Simplified Models for Edge Transitions in Rigorous Mask Modeling

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1 Simplified Models for Edge Transitions in Rigorous Mask Modeling Konstantinos Adam, Andrew R. Neureuther EECS Department, University of California at Berkeley Berkeley, CA ABSTRACT A new method is described for very accurate and rapid 3D simulation of alternating phase-shifting masks. This method for arbitrary 2D mask patterns is based on scalar imaging theory and is therefore much faster (200X or more) than rigorous 3D electromagnetic simulation. It is shown that an alternating phase-shifting mask can be decomposed into single openings and, subsequently, accurate scalar models for the single openings can be combined to give the complete mask result. The Fourier domain is found to be most suitable for the development of these accurate scalar models. A methodology for observing and modeling cross-talk between adjacent features in a phase-shift mask is introduced. The amount of cross-talk is found to be insignificant for mask technologies that are shallower than 90 o /270 o. Keywords: phase-shift mask modeling, diffraction order, complex mask transmission function, new method for scalar imaging, cross-talk in phase-shift masks, Fourier spectrum 1. INTRODUCTION In order for the semiconductor industry to successfully exploit optical lithography to its ultimate limits, engineers have used amazing ingenuity in all aspects of the lithographic design process, ranging from resolution enhancement techniques in the illumination and the photomask to new optical and resist materials for shorter wavelengths. In the photomask arena, techniques that a few years ago were considered too exotic, such as optical proximity correction (OPC) [2] and phase-shifted masks (PSM) [8], are steadily finding their way into mainstream lithographic technology. The task of understanding the impact of a whole series of parameters that affect the lithographic performance in complex and intertwined ways is simply daunting. Yet, this is exactly where simulation comes into the fore as a powerful auxiliary tool able to quickly, accurately and systematically assess the impact of the many possible design factors. One of the major challenges is that the near fields from phase-shifted masks are much more complex than those in a thin-mask model, as depicted in Figure 1. P x EMF y z x diffraction spectrum objective aerial image CMTF Figure 1. The two distinct simulation approaches for the aerial image calculation of an alt. PSM: In the near electromagnetic field (EMF) is found through rigorous simulation of the exact mask structure (with TEMPEST) and used for the aerial image calculation (with SPLAT), while in a scalar complex mask transmission function (CMTF) is assumed and used for the aerial image calculation (with SPLAT)

2 Simulation has been very successful in the design and optimization of phase-shifting masks. The infamous intensity imbalance problem of an alt. PSM was first discovered by Wong using simulation [14], [15] and later verified experimentally [9]. The optimization of the trench design through simulation was demonstrated by Friedrich et al. [3]. Hotta et al. included the spatial frequency spectrum view in the analysis of an alt. PSM and also revealed the role of the Chromium-based absorption layer [6]. One of the remaining major challenges is the 3D simulation of large non-periodic mask layouts and the considerable computational effort that is involved. The challenge is to not only get the near fields, but to modify the mask in such a way that the resulting spectrum meets the design goals. A new method that leads to alternative thin-mask models of alt. PSMs is demonstrated, such that the spectral agreement with the rigorously calculated near fields is better than 99% in the band of interest, and consequently very fast and very accurate 3D photomask simulations are possible. The rigorous near field solution is calculated using TEMPEST [14], [10] and the aerial images resulting from either the thin-mask models or the near field solutions are calculated using SPLAT [13]. The two approaches produce the same aerial image when their respective spectra match at the lower spatial frequencies. Hopkins approximation, which is inherent in the equations of SPLAT, is valid in all cases, since the diffraction orders do not materially depend on the incident angle of the illumination. The range of validity of Hopkins approximation has been investigated by many authors and it was recently shown by Pistor et al. [11] that it holds true for typical lithographic imaging of even very deep trench alt. PSMs (at least up to 90 o /270 o ) and advanced masks with OPC. Two very important themes throughout this paper are the use of the through-the-lens (TTL) diffraction spectrum as the vehicle for analysis and also the decomposition of an alt. PSM into a set of single-opening masks, in order to isolate the contribution of each individual phase-well on the Fourier spectrum. This article progresses as follows: First, building upon an example that illustrates the discrepancy between the two simulation approaches of Figure 1, we introduce and develop the accurate scalar CMTF method. Then, a very accurate way of modeling the cross-talk between adjacent features in an alt. PSM is developed. Finally, the extension of the accurate scalar CMTF method in three dimensions is demonstrated and benchmarking results that establish the validity of the method for 3D simulations are presented. 