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Calibration is indispensable for the sensor operation. Without calibration, the generally inhomogeneous spatial illumination profile and the phase aberrations of the system would by far dominate the subtle intensity and phase profile changes caused by dimensional variations of typical nanostructures. Our calibration routine comprises two steps: first, referring to a well-known, typically unstructured target, and second, removing defocus and tilt phase terms by means of a Zernike decomposition of the phase map. In view of the subsequent polarimetric analysis, the latter step is especially important because it establishes correct relative phase relationships in a series of measurements.
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Per wavelength
$ \lambda $ and propagation angle$ \left({NA}_{x},{NA}_{y}\right) $ , the interaction between the object and reference arm is a simple two-beam interference:$$ {I}_{2\mathrm{b}}={I}_{\mathrm{o}\mathrm{b}\mathrm{j}}+{I}_{\mathrm{r}\mathrm{e}\mathrm{f}}+2\sqrt{{I}_{\mathrm{o}\mathrm{b}\mathrm{j}} \; {I}_{\mathrm{r}\mathrm{e}\mathrm{f}}} \; \mathrm{cos}\left(\frac{4\pi z}{\lambda }+{\phi }_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\phi }_{\mathrm{o}\mathrm{b}\mathrm{j}}\right) $$ (1) In this equation,
$ z $ is the scan position of the reference arm, and$ \Delta \phi =4\pi z/\lambda +{\phi }_{\mathrm{r}\mathrm{e}\mathrm{f}}-{\phi }_{\mathrm{o}\mathrm{b}\mathrm{j}} $ describes the phase difference between the reference and test waves. The phase$ {\phi }_{\mathrm{o}\mathrm{b}\mathrm{j}}={\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}+{\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{o}\mathrm{b}\mathrm{j}} $ consists of two summands: the actual target phase$ {\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ , and a general phase term$ {\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{o}\mathrm{b}\mathrm{j}} $ which considers all constant phase shifts associated with the object-arm path through the optical system, such as the aberrations of the microscope objective in the object arm. Phase$ {\phi }_{\mathrm{r}\mathrm{e}\mathrm{f}} $ is defined analogously, with the reference mirror acting as the target:$ {\phi }_{\mathrm{r}\mathrm{e}\mathrm{f}}={\phi }_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}+{\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{r}\mathrm{e}\mathrm{f}} $ . Intensity$ {I}_{\mathrm{o}\mathrm{b}\mathrm{j}} $ contains the intensity response of the target$ {I}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ itself, and a unitless factor$ {\Sigma }_{\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}} $ which accounts for the spatial variation of the illumination profile in the Fourier plane:$ {I}_{\mathrm{o}\mathrm{b}\mathrm{j}}={\Sigma }_{\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}} \cdot {I}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ . In a complete analogy,$ {I}_{\mathrm{r}\mathrm{e}\mathrm{f}} $ is defined as$ {I}_{\mathrm{r}\mathrm{e}\mathrm{f}}={\Sigma }_{\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}} \cdot {I}_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} $ . We want to emphasise that all quantities are angle- and wavelength-dependent, whereas only$ {\Delta }\phi $ depends on$ z $ .The integral over all two-beam interference terms
$ {I}_{2\mathrm{b}} $ from Eq. 1 within the spectral range of the source corresponds to the white-light interference signal$ I\left(z\right) $ per angle (see Fig. 5a). To remove spatially inhomogeneous coherence effects, each measured interferogram is normalised by its contrast$ C $ , which is defined as$ C=\left({I}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{I}^{\mathrm{m}\mathrm{i}\mathrm{n}}\right)/\left({I}^{\mathrm{m}\mathrm{a}\mathrm{x}}+{I}^{\mathrm{m}\mathrm{i}\mathrm{n}}\right) $ . The maximum and minimum intensity values$ {I}^{\mathrm{m}\mathrm{a}\mathrm{x}} $ and$ {I}^{\mathrm{m}\mathrm{i}\mathrm{n}} $ are determined by fitting envelopes to the white-light signal. Subsequently, the FFT is applied. When evaluated at a single wavelength, the complex result17 of the FFT is:$$ {\tilde {I}}_{1}=2{\Sigma }_{\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}}\sqrt{{I}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}{I}_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}} \cdot {\mathrm{e}}^{\mathrm{i}\left({\phi }_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}+{\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{r}\mathrm{e}\mathrm{f}}-{\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}-{\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{o}\mathrm{b}\mathrm{j}}\right)} $$ (2) The first two summands from Eq. 1 disappear because they are independent of
$ z $ .As we are only interested in the target properties, we have to extract
$ {I}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ and$ {\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ from Eq. 2. For this purpose, we perform a second measurement on a calibration target and obtain$$ {\tilde {I}}_{2}=2{\Sigma }_{\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}}\sqrt{{I}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}{I}_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}} \cdot {\mathrm{e}}^{\mathrm{i}\left({\phi }_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}+{\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{r}\mathrm{e}\mathrm{f}}-{\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}-{\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{o}\mathrm{b}\mathrm{j}}\right)} $$ (3) from the FFT (at the same angle and wavelength as before). Next, we divide Eq. 2 by Eq. 3 to eliminate the influence of the reference arm, that is, quantities
