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Abstract
Plasmonic lithography can make the evanescent wave at the mask be resonantly amplified by exciting surface plasmon polariton (SPP) and participate in imaging, which can break through the diffraction limit in conventional lithography. It provides a reliable technical way for the study of low-cost, large-area and efficient nanolithography technology. However, there is also a phenomenon in plasmonic lithography similar to the forbidden pitch in conventional projection lithography. In this paper, combined with the imaging model and the optical transfer function (OTF), the theoretical analysis points out the reasons for the existence of the phenomenon of forbidden pitch in plasmonic lithography. Taking the 365 nm wavelength Ag thin film-based superlens imaging lithography as an example, the positions of the forbidden pitches of the 1:1 mask, the bright-field mask and the dark-field mask are calculated separately, and the key factors affecting their positions are pointed out. Simulation is carried out through commercial software, and the correctness of theoretical analysis is verified. Finally, we summarize and give some possible suggestions for solving this problem, including exploring better illumination methods, avoiding the patterns with forbidden pitch in the design, or by adding assistant feature to the design.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a supplementary lithographic technology, plasmonic lithography is an alternative candidate for the next-generation lithography. Comparing to the conventional lithography, plasmonic lithography can break through the diffraction limit by utilizing near-field imaging that contains high-frequency information [1]. It has been demonstrated that even with incident light at 365 nm wavelength, the optical resolution of ∼22 nm (∼1/17 light wavelength) can be achieved with a single exposure, which is potential for further improvement [2,3]. This method has received a lot of attention as it provides an alternative and reliable way to investigate low-cost, large-area and time-efficient nanolithography [4]. Different with conventional projection lithography in physical principle, the imaging area is localized in the near field, and the evanescent wave containing the high frequency objective information takes part in imaging. The imaging lenses are no longer ordinary optical lenses but metal-dielectric films, depends on which, evanescent wave can be resonantly amplified and involved in the imaging [5]. However, we have found in our recent studies that the forbidden pitch phenomenon exists in plasmonic lithography, which is similar to that in conventional projection lithography [6,7].
In conventional lithography, for a densely distributed periodic pattern, such as a typical line/space pattern or a dense contact hole pattern, the light intensity contrast in the imaging plan does not increase monotonically as the pitch of the pattern increases, but a local minimum occurs [8]. If the pitch of a certain pattern on the mask corresponds to this local minimum, the lithographic process window (PW) will be small and the quality of the image on the wafer will be poor, hence these periodic patterns are called forbidden pitch. Designers are often asked to avoid having such patterns in their designs or to solve the problem of limited process windows by adding assistant feature in the design [9].
Based on the optical imaging theory of conventional lithography, it is possible to explain the reason of the forbidden pitch phenomenon. And under certain conditions, the forbidden pitch can be derived through theoretical calculations [10,11]. For example, for a DUV lithography system illuminated by an on-axis point source, the forbidden pitch for a line/space pattern occurs when the second diffraction order of the mask is just about to be collected by the projection system, and the corresponding pitch can be found out as 2$\frac{\lambda }{{NA}}$, where λ is the wavelength of the illumination source and $NA$ is the numerical aperture (NA) of the projection lens system in the imaging area. However, in practice, off-axis illumination is often used to improve resolution and the light source is of a certain size, which makes the forbidden pitch correspond not to a fixed pitch, but to a small range of pitches.
Interestingly, a similar phenomenon exists in plasmonic lithography, where the optical imaging contrast drops significantly in a certain range of pitches and shows a certain regular. The discovery and theoretical explanation of this phenomenon is of great interest for further understanding of plasmonic lithography. Besides, it would be valuable to propose the corresponding solutions for the application of plasmonic lithography in the integrated circuit (IC) industry.
The article comprises three sections. The first section presents the theoretical analysis. Based on the imaging model and optical transfer function (OTF), the forbidden pitch phenomenon in plasmonic lithography is analyzed theoretically. The second section gives the simulation result with Commercial software Comsol Multiphysics 6.0, based on which the correctness of the theoretical analysis is verified. In the final part, some conclusion and possible suggestions are given for solving the forbidden pitch problem in plasmonic lithography.
