2.1.

Frequency Doubling

In this paper, we explore the benefits of phase-shift masks for printing dense features near the resolution limit. The resolution limit is given by

Fig. 2

(a) Amplitudes of 0 and $\pm 1$ scattered orders for phase-shift mask, (b) electric field at wafer, and (c) intensity image.

2.2.

Minimum and Maximum Pitch

Since this process is based on two-beam imaging of the $\pm 1$ orders, it cannot be used to print all pitches. Clearly, the pitch must be large enough that the $\pm 1$ orders pass through the pupil. However, the pitch must be small enough that the $\pm 3$ orders miss the pupil; we will allow the $\pm 2$ orders into the pupil since the ideal phase-shift mask suppresses all even diffracted orders. Assuming conventional illumination with partial coherence radius $\sigma$, each diffraction order will be a shifted copy of the illumination pattern, centered about normalized spatial frequency $\frac{1}{4k_1}$, where the factor of 4 is due to frequency-doubling and the definition of $k_1$ in terms of half-pitch. By solving for $k_1$, where the $\pm 1$ orders start to leave the pupil and where the $\pm 3$ orders start to enter the pupil, we obtain the following condition on feature size:

2.3.

Efficiency Gains for Lines and Spaces

To quantify the theoretical efficiency gains of phase-shift masks over absorber masks, we consider two idealized mask designs, each composed of equal regions of alternating reflectance $R_1$ and $R_2$: $(R_1,R_2)=(\mathrm{1,0})$ for the absorber mask and $(R_1,R_2)=(1,-1)$ for the phase-shift mask. We will use each mask type to print a line–space pattern with wafer pitch $p_{\mathrm{wf}}=25\mathrm{nm}$ using $\lambda =13.5\mathrm{nm}$, $\mathrm{NA}=0.33$, and magnification $m=4$, meaning that this feature has $k_1=\frac{25\mathrm{nm}}{2}\frac{0.33}{13.5\mathrm{nm}}\approx 0.3$. For the absorber mask, we use dipole illumination with a mask pitch $p_{\text{mask}}{=\mathit{mp}}_{\mathrm{wf}}$, while, due to frequency doubling for the phase-shift mask, we use on-axis illumination with a mask pitch $p_{\text{mask}}=2{\mathit{mp}}_{\mathrm{wf}}$. Using Fourier diffraction methods as outlined in Smith,¹⁰ we may model the reflection function with arbitrary complex reflection coefficients $R_1$ and $R_2$ as

We then decompose the reflection function by its Fourier transform, representing decomposition into plane waves. Due to the periodic nature of the pattern, the Fourier transform only contains spatial frequencies that are integer multiples of $\frac{1}{p}$:

From these formulas, we calculate the 0 and $\pm 1$ order diffraction efficiencies for the two masks, denoted $a_j^{\mathrm{Abs}}$ and $a_j^{\mathrm{PSM}}$ for the absorber and phase-shift mask, respectively:

Modeling the illumination poles as mutually incoherent delta-functions, we then compute the aerial images for the absorber and phase-shift masks as shown in Fig. 3. For the phase-shift mask, the on-axis illumination keeps the diffraction pattern centered in the pupil, allowing the 0 and $\pm 1$ orders to pass through. In contrast, each of the two illumination poles for the absorber mask shifts the diffraction pattern, allowing only the 0 and 1 or 0 and $-1$ orders to pass through. Due to symmetry, we need only consider one illumination pole for the absorber mask, since both poles create identical images that add incoherently. Finally, note that since all the diffraction efficiencies are positive and real, all waves are in phase at $x=0$, meaning that for either mask the maximum intensity is simply given by

Fig. 3

(a) Dipole illumination for absorber mask (blue) and conventional illumination for phase-shift mask (red), (b) pupil plane plot of one illumination pole for each mask, and (c) aerial images from each mask.

2.4.