$ {I}_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} $ ,$ {\phi }_{\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} $ , and$ {\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{r}\mathrm{e}\mathrm{f}} $ . Furthermore, the illumination factor$ {\Sigma }_{\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}} $ and the general phase contribution$ {\phi }_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h},\mathrm{o}\mathrm{b}\mathrm{j}} $ from the path through the object arm vanish:$$ \frac{{\tilde {I}}_{1}}{{\tilde {I}}_{2}}=\frac{\sqrt{{I}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}}{\sqrt{{I}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}}} \cdot {\mathrm{e}}^{\mathrm{i}\left({\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}-{\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\right)} $$ (4) The calibration target is simple and well-known, such that
$ {I}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}} $ and$ {\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}} $ can be removed from Eq. 4, using either simulation or additional measurements. For the model-based reconstruction presented in this study, we use a combination of both approaches. Our calibration target is an unstructured plane silicon wafer. We measure$ {I}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}} $ in the same way as the intensity distributions shown in Fig. 4a and multiply the results from Eq. 4 with$ \sqrt{{I}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}} $ to obtain the preliminary measurement results, E:$$ E=\frac{{\tilde {I}}_{1}}{{\tilde {I}}_{2}} \cdot \sqrt{{I}_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}}=\sqrt{{I}_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}} \cdot {\mathrm{e}}^{\mathrm{i}\left({\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}-{\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\right)} $$ (5) The phase
$ {\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}} $ is considered in the simulations. In a complete analogy to Eq. 5, the phase difference$ {\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}-{\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ (instead of the target phase$ {\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ alone) is used to calculate the Mueller matrix.We want to emphasise that it is not mandatory to use a silicon wafer as a reference target; this was nothing more but a convenient choice because our test targets are always surrounded by unstructured silicon. As long as the reference target is well known, any sample can be used for calibration.
Another important aspect of the proposed measurement strategy is the relative positioning of the test and calibration targets with respect to the sensor, both in terms of defocus and tilt. Minor differences from target to target are inevitable and lead to falsified phase signals
$ \phi ={\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}-{\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ . We correct such positioning errors by performing a Zernike decomposition18 of each Fourier-plane phase map (that is,$ \phi \left({NA}_{x},{NA}_{y}\right) $ at all angles simultaneously, but only one wavelength at a time). The two tilt terms are then entirely removed, whereas the rotationally symmetric contributions are partially subtracted according to the defocus phase19:$$ {\phi }_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{c}\mathrm{u}\mathrm{s}}=\frac{4\pi }{\lambda } \cdot \Delta z \cdot \mathrm{cos}\alpha $$ (6) In Eq. 6,
$ \Delta z $ is the geometrical defocus along the optical axis, and$ \alpha $ is the angle of incidence (see Fig. 3b). Finally, the phase map is re-assembled using the modified Zernike coefficients, and the results per angle and wavelength are inserted into Eq. 5, where they replace the erroneous original phase values$ {\phi }_{\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{b}}-{\phi }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}} $ .Fig. 6 illustrates the individual steps of the described calibration procedure, using the silicon line grating shown in Fig. 8 as the test sample and the neighbouring plane silicon substrate as the reference sample. In the first two columns, the corresponding (uncorrected) measurement results,
$ {\tilde {I}}_{1} $ from Eq. 2 and$ {\tilde {I}}_{2} $ from Eq. 3, are depicted for all NA coordinates at an arbitrarily chosen wavelength of 628.1 nm. The inhomogeneous illumination profile of the multimode fibre causes both absolute-square distributions to drop significantly towards large NA values. The phase distributions vary with high spatial frequency and large amplitude, which is due to the dominant aberrations of the microscope objectives. All such setup influences vanish when the two measurement results are combined according to Eq. 4, as shown in Fig. 6c. Subsequently, the absolute square is multiplied by the pre-characterised silicon intensity distribution (see Eq. 5) and the phase map is corrected in terms of defocus and tilt, providing the final calibrated results shown in Fig. 6d.Fig. 6 Step-wise illustration of the calibration procedure.