2. BACKGROUND The simulation of an image intensity distribution (aerial image) that is projected by an optical system through a photomask onto (and into) the resist-coated semiconductor wafer can be performed following one of the two distinct paths shown in Figure 1. In the more computationally laborious path of Figure 1 the exact three-dimensional geometry of the photomask is used for a rigorous calculation of the electro-magnetic field (EMF) that is being established as a result of a certain illumination. This complex (amplitude and phase) near EMF, which can be thought of as the vector complex mask transmission function (VCMTF), upon further propagation down the optical system becomes the complex diffraction pattern (Fourier spectrum of the near EMF). Note that the diffraction spectrum is continuous for a completely isolated mask structure and discrete for periodic mask structures. Typically, most simulators use periodic boundary conditions and subsequently this makes the spectrum discrete. The frequency separation of these complex discrete diffraction orders depends solely upon the mask period The discrete diffraction orders for a mask such as the one in Figure 1 (independent of y, periodic in x) correspond to a discrete set of plane waves that are equally separated in k-space by: k x,step =2π/P x (P x is the mask period in x), but their angular separation is not equal and the angles of the discrete plane waves are given by: θ x,n =arcsin(nk x,step /k 0 )

3 In the case of a discrete spectrum, these orders are referred to as through-the-lens (TTL) diffraction orders. In photolithographic imaging for IC, the objective lens acts as a low-pass filter and only the center part of the diffraction spectrum is collected and contributes in the aerial image formation. Specifically, the collected plane waves have wavenumbers such that: 2 2 k x + k 2π y NA ( + σ) λ R (1) where λ is the illumination wavelength, NA is the numerical aperture of the projection optic, σ is the partial coherence factor of the illumination and R is the reduction factor of the optical system. In the faster simulation approach of Figure 1 the rigorous calculation of the photomask EMF is omitted and, instead, a scalar complex mask transmission function (SCMTF) is assumed, typically coinciding with the photomask geometry in x, y. The limits of validity of this approach have been investigated extensively by many authors and it has been established that it breaks down when either the minimum feature on the photomask or the vertical structure of the photomask is comparable to the illumination wavelength. Unfortunately, this is true for advanced photomasks with OPC or alternating PSM and the simulation path of Figure 1 leads to inaccurate results. It suffices to say that the accuracy of this approach is contingent upon how closely the spectrum resulting from the scalar CMTF matches the spectrum from the rigorously calculated CMTF (the near EMF solution), as depicted pictorially in Figure 1. An example of a CD=400nm (4X), 1:1 line/space, 0 o /180 o alt. PSM is shown in Figure 2 and the resulting EMF solution calculated with TEMPEST for λ=193nm TE-excitation 2 is shown in Figure 2. The amplitude and phase of the near z 180deg y x 400nm (4X) (c) vector CMTF scalar CMTF (d) vector CMTF scalar CMTF (e) vector CMTF scalar CMTF 180deg Figure 2. The intensity imbalance problem of a 0 o /180 o alt. PSM. Mask geometry, amplitude of the near EM field solution from TEMPEST (λ=193nm), (c) amplitude and (d) phase of the near EMF solution at a cut-line located 50nm below the absorption layer compared with the assumed scalar CMTF, (e) the resulting aerial images of the rigorous (vector) and scalar CMTF for an imaging system with NA=0.7, R=4, σ= Throughout this paper the illumination is normally incident on the photomask and consequently TE and TM excitations are indistinguishable. The convention used here is such that the TE has the E-field parallel to the y-axis and the TM parallel to the x-axis

