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With the help of simulations we study the benefits of using coherent, phase-structured illumination to detect the overlay error in resist gratings fabricated by double patterning. Evaluating the intensity and phase distribution along the focused spot of a high numerical aperture microscope, the capability of detecting magnitude and direction of overlay errors in the range of a few nanometers is investigated for a wide range of gratings. Furthermore, two measurement approaches are presented and tested for their reliability in the presence of white Gaussian noise.
© 2015 Optical Society of America
1. Introduction
Since the first introduction of the double patterning technology by Maenhoudt and coworkers in 2005 [1], this technique and its follow-up improvements have become a widely used standard method for semiconductor lithography. Using double exposure, the grating design can be split up in two structures at a much more relaxed pitch of twice the final pitch. Apart from the selection of the exact patterning and etching parameters, the overlay of both structures is of crucial importance [2–5]. In this work, we focus on resist line gratings with line widths between 50 and 300 nm denoted as critical dimension (CD). The grating period is set to twice the CD and the height is equal to the CD, ensuring an aspect ratio of 1 for a better comparison between the different CD values (Fig. 1). The overlay error studied here is typically encountered for the two most common methods used for double patterning, the litho-etch-litho-etch (LELE) and the sacrificial self-aligned spacer method. More information about both methods can be found for example in [5] and a sketch of the LELE method is shown in Fig. 1(b).
Fig. 1 Studied grating: (a) Cross-section of the simulated resist grating on top of a bottom anti-reflective coating (Barc) of 40 nm thickness on a silicon substrate, (b) Sketch of the LELE-method that can be used for the production of the gratings, (c) Definition of studied overlay error xs.
The simulated grating is periodic in x-direction and infinite in y-direction. The indices are defined as follows: xs denotes the overlay error or shift of the second grating, and xi and yi denote the x, y-coordinates in the image plane.
Resist gratings pose the problem of having to rely on larger wavelengths for non destructive measurements, as most resist types will be damaged if DUV-light is used. Taking into account the allowed tolerances of only a few nanometers, the combined problem is to find out if it is possible to detect values of xs in the range of 2–15 nm when using (near-) visible light. A very common technique for optical inspection of wafers is scatterometry [6, 7]. Scatterfield microscopy uses carefully selected illumination and focus settings to detect defects below the diffraction limit [8]. Also the addition of white-light interferometry has shown promising results, especially regarding the depth sensitivity [9, 10]. Besides these intensity based methods, recent studies reveal a great potential of detecting small defects by utilizing the phase distribution along the illumination spot [11, 12]. Similarly, dark beam microscopy [13–15] showed promising results in detecting nano-scale features by evaluating the intensity distribution in the far field. To this end, a phase singularity was added to the illumination by a π-phase mask. So far, most of these methods focus on characterizing either isolated defects or the exact geometry of the grating. Very little emphasis has been put on asymmetric defects and / or asymmetric grating geometries. Our previous studies showed that by evaluating the phase information of phase-structured illumination, we were capable of detecting nanometer sized asymmetries in the bottom rounding of trenches [16].
In the next section, we will briefly introduce the simulated setup. Afterwards, the effects and advantages of phase-structured illumination in comparison to uniform phase illumination will be studied. Particular attention will be payed to the influence of grating structure and illumination parameters on the ability to detect the overlay error xs. In section 4, we will investigate the practical implementation and define measurement routines. Subsequently, the results will be summarized in the last section. It should be noted that all discussions in this paper are based on rigorous simulations.
2. Simulation setup
As setup, a bright-field microscope with an objective with a NA of 0.7 was used, see Fig. 2. The position of the spot on the grating in x-direction is denoted by xf. In z-direction, the spot is focused on top of the line grating.
Fig. 2 Schematic setup: A reflection microscope with a tightly focused beam (NA = 0.7) is used with the addition of a phase plate and a linear polarizer in the illumination path. For normal incidence, the electric field vector is perpendicular to the grating lines for TM polarization and parallel for TE polarization.
