Quantitative analysis and modeling of line edge roughness in near-field lithography: toward high pattern quality in nanofabrication

2.1 Effects of the decaying feature on ILS and photoresist contrast

A schematic of the near-field nanopatterning process with a plasmonic contact probe is shown in Figure 1A. Compared with conventional optical lithography, whose pattern is generated by diffraction of the far field, the lateral dimension of the pattern in NFL is determined by laterally propagating QSWs and SPPs with large decaying features. Because LER is related to ILS and ILS can be evaluated by field distribution in the xy-plane without considering the decay rate in the z-direction, we adopt the evanescent mode of the QSW and SPP for LER calculation. Accordingly, a critical step to fully capture the effect of near-field distribution on the pattern quality is to quantify the PR PSF. Extraction of the PR PSF from spot-mapping patterns using a calibration process has been demonstrated, and it plays an important role in the pattern profile prediction [5], [6]. However, LER is related to the edge roughness of the line patterns, which means that LER generation is mainly affected by the outermost edge field distribution at the exit of the plasmonic BNA. For a plasmonic BNA, as the quality of the focused beam spot at its exit plane is strongly dependent on the polarization direction of the illumination laser [40], [41], a transverse magnetic polarized laser at 405 nm was employed to obtain extremely high transmission and optical resolution, as shown in Figure 1A–C. Hence, the transmitted intensity at the maximum width in the y-direction determined by the surface waves plays an important role in LER generation along the x scanning direction.

Figure 1:

Schematics of the near-field patterning process using a scanning plasmonic BNA.

(A) Dose calibration using spot-mapping patterns to extract the decay characteristics of the evanescent field. LER analysis of the line patterns was carried out with different CDs by adjusting the exposure-dose distributions on the photoresist surface. The optical source, which is a linearly x-polarized laser with 405 nm wavelength, is incident on a metallic BNA. (B) Geometry of a plasmonic BNA, which is perforated at the tip apex of the near-field patterning probe in (A). The outline dimension is O_x=O_y=150 nm with various ridge-gap sizes (g) from 10 to 30 nm. (C) Calculated field distribution excited by a plasmonic BNA on the photoresist surface and geometry of the photoresist PSF.

For a large pattern where the pattern size is approximately half of the incident wavelength, decay rate q of the QSW is higher than that of the spherical wave. However, for a pattern size that is smaller than 0.1 of the wavelength, the pattern is mainly recorded by the intensity near the aperture. In the vicinity of the aperture, decay rate q of the QSW converges to 1, which implies that the decay rate of the QSW is the same as that of the spherical wave. Consequently, the QSW intensity decreases at 1/ρ². We further revise the analytical formula of the PR PSF reported in [5] as follows:

where the PR PSF is denoted by D_psf(ρ, φ). It is a function of the lateral length  ρ=(xm2+ym2)1/2 and φ=cos⁻¹(x_m/ρ). y_m is half of the top critical dimension (CD) of the maximum width in the y-direction, and x_m is its corresponding coordinate in the x-direction, as shown in Figure 1C. The cosine term originates from the dipole radiation of the local plasmon, where A_QSW and A_SPP are the field amplitudes of the evanescent mode of the QSW and SPP, respectively. ϕ–δ is a possible phase delay between the QSW and SPP (more details of the derivation process of Eq. (1) are shown in Supplementary Section S1).

Because of the inherent characteristic of the evanescent waves, the local dose distribution D_psf(ρ, φ) can be expressed with an exponential decay function [2], [42]. To obtain a simple analytic decay function for the PR PSF, we assume that the decay constant at the edge position is approximated by the linear function β(ρ)=a+bρ, where a is the decay constant at ρ=0, and b is a dimensionless parameter [2]. The constants a and b depend on the spatial distribution of D_psf(ρ, φ), and can be obtained by fitting the local dose at the edge position of Eq. (1). For line patterns, owing to the optical proximity effects, i.e. the overlap of the PR PSFs, the decay constant can be extracted using the convolutional relationship between the PSF and LSF (see Eq. (S5)). By comparing the exposure dose profiles of the line patterns in far-field lithography (FFL) and NFL, as schematically shown in Figure 2, the rapid decay of the evanescent field at the edge position can result in a decreased ILS. It should be noted that the small ILS caused by the decay of the evanescent field can cause the LER-generation region to be larger than that in FFL.

