1. Introduction
The study of electromagnetic scattering for line structures has great importance in real application. The line structures are usually formed in a periodic manner, which is known as gratings. They can be found in optical elements such as diffraction gratings. In photolithography process for integrated circuits fabrication, the alignment marks and overlay targets that are formed on the wafer are also consists of line structures [
11. C. B. Tan, S. H. Yeo, H. P. Koh, C. K. Koo, Y. M. Foong, and Y. K. Siew, “Evaluation of alignment marks using ASML ATHENA alignment system in 90 nm BEOL process,” Proc. SPIE 5038, 1211–1218 (2003). [CrossRef]
,
22. S. H. Yeo, C. B. Tan, and A. Khoh, “Rigorous coupled wave analysis of frontendofline wafer alignment marks,” J. Vac. Sci. Technol. B 23, 186–195 (2005). [CrossRef]
]. From the perspective of electromagnetic modeling for line structures, there are several available rigorous methods. For example, the finite difference time domain (FDTD) [
3] method, which is a popular modeling technique for the study of electromagnetic scattering. The basis in FDTD method is relied on the discrete grid space and timestepping algorithm to find the electromagnetic fields everywhere in the computational domain. Another method, which is dedicated more for diffraction gratings is rigorous coupled wave analysis (RCWA) [
44. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]
,
55. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled wave analysis for surfacerelief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995). [CrossRef]
]. In RCWA, the line structures in the gratings are treated as stratified layers and the solution is obtained by solving a set of simultaneous equations.
A common feature shares by most of the rigorous methods is that there is a lack of understanding on how the total electromagnetic solution is constructed based on the existence physical field. There are only two identified fields, namely the incident and scattered fields. There are no clues about each individual physical mechanism that contribute to the scattered field. The geometrical optics field such as reflected and transmitted fields is indistinguishable from the scattered field. Similarly, one cannot predict the scattered field based on the physical structure of the obstructer since the methods rely heavily on numerical analyses. In view of that, the high frequency approximation approach with electromagnetic ray tracing is preferable in this concern.
In this paper, an electromagnetic ray tracing (ERT) model is established to study the scattering phenomenon of line structures formed on substrate. This approach clarifies and enhances understanding of every single electromagnetic field that contributes to the final solution at an observation point above the line structures. It takes into account the existence of both geometrical optics and diffracted fields. The details of the ERT model are discussed in Section 2, which follows by the numerical results presented in Section 3. All the formulations are devoted for transverse electric (TE) polarization, where the electric field has its only component in the z direction. A time harmonic of exp(jwt) is assumed and suppressed throughout.
2. Modeling approach
Fig. 1. Decomposition of a single line structure formed on substrate into individual model.
2.1 Single impedance wedge
An impedance double wedge for the single line structure is decomposed into two impedance wedges as shown in
Fig. 2. The total solution for the impedance double wedge can be obtained by combining all the fields that are contributed from the two impedance wedges. This section aims to study the electromagnetic scattering of a single impedance wedge by identifying all the associated geometrical optics and diffracted fields. The electromagnetic fields that propagate into the internal part of the impedance wedge are not considered. Only the external fields are included in the analysis. On the other hand, the normalized surface impedance,
η¯ used for the wedge is obtained from the effective refractive index,
n
_{e} which is discussed in Section 2.3.
Fig. 2. Decomposition of an isolated impedance double wedge into two impedance wedges. The notation, η¯ denotes the normalized surface impedance for the wedges.
For an isolated impedance wedge with interior angle of (2
p̄)
π, which is illuminated by an incident plane wave to its top surface, S
_{T} the surrounding zones can be separated into three regions. These regions are constrained by the incident shadow boundary (ISB) and reflection shadow boundary (RSB) as shown in
Fig. 3. A summary of the field that contributes to each region is given below.
Fig. 3. Geometry of a single impedance wedge with plane wave incidence.
In region I, the total field is a combination of the incident, reflected and diffracted fields. The reflected field is excluded from region II as it is beyond the reflection shadow boundary. In region III, the diffracted field is the only field that exists. It can be seen that diffracted field exist in every region to ensure continuity of the total field across different shadow boundaries. The geometrical optics field gives abrupt changes of the total field across the shadow boundaries.