Efficiency Gains for Contacts

The contact array pattern is the two-dimensional version of the line–space pattern. For these special cases of $(R_1,R_2)=(\mathrm{1,0})$ and $(R_1,R_2)=(1,-1)$ for the two masks, we obtain a separable form of the 2-D reflection function

This yields the formulas for the contact array diffraction efficiencies in terms of line–space diffraction efficiencies:

Similarly to the line–space pattern, the on-axis illumination for the phase-shift mask keeps the diffraction pattern centered in the pupil, now allowing a total of nine orders to pass through. Again due to symmetry, we need only consider a single pole of the absorber mask, which shifts the diffraction pattern, allowing only four orders to pass through. Since again all diffraction efficiencies are positive and real, we apply the same calculation for peak intensity, resulting in the phase-shift mask’s peak being brighter by a factor of

The differences between imaging with the two masks are shown in Fig. 4.

Fig. 4

(a) Quadrupole illumination for absorber mask (blue) and conventional illumination for phase-shift mask (red), (b) pupil plane plot of one illumination pole for each mask, and (c) aerial image cross-sections from each mask.

2.5.

Summary of Efficiency Gains

The efficiency gains for the idealized phase-shift mask are shown in Table 1. Note that these gains are due both to the larger reflective area on the phase-shift mask as well as to the inefficiency of dipole and quadrupole illumination patterns, which shift significant scattered orders outside of the pupil, reducing the power incident at the wafer.

Table 1

Comparison of printing near minimum pitch using absorber and phase-shift masks. Analysis based on TMM. λ=13.5  nm, NA=0.33, and m=4. Dipole illumination poles placed at (±12pwf,0)=(±150  nm,0)=(±0.82,0)NAλ. Quadrupole illumination poles placed at (±12pwf,±12pwf)=(±172  nm,±172  nm)=(±0.57,±0.57)NAλ.

3.1.

Multilayer Mirror Structure

Now that the theoretical advantages of phase-shift masks have been quantified, we turn our attention to one possible realization of phase-shift masks for EUV lithography: etched multilayer phase-shift masks. All EUV masks employ a multilayer mirror substrate due to the need for high reflectivity at EUV. All simulations in this paper are based on a multilayer design of Mo-Si bilayers with a $d$-spacing of $d=6.95\mathrm{nm}$ and a Mo duty-cycle of $\gamma =0.4$, with a $t_{\mathrm{cap}}=2\text{-}\mathrm{nm}\text{-}\mathrm{thick}$ Ru capping layer, operating at a wavelength of $\lambda =13.5\mathrm{nm}$ with off-axis illumination 6 deg from normal. More complicated multilayer stacks accounting for such effects as inter-diffusion layers or interface roughness are not considered in this paper but may be a subject for future work. For this multilayer mask structure, over the angular range of 2 deg to 10 deg (roughly corresponding to the mask-side acceptance angles), the reflectivity is fairly uniform and always over 0.7. The phase response on the other hand has significant nonuniformity, with a phase shift of over 0.1 waves over this range of angles. Figure 5 shows the multilayer angular response, calculated using $n$ and $k$ values from the Center for X-Ray Optics (CXRO) database,¹¹ using the transfer-matrix method.¹²

Fig. 5

(a) Multilayer stack design and (b) reflected intensity and phase versus angle. $n$ and $k$ values drawn from CXRO database for $\lambda =13.5\mathrm{nm}$. Values used: Si: $n=0.99900154$, $k=0.0018265$; Mo: $n=0.923791$, $k=0.0064358$; and Ru: $n=0.8864$, $k=0.017066$.

3.2.

Phase Shift from Etching

As discussed in Sec. 2, the ideal phase-shift mask design has two regions with equal reflectivity and a relative phase shift of $\pi$. We denote the duty-cycle of the unetched pattern $D$ and the etch depth in bilayers $N_{\mathrm{Etch}}$. In dimensions of length, the width of the unetched region is $w=D{p}_{\mathrm{mask}}$ and the etch depth is $z_{\mathrm{Etch}}=t_{\text{Cap}}+d{N}_{\mathrm{Etch}}$. Again using the transfer-matrix method, we compute the reflectivity and phase-shift for each integer value of $N_{\mathrm{Etch}}$ from 0 to 20 bilayers, as shown in Fig. 6. Note that the reflectivity is nearly constant over this range, while the phase obeys a linear relationship with $N_{\mathrm{Etch}}$. After an etch depth of 20 bilayers, a $\pi$ phase-shift is achieved. However, we will later see that this large etch depth, ($z_{\mathrm{Etch}}=141\mathrm{nm}$) is far from the regime of small vertical and large horizontal dimensions where the thin-mask and transfer-matrix methods are accurate. This introduces significant electromagnetic edge effects, which we will later compensate for by adjusting $N_{\mathrm{Etch}}$ and $D$.