a Fourier-transformed intensity at an arbitrarily selected wavelength of 628.1 nm, measured on an exemplary silicon line grating; b corresponding reference data obtained on a plane silicon wafer; c result of the combination of both measurements according to Eq. 4; d final result after additional corrections; e corresponding simulation results for comparison.For comparison, the absolute square and phase were also simulated using the nominal grating parameters as determined by additional SEM measurements (see Table 1). The corresponding results are presented in Fig. 6e. Except for a global offset in phase, the calibrated measurements coincide well with the simulations. This statement holds true even for the largest NA values, which implies that the setup imperfections (including the performance of the microscope objectives and beam splitters) are very well corrected. It should be noted that the mentioned phase offset does not influence the Mueller matrix as long as it is the same throughout the entire measurement set. The RMS between the measurement and simulation amounts to 5.0% in the case of the absolute square and 5.2% in the case of the phase, provided that the mean value is subtracted from each phase distribution before relating them to each other. The RMS values are slightly larger than those obtained during the pure intensity measurements shown in Fig. 4, but this was to be expected because the measurement procedure requires additional steps and is hence more complex and error-prone. In addition, the target modelling is more elaborate.
Parameter SEM AFM Scatterometry Top-cd (nm) 51.2 ± 2.9 − 51 Height (nm) 70.1 ± 3.7 70.9 ± 1.1 70 Sidewall angle (°) 77.5 ± 5.3 − 81 Table 1. Parameter values of the silicon line grating obtained directly from SEM and AFM measurements, and indirectly from scatterometric measurements via model-based reconstruction. The specified uncertainties are the standard deviations obtained from averaging over ten measurement sites per parameter. In AFM measurements, lateral dimensions and sidewall shapes are always affected by the tip shape, which is why we omitted the corresponding values.
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The determination of the Jones matrix
$ J $ of the sample requires four measurements with different angular positions of the polariser and analyser. Angles$ {\theta }_{P} $ and$ {\theta }_{A} $ are defined relative to the orientation of a line-grating target: 0° (90°) corresponds to a polarising or analysing direction perpendicular (parallel) to the grating lines in the Fourier plane of the microscope objective. Because the grating lines are oriented along the$ y $ -axis (see Fig. 1), 0° (90°) is denoted by the subscript$ x $ ($ y $ ).The most obvious choice of angle combinations
$ \left({\theta }_{P},{\theta }_{A}\right) $ is (0°, 0°), (0°, 90°), (90°, 0°), and (90°, 90°). However, the second and third settings with crossed polariser and analyser lead to sharp phase jumps of$ \mathrm{\pi } $ from quadrant to quadrant in the Fourier-plane phase distribution, which is problematic from an experimental point of view. The transitions are typically smeared out in the measurement, thus complicating the calibration and especially phase correction. Therefore, we changed both polariser directions by 30° while maintaining the analyser directions. All of the resulting combinations (30°, 0°), (30°, 90°), (120°, 0°), and (120°, 90°) generate smooth intensity and phase distributions in the Fourier plane. In the same order, the corresponding measurement results are denoted as$ {E}_{x}^{\left(1\right)} $ ,$ {E}_{y}^{\left(1\right)} $ ,$ {E}_{x}^{\left(2\right)} $ , and$ {E}_{y}^{\left(2\right)} $ . We want to emphasise that this specific choice of angle is arbitrary at the moment. Other angles would have been possible as well, and it might be beneficial to optimise the angular settings as a function of the target in the future.For the calculation of the Jones matrix, the measurement results are transferred from the