2.1.1 Incident field
with all the angles, ϕ are measured from the top surface, S_{T} and the origin is located at the edge of the wedge. The wavenumber in the vacuum and distance from the origin to the field point are denoted by k and ρ, respectively.
2.1.2 Reflected field
In region I, the incident field that reaches the top surface, S
_{T} as shown in
Fig. 3 will be reflected and the amplitude will be attenuated. At a distance of
ρ from the origin as illustrated in
Fig. 3, the field at observation point
P(
ρ,
ϕ) caused by the reflection from the top surface, S
_{T} of the impedance wedge is represented as
A particular observation point corresponds to a unique reflection point on the impedance wedge surface. The location of this reflection point is obtained using direct geometrical derivation. The reflection coefficient for TE polarization is given by
and the effective refractive index, n
_{e} is given in Section 2.3.
2.1.3 Diffracted field
When the incident field illuminates the impedance wedge as shown in
Fig. 3, the diffracted field in any of the regions is expressed as
where U^{i}(Q_{E}) is the phase reference point to the origin. When the phase reference point and the origin are coincided at the edge of the wedge, the expression simplifies to
The spreading factor, ρ
^{1/2} suggests that the diffracted field is emanating in the cylindrical wave form from the edge, which is acting as a line source. The Maliuzhinets diffraction coefficient for transverse electric polarization, D^{TE} (ϕ, ϕ_{o}) is expressed as
where
RTE,ST
and
RTE,SV
represent the reflection coefficients for surfaces S
_{T} and S
_{V}, respectively. The notation, Ψ(∙) is an auxiliary function expressed in terms of Maliuzhinets functions [
88. A. V. Osipov and A. N. Norris, “The Malyuzhinets theory for scattering from wedge boundaries: a review,” Wave Motion 29, 313–340 (1999). [CrossRef]
] whereas
F(∙) denotes the transition function [
77. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974). [CrossRef]
]. Since the diffracted field discussed here is obtained without any alterations to its ray path, hereafter, they are referred as direct diffracted field.
2.2 Line structures on substrate
In Section 2, the edge of a line structure without underneath substrate is modeled as a single impedance wedge. All the associated electromagnetic fields are discussed. However, the electromagnetic solution becomes more complex when the line structures are formed on a layer of substrate. In addition to the geometrical optics and direct diffracted fields discussed in Section 2, it requires a detailed analysis of the electromagnetic waves that induced in conjunction with the existence of the substrate. For an observation point above the single impedance wedge with inclusion of the underneath substrate, the additional electromagnetic fields are reflected and diffractedreflected fields. They are discussed in this section. Similarly, the total solution of a line structure formed on substrate is obtained by combining all the electromagnetic fields contributed from each impedance wedge with underneath substrate. For ease of illustration, the half line structures are shown in the figures presented in this section, instead of using the impedance wedge model.
2.2.1 Reflected field from substrate
Referring to
Fig. 4, the direct reflected field from the substrate,
n
_{2} at surface S
_{B} contributes to the total field at point B
_{1}. At this point, the reflected field caused by reflection from the point Q
_{r} is represented using
where
ρ_{Qr} is the distance measured from the origin to the point Q
_{r}, and
ϕ_{Qr} is the angle measured from the top surface, S
_{T}. The reflection coefficient at surface, S
_{B} for TE polarization,
RTE,SB
is similar to Eq. (
3), except that the used refractive index is different. Here, the exact refractive index of the substrate,
n
_{2} is used. The term, exp(
jkρ_{r}) is used to correct the phase shift due to the movement of origin from the edge of the wedge to the point Q
_{r} as illustrated in
Fig. 4.
