1. Introduction

There is increasing interest in developing nanolithography to fabricate ultra-small features on a variety of integrated optical and electronic devices [1–7]. As derivatives of photolithography – the most widely used manufacturing approach over the last several decades – optical near-field lithographic techniques were developed to overcome the diffraction limit of light. Beyond the far-field cutoff of spatial frequencies, the higher frequency support of an evanescent field through subwavelength apertures was exploited to achieve finer features as small as a few tens of nanometers in size [1]. However, a major technical concern raised was extremely low transmission through apertures much smaller than the wavelength of the exposing light [2]. The advent of optical nanoantennas [3, 4], and metallic subwavelength apertures specifically tailored to couple surface plasmons (SP), conferred substantial benefits over conventional aperture-based sources in terms of drastic increases in near-field throughput [3–7]. Nevertheless, the rapid decay of the evanescent field from the exit plane of such nano-sources could still be problematic [3, 4] due to shallow exposure depths available for only a few tens of nanometers in photoresists. Since the lithographic system setup and process have to be optimized strictly, this often renders lithographic processes hard to control and may result in poor contrast of fabricated patterns, eventually limiting the potential of nano-lithographic applications.

The entire process of photolithography, while technically complicated, can largely be divided into two processes: exposure and development [8]. Given that a photoresist layer is formed with a certain initial thickness, the process parameters to be controlled are essentially exposure dose and developing time, depending on the use of a particular photoresist characterized by exposure contrast and dissolution rate in development. The optimization usually focuses on determining the exposure dose window, whereas the developing time can be readily set by the maximum thickness to be cleared. In near-field lithography, however, exposure and development conditions yielding the optimal result differ [1, 4] from those required to transfer relatively large patterns with exposure light that can propagate through a thicker layer of the same photoresist without appreciable attenuation. Moreover, defining and optimizing relevant process parameters is even more challenging in practice, since exact theoretical predictions and non-perturbative measurements of the near-field distribution of metallic nanoantennas are difficult [9, 10] due to their high sensitivity to geometry and the local environment.

The intensity threshold method has been frequently used to estimate photoresist profiles by finding the isoexposure contour corresponding to the threshold dose of a photoresist [4, 11]. However, the method was found to yield a considerably inaccurate prediction when forming structures on the subwavelength scale [12] due to the oversimplified assumption of a stepwise photoresist response around the threshold. Recently, an improved model that takes a finite contrast of the photoresist response into account was suggested [13], enabling a simple analysis of photoresist profiles produced by the nanoscale exposure localized in both the lateral and longitudinal directions. The effects of photoresist contrast as well as exposure decay length were evaluated based on the patterning depth and width [13, 14]. Experimental verification was also performed with several different exposure doses [15], resulting in some discrepancy with the model, but qualitative consistency with the prediction.

We recognized, however, that the previous model has a fundamental flaw in the way it determines the photoresist profile. First, it does not account for saturation of the development rate above the dose-to-clear. Consequently, it might lead to erroneous removal depths even exceeding the initial thickness, which is obviously unphysical. A potential pitfall also exists in interpreting the lithographic process features. The term initial thickness used in the papers should not literally mean the physical thickness of a prepared photoresist layer. Instead, it reasonably indicates the removal thickness expected at the end of the developing process, assuming the given photoresist is fully exposed to at least the clearing dose level. Once photoresist is exposed, the pattern profile is determined by developing process, which is composed of the two important parameters; dissolution rate and developing time. Because dissolution rate is determined from the result of developer-to-photoresist interaction and developing time is a processing parameter, there should be a normalized parameter that accounts both. Therefore, we introduce a synthesized parameter of nominal developing thickness (NDT) to represent both of a dissolution rate and a developing time.