Figure 2:

Schematic diagram showing the difference in LER generation in far- and near-field optical lithography.

The generated pattern profile depends on the threshold dose D_th and the clearing dose D_cl, which are defined as the minimum exposure doses for PR removal and the clearing doses for complete PR removal, respectively. The blue areas represent the LER generation regions. Its most significant contributor is the ILS, shown as red dotted lines. ILS is defined as the quantified profile steepness at the line edge position (i.e. the position of the full width at tenth maximum (FWTM) of the generated line profile). Note that the evanescent field can induce small ILS and thereby lead to a larger LER in NFL.

The PR contrast, which is simultaneously affected by the exposure and post-exposure processes, plays a paramount role in LER generation because it determines the PR residues at the edge positions. However, as the near-field around the plasmonic nanoaperture rapidly decays, the PR contrast in NFL has been rewritten as a function of the decay property of the exposure beam and the absorption coefficient of the used PR [2]. Thus, to further study the LER generation mechanism in the NFL, the generated PR contrast needs to be quantitatively compared with that in FFL. However, in practical applications, no theoretical analysis or experimental measurement methods can be directly applied to separately estimate the PR contrast in the near and far fields. To solve this problem, on the basis of the common theory about PR contrast [2], [39], we divide the PR contrast in NFL into two categories, namely that associated with the far-field PR contrast γ_far (which depends on the energy loss caused by the PR absorption that exists in all lithography techniques), and the evanescent-field-induced PR contrast γ_decay (which depends on the decay constant (β) of the near field). Then, the PR contrast generated by NFL, γ_near, can be theoretically expressed as

Equation (2) indicates that the effect of the evanescent field decay on the PR contrast can be separately determined (the derivation of Eq. (2) is described in detail in Supplementary Section S2). Because γ_decay is much less than γ_far, it yields a low γ_near. Generally, a larger PR contrast represents a higher pattern quality in the PR film [30]. Therefore, γ_decay can provide further support to our prediction that near-field decay will induce a large loss in the PR contrast, thereby increasing the LER. To confirm the validity of the analytical formula, we will use it to fit the measured PR contrast extracted from the PR contrast curve obtained from the experimental data. On the basis of the theoretical analysis, because the decay characteristics of the evanescent field can induce a small ILS and PR contrast loss in NFL, the causes of LER in NFL are essentially different from those in the other techniques.

2.2 LER modeling in NFL

A central issue in this work is finding an appropriate mathematical model to estimate the feature variation at the line edge position and thereby performing quantitative characterization of LER. Taylor series approximation is used to derive the local exposure-dose distribution D(y), which can be expressed as D(y)=D(yi)+∂D(y)∂y|yi(y−yi)+⋯, where Δy_i=(y–y_i). Here, y is the local position with a nominal CD, and y_i is the developed edge position measured at the ith point at the line edge (shown in Figure S3). For small values of Δy_i, i.e. smaller than 50 nm (larger than the measured LER value), the local dose distribution D(y) can be a good approximation of the Taylor series expansion in the first order with an error of 8%. By applying the threshold dose condition to the exposure dose at y_i, we can obtain the resulting change Δy_i at the PR edge position, which can be approximately expressed as

where D(y_i)=D_th is the exposure dose at the feature edge, and ∂D(y)∂y|yi is the local gradient of the exposure dose at the edge position. ILS is calculated as a function of (1D(yi)∂D(y)∂y|yi). ΔD_ex denotes the dose fluctuation (D(y)–D(y_i)) at the line edge position. For some variations in exposure dose σDex, σLER=σDexDth(1  ILS); thus, the resulting LER can be expressed as follows:

Here, D_nor is the normalized exposure dose, and γ_near represents the PR contrast at the line edge position (more details of the derivation of Eqs. (3) and (4) are shown in Supplementary Section S4).

