevertheless it is very interesting to report here, as resist collapse requires a very high aspect ratio (typically greater than 1.5:1) and hence such collapse with EIL has never been thought possible earlier. The AFM scans in Fig. 13 demonstrate aspect ratios as high as 1.8:1, where better process control has resulted in lines that have not collapsed.

It is necessary to also point out here that resist collapse is also due to the amplification of the imperfect minima (nulls) in the interference pattern. Conventionally with interference lithography, imperfect nulls only contribute to the background exposure; however, as the effective reflection coefficient from the EGM amplifies/enhances all signals (even the imperfect minima of the interference pattern), and redistributes energy to the bottom of the resist, collapse takes places. For future work, and for application of the method for nanofabrication, resist collapse should be avoided—this can be achieved by careful control of the interfering beams to improve the nulls in the interference pattern, together with good process control (exposure dose, development time and low surface-tension solvent evaporation).

8. Conclusion

Ultra-high NA patterning has long been a critical outstanding problem due to the physical limits presented by evanescent fields. We have suggested coupling of the evanescent fields to surface states as a means to control the field intensity and profile in a photoresist cavity. Through the use of the Fresnel reflection equation for TE and analysis of fields, we have presented a practical way to couple the evanescent field into a waveguide-like underlayer to image super-resolved structures with high aspect ratios. We show how a high aspect ratio may be achieved at NA = 1.85 and λ = 193 nm in photoresist by using a layer of sapphire deposited on SiO₂. The method encourages use of solid-immersion systems and paves the way for feasible ultra-high-NA imaging and patterning.

An experimental demonstration of the method at λ = 405 nm shows imaging at an ultra-high NA of 1.824. While the imaging depth for a thick resist was measured to be only ~30-40 nm maximum for a set dosage, the use of a hafnium oxide on silicon dioxide stack served to increase this to the range of 90-105 nm. Careful monitoring of the refractive index and thickness of the hafnium oxide film is expected to result in a better design that is likely to give image depths of ~160 nm under the same circumstances.

Such results are the first of their kind to be reported, and with the use of a more robust dry or immersion interference lithography system such as the Amphibian^TM [10] this could also be demonstrated at λ = 193 for a NA of 1.85, resulting in sub-30 nm high-aspect-ratio imaging at low cost over large areas.

9. Appendix

In this section an analysis is presented of the Fresnel reflection equations for both the TM and TE polarizations of light in the near-field reflection regime relevant to this paper. The analysis is used to deduce which substrates would allow a reflection larger than unity in the evanescent regime. Further details and derivation can be found in Ref [30].

The various terms and relationships used or mentioned in this section are listed in Table 2, where l is used to refer to a particular layer. The analysis is limited to non-magnetic media due to the lack of availability of magnetic materials at optical frequencies and is also limited to the magnitude of the Fresnel reflection, and not its phase. Media 1, 2 and 3 are indicated in Fig. 2 and these will also be referred to in the text. The equations relating the transmission and reflection coefficients of the x, y and z components of the E-field for the TM and TE polarizations at the interface between two media ${\epsilon }_a$ and ${\epsilon }_b$ are tabulated in Table 3 and Table 4 [31]. Here, transmission denotes the transmitted amplitude into medium b from medium a and reflection denotes the reflected amplitude back into medium a from medium b.

Table 2. Definitions and identities used in this chapter

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Table 3. E-field Fresnel transmission and reflection coefficients at an interface between medium a and b for TM polarization. The transmitted field, E_t, and reflected field, E_r, are considered in total (no additional subscripts) and by their x or z components (with x or z subscripts respectively).

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Table 4. E-field Fresnel Transmission and Reflection coefficients at an interface between medium a and b for TE polarization. The total field E only has a y component E_y in this case.

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TM Fresnel reflection

Firstly, the TM Fresnel reflection equation is analysed for conditions that will give rise to a reflectivity r > 1, which is required for the resonance-assisted lithography technique to be effective. The TM Fresnel equation can be derived as (see [31] for details)

(2)$$r_{23,TM}=\frac{{\epsilon }_3{k}_{z,2}-{\epsilon }_2{k}_{z,3}}{{\epsilon }_3{k}_{z,2}+{\epsilon }_2{k}_{z,3}}.$$

For a reflection with larger than unity magnitude, $\left|r_{TM}\right|>1$,

(3)$$\left|r_{23,TM}\right|=\frac{\left|{\epsilon }_3{k}_{z,2}-{\epsilon }_2{k}_{z,3}\right|}{\left|{\epsilon }_3{k}_{z,2}+{\epsilon }_2{k}_{z,3}\right|}>1.$$

A finite loss in the photoresist is first assumed, such that ${\epsilon }_2$ has both non-zero real and imaginary components. Expanding Eq. (4) in its complex form and rearranging gives [30],

(4)$$\begin{array}{c}{k^{\prime }}_{z,3}\left({{\epsilon }^{″}}_3{k^{″}}_{z,2}{{\epsilon }^{\prime }}_2-{{\epsilon }^{\prime }}_3{k^{\prime }}_{z,2}{{\epsilon }^{\prime }}_2-{{\epsilon }^{″}}_3{k^{\prime }}_{z,2}{{\epsilon }^{″}}_2-{{\epsilon }^{\prime }}_3{k^{″}}_{z,2}{{\epsilon }^{″}}_2\right)>\\ {k^{″}}_{z,3}\left({{\epsilon }^{\prime }}_3{k^{″}}_{z,2}{{\epsilon }^{\prime }}_2+{{\epsilon }^{″}}_3{k^{\prime }}_{z,2}{{\epsilon }^{\prime }}_2+{{\epsilon }^{″}}_3{k^{″}}_{z,2}{{\epsilon }^{″}}_2-{{\epsilon }^{\prime }}_3{k^{\prime }}_{z,2}{{\epsilon }^{″}}_2\right)\end{array}.$$