In implementing near-field lithography, finding an appropriate process to achieve this nominal thickness is difficult because of the evanescent nature of the exposure dose. This study builds on previous work in several respects. First, a more elaborate model is formulated to predict the photoresist profile with evanescent light exposure followed by development for a designated process time. From an aerial image of localized exposure dose, the dissolution rate distribution in a finite-contrast photoresist and the photoresist-developer interface marching level are modeled to derive an analytic formula that describes the end-point geometry of photoresists. Second, the impacts of relevant process control parameters are explored by assessing the geometric features of the photoresist including depth, width of the photoresist cross-section. The role of developing time was found to be two-fold as it affected the depth as well as the width of the engraved pattern. A comparison of our results to those of methods based on non-decaying exposure and the previously reported approach demonstrates qualitative and quantitative improvements in the prediction of lithographic features. Finally, based on the model proposed in this work, we determine the near-field distribution produced by a bowtie-shaped nanoaperture in a metal film, precisely controlling the parameters of near-field lithography.

2. Theoretical model for evanescent-field optical lithography

We develop a simple one-dimensional (1-D) development model for a photoresist exposed to evanescently decaying light fields that are localized to the nanometer scale. While the most accurate simulation of photolithography requires the entire fabrication process to be considered comprehensively [12], we make several assumptions in order to identify the critical process parameters and understand the behaviors characteristic to near-field lithography. First, the spatial distribution (aerial image) of exposed light in the photoresist is approximated by the irradiance at the entrance plane of the photoresist multiplied by a factor that accounts for the light’s evanescent decay along the direction of the photoresist depth with the exponential decay length. Second, we consider a positive-type photoresist with a dissolution rate that can be determined directly from the aerial image of exposure dose and is not affected by chemical diffusion or mechanical shrinkage due to post-exposure processes. The fundamental parameter characterizing the photoresist is its finite contrast with the maximum dissolution rate above the clearing dose. Finally, the removal of a photoresist is assumed to proceed only toward increasing depth, and the developing time physically limits the end point of the development process. The details of the theoretical model are described in the following subsections.

2.1 Near-field exposure

Near-field intensity distributions in the vicinity of metallic subwavelength apertures or plasmonic masks are complicated, and have to be calculated numerically via the finite-difference time-domain (FDTD) method, rigorous coupled-wave analysis (RCWA), and so on. The localized near-fields, however, usually have a smooth varying envelope on the nanometer scale in the lateral plane and decay exponentially along the direction normal to the exit plane [1, 6, 7, 14]. We assume an exposure dose $E_{\text{ex}}(x,y;z)$to be Gaussian in the transverse direction (x,y) and decaying exponentially in the direction of increasing depth (z) in the photoresist, in the form of

2.2 Photoresist parameters

A positive photoresist spun onto a substrate is basically a mixture of an alkaline soluble resin and a photosensitive dissolution inhibitor, often referred to as a photo-active compound (PAC) [8]. The PAC prevents the resin from being dissolved in an alkaline solution unless the photoresist is exposed to ultraviolet (UV) light of sufficient energy. Above a certain threshold dose $E_{\text{th}}$of exposure, however, the PAC undergoes photochemical change to render the resin soluble in alkaline developers. The dissolution rate $v_D$of the resin can be further enhanced with increasing exposure dose up to a second critical clearing dose $E_{\text{cl}}$, above which the dissolution rate saturates to its maximum at$v_{\text{cl}}$. The differential dissolution rate between the exposed and unexposed photoresist permits the transfer of optical exposure patterns into positive development profiles. Such a chemical response can usually be specified by the photoresist contrast

Here the dark erosion rate (a dissolution rate for unexposed ($E_{\text{ex}}<E_{\text{th}}$) photoresists) is assumed to be negligible, as is often the case in well-optimized experiments.

2.3 Developing

We model the development process to remove the locally exposed photoresist at the dissolution rate $v_D$, prescribed by the exposure dose distribution $E_{\text{ex}}(x,y;z)$, within a given time constraint ${\tau }_D$. In this study the photoresist removal is assumed to proceed one dimensionally and only in the direction of increasing depth z.

The eliminated thickness at the end point of development carried out for ${\tau }_D$ is determined by constructing the isometric surface $S_E(x,y)$ that is defined by [11]

$T_0=v_{\text{cl}}{\tau }_D$is the nominal developing thickness (NDT) that would have been removed when the same photoresist was completely exposed to the clearing light dose over its entire volume and subjected to the developing process for the time span ${\tau }_D$. We take $T_0$ as a convenient measure for the developing process span because it is physically more meaningful than using ${\tau }_D$ as an absolute developing time.