4 EMF at a cut-line located 50nm below the end of the 80nm-thick Chromium absorbing layer is shown in Figure 2(c) and 2(d) respectively. The amplitude and phase of the scalar CMTF for the scalar simulation is also shown for comparison. Finally, the two distinct aerial images resulting from the vector CMTF and the scalar CMTF for imaging through a system with NA=0.7, R=4 and σ=0.3 are shown in Figure 2(e). This kind of discrepancy between the two aerial images is usually attributed to vertical or deep trench EM effects, that are impossible to model in scalar diffraction. In the remaining of this paper, scalar models capable of including these effects in imaging will be shown. 2. NEW METHODOLOGY FOR VERY ACCURATE SCALAR-BASED SIMULATIONS 2.1 Decomposition of an alt. PSM into single-opening masks Consider the decomposition of an alt. PSM shown in Figure 3 [6], [12]. The equation implied here is assumed to hold true for the near EM fields everywhere in the mask domain, but particularly at a cut-line that is located a few nm below the absorbing layer 3. However, one can easily verify that although this equation holds exactly for the geometry (ε(x,z)=ε 1 (x,z)+ε 2 (x,z)-ε 3 (x,z)), it does not hold for the near field solution. One should not confuse the superposition principle that follows from the linearity of Maxwell s equations with respect to E and H with this degenerate superposition of domains, which is not correct in general. Having made this disclaimer, there is a multitude of cases of practical interest, where the decomposition proposed in Figure 3 is indeed accurate to better than 0.5%, especially for the diffracted fields at cut-lines below the absorbing mask layer. This decomposition is accurate for shallow-trench alt. PSM technologies, but it breaks down for deep-trench technologies due to cross-talk between adjacent features [6]. z ε, E, H 180deg = + ε 1, E 1, H 1 ε 2, E 2, H 2 ε 3, E 3, H 3 - y x Figure 3. Decomposition of an alt. PSM into single opening masks For the 400nm (4X), 1:1 line/space pattern considered in Figure 2 the decomposition of the 0 o /180 o alt. PSM according to Figure 3 is shown in Figure 4. In Figure 4 the EM field of the original mask domain is decomposed into the first two single-opening domains shown in Figure 3. In Figure 4 the EM field at a cut-line 50nm below the absorbing layer is shown for both the original alt. PSM mask (both openings are present simultaneously) as well as the superposition of the two openings. The agreement between the two is almost perfect, but upon close inspection it is clear that the near fields differ slightly at the dark regions. This is because the third term is also needed. Adding it gives an accuracy better than 0.5% everywhere. Finally, Figure 4(c) depicts on the complex plane the three TTL diffraction orders (k -1, k 0, k +1 ), for both the original alt. PSM mask and the decomposition shown in Figure 4. Note that the non-zero k 0 is responsible for the imbalance effect and that the small difference of the two methods in k 0 is fully attributed to the residual background transmission, which was not accounted for. 3. Throughout this study all EM field cut-lines are taken at 50nm below the absorbing layer. The angular spectrum of the near EM field is changing with distance z according to [4], but at 50nm below the absorbing layer this change is insignificant

5 = + (c) k +1 k 0 k -1 Figure 4. Decomposition of the EM field of Figure 2 into the first two terms of Figure 3, comparison of the near field after the decomposition shown in with the near field of the exact alt. PSM geometry of Figure 2 at a cut-line located 50nm below the absorption layer, (c) comparison of the through-the-lens (TTL) complex diffraction orders of the EM fields shown in 2.2 Identifying the discrepancy between scalar and vector theory Figure 5 shows the EM field solution of a single 180 o opening under TE plane wave excitation. The amplitude of the rigorously calculated diffracted field at a cut-line 50nm below the absorbing layer is shown in Figure 5. Also shown in Figure 5 are the amplitude of the original scalar CMTF, which is just a unit-amplitude, 400nm-wide (4X) rect function properly centered, and the amplitude of the modified scalar CMTF, which has resulted from a systematic procedure that will be explained in the next section. For now, note that the modified (adjusted) scalar CMTF in this case is also a rect function with reduced width, slightly increased amplitude and slightly different phase than the original scalar CMTF. The result of this modified scalar CMTF on the Fourier spectrum is depicted in Figure 5(c) and (d). The magnitude of all propagating diffraction orders vs. the wavenumber k x is shown in Figure 5(c). Notice that the spectrum of the modified scalar CMTF matches the spectrum of the vector (rigorous) CMTF to better than 1% accuracy up to k x ~0.012, which is well beyond the objective lens collection ability. Figure 5(d) illustrates nicely the effect of properly modifying the scalar CMTF, where it is shown that the TTL diffraction orders can be relocated on the complex plane, so that they perfectly match (to within 0.5%) the orders resulting from the rigorous calculation. A similar adjustment can be independently performed for a single 0 o opening, which will result in a modified scalar CMTF for that opening. Putting together the two modified scalar CMTFs for the 0 o and the 180 o openings results in a scalar simulation of the aerial image that perfectly matches (to within <0.5%) the aerial image using the vector CMTF. This is shown in Figure 6.