Equation (4) is a general form including resist losses, but is not sufficiently simple to allow us to get any idea of the material properties required of substrate 3 to achieve resonance-assisted enhancement, which is the primary goal of this exercise. However, note that the loss or extinction coefficient in conventionally used photoresists is more than 10 times smaller than its refractive index, and close to a 100 times smaller than the resist index in several cases, so we can make the reasonable assumption that ${{\epsilon }^{″}}_2\ll {{\epsilon }^{\prime }}_2$ to proceed further. Under this assumption, which also implies that ${k^{\prime }}_{z,2}\approx 0$, since operation is in the evanescent regime of the photoresist, Eq. (4) then simplifies to

(5)$${{\epsilon }^{″}}_3{k^{\prime }}_{z,3}>{{\epsilon }^{\prime }}_3{k^{″}}_{z,3}.$$

Interestingly, Eq. (5) is now independent of the optical properties of the photoresist. However, this is so only if it is a lossless photoresist operating in the evanescent regime (so that n₂ < NA). If the conditions in the following paragraph are satisfied, then a method to achieve a reflection greater than unity in the evanescent regime will be found.

If ${{\epsilon }^{\prime }}_3<0$ and ${{\epsilon }^{″}}_3\ge 0$, then the right hand side of Eq. (5) is always negative and its left hand side is always positive, therefore satisfying it. If ${{\epsilon }^{\prime }}_3\ge 0$ and ${{\epsilon }^{″}}_3>0$, then Eq. (5) may still be satisfied if material 3 is sufficiently lossy. This indicates that the underlying enhancer could either be a plasmonic material (i.e. a metal, with ${{\epsilon }^{\prime }}_3<0$) or a positive dielectric with significant loss. These are important solutions and reinforce the prior knowledge that plasmonic metals [17–19] are possible candidates, but that other surface states such as the SEP [20–22] may also be used.

However, one may go beyond the well-known SPP and SEP solutions and a closer look reveals that Eq. (5) is also satisfied by a material with ${{\epsilon }^{\prime }}_3>0$ and ${{\epsilon }^{″}}_3<0$ provided ${k^{\prime }}_{z,3}>0$ and ${k^{″}}_{z,3}<0$. This requires a material with a negative loss (i.e. a gain medium requirement). Figure 16 shows the use of a complex plane plot with relevant quantities to illustrate this particular scenario. The relevant quantities are labeled and depiction of the figure is explained as necessary in the caption of Fig. 16, for example ${\epsilon }_3$ is labeled as Vector C and denoted by a dotted arrow.

 

Fig. 16 Complex-plane plot showing that Eq. (5) is also satisfied if the substrate ε₃ has a positive real part ${{\epsilon }^{\prime }}_3$ and a negative imaginary part ${{\epsilon }^{″}}_3$, depicted by Vector C. Although, this results in an argument (Vector B) that has two possible Roots (to compute the z-wave number k_z), namely Root 1 (Vector D) and Root 2 (Vector A); the correct solution is in fact Root 2 (Vector A) as this allows $k_{z,3}^{}$ to have a positive real part $k_{z,3}^{\prime }$ and a negative imaginary part $k_{z,3}^{″}$. This figure indicates the relative positions of the vectors required to achieve the desired solution.

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The quantity $k_0^2\left({{\epsilon }^{\prime }}_3+i{{\epsilon }^{″}}_3-N{A}^2\right)$ (Vector B) is the argument for which the roots are to be determined to compute k_z,3 so that the gain-medium requirement may be satisfied; this argument is denoted as a solid arrow. Like any complex number this quantity also has two possible roots (Vector A and Vector D), only one of which satisfies the gain medium requirements. The desired root is Root 2 as labeled in Fig. 16 as it is the only correct solution that allows Eq. (5) and the gain medium requirements to be satisfied.

The enhancement shown above using metals (or lossy dielectrics) is limited to the TM polarization. The TM polarization suffers from an increasingly lower contrast at higher NAs [3]. In addition, as we will show in a later section, the material requirements for NAs as high as 1.85 and possibly larger are met much more easily for the TE polarization of light, which is the industry’s preferred polarization for lithography. With this we proceed to solve the TE Fresnel reflection for a reflectivity > 1.

TE Fresnel reflection

It is well known that metals and lossy dielectrics are not a possible solution to support TE surface waves for evanescent field enhancement as the TE polarization lacks a normal component of the electric field. The following analysis confirms this result but we also seek new gain-like solutions. The TE reflection coefficient at the ${\epsilon }_2:{\epsilon }_3$ boundary is [31]

(6)$$\left|r_{23,TE}\right|=\left|\frac{k_{z,2}-k_{z,3}}{k_{z,2}+k_{z,3}}\right|>1.$$

Rearranging with real and imaginary terms and simplifying [30] yields

(7)$${k^{\prime }}_{z,2}{k^{\prime }}_{z,3}+{k^{″}}_{z,2}{k^{″}}_{z,3}<0$$

which, upon invoking the low loss resist assumption, gives

(8)$${k^{″}}_{z,2}{k^{″}}_{z,3}<0.$$

Of course it is known that ${k^{″}}_{z,2}$ must be positive owing to the fact that the photoresist is a real medium operating in the evanescent regime. This directly implies that for Eq. (8) to be satisfied ${k^{″}}_{z,3}$ must be negative. This is the solution for a growing evanescent wave in the substrate which again implies gain in the underlying medium. While a gain medium below the photoresist may seem at first unfeasible, there is a practical effective-gain-medium solution, involving the use of composite or multi-layered media, thanks to the evanescent nature of the waves, as used in this paper.

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