2. Theoretical analysis
2.1 Mask full-pattern imaging and the forbidden pitch phenomenon
For a complete lithographic process, there will be a variety of patterns with different pitches in a complete mask layout. Taking the periodic line/space pattern as an example, there will be a series of 1:1 periodic pattern with different critical dimension (CD), as shown in Fig. 1(a), and a series of 1:2, 1:3, 1:4 etc. patterns with the same CD but different pitches, as shown in Fig. 1(b) and (c). In a single exposure process, all the patterns on the mask are taken into account and the presence of forbidden pitch reduces the common lithography process window. Therefore, it is important to analyze the imaging contrast of the different pitches of the pattern, to identify and solve the problem of forbidden pitch for the actual lithography process.
Fig. 1. Examples of different types of mask patterns: (a) 1:1 periodic pattern with different CD, (b) bright-field mask: the same CD (black) but different pitches, and (c) dark-field mask: the same CD (white) but different pitches.
Plasmonic lithography mainly includes plasmonic imaging lithography, interference lithography and direct writing lithography [4]. For masks containing a variety of different patterns, the plasmonic imaging lithography solution should be used in order to get all patterns imaged accurately on the wafer. Since superlens is designed to provide a wide optical transmission passband, it is mainly used for imaging lithography. While hyperbolic metamaterial (HMM) has a narrower passband and is mainly used for interference lithography [12]. Considering that the identical metallic material can play different roles in different models and wavelengths [12,13], the following analysis and simulation are based on superlens imaging model with layers of Silver-Photoresist-Silver, and monochromatic source with 365 nm wavelength in the ideal situation. Thus, the superlens based imaging lithography is discussed in this article.
2.2 Imaging model
As shown in Fig. 2, taking superlens imaging lithography as an example, the mask pattern is periodically distributed line/space along the x-direction, and infinitely long along the y-direction. Considering that TM waves are more effective than TE waves for imaging [14], only the TM wave case is analyzed here. Namely, polarized transverse magnetic wave with y-component is utilized for incidence.
Following the mathematical model of conventional imaging lithography, the electric field at the center of the photoresist z = z’ can be approximated as [15]
The above derivation process is to solve the three components of the electric field ${E_x}{\; }$and ${E_z}$ respectively, and finally get the total light intensity I. Another idea is to obtain the total electric field ${\boldsymbol E}$ and light intensity I at the image plan directly from the optical transfer function of the total electric field $OT{F_E}$. In this case, the optical transfer function of the total electric field can be expressed as
where ${E_{z^{\prime}}}({{k_x}} )= \sqrt {{E_{x,z\mathrm{^{\prime}}}}^2 + {E_{z,z\mathrm{^{\prime}}}}^2} $ and is the total electric field of image plan. ${E_{inc}} = \sqrt {{E_{x,inc}}^2 + {E_{z,inc}}^2} $ and is the total electric field of the incident plan. Hence, the light intensity in the imaging plan of the superlens can also be expressed asAccording to the imaging process in k-space described on the right side of Fig. 2, when the optical transfer function of the total electric field $OT{F_E}$ is obtained, the transfer efficiency of the superlens imaging system for electric fields with different ${k_x}$ can be intuitively analyzed. Thus, the electric field distribution at the image plan in k-space can be obtained directly. This method is more convenient for the subsequent analysis of light intensity contrast at the image plan.