6 y z x (c) (d) Figure 5. Electromagnetic field amplitude of a single 180 o phase-shifted opening, amplitude of the EM field at the cut-line shown in compared with the original scalar CMTF (thin mask) and the adjusted scalar CMTF (adjusted thin mask), (c) magnitude of diffraction orders vs. k x and (d) the TTL complex diffraction orders on the complex plane for the three CMTFs shown in vector CMTF scalar CMTF (c) (d) Figure 6. Amplitude and phase of the rigorous vector CMTF of the complete alt. PSM shown in Figure 2 and the modified scalar CMTF. Note that the smaller opening corresponds to the 180 o phase-well. (c) The TTL complex diffraction orders of the two CMTFs in, are aligned and consequently the aerial images in (d) are identical

7 2.3 Systematic manipulation of the Fourier spectrum via the accurate scalar CMTF method In this section we examine in depth how the attributes of the scalar CMTF can be adjusted to lead to a Fourier spectrum that closely matches the spectrum resulting from the vector CMTF. As shown in Figure 7, we are considering a single 180 o phase-well of CD=600nm (4X) with 50nm of isotropic underetch. The scalar CMTF (shown in Figure 7) is a properly centered rect function, but its width L x0, amplitude T 0 and phase φ 0 are varied until the best spectral match with the vector CMTF is achieved. The residual transmission of the background with amplitude T b and phase φ b are treated as constants, whose values are related to the thickness and the optical properties of the Chromium-based absorption layer. 600nm (4X) 180deg well Scalar CMTF T 0,φ 0 L x0 T b,φ b mask position Figure 7. Single 180 o phase-well of CD=600nm (4X) with 50nm of isotropic underetch. The mask period is 4xCD=2.4µm. The rect-cmtf model, where its width L x0, amplitude T 0 and phase φ 0 are varied until the best spectral match with the rigorous vector CMTF is achieved. T b and φ b are the amplitude and phase of the residual background light. Figures 8 and 8 show the effect of reducing L x0, while keeping T 0 and φ 0 constant. The spectrum of the vector CMTF is also shown for comparison. As expected from Fourier theory, reducing L x0 results in a broader amplitude spectrum (Figure 8). The behavior of the seven TTL diffraction orders (up to +/- 3 for NA~0.8) on the complex plane as we vary L x0 is shown in Figure 8. Figures 8(c) and 8(d) depict the effect of reducing T 0, while keeping L x0 and φ 0 constant. Note that varying T 0 does not change the overall shape of the spectrum (the nulls in the amplitude spectrum of Figure 8(c) remain unchanged), but simply scales the amplitude level of the spectrum. Figures 8(e) and 8(f) illustrate the effect of varying φ 0, while keeping L x0 and T 0 constant. The amplitude spectrum of Figure 8(e) is completely unaffected by the changes in φ 0, but the phase of the diffraction orders is changing in liaison with φ 0, in such a way that increasing φ 0 results in a counter-clockwise rotation of the spectrum, as shown in Figure 8(f) for the TTL diffraction orders. In summary, the adjustment of the three parameters of the rect-cmtf model (L x0, T 0, φ 0 ) should proceed as follows: First, we adjust (usually reduce) L x0 from its originally designed mask value, CD=600nm (4X) in this case, until the overall shape of the scalar CMTF amplitude spectrum matches that of the rigorous vector CMTF spectrum. A good measure of the degree of spectral matching is to look at the location of the nulls for the two spectra. Next, T 0 (originally is 1) should be raised or reduced accordingly, until the same TTL amplitude spectra are achieved. In practice, this model matched the amplitude spectra of all mask geometries that were considered well beyond k x =0.01, i.e. it is more than adequate for lenses of NA up to 1. Finally, φ 0 (originally set to the true phase of the EM field emerging from the phase-well with respect to an unetched