Substituting the expressions for $\eta (E_{\text{ex}})$from Eq. (3) and $E_{\text{ex}}(x,y;z)$ from Eq. (1) into Eq. (5), we find the analytic solution for the development profile $S_E(x,y)$:

The determinant parameters involved are $T_0,$ $\beta ,$ $E_{\text{th}},$ and γ, where $\gamma ={[\ln (E_{\text{cl}}/E_{\text{th}})]}^{-1}$ is an alternative parameter introduced for mathematical convenience, representing the photoresist contrast as $\mathrm{\Gamma }=\gamma \ln 10\approx 2.3\gamma$. The symbol e denotes Euler’s number (approximately 2.718). Given the exposure dose distribution at the entrance plane $E_{\text{0}}(x,y)$, the photoresist removal thickness $S_E$ as a function of position $(x,y)$ can then be evaluated.

3. Numerical calculation of lithography profiles

Based on the analytic formula we derived in Eq. (6), developed profiles of a photoresist with evanescent field exposures were calculated. We used lithographic parameters similar to those of previous theoretical [13] and experimental [15] studies: a positive photoresist was assumed to have a finite contrast Γ of 3, with threshold and clearing doses of $E_{\text{th}}$ = $20{\text{mJ/cm}}^{\text{2}}$ and $E_{\text{cl}}$ = $43{\text{mJ/cm}}^{\text{2}}$, respectively. Localized exposing light in a Gaussian had a FWHM diameter $W_{\text{FWHM}}$ of 60 nm at the top surface of the photoresist, and decayed rapidly with a decay constant β of 30 nm. The exposure dose $E_p$ (as measured at the peak of the Gaussian distribution) varied between $E_{\text{th}}$ and 10$E_{\text{th}}$. The NDT ($T_0$) proportional to the development time was taken as a process parameter that can be arbitrarily set regardless of the physical thickness of the initial photoresist layer.

3.1 Effect of peak exposure dose

We calculated the topography of the photoresist developed with an NDT ($T_0$) set as 50 nm for several different peak exposure doses. The cross sections of the photoresist along the xz plane through the center of exposure are shown in Fig. 1 . Consistent with previous studies, the depth and width of the photoresist hollows increased with peak exposure dose because the higher peak exposure allows the deeper and wider region in the photoresist to receive an exposure dose above the threshold, as depicted in Fig. 1(a). Compared with the result in Fig. 1(b) for a far-field exposing light that is not attenuated (i.e., equivalent to the case when$\beta \to \infty$), an evanescent field of low exposure dose will lead to profiles with some disadvantages, such as relatively shallow depths and gentle sidewall angles, as shown in Figs. 1(c-d). With far-field exposures above the clearing dose, the maximum removal depth is limited fundamentally by the developing time or more directly by the NDT ($T_0$). While this is also the case for the near-field exposure, the spatially diminishing exposure further reduces the removal depth appreciably less than does the given NDT ($T_0$). However, we note that the quantitative estimation of such topographic aspects reveals discrepancies between the previous model [13] shown in Fig. 1(c) and the model in this study shown in Fig. 1(d).

 

Fig. 1 Cross-sectional topography of a photoresist exposed with several different peak doses and developed for the same NDT of 50 nm. (a) Exposure dose distributions on the top surface of the photoresist. The dashed lines represent the levels of the threshold (lower) and clearing (upper) doses. (b) Photoresist profiles with far-field exposures assuming no attenuation along the depth. Near-field-exposed profiles obtained from (c) the previous model by E. Lee et al. [13] and (d) the analytic formula in this paper.

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3.2 Effect of developing time

In order to assess the role of developing time in forming a lithographic feature, photoresist profiles with a peak exposure dose of $E_p=5E_{\text{th}}$ were calculated for a set of different NDTs. As shown in Fig. 2 , the topography of a dimple feature was found to change in depth, width, and sidewall slope, depending on the process parameter, namely the NDT ($T_0$). To understand the physics involved it is helpful to examine two characteristic length scales of an evanescent exposure in relation to the NDT. One is the isoexposure depth $S_{\text{th}}(E_0)=\beta \ln (E_0/E_{\text{th}})$, at which the exposure attenuates to the threshold dose, and the other is $S_{\text{cl}}(E_0)=\beta \ln (E_0/E_{\text{cl}})$, which is defined in terms of the clearing dose.