2.3 Optical transfer function (OTF)
Combined with equations (5) or (7), it can be found that the light intensity distribution at the image plan depends on the mask pattern and the optical transfer function (OTF) of the superlens. From the perspective of the spatial frequency, the mask pattern determines the range of the spatial frequency, while OTF determines which spatial frequencies can be transmitted to the image plan and transfer efficiency of the certain spatial frequency. In conventional projection lithography, the pupil function (PF) acts similarly to the OTF here, and it has only two values, which are 1 within the cutoff frequency and 0 beyond the cutoff frequency. Actually, it acts the role of a low-frequency filter [9]. Therefore, high-frequency evanescent waves cannot participate in imaging, so that the details of the objective of the mask pattern cannot be reimaged in the imaging plan, which is the fundamental reason for the diffraction limit of conventional lithography. However, for plasmonic lithography, by exciting surface plasmon polariton (SPP), evanescent wave carrying high frequency objective information of the mask is resonantly amplified and participates in imaging, thereby breaking through the diffraction limit and improving the resolution of imaging [13].
Solved by Rigorous Coupled Wave Analysis (RCWA) [17–20] and the eigenmatrix method [21], the OTF of the superlens imaging system is shown in Fig. 3. As TM wave is adopted, the optical transfer functions of magnetic field ${H_y}$, electric field components ${E_x}$ and ${E_z}$ and total electric field ${\boldsymbol E}$ are calculated respectively. The wavelength λ is 365 nm, and the substrate of the mask is quartz glass, with permittivity of 1.46. Along the positive direction of light propagation, the mask Cr grating layer, PMMA spacer layer, Ag film, photoresist PR and reflective Ag layer are sequentially distributed. The corresponding thickness of each layer is 50 nm, 50 nm, 20 nm, 40 nm and 50 nm. And the corresponding permittivities of each layer is -8.55 + 8.96i, 2.25, -2.4 + 0.25i, 2.56 and -2.4 + 0.25i, respectively. As evanescent wave participates in imaging process, the property of the optical transfer function of the superlens is different with the conventional optical lens. As shown in Fig. 3 of the OTF curve, different values of ${k_x}$ characterizes different elements of light waves, as follows:
When ${k_x} < n{k_0}$, ${k_z} = \sqrt {{{({n{k_0}} )}^2} - {k_x}^2} > 0$ and is a real value, the corresponding light wave is a transmission wave;
When ${k_x} = n{k_0}$, ${k_z} = \sqrt {{{({n{k_0}} )}^2} - {k_x}^2} = 0$, the corresponding wave is in complete grazing incidence, and the absolute value of OTF tends to zero;
When ${k_x} > n{k_0}$, ${k_z} = i\sqrt {{k_x}^2 - {{({n{k_0}} )}^2}} $ is an imaginary value, and the corresponding light wave is an evanescent wave.
$n$ is the refractive index of the mask incidence medium, and the relationship with the dielectric constant ε is $n = \sqrt \mathrm{\varepsilon } $; ${k_0}$=$\frac{{2\pi }}{\lambda }$ is vacuum wave vector.
Through Fig. 3, it can be found that the OTF curve of the superlens imaging system is not a simple low-frequency filter, but also has a certain transmission efficiency in the high-frequency evanescent wave region, which intuitively explains the reasons for the improved optical resolution of the system.
In the lithography process, for a one-dimensional infinite distributed periodic line/space pattern, the imaging contrast ratio $\frac{{{I_{max}} - {I_{min}}}}{{{I_{max}} + {I_{min}}}}$ largely reflects the imaging quality and is directly related to the size of the process window in the lithography process [9]. Since the high-frequency diffraction orders can restore the detailed information of the object (mask), more high-frequency information is transmitted to the image plan with greater efficiency is the key to improving the contrast of imaging [22]. This is also the research idea in plasmonic lithography on how a fixed periodic pattern can improve its imaging contrast, such as using a high refractive index incident substrate or achieving mask spectrum shifting through off-axis illumination, so that ±1 or ±2 diffraction orders fall to the higher transmission efficiency range [23–26]. But this is only a measure to improve the contrast of imaging for a fixed mask pattern. As described in section 2.1, in a complete mask layout or the whole set of lithography process, there will be a variety of patterns.