8 clear background) is adjusted accordingly, so that the spectrum of the scalar CMTF is rotated by the proper amount that will overlap it with the vector CMTF spectrum. k 0 k +1 k +3 k -3 k +/-2 k -1 (c) (d) (e) (f) Figure 8., (c), (e) Magnitude of diffraction orders vs. k x of the rigorous vector CMTF and three rect-cmtfs, (d), (f) the seven through-the-lens (TTL) diffraction orders (for NA~0.8). In, T 0 =1, φ 0 =0 o are kept constant and L x0 =600, 550, 500nm. In (c), (d) L x0 =600nm, φ 0 =0 o are kept constant and T 0 =1, 0.9, 0.8 and in (e), (f) L x0 =600nm, T 0 =1 are kept constant and φ 0 =0 o, 10 o, 20 o Models for the scalar CMTF other than the rect-cmtf were also evaluated. One that yielded successful spectral matching at approximately half the bandwidth of the rect-cmtf, was a model that attempted to replicate the edge-diffraction effects present to the mask geometry by adding local rect functions close to the edges. Although such a model is intuitively very satisfactory, it was found to suffer from narrower bandwidth performance compared to the rect-cmtf, reminding us that edge-diffraction in the photomask is not localized near the edges, but rather extents over most of the feature opening.

9 3. DEEP PHASE-WELL AND CROSS-TALK EFFECTS Simulation can give insight into the complex electromagnetic effects of edges and cross-talk between edges. In this section we extend our geometry decomposition technique to isolate and quantify these phenomena. 3.1 Scattering off of a 90 o air/glass discontinuity Figure 9 depicts the rigorous EM solution for the instantaneous field being established, when a 90 o air/glass discontinuity is illuminated from the top by a linearly polarized, monochromatic and normally incident plane wave. The bottomleft part of the plot is the air region with refractive index n=1 and the remaining part indicated by the dotted line is the glassfilled region with refractive index n=1.563, at λ=193nm. Note the inherent with the simulation program periodicity along the x-axis, which unavoidably leads to simulating a periodic array of 90 o air/glass discontinuities. However, this periodicity does not pose a significant limitation, since the dimensions chosen are sufficiently large compared to λ to mimic an isolated discontinuity. Observe from Figure 9 that the reduced phase velocity of light in glass causes the wave in the glass region to lag compared to the wave in the air region and also the fact that there exists a finite dark region to the left and to the right of the air/glass interface. E y (TE) λ=193nm µm µm z y x n=1 n= o µm µm Figure 9. Instantaneous EM field being established when a 90 o air/glass discontinuity is illuminated from the top by a normally incident TE plane wave at λ=193nm, the instantaneous scattered field (at the same instant as ) due to the 90 o corner In order to gain more insight on the effect of the corner itself, the instantaneous field (at the same instant!) of a glass/ air interface with no discontinuity along the x-axis was subtracted from the left-half of the original solution and the instantaneous field of a uniform plane wave travelling in glass was subtracted from the right-half of the original solution. The result of this process is the strange looking plot of the scattered field from the corner, depicted in Figure 9. There are two very interesting observations to be made on this plot: First, a 90 o corner at an air/glass interface results in a scattered wave that is cylindrical in nature with a non-uniform radiation pattern, where most of the energy is directed in the forward direction (direction of the incident field). Second, inside the glass side of the interface, but below the location that the instantaneous incident field has time to propagate there exists a region of non-zero scattered field. Clearly the source of this field cannot be the corner discontinuity itself, since the scattered field from the corner has not had time to directly propagate in glass to that point. In order to satisfy the boundary conditions of Maxwell s equations everywhere along the vertical interface, a non-zero field should exist on the glass side, although there is no time for either the incident or the scattered field to reach that location. In a way, the speedier field on the air side is depositing elementary Huygens spherical sources [5] that radiate into the glass side. Since