Fig. 6

(a) Schematic of pattern dimensions and (b) reflected intensity of etched region and phase shift between regions versus $N_{\mathrm{Etch}}$.

3.3.

Anamorphic Versus Isomorphic Magnification

Unlike traditional lithography tools, the next-generation of EUV lithography tools will use anamorphic magnification, which applies a different magnification to the two directions to minimize mask shadowing effects.^13,14 We will analyze the performance of etched multilayer phase-shift masks using both current-generation 0.33 NA isomorphic $4\times$ magnification, as well as next-generation 0.55 NA anamorphic $4\times /8\times$ magnification. An example of printing isomorphic contacts using both mask technologies is shown in Fig. 7.

Fig. 7

(a) 0.33 NA isomorphic $4\times$ magnification to print $p_{\mathrm{wf}}=36\mathrm{nm}$ contacts. (b) 0.55 NA anamorphic $4\times /8\times$ magnification to print $p_{\mathrm{wf}}=22\mathrm{nm}$ contacts. Left to right: mask geometry, mask-side pupil, wafer-side pupil, and aerial image.

4.1.

Multilayer Dispersion of Diffracted Waves

One source of asymmetry is ray-optical shadowing, whereby one sidewall is illuminated by incident light rays and the other sidewall casts a shadow as shown in Fig. 8(a). The angular dispersion of the multilayer is another source of asymmetry. As shown in Fig. 5, plane waves with different angles of incidence relative to normal acquire a different amplitude and phase upon reflection from the multilayer mirror. This applies to diffraction orders, which propagate through the multilayer at different angles, particularly affecting the relative phase of the orders. This angular dispersion is a source of asymmetry between the shadowing and nonshadowing orientations because the angle of the $j$’th diffracted order relative to the multilayer surface normal (${\theta }_j$) is different for the two orientations and is given by

Fig. 8

(a) Nonshadowing orientation and (b) shadowing orientation. Top to bottom: Incident ray diagram; diffracted angles at 0.33 NA isomorphic $4\times$ magnification, $p_{\mathrm{wf}}=25\mathrm{nm}$; diffracted angles at 0.55 NA anamorphic $4\times /8\times$ magnification, $p_{\mathrm{wf}}=15\mathrm{nm}$.

4.2.

Pitch-Dependence

Thin-mask and transfer-matrix methods are only accurate in the regime of small vertical and large horizontal dimensions (approximately vertical dimensions $<\frac{\lambda }{2}$ and horizontal dimensions $>2\lambda$),⁹ which are not valid assumptions for these features as the etch depth is on the order of $10\lambda$. Using rigorous FDTD analysis, we may quantify the deviation of the amplitudes of the diffracted electric fields as a function of etch depth from the thin-mask transfer-matrix prediction, which is calculated by computing the $R_1$ and $R_2$ values for the etched and unetched multilayer stacks as shown in Fig. 6 and then calculating diffraction efficiencies from Eq. (1). This is shown in Fig. 9, where we see that decreasing the pitch causes further deviation from the TMM. Again, we use the convention that the incident intensity on the mask is normalized to 1. Although the TMM predicts that the zero order is minimized at $N_{\mathrm{Etch}}=20$ bilayers (when the $\pi$ phase-shift occurs), the FDTD simulation shows that, as the pitch decreases, a deeper etch is required, as shown in Fig. 10. This effect has also been observed in DUV phase-shift masks^4,5 and is likely due to electromagnetic edge effects rather than angular dispersion of the multilayer. As the pitch decreases, the amplitudes of the $\pm 1$ orders become attenuated, likely due to a combination of edge effects and a lower multilayer reflectivity at higher angles.