$ x $ -$ y $ to a$ p $ -$ s $ coordinate system via$$ \left(\begin{array}{c}{E}_{p}^{\left(j\right)}\\ {E}_{s}^{\left(j\right)}\end{array}\right)=\left(\begin{array}{cc}\mathrm{cos}\varphi & \mathrm{sin}\varphi \\ -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right) \cdot \left(\begin{array}{c}{E}_{x}^{\left(j\right)}\\ {E}_{y}^{\left(j\right)}\end{array}\right) $$ (7) where the index
$ j\in \left\{\mathrm{1,2}\right\} $ represents the two polariser angles$ {\theta }_{P}^{\left(j\right)} $ and$ \varphi $ is the azimuth angle in the Fourier plane, as shown in Fig. 3a. The Jones matrix$ J $ of the sample relates the initial polarisation state of the illumination to the results of Eq. 7:$$ \left(\begin{array}{cc}{E}_{p}^{\left(1\right)}& {E}_{p}^{\left(2\right)}\\ {E}_{s}^{\left(1\right)}& {E}_{s}^{\left(2\right)}\end{array}\right)=J \cdot \left(\begin{array}{cc}{E}_{p,\mathrm{i}\mathrm{n}}^{\left(1\right)}& {E}_{p,\mathrm{i}\mathrm{n}}^{\left(2\right)}\\ {E}_{s,\mathrm{i}\mathrm{n}}^{\left(1\right)}& {E}_{s,\mathrm{i}\mathrm{n}}^{\left(2\right)}\end{array}\right) $$ (8) with
$$ \left(\begin{array}{c}{E}_{p,\mathrm{i}\mathrm{n}}^{\left(j\right)}\\ {E}_{s,\mathrm{i}\mathrm{n}}^{\left(j\right)}\end{array}\right)=\left(\begin{array}{cc}\mathrm{cos}\varphi & \mathrm{sin}\varphi \\ -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right) \cdot \left(\begin{array}{c}\mathrm{cos}{\theta }_{P}^{\left(j\right)}\\ \mathrm{sin}{\theta }_{P}^{\left(j\right)}\end{array}\right) $$ (9) By inversion of the illumination matrix on the right-hand side of Eq. 8, the Jones matrix is calculated as follows:
$$ J=\left(\begin{array}{cc}{E}_{p}^{\left(1\right)}& {E}_{p}^{\left(2\right)}\\ {E}_{s}^{\left(1\right)}& {E}_{s}^{\left(2\right)}\end{array}\right) \cdot {\left(\begin{array}{cc}{E}_{p,\mathrm{i}\mathrm{n}}^{\left(1\right)}& {E}_{p,\mathrm{i}\mathrm{n}}^{\left(2\right)}\\ {E}_{s,\mathrm{i}\mathrm{n}}^{\left(1\right)}& {E}_{s,\mathrm{i}\mathrm{n}}^{\left(2\right)}\end{array}\right)}^{-1} $$ (10) Clearly, the determinant of the illumination matrix must be different from zero for this step, but this can easily be assured by the choice of polariser angles. We want to emphasise again that the correct relative phase relationships between all four measurements are guaranteed by the sensor calibration. The global phase remains unknown, but a common offset does not change the Mueller matrix M. Using the result from Eq. 10, M follows from20
$$ M=\left(\begin{array}{cccc}{m}_{11}& {m}_{12}& {m}_{13}& {m}_{14}\\ {m}_{21}& {m}_{22}& {m}_{23}& {m}_{24}\\ {m}_{31}& {m}_{32}& {m}_{33}& {m}_{34}\\ {m}_{41}& {m}_{42}& {m}_{43}& {m}_{44}\end{array}\right)=A\left(J\otimes {J}^{*}\right){A}^{-1} $$ (11) where
$ {J}^{*} $ denotes the complex conjugate of$ J $ ,$ \otimes $ is the Kronecker tensor product, and matrix A is defined as$$ A=\left(\begin{array}{cccc}1& 0& 0& 1\\ 1& 0& 0& -1\\ 0& 1& 1& 0\\ 0& -\mathrm{i}& \mathrm{i}& 0\end{array}\right) $$ (12) Because our setup is not capable of measuring absolute intensities, we normalise all Mueller matrices to their respective first element,
$ {m}_{11} $ . It should also be noted that all Mueller matrices are depicted in a$ p $ -$ s $ coordinate system, whereas the intensity and phase distributions shown in Figs. 1, 4, and 6 are displayed in the laboratory$ x $ -$ y $ coordinate system. We could have transformed the Mueller matrices to the$ x $ -$ y $ coordinate system as well, but this additional step is unnecessary for model-based reconstruction. Finally, it is important to keep in mind that we assumed the optical system and the sample to be non-depolarising. Consequently, Mueller matrix M from Eq. 11 is actually a Mueller-Jones matrix with only seven independent elements20. -