10 the velocity by which these Huygens sources are deposited is faster than the phase velocity of the disturbance that they cause inside the glass region, the end result is a shock wave front of Cherenkov radiation 4 [7]. The angle by which this plane wave is tilted is theoretically given by: φ v arc p, glass n = sin = arcsin air = v p, air n glass o which is in excellent agreement with that found from our simulation. This leaking field is responsible for energy being sucked into the higher refractive index material and is also interfering with the cylindrical scattered field from the corner at the glass side to produce the ripples present in the plot of Figure Modeling the cross-talk between phase-wells Now that we have a much better understanding of what happens at the air/glass discontinuities of the etched phasewells of alt. PSMs we can propose a revision to the alt. PSM decomposition model of Figure 3 that includes the cross-talk between phase-wells and it is shown in Figure 10. Two extra terms are included in this model, namely CT 12 and CT 21, that represent the amount of cross-talk that the first phase-well (shallow) is causing on the second (deep) and the second on the first respectively. y z ε, E, H well 1 x 180deg well 2 = + + ε 1, E 1, H 1 ε2 ε 2, E 2, H 2 ε 3, E 3, H 3 + CT 12 CT 21 - Figure 10. Accurate decomposition of an alt. PSM The calculation of the cross-terms CT 12 and CT 21 can be performed using the simulation experiment that is pictorially depicted in Figure 11. In order to determine the portion of scattered light from the corners of the shallow phase-well that goes through the opening of the deep phase-well, we proceed as depicted in Figure 11: First, we run a simulation in which the opening of the shallow phase-well is shut with the absorption layer but the phase-well itself is left intact. Then, from the resulting EM field solution of this geometry we subtract the EM field solution coming from a simulation in which only the single opening of the deep phase-well is present. Finally, we have to run the two remaining simulations shown in Figure 11 and CT 12 CT 21 ε = 4, E 4, H 4 - = ε 4, E 4, H 4 ε 2, E 2, H 2 ε 5, E 5, H 5 ε 3, E 3, H ε 6, E 6, H 6 ε 1, E 1, H 1 ε 7, E 7, H 7 ε 3, E 3, H 3 Figure 11. Calculation of the cross-terms: CT 12 represents the portion of scattered light from the corners of the shallow phase-well that goes through the opening of the deep phase-well and CT 21 represents the portion of scattered light from the corners of the deep phase-well that goes through the opening of the shallow phase-well 4. Strictly speaking, Cherenkov radiation involves relativistic motion of elementary particles. Nevertheless, the physical phenomenon that is observed here follows the same principle

11 subtract/add the EM field solutions due to the finite residual transmission of the absorption layer. The same simulation steps are followed in order to calculate CT 21. Note that calculating CT 12 and CT 21 as shown in Figure 11 is consistent with the discussion of section 2.1, since ε 4 (x,z)-ε 2 (x,z)-ε 5 (x,z)+ε 3 (x,z)=0 and also ε 6 (x,z)-ε 1 (x,z)-ε 7 (x,z)+ε 3 (x,z)=0, and consequently the cross-terms CT 12 and CT 21 can be viewed as perturbation terms necessary to restore the accuracy of the mask decomposition of Figure 3. Using the above model, we have calculated the cross-talk of four possible alt. PSM technologies, namely 0 o /180 o, 90 o /270 o, 180 o /360 o and 270 o /450 o, under TE illumination 5. The results of these calculations are depicted in Figure 12, where both the near field correction terms CT 12 and CT 21 are shown along with their respective spectra in each case. It is worth to note that the deeper phase-well is contributing more cross-talk to the shallower (i.e. CT 21 >CT 12 ) in every technology and that the cross-talk increases for deeper phase-well technologies. Both these results are intuitively satisfactory. Observe also from the spectra plots that most of the cross-talk energy is in specular directions and does not go through the optical