 

Fig. 2 Cross-sectional profiles of a photoresist developed for several different processing times corresponding to NDTs of (a) 15 nm, (b) 25 nm, (c) 35 nm, (d) 50 nm, (e) 70 nm, and (f) 90 nm. The profiles based on our analytic model are represented by blue lines, whereas those obtained by the previous model and the non-decaying far-field model are represented by red and green lines, respectively. Dashed lines in each panel indicate isoexposure contours at the threshold dose (lower curve) and the clearing dose (upper curve).

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Until $T_0$ reaches the isoexposure depth $S_{\text{cl}}$ (within which the dissolution rate is the highest at its saturated value), the lowest depth of a profile $S_E(x,y)$ is kept identical with $T_0$ as in Figs. 2(a-b). When $T_0$exceeds$S_{\text{cl}}$as in Figs. 2(c-f), the dissolution gradually slows, and the actual removal thickness $S_E(x,y)$ always becomes smaller than the nominal process thickness $T_0$. In principle the deepest profile achievable with the evanescent exposure is determined fundamentally by the isoexposure depth $S_{\text{th}}$. However, Figs. 2(d-f) suggest that this is the case provided that the developing process is carried out when the value of $T_0$ is sufficiently larger than $S_{\text{th}}$.

Note that the forefront of our photoresist profile is always limited by $T_0$ of the developing process, whereas the previous model [13] misleads to the depth exceeding $T_0$ for a small $T_0<S_{\text{cl}}$, which is obviously unphysical, as shown in Fig. 2(a). For a moderate level of $T_0\approx S_{\text{th}}$, the previous model then underestimates the removal depth achievable with a given set of process parameters.

3.3. Minimum NDT for optimizing the near-field lithographic features

We investigated in more detail how the lithographic process parameters affect the photoresist topography. Figure 3(a) reveals that the higher level of exposure dose with respect to the threshold dose results in a greater depth and width of the photoresist hollow, and this finding is consistent with the predictions and observations of previous studies [13]. With a sufficiently large NDT ($T_0$ = 100 nm), the profile depth in Fig. 3(a) increases logarithmically with the dose, which is simply given by $S_E(E_p)=\beta \ln (E_p/E_{\text{th}})$ from Eq. (6) for $T_0\to \infty$.

 

Fig. 3 Pattern profile (solid lines) as a function of (a) peak exposure dose and (b) photoresist contrast for several different NDTs indicated in the legend. The expectations of the previous model (fine dashed-lines) are also plotted as dotted lines in each graph. The depth at threshold dose and clearing dose are also plotted as bold dashed line and bold black dot-dashed line, respectively.

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Figure 3(a) clearly demonstrates the difference between the photoresist features predicted in the two models. Apparently the previous model underestimates the removal depth by up to ~20% unless the NDT parameter (accordingly, the developing time) appreciably limits the completion of the developing process for the maximum possible depth allowed by the threshold-dose isoexposure forefront. This is the case shown in Fig. 3(a) when $T_0\approx S_{\text{th}}$ or $T_0>S_{\text{th}}$ with a moderate exposure dose. Conversely, in underdevelopment ($T_0<S_{\text{cl}}$) the prediction fails to provide meaningful results, leading to an overestimate of removal depths that are physically inconsistent with the definition of either the NDT or initial thickness.

In the previous section we showed that a developing process with sufficiently large NDT ($T_0>>S_{\text{th}}(E_p)$) is desirable for a given exposure dose. However, too long of a developing time does not deepen or narrow the photoresist profile, but degrades the topography. When including a dark erosion process [8], a number of technical factors would affect such overdevelopment in practice. In order to estimate the minimum developing time required to avoid the drawback of overdevelopment as well as underdevelopment, we characterized the photoresist topography as a function of NDT.

In Fig. 3(a) the profile depth follows the constraint imposed by NDT (solid line) up to a certain point, and then asymptotically approaches the maximum at the threshold-dose isoexposure depth (black dashed-line), depending on the exposure dose level. We can estimate the minimum NDT by limiting the tolerance to $f_{\text{res}}$ (the ratio of the residual thickness to the maximum possible depth), so that the removal thickness $S_E(E_p)$ in Eq. (6) should be greater than $(1-f_{\text{res}})S_{\text{th}}(E_p)$.