Fig. 4. Plane wave reflection from the bottom surface, S_{B}
2.2.2 Diffractedreflected field
In
Fig. 5, the total field at point B
_{II} (x
_{BII}, y
_{BII}) involves reflection of the diffracted field from the substrate,
n
_{2} at surface S
_{B}. Here, this particular electromagnetic field is called as diffractedreflected field. The diffractedreflected rays are treated to emanate from an image source at point O’. These reflected rays are characterized by a spreading factor of (
ρ^{i}+
ρ^{r})
^{1/2}, where
ρ^{i} is the distance from the wedge origin to the point of reflection at the substrate surface, S
_{B} and
ρ^{r} is the distance from the point of reflection, Q
_{r} (x
_{r1}, y
_{r1}) to the point of observation, B
_{II} (x
_{BII}, y
_{BII}). Existence of these rays ensures continuity across Regions I & II and Regions II & III. If it is neglected, the amplitude of the total field can show a significant disruption across different boundaries. Study of the geometrical optics field in different regions is similar to Section 2.1. With this, the diffractedreflected field at observation point, B
_{II} (x
_{BII}, y
_{BII}) is expressed as
The diffraction coefficient
D_{TE} (
ϕ,
ϕ_{o}) is obtained from Eq. (
5). On the other hand, the reflected point, Q
_{r} (x
_{r1}, y
_{r1}) for a particular diffracted ray is required to determine the change of the phase and its attenuated amplitude. By solving the following quadratic equation,
the location of the reflected point, x
_{r1} for an observation point at B
_{II} (x
_{BII}, y
_{BII}), is obtained. Similarly, the reflection coefficient for the surface, S
_{B},
RTE,SB
is obtained from Eq. (
3) but with the use of exact refractive index,
n
_{2} for the substrate.
Fig. 5. Diffracted rays are reflected from the substrate surface, S_{B}.
When an isolated single line structure is considered, the diffractedreflected field in
Fig. 5 is not restricted by any adjacent structures. However, this is not the case for twoline structures. In this work, when twoline structures are considered, the diffractedreflected field emanating from the edge at point B as illustrated in
Fig. 6 is truncated at the cutoff point C
_{II}. This implies that for observation level at y=h1, the diffractedreflected field is considered from x=
l_{s}/2 to x=C
_{II}. The selection of cutoff point C
_{II} is based on the point C’
_{II}, which is located exactly below the origin. By doing so, it assumes that the weightage of the diffractedreflected field to the total field at point C
_{II} is relatively small. If it is nonnegligible, it should show a clear discontinuity at the cutoff point, C
_{II} in the amplitude profile. The discontinuity is supposed to be compensated by the secondary diffracted field emanating from the edge at point C. This secondary diffracted field is induced by the reflecteddiffracted field that follows the ray path BC’
_{II}C. On the other hand, the same approach is applied analogously to determine the cutoff point for the diffractedreflected field emanating from the edge at point C of another single line structure.
Fig. 6. Cutoff point for diffracted field emanating from edge at point B.
2.3 Effective refractive index
Table 1. Type of field and its associated refractive index used in the ERT model 

The effective refractive index is obtained from the amplitude of the film stack reflection coefficient, R
_{f} using [
1010. H. A. Macleod, ThinFilm Optical Filters (Institute of Physics Publishing, Philadelphia, US, 2001). [CrossRef]
]
where η_{o} is the vacuum tilted admittance. The optical admittance of the multilayer, Y which is given by the ratio, C/B is deduced from the film stack characteristic matrix,
where δ
_{1} and η
_{1,2} are the phase thickness and the tilted admittance for the respective layer.
3. Results and discussion
Fig. 7. Isolated line structure formed on substrate.
3.1 Individual electromagnetic field for single line structure
Figures 8(a) and
8(b) show each individual field that contributes to the electromagnetic scattering of an isolated single polysilicon line structure formed on silicon substrate. The selection of line width, x=2 µm is arbitrary and mainly for elaboration purpose. In
Fig. 8(a), all the geometrical optics fields that contribute to the total solution are presented. This includes the (i) incident field which has unit amplitude, (ii) reflected field from the top surface, S
_{T} of the line structure and (iii) reflected field from the substrate surface, S
_{B}. The reflected field from the top surface, S
_{T} is exist in the region of x<1 µm whereas the reflected field from the substrate surface, S
_{B} is exist in the region of x>1 µm. The amplitude of both reflected fields show a clear step jump at the shadow boundaries, i.e. x=±1 µm. These step profiles are expected to be compensated by the direct diffracted and diffractedreflected fields. The (i) direct diffracted field shown in
Fig. 8(b) exists across all the x distance. Its amplitude decays exponentially along the increasing direction of x>1 µm. In the region of x<1 µm, its amplitude gives a “ripple” trend. This can be explained by considering the direct diffracted field is constructed based on the concurrent diffraction from two different edges at x=±1 µm and interaction of these fields can occur. On the other hand, the (ii) diffractedreflected field shown in
Fig. 8(b) only exist in the region x>1 µm, with the amplitude decay exponentially from the increasing x>1 µm distance.