Fig. 9

Amplitude of 0, $\pm 1$ orders versus $N_{\mathrm{Etch}}$ for (a) 0.33 NA and (b) 0.55 NA nonshadowing features.

Fig. 10

Etch depth that minimizes zero-order amplitude versus wafer pitch, nonshadowing features.

The different etch depths required to print different pitches may limit what features can be printed in a single exposure because it is not feasible to include multiple etch depths on the same mask. This effect may be overcome at 0.33 NA, since the difference in optimal etch depth between 25 ($k_1\approx 0.3$) and 50 nm ($k_1\approx 0.6$) features is only about one bilayer. However, the effect becomes much more concerning at 0.55 NA, where the difference in optimal etch depth is roughly four bilayers. These feature sizes were chosen based on $\sigma =0.2$ and the criterion $\frac{1}{4(1-\sigma )}<k_1<\frac{3}{4(1+\sigma )}\Rightarrow 0.3<k_1<0.6$. Therefore, the two features cover the entire printable range at $\sigma =0.2$ and include both the cases of three-beam imaging of the 0, $\pm 1$ orders ($k_1\approx 0.3$) and five-beam imaging of the 0, $\pm 1$, and $\pm 2$ orders ($k_1\approx 0.6$).

4.3.

Orientation Dependence

Carrying out similar analysis in the shadowing orientation, in Fig. 11, we can see that, just as in the nonshadowing orientation, smaller features on the mask require a deeper etch to minimize the zero order. In this orientation, the $\pm 1$ orders become asymmetric, an effect which becomes more severe at smaller mask pitches. However, due to the anamorphic magnification of the 0.55 NA system, features at the same $k_1$ in the shadowing orientation are larger by a factor of 1.2 on the mask than at 0.33 NA, causing them to experience somewhat less deviation from the thin-mask prediction. This stands in contrast to the nonshadowing orientation (Fig. 9), where at 0.55 NA, features at the same $k_1$ are smaller by a factor of 0.6 on the mask than at 0.33 NA, causing them to experience significantly more deviation from the thin-mask prediction.

Fig. 11

Amplitude of 0, $\pm 1$ orders versus $N_{\mathrm{Etch}}$ for (a) 0.33 NA and (b) 0.55 NA shadowing features.

5.1.

${\mathit{EPE}}_{\mathit{max}}$: Maximum Uncorrectable Edge Placement Error

To quantify the patterning performance of etched multilayer phase-shift masks, we introduce a metric referred to as the maximum uncorrectable edge placement error (EPE), or simply ${\mathrm{EPE}}_{\max }$, which tracks the maximum deviation of all edges from their nominal locations after correcting for an average shift of the entire pattern at best focus and exposure threshold. Note that for the line–space pattern, due to frequency doubling, two lines are printed in each electric-field period; since two edges are considered in each line, a total of four edge points are used to compute ${\mathrm{EPE}}_{\max }$. In contrast for the contact-array pattern, frequency doubling in both horizontal and vertical directions introduces four contacts in each electric-field period; since two horizontal and two vertical edges are considered for each contact, a total of 16 edge points are used to compute ${\mathrm{EPE}}_{\max }$.

As shown in Fig. 12, to calculate this metric, we (a) calculate image edge location, $x_i(d,t)$ for each $(i,d,t)$ (edge index, defocus, and intensity threshold); (b) subtract off the nominal edge locations $x_i^{\mathrm{nom}}$ to obtain: ${\mathrm{EPE}}_i(d,t)\triangleq x_i(d,t)-x_i^{\mathrm{nom}}$; (c) calculate the average pattern shift at each $(d,t)$: $\mathrm{\Delta }x(d,t)\triangleq {\mathrm{mean}}_i[{\mathrm{EPE}}_i(d,t)]$; (d) find the best focus, intensity threshold, and pattern shift by minimizing: $(d^{*},t^{*})\triangleq {\mathrm{argmin}}_{(d,t)}{\max }_i|{\mathrm{EPE}}_i(d,t)-\mathrm{\Delta }x(d,t)|$, then defining $\mathrm{\Delta }{x}^{*}\triangleq \mathrm{\Delta }x(d^{*},t^{*})$; (e) calculate uncorrectable EPE for each edge: $|{\mathrm{EPE}}_i(d,t)-\mathrm{\Delta }{x}^{*}|$; and (f) calculate maximum uncorrectable EPE: ${\mathrm{EPE}}_{\max }(d,t)\triangleq {\max }_i|{\mathrm{EPE}}_i(d,t)-\mathrm{\Delta }{x}^{*}|$.