For exposures less than dose-to-clear (i.e., $E_p<E_{\text{cl}}$ or $E_p<E_{\text{th}}{e}^{1/\gamma }$), the developing process is required to provide

For exposures greater than dose-to-clear (i.e., $E_p>E_{\text{cl}}$ or$E_p>E_{\text{th}}{e}^{1/\gamma }$), the required NDT depends on the exposure dose as well, appearing in a somewhat complicated form given by

Generally, high-contrast photoresists are preferred for transferring exposure aerial images into fine patterns with large depths and steep edges. Since the contrast of a photoresist varies with the photoactive material used, baking procedures involved, exposure wavelength employed, and so on, understanding the impact of photoresist contrast on near-field lithography will help optimize such technical factors.

We characterized the photoresist profile (formed with an exposure dose four times the threshold) as a function of its contrast Γranging from one to ten. As shown in Fig. 3(b), it is also true for evanescent exposures that higher contrast permits deeper and steeper patterns given the same NDT. However, the apparent shortcomings of low-contrast photoresists can be remedied by extending the NDT (or equivalently the developing time). To put it differently, the photoresist contrast makes no fundamental difference to the best achievable topography.

The best topography with a given exposure dose is theoretically independent of a photoresist contrast, and can be readily achieved by implementing the developing process with an NDT fit to a particular contrast, as demonstrated by the results in Fig. 3(b). We also noticed that the analytic form of the previous model [13] becomes invalid for fairly high contrasts since the NDT parameter is ignored as the contrast approaches infinity.

4. Near-field lithography experiment for profiling of isointensity contour

A near-field scanning optical microscope (NSOM) has been widely used to measure the near-field distribution produced by a nanoaperture in a metal film [10, 16, 17]. In practice, however, such a method strongly perturbs the field distribution due to the metallic tip of the NSOM, which measures only the relative intensity at the local area. In the previous section we proved theoretically that the lithography profile exposed by a near field quantitatively agrees with its distribution matching to a threshold dose under proper development process conditions. Based on the exposure model, we performed experiments to determine the near field distribution of a bowtie-shaped aperture generating a bright near-field distribution on a sub-wavelength scale. In this section we investigate the NDT conditions for developing the photoresist toward the threshold dose contour $S_{\text{th}}$, and quantitatively analyze the shape of the field distribution, comparing it with the results of FDTD calculations.

4.1 Experimental setup

To measure the field distribution close to the surface of a nanoaperture, we placed an optical near-field scanning probe in close contact with the photoresist to be exposed. The nanoaperture was fabricated on a probe holding a pyramidal tip (height 5 μm) made of a 400-nm-thick silicon nitride (Si₃N₄, n=2.07) membrane. The tip apex has a flat surface and the diameter is 3 μm. On the nitride membrane, a 150-nm-thick aluminum film was deposited by ion-assisted thermal evaporation, and the root-mean-square (RMS) surface roughness was about 1.5 nm.

The bowtie-shaped aperture was milled by a focused ion beam (FIB) (SII SMI3050). Before the milling of the aperture, a 2 μm × 2 μm area and 300-nm-thick silicon nitride film was milled at the top using a 12 pA current ion beam to yield a 100-nm-thick Si₃N₄ membrane. The surface of the Si₃N₄ membrane was coated with a 2-nm-thick layer of Platinum (Pt) for precise milling of the nitride film, minimizing the charging effect. We milled the aperture from the Si₃N₄ membrane side in order to form sharp ridges at the bottom metal surface [18]. The bowtie-shaped nanoaperture with a 150 nm × 150 nm outline size and a 20 nm ridge gap distance was milled through the 100-nm-thick Si₃N₄ and 150-nm-thick aluminum layers under 30 kV acceleration voltage, 1 pA current, and 5 μs dwell time. The fabricated bowtie aperture is shown in Fig. 6(c) .

 

Fig. 6 Cross-section of the developed pattern profile (under-filled black solid lines) and FDTD calculation results (dotted-lines) for the (a) xz-plane and (b) yz-plane. A scanning electron micrograph of bowtie-shaped nano aperture in experiment is shown in (c).