For the structured illumination, a π-phase plate is introduced in the entrance pupil of the microscope. This results in a dumbbell-type intensity distribution in the focus plane that is reflected from the sample and then imaged onto the camera. The π-phase shift in one half of the spot is mostly conserved but shows high sensitivity to asymmetries in the grating, see Fig. 3.
Fig. 3 Intensity and phase distribution of the reflected light in the image plane for a grating with CD = 100 nm, λ = 405 nm, xs = 0 nm and TE polarization. The focus position xf of the illumination is set to the middle of the top of the grating line.
The dumbbell spot can mathematically be described by the LP11-mode. For the evaluation of the behavior of said spot for different values of CD, polarization (TE / TM) and used illumination wavelength λ, simulations were performed using the rigorous coupled wave analysis (RCWA) [17]. We studied wavelengths between 305 and 505 nm, as those do not damage the resist.
3. Simulation results
In this section, we will study the influence of illumination (wavelength and polarization) as well as grating structure parameters on the ability to observe the shifts xs. Using simulations, we are able to directly analyze the impact of xs on the intensity and the phase distribution.
For the evaluation, we will use cross-sections of the intensity and phase distributions at yi = 0 nm. We will also set the center of the spot to the middle of a grating line and define this as focus position xf = 0 nm. As we are interested in the possibility of detecting the overlay error, we will study the difference in intensity ΔI and phase Δφ between cases with introduced shift and perfect overlay (xs = 0):
3.1. Detection of asymmetry
First, we discuss whether phase-structured illumination helps in detecting asymmetries. Here, an important point is not only the ability to detect the absolute value of the shift (|xs|), but also the direction. Since a uniform phase distribution leads to a symmetric Airy disc intensity distribution, the detection of the direction of the asymmetry might not be possible, whereas an asymmetric distribution in intensity and phase might help to detect the direction. Therefore, we selected the following case for analysis: A grating with CD of 150 nm with shifts of xs = ±10 nm was studied under TE polarized illumination (λ = 405 nm). Comparing the differences between intensity of an asymmetric grating and a symmetric one (xs = 0 nm), see Fig. 4, one can draw the following conclusions:
Fig. 4 Normalized intensity differences ((a) and (b)) and absolute intensity differences ((c) and (d)) between an asymmetric grating with xs-values between ±10 nm and a symmetric one for symmetric (left) and asymmetric (right) intensity distributions of the illumination spot. All gratings have a CD of 150 nm and are illuminated by TE polarized light (λ = 405 nm). The intensity values were rescaled to values between [0,1], where 1 defines the global maximum for xs = 0 nm.
First of all, it has to be stated that not only the structure size is below λ/2, but also the shifts xs are in the range of λ/40. One can see that the intensity changes ΔI are larger if a phase plate is inserted. The other striking fact is that the value of ΔI changes sign depending on the direction of the introduced shift, which even holds true for the symmetric spot (no phase plate). And although the absolute intensity changes look symmetric for a certain absolute value of xs, there are small differences even for the symmetric spot and larger ones for the phase-structured spot.
3.2. Profit of using phase information
So far, we have evaluated the advantage of inserting a phase plate to generate a dumbbell spot when analyzing the intensity. As pictured in Fig. 3, the phase plate also leads to a distinct phase distribution. The ability to detect the asymmetry using the change in phase is studied next. For now, we will focus solely on the simulated phase changes. In order to include limitations introduced by the phase measurement, we will limit our study to areas with more than 5 % of the maximum intensity, as areas with lower intensities are more prone to phase errors. The influence of the shift xs on the phase signal is compared in Fig. 5 for a grating structure of CD= 150 nm and TE polarized light (λ = 405 nm) for an unstructured and phase-structured spot. The comparison reveals that if phase-structured illumination is used, the signal strength increases by 33 % if intensity is concerned and by 50 % if phase is concerned.