According to Eq. (4), we know that there are several sources of LER in NFL; the exposure dose variation, PR material properties, and aerial image quality all contribute to the total LER of the final pattern feature. Note that the causes of LER generation in NFL are different from those in EUV lithography, where the photon shot noise (PSN) is the significant contributor to LER. The PSN contribution to LER can be ignored in NFL, as the number of the photons absorbed near the edge feature in NFL is over 25 times more than that in EUV lithography when the PRs used have the same sensitivity. The exposure variations can be considered as random fluctuations in the patterning process, which can be modeled using Poisson statistics, and yield the standard 1/Dose dependence observed in essentially all LER models [30].

3.1 Decay characteristics extraction in the evanescent field

The pattern feature is determined by the exposure-dose distribution on the PR depending on the complicated decay characteristics of the evanescent field. Thus, the decay constant of the evanescent field can be extracted from the experimental result. Spot-mapping patterns were recorded on a positive PR (Dongjin Semichem, DPR i-7201) using a direct-writing laser lithographic system and a NFL system with a nanoscale bowtie aperture. A 100-nm-thick PR was deposited on a Si wafer using the spin-coating method. The spot-mapping patterns were generated with exposure times ranging from 1 to 12 ms under a fixed laser power, followed by cold KOH development (at a temperature of approximately −10°C). The developed feature sizes of the spot-mapping patterns were measured using atomic force microscopy (AFM; Park systems, XE-100) in a noncontact mode. The AFM image of the generated spot pattern is shown in the inset in Figure 3A.

Figure 3:

Normalized photoresist PSFs and photoresist contrast curves.

(A) Measured curve of the normalized photoresist PSFs obtained by analyzing the features of the (solid circles) spot-mapping patterns and (solid line) fitted curve of the normalized photoresist PSFs. Inset: AFM image of the generated spot pattern. y_max is the maximum width in the y-direction, ρ is the lateral length, and β(ρ) is the corresponding decay constant with respect to ρ. (B) Photoresist contrast curve of (solid line with squares) far-field lithography, (solid circles) measured photoresist contrast curve of NFL, and (solid line) fitted photoresist contrast curve using the analytic formula of the photoresist contrast for NFL.

Comparison with the calculated field distribution shown in Figure 1C revealed that the spatial distribution of the near field through a plasmonic BNA could be quantitatively mapped using the spot patterns. The PR PSF was defined as a function of the lateral length ρ, which was normalized to D_th to compare the measured PR PSFs without the effect of different PR sensitivities under these exposure energies (see Eq. (S4)). The plot of the normalized PR PSFs versus the spot lateral length (ρ) is shown in Figure 3A. The solid line in Figure 3A is the theoretical D_psf profile generated by the plasmonic BNA. The residual errors between the measured PR PSFs and the fitted values were estimated to be within ±0.002. It shows that the proposed PR PSF formula agrees well with the experimental data. From the fitting process in Eq. (1), we obtained A_QSW=91.5763, A_SPP=−0.4387, and the phase delay term of cos(ϕ–δ)=0.4890. The parameters a and b of the decay constant β were fitted as 17.0700 nm and 0.4148 from the PR PSF curve. Then, according to the estimated decay constant β(ρ) determined by the PR PSF, the evanescent-field-induced PR contrast γ_deacy can be obtained (see the equations in Supplementary Section S2). The PR contrast curves generated by FFL and NFL are shown in Figure 3B. The remaining width was the estimated top CD of the nonexposure linewidth for a positive PR, and it is analogous to the remaining thickness of PR contrast in the conventional FFL. By normalizing the remaining width and using the threshold dose model, we obtained the measured PR contrast as γ=(ln(D_cl/D_th))⁻¹, and its value could be extracted from the PR contrast curve. The far-field PR contrast was estimated to be γ_far~3.7845. On the other hand, the value of the near-field PR contrast was not fixed. It varied with different lateral lengths, 0.1703≤γ_near≤0.4595. Figure 3B also shows that the estimated results obtained from Eq. (2) could fit well with the experimental results, with residual errors within ±0.07. These results indicated that the near-field PR contrast γ_near was significantly smaller than the far-field γ_far because of the loss induced by the rapid decay of the near-field. More importantly, the evanescent-field-induced PR contrast γ_decay implies that one of the most significant contributors to the LER in near-field nanopatterning is the PR contrast.