system. E y (TE) well 1 180deg well 2 1:1 line/space CD=100nm (1X) (c) (d) Figure 12., Amplitude of CT 12 and CT 21 for four possible alt. PSM technologies, namely 0 o /180 o, 90 o /270 o, 180 o /360 o and 270 o /450 o, under TE illumination. (c), (d) Magnitude spectra of CT 12 and CT 21 An example of how accurate this decomposition is, is shown in Figure 13, where the EM field solution for a 180 o / 360 o alt. PSM is computed in three different ways: First, simulating the complete mask structure with both openings present, E y (TE) well 1 180deg well 2 1:1 line/space CD=100nm (1X) Figure o /360 o alt. PSM example of the accurate decomposition method. The EM field solution, by simulating the complete mask structure with both openings present, is compared with the EM field solution obtained by the decomposition method of Figure 3 (no cross-talk terms) and the accurate decomposition method of Figure 12 and Figure 13 (cross-talk modeling) 5. TM excitation causes similar cross-talk

12 then decomposing the mask using the original method shown in Figure 3 and finally using the accurate mask decomposition shown in Figures 10 and 11. Clearly, ignoring the cross-talk in this case is inaccurate. The agreement of the model with the exact geometry is spectacular, better than 98% everywhere in the solution! 4. APPLICATION OF THE ACCURATE SCALAR CMTF METHOD IN 3D SIMULATIONS So far we have demonstrated the accurate scalar CMTF method for 2D simulations of alt. PSM, i.e. mask geometries that are independent of one dimension. One is justifiably left wondering what the practical use of such a method is, since the adjustment of the rect-cmtf model (and also any other model) of this method involves decomposing the original alt. PSM into its single-opening terms (as in Figure 3) and also many other terms that synthesize the cross-talk (as in Figure 11). But since all of these terms require rigorous 2D EM field simulation in order to acquire the spectrum of the vector CMTF that we are trying to match with the scalar CMTF model, we end up running multiple 2D simulations, when running just a single 2D simulation of the original complete mask geometry would be faster and less laborious! However, the true power of the accurate scalar CMTF method lies in 3D simulations, as shown in the following example. Figure 14 depicts a 90 o /270 o, CD=150nm (1X), 1:1.5 dense contact hole alt. PSM with 50nm of isotropic underetch. The calculation of the aerial image of this mask under completely unpolarized illumination, with λ=193nm, NA=0.7, σ=0.3 and R=4 requires, according to the simulation path of Figure 1, the rigorous EM field solutions under both TE and TM plane wave excitations 6. However, due to the 90 o rotational symmetry of this mask it suffices to run only one 3D simulation. Although a discretization of ~25 cells per λ is sufficient for most practical cases, in this example the 3D mask domain was discretized with 40 cells per λ, in order to achieve indisputable accuracy of the near EM fields of better than 0.5% and make the benchmarking comparisons with the scalar method. Consequently the processing time and memory requirements rose dramatically. TEMPEST ran for approximately 55hrs on a single 550MHz processor of the ~100 CPU Millennium cluster at UC Berkeley [16] and used up to 1.82Gb of memory. Alternatively, the results can be obtained via the accurate scalar CMTF (c) E y (TE) o 3 90 o 3µm (4X) o 270 o µm (1X) y x 3µm (4X) µm (1X) Figure 14. The mask geometry of a 90 o /270 o, CD=150nm (1X), 1:1.5 dense contact hole alt. PSM with 50nm of isotropic underetch. Contours at 30% intensity of the aerial image under TE excitation at λ=193nm, imaged with NA=0.7, σ=0.3 and R=4. Three images are compared, namely the one resulting from full 3D rigorous vector simulation, the one from a conventional scalar simulation and the one from the accurate scalar CMTF method (c) The same three images as in at the cut-line that runs through the center of the contact holes 6. The two orthogonal excitations (TE and TM) are mutually incoherent in the case of completely unpolarized light [1]. Therefore, the aerial image for unpolarized excitation is the sum of the aerial images for the TE and TM excitations