Fig. 5 Comparison of influence of shift xs in phase for non-structured and structured phase distribution in the measurement spot. Areas with less than 5 % of the maximal intensity (marked gray) were not used for evaluation. The studied grating has a CD of 150 nm and is illuminated with λ = 405 nm and TE polarization.
3.3. Influence of structure size
So far, all results were obtained for one grating size. However, a study concerning the minimal needed structure size will help to better describe the potential as well as the limitations of our method. Therefore, we will examine the signal behavior depending on structure size (CD) with regard to wavelength. Illustrated in Fig. 6 is the maximum of the absolute value of ΔI and Δφ as a function of CD for wavelengths λ ranging from 305 to 505 nm. The value for xs was set to 0.05 CD in all cases to ensure comparable results.
Fig. 6 Threshold of CD for detectable signals in intensity and phase depending on wavelength. Dashed lines denote no phase plate, solid lines denote phase plate.
Two conclusions can be drawn from these results for both intensity and phase measurement: First, there is a minimum CD value below which the method collapses. This minimum value decreases with decreasing wavelength as expected and does not depend on whether phase-structured illumination is used or if intensity or phase is taken as measurement quantity. However, the overall growth of the signal with increasing CD is much stronger if a phase plate is present, which allows better measurements for small CDs.
The second interesting fact is that there is also a CD with maximal signal strength. This peak occurs for CD-values just below λ/2. If the CD is further increased, the signal strength is again decreasing. It may be noted that this behavior is observed mainly for the case of a present phase plate. It is due to the fact that for structure sizes above λ/2, the intensity profile is no longer characterized by the LP11-mode, but by an overlap due to higher diffraction orders leading to lower sensitivity in this case. Similar conclusions apply to the phase distribution that no longer exhibits one phase jump of π but a more complex shape. In the case of no phase plate there is no obvious maximal CD, as the overall intensity distribution along the spot is constant in the range of the studied CDs. The same holds true for the phase distribution.
3.4. Influence of used polarization
Apart from structure size and wavelength, the polarization state of the illuminating light can be used to enhance the detection of the shift. When comparing results for TE and TM polarized light, one finds that the influence is also a function of structure size and wavelength, hence one is able to find cases where one polarization state is preferable or where the difference in polarization is virtually non existing. Despite these partially unsatisfactory results, it should be noted that even for cases where one polarization state yields better results, the other polarization state still enables a sufficient measurement.
4. Measurement strategies
In this section, measurement routines relying on the phase information will be studied. The results presented so far help to get insight into the theoretical limits of the proposed method, as the used quantities ΔI and Δφ allow us to determine whether a certain shift xs can be detected. In practice, these quantities are ill suited as one would have to have access to a perfect structure (xs = 0 nm) or use a simulation result as reference. Also, we have assumed that our spot is focused at xf = 0 nm. This assumption is not easily applicable in practice, since the structures themselves are smaller than the resolution limit posed by the Rayleigh-criterion and therefore are not resolved.
Two measurement strategies will be discussed here, one making use of the Fourier transformation of the signal, the other one uses two measurements with a defined focus spot separation of one grating period. Taking the grating period as a priori information is justified, since this quantity is well defined in practice.
Both methods allow a detection of the shift xs without knowledge about the focus position xf on the grating by using information from laterally scanning over the sample. The effect of different spot focus positions xf on the phase is shown by cross-sections along xi in the image plane in Fig. 7 for both measurement quantities (phase and Δφ) and a grating with CD = 150 nm, xs = 10 nm, illuminated by TE polarized light with λ = 405 nm. For simplicity, all further discussions in this sections use this grating geometry.
Fig. 7 Dependence of phase distribution on xf: (a) shows the phase distribution of the cross-sections over xf, (b) shows the value of Δφ for the self-reference approach for different xf – the y-axis denotes the first focus position.