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The probe was affixed to the laser direct writing system coupled with a 405 nm wavelength diode laser (CrystaLaser, BCL-025-405S). The aperture was illuminated with a laser beam focused by a high NA (0.8) objective lens (Nikon CFI LU Plan Epi ELWD 100×). The focused beam spot size was assumed to be about 625 nm (λ/ NA_eff) based on the effective NA (NA_eff=0.65) of illumination determined by the pyramidal tip geometry. The laser beam was polarized along the direction of the ridge gap. The laser power was measured at the plane between the objective lens and optical probe. The exposure dose was adjusted by changing the exposure time. In this experiment, the exposure time was varied in the range of several tens of milliseconds, and the error in exposure time was less than 0.2%. The 140-nm-thick positive-type photoresist (Donjin Semichem, DPR-i7201 diluted) was spun at 2000 rpm for 40 s on the 10 mm × 10 mm diced silicon wafer. The film was soft-baked at 100 °C for 100 s before exposure. Development of the photoresist after the exposure process was carried out with developing solution (AZ MIF 300), and the three-dimensional topography of the exposure profile in the photoresist was measured using an atomic force microscope (Park Systems, XE-100) with a high resolution sharp silicon tip (Nanosensors, SSS-NCHR, typical tip radius ~2 nm) in tapping mode.

4.2 Determination of the minimum NDT required for developing toward the threshold dose contour

The standard developing solution has a high saturated dissolution rate around several tens of nanometers per second (~50 nm/s). Since the development time is only a few seconds under the high dissolution rate, it is hard to control the NDT with several tens of nanometers. To determine the NDT with several tens of nanometers, we reduced the dissolution rate of the development process to 4 nm/s by diluting the developing solution with deionized water. The developing process was performed at 30 ± 0.1 C˚ and 30% relative humidity.

To determine the minimum NDT of the pattern profile, spot patterns were recorded on the same substrates with the same exposure dose. The laser power illuminated at the aperture was set at 120 nW with an exposure time of 30 ms. Since the NDT is given by $T_0=v_{\text{cl}}{\tau }_D$, we adjusted the NDT by changing the developing time. To determine the minimum NDT for the spot patterns recorded with the nano-aperture, we measured the maximum depth of the spot patterns for different NDTs.

Figure 4 shows the maximum depth of developed spot patterns with respect to NDT. The asymptotically approaching depth ($S_{\text{th}}$) is given as 48.7 nm, with the averaged value of the maximum depths at large NDT of 200 nm, 400 nm, and 800 nm. The experimental results were fitted to the calculation using Eq. (6) by setting the decay constant equal to 40 nm and contrast of photoresist Γ =3. From the fitted curve we obtained a minimum NDT of 145 nm with a tolerance of ~1% of the maximum depth of S_th for the isoexposure contour at the threshold dose.

 

Fig. 4 Developed photoresist maximum depth with respect to NDT. As NDT increases, the pattern depth increases until reaching the contour of threshold dose (S _th). In this experiment the maximum depth of S _E was expected to approach the 48.7 nm (red dotted-line). Above a NDT of 145 nm (dashed arrow), the pattern depth was expected to be similar to that of isoexposure at the threshold dose with a tolerance less than 1%.

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According to the lithography model proposed in this work, the NDT should be larger than the minimum NDT to facilitate the development process toward the threshold contour. In practice, however, the developing is actually influenced by dark erosion, including dissolution of unexposed and exposed regions under the threshold dose level. Even though the dark erosion rate of a conventional photoresist is about 1% of the dissolution rate [19], it affects the pattern profile as NDT increases. To minimize the effect of dark erosion, the NDT should be given by a little bit larger than PR thickness to be removed (in general, initial thickness) in a conventional lithography. There is trade off, however, between removal depth and broadening of pattern width in near-field lithography. For the purpose of minimizing the effect of dark erosion, one should consider the rate of dark erosion and tolerance ($f_{\text{res}}$) to optimize NDT during developing process.