Still taking the periodic line/space pattern as an example, there will be different CD for 1:1 periodic line/space patterns, and there will be a series of 1:2, 1:3, 1:4 line/space patterns with the same CD but different pitches. Hence it is necessary to analyze the imaging contrast for different periodic patterns, and to propose a common process window. Combined with the OTF curve shown in Fig. 3, when the light source is incident with a TM wave and the pitch of the mask pattern changes, the position of the diffraction order in k-space changes accordingly. The spatial frequency of the mask diffraction order can be expressed as
2.4 Forbidden pitch in plasmonic lithography
With normal incidence, when the mask pitch p equals to ${P_0}$, the position of each diffraction order in the OTF curve is shown by the green arrow in Fig. 3, and the zeroth order is represented by a black arrow. As the size of the pitch p increases, the position of each diffraction order moves closer to the origin of the horizontal axis, as shown by the orange arrow. The position of the zeroth order is unchanged which is still represented by a black arrow. Combined with the characteristics of the OTF of the electric field ${\boldsymbol E}$, which is presented with black curve in Fig. 3, in the process of increasing the pitch, the corresponding absolute coordinate of the diffraction orders decreases. When the position of the wave vector of the diffraction order approaches to point A in the curve, the transmission efficiency reaches a local maximum value, and the corresponding imaging contrast will be a local maximum value. With the increasing of the pitch p, the wave vector of the diffraction order reaches the point B in the curve, the wave vector ${k_x}$ is
the corresponding mask pitch is In this case, the transfer efficiency is very small, and the image contrast decreases significantly. With the increasing of the pitch p, the image contrast reaches a local maximum again until the wave vector of another higher diffraction order reaches point A. Therefore, in the process of increasing the mask pitch, the contrast of the image plan is not monotonously increased, but there will be a trough region, and this region is very similar to the forbidden pitch in conventional lithography, which we call the forbidden pitch in plasmonic lithography. And if the range of the size of the pitch is large enough, multiple forbidden pitches will occur.The location where the forbidden pitch occurs in the light intensity contrast - pitch graph is closely related not only to the shape of the OTF curve, but also to the mask type. The position of forbidden pitch of 1:1 mask and dark-field mask is basically $m\frac{\lambda }{n}$, and the range is relatively small, but the forbidden pitch of bright-field mask corresponds to a relatively long region. In general, the position of the forbidden pitch can be quantitatively solved and represented by Eq. (10), and it can be seen that the incident wavelength λ and the refractive index of the incident substrate n are closely related. And the specific performance can be seen in the subsequent simulation experiments.
Similarly to conventional lithography, changes in the source conditions can cause changes in where the forbidden pitch appears. The source conditions include the type of light source, the wavelength, off-axis angle, coherence, etc. When off-axis illumination or even other more complex illumination techniques are adopted, the forbidden pitch no longer corresponds to a fixed pitch, but to a small range of pitches, making it difficult to specify accurately where the forbidden pitch appears. In addition, unlike conventional DUV/EUV lithography, the imaging system (configuration) of plasmonic lithography is diverse and the number of film layers, film thickness and materials may change. However, the characteristics of the plasmonic imaging lithography system are maintained unchanged. Hence, the characteristics of its OTF will not change much, that is, the transmission passband of the OTF is wider compared to plasmonic interference lithography but it cannot involve all high-frequency orders in imaging. And the transfer efficiencies of different spatial frequencies are different. Therefore, when the ±1 or ±2 orders, which account for most of the energy, are less efficient, the contrast decreases more significantly and the forbidden pitch occurs.
The next step is to verify the validity of the theoretical analysis by means of simulations and to find the most important factors influencing the position of the forbidden pitch.
3. Simulation verification
In this section, a series of numerical simulation experiments are set up to verify the phenomenon of the forbidden pitch in plasmonic lithography. The simulation software is COMSOL Multiphysics 6.0, which is based on the finite element method (FEM), and the configuration is given in Fig. 4.