Fig. 8. Amplitude of the individual electromagnetic field that contributes to the total solution for the scattering of an isolated single polysilicon line structure. (a) Geometrical optics field (i) incident field, (ii) reflected field from the top surface of the line structure, S_{T} and (iii) reflected field from the substrate surface, S_{B}. (b) Diffracted field (i) direct diffracted field and (ii) diffractedreflected field.
A step by step combination of all the geometrical optics and diffracted field is given here. In
Fig. 9(a), the amplitude of the total geometrical optics field is shown. It is the combination of all the geometrical optics fields given in
Fig. 8(a). The total geometrical optics field shows a step jump from 0.85 to 1.5 at the shadow boundaries, x=±1 µm. By adding in the direct diffracted field, the trend of constant amplitude is no longer exist. The amplitude is decaying smoothly in the direction of increasing x distance from x=±1 µm, as shown in
Fig. 9(b). However, there is still a discontinuity at the shadow boundaries, x=±1 µm, which implies that some fields are still missing. The remedy to this discontinuity is by adding in the diffractedreflected field. The final and total amplitude is given in
Fig. 9(c). It is observed that the amplitude across the shadow boundaries at x=±1 µm is smooth. This concludes that all the prominent fields have been taken into consideration.
Fig. 9. (a). Total geometrical optics field (b) Combination of total geometrical optics field and direct diffracted field (c) Total solution for the electromagnetic scattering of a polysilicon line structure formed on silicon substrate.
Following the numerical results presented so far, it is shown that the total solution in the ERT model was obtained by constructing all the geometrical optics and diffracted fields. This is one of the key advantages of ERT model for line structure, which allows the causes to an electromagnetic disturbance to be explained. For example, there is a disturbance of the amplitude at point X in
Fig. 9(c). Throughout all the step by step constructions of the solution, it is identified that the disturbance is caused by the existence of diffractedreflected field. On the other, rigorous electromagnetic solution such as FDTD does not provide this explanation offered by the ERT model. In FDTD approach, only the final of the total electromagnetic solution is obtained. Thus, the users have no clues about how is the geometry of the obstructer or its refractive index could have impact on the final solution. The ERT model for line structures developed in this work provides the answer to this enquiry.
3.2 Accuracy of the ERT model
The aim of this section is to verify accuracy of the ERT model. The amplitude and phase of the total electromagnetic field obtained from the ERT model is compared with the reference FDTD solution [
3]. If they are matched with small errors, this indicates that the decomposition of the total field into their individual electromagnetic field, particularly all the diffracted fields is valid. The accuracy study here includes the electromagnetic scattering for a single and twoline structures, and line structures with variation in refractive index. The used parameters are ΔEz, ΔPhase, amplitude correlation coefficient (ACC) and phase correlation coefficient (PCC). The notations, ΔEz and ΔPhase (in radian unit) denote the difference of the amplitude and phase results, which are defined as,
On the other hand, the ACC and PCC results give an indication of the degree of matching of ERT model results to FDTD solution. The closer the ACC and PCC values to one, the better the matching of the ERT model with the reference FDTD solution. A perfect match gives value of one for both ACC and PCC. For a particular point, i at distance x, the ACC_{i} and PCC_{i} are defined as,
and the presented ACC and PCC results for a particular line structure are the average values of all the points within the domain of x distance.
3.2.1 Single line structure
Fig. 10. Amplitude and phase of the total field for an isolated single polysilicon line structure with width, l of (a) 8 µm and (b) 1 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.
Fig. 11. The ACC and PCC results for single polysilicon line structure on silicon substrate.
Fig. 12. Amplitude and phase of the total field for an isolated single polysilicon line structure with subwavelength width, l of (a) 0.25 µm and (b) 0.125 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.
3.2.2 Twoline structures
Fig. 13. Amplitude and phase of the total field for two identical 2 µm polysilicon line structures on silicon substrate with spacing of (a) 8 µm and (b) 1 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.