3.2 LER evaluation in the near-field nanopatterning

The main focus of our experiments was to estimate the LER versus feature size relationship and then determine an LER-reduction method based on the proposed approximate analytical solution. Because of the complicated proximity effects on the line patterns, i.e. the overlap of the PR PSFs is the main factor that determines the PR residue among features, an accurate measurement of the PR LSF is fundamental to modeling the LER in the near-field nanopatterning process. To measure the LSF in the PR, a sequence of line patterns at increasing exposure doses were generated using NFL with a plasmonic BNA while keeping the scanning speed at 0.5 mm/s. The used bowtie-shaped nanoaperture was fabricated using the focused-ion-beam milling method, which had a dimension of 150 nm×150 nm with a 20-nm gap size, as shown in the inset of Figure 4B. After the development, the top CD of the exposed line patterns was measured using AFM.

Figure 4:

LER estimation and characterization.

(A) Comparison of the (solid diamonds) measured and (solid line) calculated LSFs of the photoresist obtained using the convolutional relationship between the photoresist PSF and a line image and the calculated ILS as a function of feature size. (B) LER as a function of the top CD of the generated line pattern. (Solid circles) Measured and (solid line) predicted LERs. The error bars show the standard deviations (<3%) of the measured LERs. The inset shows the scanning electron microscopy(SEM) image of the used bowtie-shaped nanoaperture. (C) AFM images of the line patterns with various top CDs (the black arrows show the measured top CD). The generated LER is very clearly dependent on the feature size.

Figure 4A shows the results of the measured and calculated normalized PR LSFs. The calculated LSFs obtained from Eq. (S5) are in good agreement with the measured LSFs. The ILS is also plotted as a function of the feature size in Figure 4A. To confirm the feasibility of Eq. (4), the estimated LER was compared with the measured LER from the experimental pattern results. The LERs of the line patterns from the top-view images of the developed pattern profiles with various top CDs were analyzed using software developed in-house. The resulting pitch-dependent LER is shown in Figure 4B, which confirms that the calculation results of the analytical model and the experimental results match well with a small fitting error. This result shows that Eq. (4) can be used to theoretically predict the LER of NFL at various feature sizes. Figure 4B and C show that the LER of NFL increased with the decrease in the feature size because the ILS, PR contrast, and the required exposure dose decreased, resulting in serious pattern collapse, division, residuals, or height reduction. Thus, a LER reduction method is highly needed to meet the LER-reduction requirement. Note that as the feature size continuously increased by more than 150 nm, the measured LER saturated at ~9.1±0.8 nm. After the saturation point, any increase in the exposure dose had no effect on the improvement of the LER. Therefore, we believe that the saturated LER mainly originated from the limitation of the experimental conditions, such as the intrinsic material characteristics of the used PR and the developer. The proposed LER estimation method for NFL as an experiment-based model can effectively predict the generated LER.

LER reduction is another very crucial chanllenge to be addressed in advanced nanotechnology nodes. In order to achieve LER minimization in NFL, the effects of ILS on the generated LER were experimentally observed and theoretically analyzed. Because the gap size between the ridges of the plasmonic BNA determines its near-field localization, ILS can be controlled via the gap size, thereby influencing LER generation. The relative relationship of the ILS among the gap sizes of 10, 17, 20, and 30 nm were used to calculate the LER generated by these gap sizes with 1 mm/s scanning speed while keeping the development conditions constant. The comparison result of the predicted and measured LERs is plotted with the gap size and shown in Figure 5A. Figure 5B shows the AFM images of the line patterns with a 100-nm feature generated using probes with gap sizes of 10 and 20 nm. The experimental results again demonstrate that the rapidly decaying evanescent field is a dominant optical contributor to LER generation because the smaller the effect of the evanescent field on the ILS, the smaller is the LER. The results also reveal that the generated LER can definitely meet the standard of LER requirements when the NFL system produces a sufficiently good quality of the aerial image.