Both signals show the largest deviations for xi = 0 nm, hence this spot will be selected for the following evaluations.
4.1. FFT-approach
For the FFT-approach, the phase distribution at xi = yi = 0 nm is measured while scanning over the focus position xf. This signal is then Fourier transformed and the absolute value and the phase of the complex signal φ̃ are evaluated, where “∼” denotes the Fourier transformed values. In Fig. 8, the dependence of both quantities on the value of xs is shown.
Fig. 8 Change of Fourier transformed signal for various values of xs: The absolute value of the shift can be determined from |φ̃| and the direction from the phase of φ̃. For better visibility, the peak at xf = 0 for the absolute value in (a) was neglected, as it holds no information.
It can be seen that the absolute value gives information about the size of the shift xs, whereas the phase distribution of the Fourier transformed signal determines the sign of the shift. If no shift is present, this value changes from +π/2 to −π/2. For negative values of xs, the change in phase is smaller, whereas for positive values it is larger.
4.2. Self-reference approach
For the self-reference method, two measurements at two different focus positions (separated by one grating period) are evaluated. The resulting signal is similar to the signal in section 2, without relying on a reference signal of a perfect grating. Thus, making this approach more useful in practice, as a real grating might also have other minor imperfections, which may not be included in the simulation. For this method, it is necessary to redefine the measurement quantity Δφ to:
When evaluating the signal strength, a distinct dependence on the choice of xf can be observed for values of xs ≠ 0 and hence, information about xf can be obtained. The signal at xi = yi = 0 nm is shown in Fig. 9 for xs-values between ±15 nm.
Fig. 9 Δφ as defined in (2) as a function of xf for various values of xs. Below the signal the xf-position on a grating without shift is illustrated.
The following conclusions may be drawn from this: For unknown xf-position, the peak-to-valley (PV) of the signal is well suited to determine the magnitude of the shift xs. The determination of the direction of the shift is also possible. It may be worth noting that the definition of xs is itself dependent on the focus position. This becomes clear when looking at the schematic drawing in Fig. 10.
Fig. 10 Depending on whether the grating on position A or B is selected, the shift xs is denoted as positive or negative.
However, when taking the sign of the phase difference Δφ into account, the extreme values are always associated with focus positions at the top of the grating. If this value is positive, the groove on the left side is larger than half the grating period and likewise if the value reaches its minimum, the groove on the left side is smaller than half the grating period.
4.3. Noise evaluation
For both methods we study the influence of noise on the signal by adding white Gaussian noise to the simulated phase and evaluate the resulting change in signal.
For the FFT-approach, the evaluation of the absolute value of xs is not affected by the addition of noise. For the detection of the sign of xs, a noise level above SNR = 30 might cause too strong interference, rendering the signal useless, see Fig. 11.
The PV-value for the self-reference approach is very stable in the presence of this noise, showing a visible offset only for values of SNR below 30, see Fig. 12.
Fig. 12 Noise dependence of PV-values for different values of xs. Only for SNR-values below 30 the induced shift will prevent accurate measurements.
5. Conclusion
In this paper, we showed that using phase-structured illumination improves the detection of overlay errors in resist gratings fabricated by double patterning. Although some improvement can be seen for intensity evaluation, using the phase distribution enables by far more precise measurements of xs. The dependence of signal strength on structure size in combination with illumination wavelength shows that with visible light, a detection of a few nanometers overlay error is possible even for structures sizes below 100 nm. For an experimental realization, two methods were proposed with regard to their applicability in presence of white Gaussian noise and both methods showed a promising behavior. Using methods like digital holographic microscopy (DHM), one can access the complex field distribution. Recent works studying the quality of phase reconstruction for DHM showed excellent results close to the shot noise level [18, 19].
Acknowledgments
The authors would like to thank Arie den Boef and Simon Mathijssen from ASML for fruitful discussions. This work was supported by the German Research Foundation (DFG) within the DFG funding program Open Access Publishing.
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