Fig. 12

Calculation of ${\mathrm{EPE}}_{\max }$.

By tracking the positions of all edges, this metric accounts for many types of patterning errors, including CD errors, placement errors, and telecentricity errors. For any given mask design, we compute ${\mathrm{EPE}}_{\max }$ through focus and intensity threshold, set a specification limit ${\mathrm{EPE}}_{\mathrm{spec}}$, and calculate the process window or range of acceptable focus and dose errors. We can then optimize the mask design to maximize the area of the process window. Furthermore, we can co-optimize the design for printing multiple features on the same mask by maximizing their process-window overlap.

6.1.

Line–Space, NA = 0.33 Isomorphic

Below in Fig. 13, we show the process windows and optimized aerial images through focus on the 0.33 NA isomorphic system. Each row shows one $p_{\mathrm{wf}}$ and orientation, and each column shows one $N_{\mathrm{Etch}}$; the process window shown for each case uses the optimized $D$. The aerial images through focus use the optimized $N_{\mathrm{Etch}}$ and $D$. Note that for the three-beam imaging cases ($p_{\mathrm{wf}}=25\mathrm{nm}$), an extended depth of focus is achieved at the optimized $N_{\mathrm{Etch}}$. As might be expected, these $N_{\mathrm{Etch}}$ values found by optimization turn out to be the values that minimize the zero-order amplitude (Fig. 10). The process window is narrower for the larger features, due to the presence of the $\pm 2$ orders in the pupil, and is especially restrictive for the $p_{\mathrm{wf}}=50\mathrm{nm}$ shadowing feature due to the phase asymmetry in the diffracted orders as shown in Fig. 14. Another reason that the process window is narrower for these features is that the same ${\mathrm{EPE}}_{\mathrm{spec}}$ of 1 nm is applied to all features despite the factor of two difference in pitch.

Fig. 13

(a) ${\mathrm{EPE}}_{\max }$ process windows for $N_{\mathrm{Etch}}=21-24$ bilayers, each pitch and orientation; each process window calculated with optimal $D$. (b) Aerial images through focus. Solid lines are contours at highlighted threshold values and dotted lines are nominal edge positions after correcting for $\mathrm{\Delta }{x}^{*}$. Aerial images use ($N_{\mathrm{Etch}},D$) with the largest process window at ${\mathrm{EPE}}_{\mathrm{spec}}=1\mathrm{nm}$.

Fig. 14

(a) $p_{\mathrm{wf}}=50\mathrm{nm}$ scattered order amplitude and (b) scattered order phase.

Whether these process windows are acceptable for manufacturing will depend on which features must be printed on the same mask, as well as the precise specifications for each feature. Printing multiple features in the same exposure would cause a loss in process window, both because there is no single etch depth that is optimal for all features and because the exposure process windows do not align for all features. As we will later show, sensitivity to etch depth as well as overall depth of focus can be greatly improved with a central obscuration. As a topic of future work, it may also be possible to increase the process window overlap using SRAFs.

6.2.