First, for the 1:1 line/space periodic pattern with different critical dimension (CD), the size of the pitch varies to observe the changing regular of the image contrast. In this process, the refractive index of the incident medium (Glass) is changed to verify its effect on the position of the forbidden pitch in the light intensity contrast - pitch graph. For patterns in submicron dimension, the exposure condition for the positive photoresist is that the light intensity contrast ratio should be greater than 0.4 [27]; meanwhile, the exposure condition for the negative photoresist is that the light intensity contrast ratio should be greater than 0.2 [28–30]. Therefore, the target CD at the mask is obtained based on the light intensity contrast of 0.4 and 0.2 separately.
Next, according to the CD determined in the former step, keep the CD in the mask plan unchanged, and the pitch is increased by equal step size. Besides, bright-field mask and dark-field mask are used separately. For bright-field mask, the width of the opaque part is unchanged, and the dimension of the transparent part increases gradually. For dark-field mask, the width of the transparent part is unchanged, and the dimension of the opaque part gradually increases. Thus, the impact of the dimension of the pitch on the light intensity contrast is observed. Meanwhile, the impact of the refractive index of the mask substrate (Glass) on the light intensity contrast is verified for a range of dimension of the pitch.
Finally, the impact of the thickness of the PMMA, the upper-side Ag film and the thickness of the photoresist on the light intensity contrast is observed. Based on the several steps, the key factors affecting the position of the forbidden pitch in the light intensity contrast - pitch graph are determined.
3.1 1:1 mask
According to the superlens imaging configuration given in Fig. 4 and the different parameters information given in section 2.3, a two-dimensional model is established in the software COMSOL Multiphysics 6.0. The light source is assumed to be monochromatic TM wave with wavelength of 365 nm. Normal incidence is utilized, and the light transmission direction is along the positive z axis. Periodic boundary conditions are employed on the left and right boundaries of the model, and perfectly matched layers (PMLs) are used on the upper and lower boundaries. The two-dimensional normalized light intensity distribution along the field transmission direction of the x-z plan is given in Fig. 5(a). The one-dimensional curve of the normalized light intensity distribution in the center of the photoresist (dotted line in Fig. 5(a)) is given Fig. 5(b). The normalization method of Fig. 5(a) is the light intensity at all positions in the two-dimensional graph divided by the maximum of the light intensity, so the entire normalized light intensity range is [0,1]. The normalization method in Fig. 5(b) is all light intensity values in the photoresist center divided by the maximum of the light intensity on the transversal line (dotted line in Fig. 5(a)), so the normalized light intensity range is also [0,1].
Fig. 5. Schematic diagram of COMSOL Multiphysics 6.0 simulation results: (a) the two-dimensional normalized light intensity distribution along the x-z plan, and (b) the one-dimensional curve of the normalized light intensity distribution in the center of the photoresist.
Based on the simulation results of the light intensity distribution along the transmission direction of the x-z plan and the light intensity distribution in the center of the photoresist, the impact of the size of the mask pitch to the light intensity contrast in the center of the photoresist is analyzed. The different parameters of the superlens configuration are kept unchanged. For the mask configuration, only the size of the pitch is varied periodically, and the ratio between the width of the transparent area and opaque area is 1:1. The size of the pitch increases from 100nm to 400nm by equal step size for three types of mask substrate with refractive index n of 1.1, 1.46 and 1.8.
As given in Fig. 6, with the increase of the size of the pitch, the change of the light intensity contrast is not monotonous. A trough area appears which corresponds to the forbidden pitch. The locations of the trough area are around 330nm, 250nm and 200 nm respectively. According to the analysis in section 2, the diffraction order of the forbidden pitch should correspond to the region B shown in Fig. 3, where the spatial frequency of the diffraction order is ${k_x} = n{k_0}$. According to Eq. (10), the corresponding mask pitch is $p = m\frac{\lambda }{n}$. At this point the forbidden pitch appears and m is taken to be 1, so the corresponding forbidden pitch should be λ/n, which is in general agreement with the simulation results. As the pitch continues to increase, fluctuations of the contrast curve become very small as the contrast is very close to 1. In this case, it is difficult to observe multiple forbidden pitches.