To avoid pattern broadening due to dark erosion, we experimentally investigated the reasonable range of NDT by developing spot pattern samples with different NDTs. We recorded near-field spot patterns generated by a nano-aperture with a laser power of 120 nW and an exposure time of 10 ms. Developing the pattern samples within the range of 120 nm and 800 nm NDTs, we measured the maximum depths of the pattern profiles. The maximum depth of the pattern profiles was 17 nm, and the deviation of the depth was within ±1 nm when the NDT was varied from 120 nm to 800 nm. Thus, the experimental NDT value we used for the subsequent developing process of near field patterns was 500 nm, which is in the middle of the NDT range listed above and is sufficiently larger than minimum NDT from the fitting curve.

4.3 Measurement and analysis of the isointensity distribution of a bowtie-shaped nanoaperture

We measured the near-field distribution of a bowtie-shaped nanoaperture using the lithography model. We recorded several isointensity contours of the near field distribution with different exposure doses by changing the exposure time. Setting the laser power at 60 nW, we changed the exposure time from 20 ms to 60 ms in increments of 10 ms. At each exposure time we averaged five photoresist profiles measured with an AFM to obtain a three-dimensional profile. Because of the small AFM tip radius, we expected lateral and vertical profile measurement resolutions of about 2 nm and less than 0.1 nm, respectively. In the case of the profile with a depth of 41 nm, the deviation was about 2.56 nm, which was about 6% of the averaged value. The corresponding width was 121 nm in the x axis with a deviation of 6.18 nm, which was about 5% of the averaged width. The near-field distribution of the bowtie-shaped nanoaperture used in this experiment was calculated with a FDTD (XFDTD ver. 7.0). The total calculation volume was 500 nm × 500 nm × 500 nm in the xyz axis, and the mesh size was 2 nm × 2 nm × 2 nm in the xyz axis. In the analysis of isointensity surface (or contour) from FDTD calculations, we notice that the two peak at the ridges of bowtie aperture merge into a single peak when the maximum depth is about 6 ~10 nm. Figure 5 show the three-dimensional profiles of isointensity surfaces obtained with exposure times of 20 ms, 30 ms, and 50 ms for the comparison of the experimental results [(a) - (c)] and FDTD calculations [(d) - (f)]. The experimental results representing the structures of the near-field distribution of a bowtie-shaped nanoaperture agreed well with the FDTD calculations.

 

Fig. 5 (a) – (c) AFM measured three-dimensional profiles of developed patterns for different exposure times. From left (a) to right (c) in the upper row, the exposure time was 20, 30, and 50 ms, respectively, while the laser power was set at 60 nW. The isointensity surfaces from the three-dimensional near-field distribution using FDTD calculation are plotted in the bottom row (d) - (f), and the corresponding intensity was normalized to the intensity of the input plane. The intensity level of (d) - (f) are correspond to the peak intensity at the depth of 6.5, 21, and 41 nm, indicated in the inset of (a) - (c), respectively. The shape of the bowtie aperture is illustrated with white dotted lines, and white arrows indicate the polarization axis of illumination. The visualized space is 500 nm × 500 nm × 50 nm in x-y-z space.

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In the quantitative analysis of the near-field distribution, we compared the results of the measurements and the calculations with respect to the cross-sectional profiles in both axes in Fig. 6. Since the threshold dose is an inherent property of a photoresist, we evaluated the absolute intensity of the isointensity contour corresponding to a photoresist profile. We obtained photoresist profiles at different exposure doses, and the cross-sections of the profiles are depicted in Figs. 6(a) and 6(b). Referring to the threshold dose of the photoresist (40 mJ/cm² at 405 nm), we were able to determine the intensity of each isointensity contour with an error of less than 1.03%, for which the error in measurement of the threshold dose was about 1% [20] and in exposure time control is about 0.25% (minimum illumination time of the electrical shutter was about 50 μs). The intensities corresponding to the profiles in Fig. 6 were 2.0, 1.3, 1.0, 0.80, and 0.67 W/cm², as indicated in the order of the maximum depth. Note that unlike the method using NSOM, which measures the relative intensity distribution, this method measures the three-dimensional profile of the isointensity contour and also the absolute intensity value.