Line–Space, NA = 0.55 Anamorphic

In the previous section, we considered $p_{\mathrm{wf}}=25\mathrm{nm}$, 50 nm for a wafer-side NA of 0.33. In this section, we proportionally scale down these pitches for a wafer-side NA of 0.55, to $p_{\mathrm{wf}}=15\mathrm{nm}$, 30 nm, as shown in Fig. 15. The most significant difference from 0.33 NA is that at 0.55 NA different features can achieve their widest process window at $N_{\mathrm{Etch}}$ values that differ by up to four bilayers. This behavior can also be seen in Figs. 9 and 11 and arises because the anamorphic magnification leads to much smaller mask pitches in the nonshadowing direction, which then require larger values of $N_{\mathrm{Etch}}$ to minimize the zero order. These differences in optimal etch depth greatly increase the difficulty of printing these features simultaneously; however, this effect can be mitigated with a central obscuration.

Fig. 15

(a) ${\mathrm{EPE}}_{\max }$ process windows for $N_{\mathrm{Etch}}=22$ to 26 bilayers, each pitch and orientation; each process window calculated with optimal $D$. (b) Aerial images through focus. Solid lines are contours at highlighted threshold values, and dotted lines are nominal edge positions after correcting for $\mathrm{\Delta }{x}^{*}$. Aerial images use ($N_{\mathrm{Etch}},D$) with the largest process window at ${\mathrm{EPE}}_{\mathrm{spec}}=0.6\mathrm{nm}$.

Fig. 16

(a) ${\mathrm{EPE}}_{\max }$ process window for $p_{\mathrm{wf}}=36\mathrm{nm}$ contacts, 0.33 NA, $N_{\mathrm{etch}}=24$ bilayers and (b) aerial image at best focus, contours at exposure thresholds.

Fig. 17

(a) ${\mathrm{EPE}}_{\max }$ process window for $p_{\mathrm{wf}}=22\mathrm{nm}$ contacts, 0.55 NA, $N_{\mathrm{etch}}=26$ bilayers. (b) Aerial image at best focus, contours at exposure thresholds.

6.3.

Contacts, NA = 0.33 Isomorphic

At an $N_{\mathrm{Etch}}$ value that minimizes the zero-order amplitude (24 bilayers) and using an equal checkerboard design, the $p_{\mathrm{wf}}=36\mathrm{nm}$ contact array pattern enjoys a similar extended depth of focus and wide exposure latitude as the optimized line–space pattern, as shown in Fig. 16. Furthermore, the nominal intensity threshold at 0.33 NA is 0.9, or 91% of the thin-mask transfer-matrix prediction, which is based on a 20-bilayer etch depth using the transfer-matrix reflection coefficients and the TMM to generate diffraction efficiencies.

6.4.

Contacts, NA = 0.55 Anamorphic

We can achieve a similar extended depth of focus and wide exposure latitude to print 22-nm contacts in the 0.55 NA anamorphic system, using a somewhat deeper etch depth (26 bilayers) again with the equal checkerboard design, as shown in Fig. 17. Furthermore, the nominal intensity threshold at 0.55 NA is 0.98, or 99% of the thin-mask transfer-matrix prediction. The increase in brightness from the 0.33 to the 0.55 NA system is due to the reduced shadowing. Indeed, a similar effect is observed with $k_1\approx 0.3$ shadowing-orientation lines and spaces, where the intensity at 0.55 NA is 15% higher than at 0.33 NA.

6.5.

Summary of Optimized Mask Designs

Tables 2 and 3 summarize printing results for the line-space and contact-array patterns. The first row in each table is calculated using the TMM, with reflection coefficients calculated using the transfer-matrix method with a 20-bilayer etch depth into 60 bilayers of multilayer. All other rows were calculated using FDTD in Panoramic Hyperlith software. Threshold refers to the intensity threshold at best printing conditions (minimum ${\mathrm{EPE}}_{\max }$); note that the threshold is based on normalizing the incident intensity on the mask to 1. Exposure latitude is calculated as the max minus min threshold as a fraction of nominal, at best focus. ${\mathrm{N}}_{\mathrm{Etch}}$ and $D$ values for each feature maximize the process window area at the listed ${\mathrm{EPE}}_{\mathrm{spec}}$.

Table 2

Line–space summary for concurrent optimization of threshold, exposure latitude, etch depth, and duty-cycle.

Table 3

Contact array summary for concurrent optimization of threshold, exposure latitude, and etch depth.