Fig. 6. Contrast curves with pitch for 1:1 mask at different refractive indexes of the incident medium.
In addition, there is a peak at the position adjacent to the forbidden pitch with a slightly smaller pitch at the black arrow in Fig. 6, which should correspond to the region A shown in Fig. 3, because the corresponding diffraction order transfer efficiency is higher at this point.
Furthermore, according to the curves in Fig. 6, the CDs (half-pitch) corresponding to a contrast of 0.4 are found to be 88 nm, 85 nm and 80 nm, and the CDs (half-pitch) corresponding to a contrast of 0.2 are found to be 68 nm, 67 nm and 65 nm for incident medium refractive indexes of 1.1, 1.46 and 1.8, respectively.
3.2 Bright-field mask
For a bright-field mask, critical dimension (CD) represents the size of the opaque area of the mask. For the three respective sizes of the critical dimension of 88 nm, 85 nm and 80 nm, the size of the transparent part is increased with equal step size of 4 nm. In the process, the thickness of each film layer of the superlens configuration and the opaque size of the mask are maintained unchanged. The corresponding one-dimensional curve of the light intensity contrast to the different size of the pitch is given in Fig. 7(a). For three respective cases of the refractive indexes, the curve appears several peaks and trough regions. The light intensity contrast in the trough area between the first and second peaks is even less than 0.4, which is lower than contrast of 1:1 dense pattern mask. Therefore, the pitch corresponding to this trough region can be called the forbidden pitch. To be specific, when CD is 80 nm and the refractive index of the mask substrate is 1.8, the pitches range from a1 to a2 are the forbidden pitches in this case. And the pitches range from b1 to b2 and c1 to c2 are the forbidden pitches for the case that CD is 85 nm and the refractive index is 1.46, and CD is 88 nm and the refractive index is 1.1, respectively.
Fig. 7. The phenomenon of the forbidden pitch of bright-field masks with different CDs and refractive indexes of incident medium: (a) different CDs determined according to contrast of 0.4 and different refractive indexes of incident medium, and (b) different CDs determined according to contrast of 0.2 and different refractive indexes of incident medium
Similarly, according to CD determined by the contrast of 0.2, the light intensity contrast-pitch curves of different refractive indexes of mask substrates are obtained, as shown in Fig. 7(b). The phenomenon in Fig. 7(b) is similar to that in Fig. 7(a). The pitches range from A1 to A2 and B1 to B2 are the forbidden pitches for the case that CD is 65 nm and the refractive index is 1.8, and CD is 67 nm and the refractive index is 1.46, respectively. The contrast of the two pitches is lower than that of 1:1 dense pattern (0.2). When CD is 68 nm and the refractive index of the mask substrate is 1.1, the range of the forbidden pitch is small and can be represented by point C.
According to the analysis of section 2, with the increase of the mask pitch, when the first, second and other higher diffraction orders reach the region A shown in Fig. 3, the transfer efficiency is higher, and the contrast is higher at this time which correspond to the peaks in Figs. 7(a) and (b). And in the process of the arrival of these diffraction orders successively, there will be several trough regions. Light intensity contrast in these trough regions is low, especially in the first trough. When the contrast of these trough regions is lower than that of 1:1 dense pattern, the corresponding pitch range becomes forbidden pitch.
3.3 Dark-field mask
For a dark-field mask, critical dimension (CD) represents the size of the transparent area of the mask, which is the opposite of bright-field mask. Similar to the processing method of bright-field mask in section 3.2, one-dimensional curves of contrast varying with the pitch under different refractive indexes are calculated and obtained according to different CDs with contrast of 0.4 and 0.2, as shown in Figs. 8(a) and (b). And the thickness of each film layer of the superlens configuration and the transparent size of the mask are also maintained unchanged in the process.