Fig. 14. The ACC and PCC results for twoline structures of polysilicon on silicon substrate. The spacing in between the 2 µm lines is varied from 8 µm to 1 µm. A subwavelength space width of 0.5 µm is also included.
When the spacing in between the 2 µm line structures is reduced to a subwavelength width, 0.5 µm (0.8
λ), the ACC and PCC results are further reduced to 0.943 and 0.927, respectively as shown in
Fig. 14. Associated with this, the amplitude and phase results obtained from the ERT model give a relatively poor match with the reference FDTD solution as presented in
Fig. 15. There are significant mismatch of both the amplitude and phase results, particularly in the x distance of ± 0.25 µm. The maximum amplitude difference, ΔEz and phase difference, ΔPhase are 0.172 and 0.399 radian, respectively. It is concluded that the ERT model gives relatively poor accuracy for the twoline structures with subwavelength spacing of 0.5 µm (0.8
λ). In this case, the smallest allowable spacing between the two identical polysilicon line structures is identified as 1 µm (1.6
λ).
Fig. 15. Amplitude and phase of the total field for two identical 2 µm polysilicon line structures on silicon substrate with spacing of 0.5 µm obtained using the ERT model (this work) and its comparison with the reference FDTD solution.
3.2.3 Variation in refractive index
Fig. 16. The ACC and PCC results for single and twoline (equal spacing) structures across different real refractive indexes. Each single line has width of 1 µm and the observation distance, y is 0.8 λ. The imaginary part of the refractive index is taken as 0.1j.
3.3 CPU computational time and memory used
In the FDTD method, the electromagnetic field is computed everywhere in the specified domain. The domain must be sufficiently large to include the entire geometry of the structures. This is the usual case in the FDTD method where the electromagnetic field at points which are not of interest is also generated. Furthermore, a converged FDTD solution is also required. As a result of these properties, the extensive consumption of the CPU computational time and memory is unavoidable. In this paper, the amplitude and phase results of the reference FDTD solution presented in Sections 3.2.1 and 3.2.2 for each condition of line structures required CPU computational time of more than one hour using a notebook PC with Intel Core 2 Duo Processor T5450 (1.66 GHz). The memory used exceeded 100 MB. In contrast to this, the ERT model demonstrates the advantage of relatively low CPU computational time and memory used.
Figure 17(a) shows the CPU computational time of the amplitude results presented in Sections 3.2.1 and 3.2.2 for the ERT model using the same notebook PC. The CPU computational time taken for all the single and twoline structures with width or space of at least 1 µm (1.6
λ) is well below 120 s. Associated with this is the relatively low consumption of memory, which is below 10 MB as presented in
Fig. 17(b). Such a relatively low CPU computational time and memory used is due to the flexibility of the ERT model that computes only the required electromagnetic field. The solution is based on direct computation using the explicit expressions derived for each electromagnetic field, be it the geometrical optics or the diffracted fields, as discussed in Sections 2.1 and 2.2.
Fig. 17. The (a) CPU computational time and (b) memory used for the amplitude results obtained from the ERT model for single and twoline structures.
4. Conclusion
An electromagnetic scattering model for line structure was established based on ray tracing approach. This electromagnetic ray tracing (ERT) model is able to identify different electromagnetic fields that contribute to the total solution. This includes the geometrical optics field for incidence, reflection and transmission phenomenon. Associated with this is the existence of different diffracted fields. By applying this model, the particular electromagnetic field that causes a disturbance in the amplitude profile for single line structure was identified and elucidated. The accuracy of the ERT model was verified through the comparison with the reference FDTD solution. For a single 1.6 λ (width) polysilicon line structure on silicon substrate, the ERT model was able to demonstrate ACC of 0.978 with the maximum amplitude difference, ΔE of as low as 0.067. In additions to that, the amplitude and phase results for a single line structure with subwavelength line width of 0.4 λ, obtained from the ERT model was able to show a good match with the reference FDTD solution. When the ERT model was applied for twoline structures, the smallest allowable spacing between the two identical polysilicon line structures was 1.6 λ. A linear trend of improving ACC and PCC results with reducing real part of the refractive index was also observed for twoline structures with lossy material.
Acknowledgments
The authors would like to acknowledge Lap Chan and Ng Chee Mang from Chartered for their support on this work.
References and links
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