Table 4

Exposure latitude calculated at best focus, and focus latitude calculated at 10% exposure error.

8.1.

Central Obscuration

A central obscuration has been included in many designs for high-NA EUV lithography systems.¹⁶ In our analysis, this will provide substantial benefits for printing dense line–space and contact array patterns with phase-shift masks. Indeed, even using a 0.33 NA system to print these patterns, modification to include a central obscuration would most likely be indispensable. The reason is that a central obscuration greatly relaxes the requirement to completely annihilate the zero order on the mask because it will be blocked in the pupil. This means that an infinite depth of focus can be achieved even with an imperfect etch depth, which not only reduces sensitivity to mask making errors but also may enable simultaneous printing of line–space features with different optimal etch depths (although the exposure windows would still need to be equalized by SRAFs or other means).

These benefits are visualized in Fig. 19, which shows how a central obscuration substantially improves the process window and reduces requirements on etch uniformity for $p_{\mathrm{wf}}=36\mathrm{nm}$, 0.33 NA contacts. Without the obscuration (left), variations in the etch depth significantly shift the best focus location, while incomplete suppression of the zero order leads to undesirable fluctuations through focus even in the best case. By blocking the unwanted zero-order light with a central obscuration (right), through-focus variations are completely removed and errors in the etch depth result in only a slight shift in intensity. The complete removal of through-focus variation in this case is due in part to the suppression of the $(\pm \mathrm{1,0}),(0,\pm 1)$ orders on the mask by the perfectly even checkerboard design. On a real mask, errors in the dimensions of checkerboards may also introduce nontrivial $(\pm \mathrm{1,0}),(0,\pm 1)$ orders. If necessary, these orders could also be blocked by a pupil filter that blocks all five of the $(\mathrm{0,0}),(\pm \mathrm{1,0}),(0,\pm 1)$ orders. This would require a custom pupil filter with obscurations at locations specific to the pattern on the mask, which, although introducing additional sources of complexity, has been demonstrated to work experimentally.¹⁷

Fig. 19

(a) ${\mathrm{EPE}}_{\max }$ process windows without central obscuration and (b) ${\mathrm{EPE}}_{\max }$ process windows with central obscuration. Based on FDTD simulation in Hyperlith of a $p_{\mathrm{wf}}=36\mathrm{nm}$ contact array pattern, $\mathrm{NA}=0.33$, $N_{\mathrm{Etch}}=24$ bilayers.

8.2.

Subresolution Assist Features

In this paper, we optimized the process windows for different features by modifying only $N_{\mathrm{Etch}}$ and $D$ on the mask. However, further improvements may be possible using SRAFs. These could take a variety of forms: either subresolution-etched features, which would preserve the low cost and complexity of a single mask writing step, or subresolution absorber features, which would be patterned during a second mask writing step. Either type of SRAF could be used to suppress the $\pm 2$ orders when printing larger pitches or to equalize the exposure process windows for different line–space pitches and orientations. Absorber SRAFs would allow for much more flexibility in etched phase-shift mask design by enabling modulation of the amplitude; however, these benefits must be weighed against the substantial increase in cost and complexity from a second mask-writing step. It will likely still be beneficial to include a central obscuration even after designing a mask with SRAFs, due to the reduced sensitivity to mask manufacturing errors and the larger depth of focus.

8.3.

Engineered Multilayer Mirror

Others^18,19 have explored engineering broadband multilayer mirrors, which sacrifice some reflectivity in exchange for a more uniform angular response. A similar approach could be applied to engineer a multilayer mirror for use in etched phase-shift masks. The key to a good design would have a few components: first, the phase shift per nanometer should be as large as possible to minimize the total etch depth and hence minimize edge effects. Second, both the amplitude and phase of the angular response must be as uniform as possible for all angles within the mask-side NA, to minimize angular dispersion. Finally, the maximum reflectivity should be kept moderately high to maintain a high throughput; however, given the already substantial throughput gains of phase-shift masks, a slight decrease in reflectivity can likely be tolerated if the other two metrics can be substantially improved.