Figure 5:

LER reduction by controlling the gap size.

(A) Comparison results of the LER versus gap size between the (solid squares) measured and (solid line) predicted LERs. (B) AFM-measured profiles of the line patterns with a 100-nm top CD generated using two probes with gap sizes of 10 and 20 nm.

When an x-polarized light is incident on the metallic bowtie-shaped nanoaperture, which is a type of ridge aperture that exploits the gap plasmon, the field confinement factor and field distribution strongly depend on the gap size [43], [44], [45]. To further understand the relationship between the gap size and the ILS and their impact on LER generation, we obtained the distributions of the electric field intensity components with gap sizes varying from 5 to 50 nm by implementing the finite-difference time-domain method (FDTD; Lumerical Solutions Inc., ver. 8.12.590) simulations. For the numerical analysis, we used the permittivity of the Al metal film as −23.9819+4.9508i and PR (n=1.7), and set the bowtie outline as 150 nm×150 nm. The Al and PR thicknesses were 100 nm each. The total intensity distribution on the xz-plane is I_total=|E_x+E_z|², where E_x and E_z are the x and z component of the electric field, respectively. Because the ratio of the transmission amplitude of the two electric field components determines the pattern fidelity, a high ratio of |E_x|²/|E_z|² is required to generate an aerial image with good contrast [13], [46]. The calculated intensity ratios |E_x|²/|E_z|² for several plasmonic bowtie apertures with various gap sizes are shown in Figure 6A, which show that with decreasing gap size, a larger ratio of |E_x|²/|E_z|² can be obtained, thus yielding an aerial image with higher quality. To further validate the gap-size-dependent field-coupling effect, the cross-sectional view of the intensity distribution (at the center of the ridge aperture along the x-direction on the PR surface) is shown in the inset of Figure 6A.

Figure 6:

Effects of the gap size on the quality of the aerial image.

(A) Calculated ratio |E_x|²/|E_z|² as a function of the gap size. The inset shows the transmitted intensity profile through the different gap sizes at the gap center along the x-direction. (B) Calculated ratio A_QSW/A_SPP from the photoresist point-spread functions (PSFs) generated by the plasmonic BNA with gap sizes of 10, 17, 20, and 30 nm and calculated ILS as a function of the gap size at the edge position of the line patterns with a 100-nm top CD.

As the gap size becomes smaller, higher intensity can be achieved via the strong coupling of the plasmon-induced field. However, it strongly decays along the direction away from the two ridge edges of the bowtie-shaped aperture. The reason for this is that for a gap size smaller than λ/10, when the gap size becomes smaller, the SPP generation efficiency decreases, which implies that the influence of the evanescent field on the aerial image quality gradually weakens [47], [48]. To verify this conclusion, the ratio A_QSW/A_SPP, which can be approximately considered as the intensity ratio of the far- to the near-field component, is calculated from the PR PSF (i.e. Eq. (1)). The PR PSFs were obtained by the plasmonic BNA with gap sizes of 10, 17, 20, and 30 nm from the dose calibration processes. Figure 6B shows that the ratio A_QSW/A_SPP increases as the gap size decreases. Thus, the field ratio of the evanescent mode becomes smaller with the decrease in the gap size. Figure 6B also shows the calculated ILS with a 100-nm fixed top CD as a function of gap size by adjusting the PR PSFs. By comparing the variation tendency of the ratio A_QSW/A_SPP and ILS with the gap size, it is seen that they have almost the same relative relationship with respect to the gap sizes. Therefore, ILS can be enhanced by reducing the evanescent-field ratio generated by the SPPs via gap-size control, which yields low LER in the final pattern.