Fig. 8. The phenomenon of the forbidden pitch of dark-field masks with different CDs and refractive indexes of incident medium: (a) different CDs determined according to contrast of 0.4 and different refractive indexes of incident medium, and (b) different CDs determined according to contrast of 0.2 and different refractive indexes of incident medium
This case is very close to the 1:1 mask, that is, as the pitch increases, the first trough region appears around 330 nm, 250 nm and 200 nm respectively, which can be called the forbidden pitch. And as the pitch range is extended, the second, the third and more contrast trough regions can be seen, in line with the theoretical analysis of $m\frac{\lambda }{n}$ (where m is taken as 2, 3, …). However, it can be noted that the contrast changes in the second and subsequent trough regions are not very evident.
3.4 Varying the film layer thickness
From section 3.1 to section 3.3, the film thickness of the superlens is kept unchanged. The impact of the different refractive indexes of the mask substrate is verified. Through comparison, it can be found that the forbidden pitch phenomenon is most evident in bright-field masks, and the intensity contrast is even lower than 1:1 dense pattern in the process of increasing the pitch. For dark-field mask and 1:1 line/space mask, in the contrast-pitch curve, the local minimum of light intensity contrast at the forbidden pitch is relatively higher than the contrast of the 1:1 dense pattern (0.4 or 0.2), which indicates that the contrast does not increase monotonically with the pitch. Therefore, in a more strict way, the forbidden pitch of 1:1 mask and dark-field mask are regarded as the general forbidden pitch. Therefore, in this subsection, the bright-field mask is used, and the refractive index of the incident medium is kept constant at 1.46 and the CD (opaque area) of the mask is kept at 85nm. The film thickness of the superlens is changed to verify its impact on the light intensity contrast.
Figure 9(a) shows the variation of the light intensity contrast with the increasing of the size of the pitch, when the thickness of the upper-side Ag film upon the photoresist is varied. It can be observed that as the thickness of the Ag film increases from 15nm to 30nm, the overall light intensity contrast increases, but the changing regular of each curve remains basically the same, that is, there is a low contrast region between the first and second peaks. Figure 9(b) shows the change of the light intensity contrast with the variation of the size of pitch when the thickness of the photoresist is varied. When the thickness of photoresist increases from 30 nm to 45 nm, the overall contrast curves decline, but the trend of each curve remains basically the same, with the trough region still appearing between the first and second peaks. Similarly, Fig. 9(c) shows the change of the light intensity contrast with the variation of the size of pitch when the thickness of the PMMA is varied. The trend and characteristics of the whole curve are very close to the situation when the photoresist thickness is changed, that is, as the thickness of PMMA increases from 40 nm to 55 nm, the overall contrast curves decline but the trend of each curve remains basically the same, with the trough region still appearing between the first and second peaks. Unlike the case of changing the Ag film thickness, as the photoresist thickness and PMMA thickness increase, the whole contrast curve shifts downwards rather than upwards, which is related to the different properties of the media. Ag is a metal, while photoresist and PMMA are dielectrics. In conclusion, these three cases prove that although changing the film thickness can improve the contrast overall, the forbidden pitch phenomenon still exists.
Fig. 9. Forbidden pitch phenomenon when changing the thickness of the film layer: (a) at different upper-side Ag layer thicknesses, (b) at different photoresist layer thicknesses and (c) at different PMMA layer thicknesses.
Through the verification in this section, it is clear that the position where the forbidden pitch appears is closely related to the refractive index of the incident medium, as ${k_x} = n{k_0}$ corresponds to the case of complete grazing incident light wave, and the OTF tends to zero, at which point the contrast drops significantly. A change in film thickness will affect the shape and overall position of the contrast curve, but will not change the position of the forbidden pitch in the light intensity contrast - pitch graph. Therefore, to change the position of the forbidden pitch, one can use an incident medium with different refractive index. And to improve the overall image contrast, one can adjust the film thickness. In addition, changing the type of mask will result in a different forbidden pitch phenomenon. Therefore, for the same technology node, if the process conditions allow, the dark-field mask could be used to reduce the effect of the forbidden pitch.