While comparing the experimental results and the calculations, however, we found that the intensity matching to the photoresist profile decreased by a factor of 3 (from 2 to 0.67 W/cm²) for a variation in maximum depth from 6.5 nm to 48 nm, and the calculated intensity of the isointensity contour decreased by a factor of 13 (from 0.40 to 0.03) for a similar range of depth variation. The absolute magnitude of each peak is indicated in the figure, and the intensity at the entrance of the aperture was assumed to be 1 for the calculation. Therefore, the photoresist profile was properly matched with the calculated intensity of the isointensity contour by comparing the maximum depth and width.

In order to clarify the difference between the experiment and the calculation in terms of the decay rate of the near field, we noted that the size of the beam spot resulted from the confinement of the near field at the beginning of the aperture exit. We presumed that the disagreement in the decay rate was due to the difference in the beam size confined in front of the exit of the nanoaperture since the near-field intensity of a large beam slowly decreases compared with that of a small beam. The beam sizes for the first peaks of the photoresist profile and the isointensity contour of FDTD calculation in Fig. 6 (a) were 70 nm and 37 nm, respectively. Using beam size, effective wavelength (λ_eff = 253 nm) in PR and a propagation vector $k_z=\sqrt{{(2\pi /{\lambda }_{eff})}^2-{(k_x)}^2}$ in the direction of PR depth, we estimate a corresponding decay length, $\beta =\pi /\left|k_z\right|$. Assuming $k_x=2\pi /(beamdiameter)$, we obtain the decay length is 36.4 nm and 18.7 nm for the beam size of 70 nm and 37 nm, respectively. From a simple exponential decay function ($e^{-z/\beta }$), we estimate that the intensity decreased by factors of 3.3 and 10.8, respectively, for the depth variation from 6.5 to 48 nm and it is found that these computations properly explain the discrepancy in decay rates of the near fields obtained by the experiment and the FDTD calculation.

In this experiment we were unable to clarify the origin of the beam spot broadening in the x-direction by a factor of ~2. We carefully fabricated the ridge of the aperture in the rear side with FIB to minimize the edge roundness and the gap distance between the ridges of the bowtie aperture was precisely measured by SEM with uncertainty ± 2 nm. We presumed that the finite size grains (estimated 50-100 nm in diameter) in the metal film and the rounded edges of the ridge aperture may have affected the confinement of the fields induced by the surface plasmon polariton and the near field of the diffracted light at the exit of the aperture. Further work is required to clearly investigate beam size broadening in the metal nanoaperture.

5. Conclusion

We derived a simple analytic model to predict photoresist development profiles with a localized evanescent exposure that decays rapidly on the nanometer scale along the depth of a photoresist. The photoresist profiles were calculated in order to investigate their topographic dependence on lithographic parameters such as the exposure dose, photoresist contrast, and developing time. We introduced the concept of NDT to characterize the photoresist topography in terms of removal depth. From the numerical analysis of the lithographic pattern profile, the deepest and narrowest topography with a given exposure dose was found to be achievable only when a developing process was carried out with a sufficiently large NDT. Determining the appropriate NDT is difficult because it is equal to neither the initial physical thickness of a photoresist nor the desired maximum removal depth. In contrast to the analysis of the previous model [13], we revisited the effect of the photoresist contrast, which leading to the reinterpretation that, in principle, the best topography achievable with a given exposure dose is unaffected by the photoresist.

We experimentally determined the minimum NDT in order to determine the proper NDT for the development process that makes the exposure profile fit to the isointensity contour. Under the NDT condition, we measured the near field intensity distribution of a bowtie-shaped nanoaperture from the exposure profile. Using the threshold expose dose of the photoresist, this method can be used to measure not only the distribution, but also the absolute value of the intensity of each contour. We also analyzed the difference in decay rates of the near field distributions obtained experimentally and by calculation based on the initial beam size of the near field at the exit of the aperture. For maximum depth of 41 nm, we estimated the uncertainty in the measurement of the near-field profile and that in the measurement of the absolute intensity to be less than 6% and 1%, respectively. We expect the suggested method of near-field mapping based on an accurate lithography model to be useful for quantitative evaluation of nano-scale devices producing strong near-field distribution under a diffraction-limited scale, such as high transmission nanoapertures, nano probes for adiabatic concentration of light, nano antennas, and so on.