13 method, in which the rect-cmtf, which is now a function of both x, y, is determined independently for the x- and y- directions of the alt. PSM. A total of only four 2D rigorous EM simulations are required (shown in Figure 14 as cut-lines 1-4) that take just under 8min to run and utilize negligible memory (~3Mb). Note that in this example the cross-talk can be neglected (the 50nm of underetch virtually eliminated cross-talk between adjacent features) and the mask decomposition follows Figure 3. The agreement of the accurate scalar CMTF method with the full 3D rigorous EM field simulation is spectacular. The image intensity contours at 30% of the clear field virtually overlap and the same is true for the aerial images at the cut-line that runs through the center of the contact holes, as shown in Figure 14 and 14(c). For comparison, the aerial image contours and at the same cut-line of the scalar CMTF without adjustment is also shown on these plots. Three-dimensional simulation results from other dark-field alt. PSM layouts (dense line/space pattern, corners) were also used to benchmark the accurate scalar CMTF method with equally successful outcomes. The ~400X speed-up achieved in the example presented would have been approximately halved for a discretization of 25 cells per λ. The speed-up resulting from this method depends on the size of the 2D mask layout, the discretization chosen for the rigorous EM field simulations and the number of rectangles that constitute the layout of the dark-field alt. PSM. 5. CONCLUSIONS Rigorous electromagnetic simulation of the role of the mask in the imaging process has so far been limited to 2D problems and relatively small periodic 3D problems. Although this capability is enough to tackle a variety of advanced photomask design issues, it is unquestionably very restrictive. For example, rigorous simulation of a 10µm by 10µm area on an alt. PSM that contains random (non-periodic) structures, at λ=193nm, would require on the order of ~3.6Gb of memory (discretization of the ~14,000λ 3 domain with 20 cells per λ) and would run for many hours on a powerful CPU. Although the scalar theory can handle large mask domains, it has been discarded for advanced photomask technology simulations as being inaccurate. In this article, we have demonstrated a way to improve the accuracy of simulations with the scalar theory via the accurate scalar CMTF method. This method involves using scalar CMTF (complex mask transmission function) models of single openings that are tuned to provide excellent spectral matching with the rigorously calculated CMTF inside a band of interest. A simple scalar CMTF model, namely the rect-cmtf, was presented and the effect of its three parameters on the spectrum was thoroughly investigated. The accurate scalar CMTF method is capable of better than 99% agreement with the vector theory and huge computational speed-ups, on the order of 200X or better, can be realized. Model-based CAD tools, that so far rely on scalar theory for aerial image calculations, can be tuned to be very accurate. Conceivably, accuracy in par with a rigorous 3D simulation becomes feasible for large - even full mask - areas. The way to decompose an alt. PSM into many constituents that properly account for the cross-talk between phasewells was demonstrated, as an integral part of the accurate scalar CMTF method. Through this decomposition, a much deeper understanding of the physical phenomena that take place at the corners and at the edges of phase-wells of alt. PSMs arose. Specifically, the corners and edges at the bottom of the phase-wells were identified as the sources of cross-talk between adjacent phase-wells and it was found that the amount of light due to cross-talk becomes significant for alt. PSM technologies deeper than 90 o /270 o. ACKNOWLEDGEMENTS The lithography research was supported by industry and the State of California under the SMART program SM97-01.

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