3.5 Varying the permittivity of Ag
Considering the actual processing, oxidation and high temperature will change the properties of the metal Ag, so a slight perturbation to the standard Ag permittivity of -2.4 + 0.25i is made, to verify the effect of deteriorating the properties of Ag on forbidden pitch. By selecting the bright-field mask with the most obvious phenomenon and changing the permittivity of Ag by an amplitude of ±0.2, the obtained data are summarized in Fig. 10. As shown in the below figure, although the overall curve moves slightly when the permittivity changes, the forbidden pitch phenomenon is consistent with the theoretical analysis, that is, there is a low contrast region between the first and second peaks. As a result, the deteriorating properties of Ag will not fundamentally change forbidden pitch phenomenon. The forbidden pitch phenomenon has a good tolerance to the change of metal parameters caused by process conditions.
4. Conclusion
Through the above theoretical analysis, it is clear that, similarly to conventional projection lithography, the forbidden pitch phenomenon also exists in plasmonic lithography represented by superlens imaging lithography. With normal incidence of TM wave, and the monochromatic light of 365 nm wavelength, it is obtained that the forbidden pitch should appear at the complete grazing incidence case where the mask diffraction order ${k_x} = n{k_0}$, and OTF tends to 0. The corresponding mask pitch $p = m\frac{\lambda }{n}$, and m can be taken as 1, 2, 3, …, that is, more than one forbidden pitch exists.
Subsequently, for 1:1 line/space mask, the bright-field mask and the dark-field mask, the forbidden pitch phenomena are verified respectively by setting up different simulation experiments. When the refractive index of the incident medium of the mask substrate is changed, the 1D curve of the light intensity contrast changes in accordance with the theoretical analysis. Then, for bright-field mask, the forbidden pitch phenomenon is most obvious. It is used to verify the effect of the film thickness on the forbidden pitch phenomenon, and it is found that the film thickness affects the shape and overall position of the contrast curve, but does not change the position of the forbidden pitch in the light intensity contrast - pitch graph. Finally, by selecting the bright-field mask with the most obvious phenomenon and changing the permittivity of Ag by an amplitude of ±0.2, it is proved that the deteriorating properties of Ag will not fundamentally change forbidden pitch phenomenon.
In addition, this paper uses 365 nm wavelength with normal incidence, and the superlens imaging lithography is taken as an example for theoretical analysis and simulation verification. Considering the objective of mask full-pattern imaging and the background that the same metal can play different roles in different schemes or wavelengths, the available options are inherently limited. And Ag film superlens at the wavelength of 365 nm is the most common configuration of plasmonic imaging lithography at present, hence it is representative for analysis and validation. Subsequently, with the development of plasmonic lithography, especially imaging lithography, the multilayer configuration with alternating metal and dielectric, the different wavelengths, and the different materials can be studied in order to be more comprehensive. In the future, we will further theoretically and experimentally develop shorter wavelength light sources such as DUV, to continue to improve the imaging resolution of plasmonic lithography.
Overall, as a supplementary lithographic technology, plasmonic lithography is an alternative candidate for the next-generation lithography. The discovery of the phenomenon of forbidden pitch in plasmonic lithography is of great significance for the further understanding and development of this lithography technology, as it is a problem that must be solved in the process of moving towards the industrial application of integrated circuits. With the improvement and updating of the experimental conditions of the lithographic station, we will further carry out experimental verification, and it is greatly welcomed that our theoretical results can be verified by experiments of other research groups. Taking the conventional projection lithography as a reference, plasmonic lithography can also be solved by avoiding forbidden pitch patterns or by adding assistant feature in the design. And it would also be a good idea to try to explore more lighting options and optimize the light source to solve the problem of forbidden pitch.
Funding
Scientific Research Foundation of the University of Chinese Academy of Sciences (118900M032); Guangdong Province Research and Development Program in Key Fields (2021B0101280002); A high-level innovation research institute from Guangdong Greater Bay Area Institute of Integrated Circuit and System (2019B090909006); The construction of new research and development institutions (2019B090904015); Guangzhou City Research and Development Program in Key Fields (202103020001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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