As already mentioned, our first test target for model-based feature reconstruction is a dense silicon line grating with a pitch of 150 nm. Considering that our sensor operates in the visible range of the spectrum, this structure is already in the sub-wavelength regime. Fig. 8 shows an SEM cross-sectional image of the grating. Note that the tilt angle of the stage (51.3°) causes perspective distortion and makes the line height appear smaller than it really is. The total target area is 200 × 200 µm2, and we employed a field stop to limit the diameter of the illumination spot to approximately 70 µm.
Fig. 8 SEM cross-sectional image of the silicon line grating.
The measurement was performed on an FEI Helios NanoLab 600. The blue grating profile illustrates the reconstructed parameter values. Note that the profile is compressed in its height according to the tilt angle of the stage.The measured Mueller matrices are shown in Figs. 9a, 10a. In Fig. 9a, the wavelength is fixed at an exemplary value of 546.4 nm, and all matrix elements at all angles are displayed simultaneously at this single wavelength. The wavelength dependence can then be assessed from Fig. 10a, where only the Mueller matrix element
$ {m}_{13} $ at all angles is shown at several wavelengths in the range 520–680 nm. The selection of the matrix element and the wavelengths in the specified range are arbitrary and simply made to reduce the amount of data for the plots. Note that all matrix elements are normalised to the first element,$ {m}_{11} $ . Outside of the wavelength range mentioned previously, the light intensity is relatively low (see Fig. 4b). Consequently, we are faced with poor signal-to-noise ratios in these regions, but this limitation could, in principle, be overcome by switching to a different light source.Fig. 9 Mueller matrix, all 16 matrix elements at all angles, but at a single wavelength only (546.4 nm).
a Measurement; b simulation using the optimum set of parameter values obtained from the model-based reconstruction; c difference between measurement and simulation. Note the rescaled colour bar in c.Fig. 10 Mueller-matrix element
$ {\mathit{m}}_{13} $ at all angles and at different wavelengths.a Measurement; b simulation; c difference between measurement and simulation. Again, note the rescaled colour bar in c.The scan of the reference arm was performed with a step size of 25 nm, and a total of 600 images were recorded. This parameter combination corresponds to a scan length of 15.0 µm, which enables the measurement of 14 wavelengths in the desired range. From additional simulations, this value is known to be sufficient in most cases to achieve convergence of the associated parameter uncertainties.
To retrieve the grating parameters, we compare the measured Mueller matrices to a library of simulated ones2. For library generation, all grating parameters are varied around their nominal values, and the corresponding Mueller matrices are calculated at all relevant wavelengths and angles. We determined the nominal values of the silicon line grating by performing additional measurements using a scanning electron microscope (SEM) and an atomic force microscope (AFM).
The SEM image shown in Fig. 8 suggests that the grating profile can be described well by three parameters: the line width (or critical dimension cd) at the top of the line, the line’s height, and the sidewall angle; see also Fig. 7. Clearly, the lines feature a certain roughness, and the parameter values are expected to vary from measurement site to site. Because our scatterometric approach averages over many grating lines at once, we also averaged the SEM and AFM measurement values over ten positions per parameter. The corresponding mean values and their standard deviations are summarised in the second and third columns of Table 1.
The standard deviations listed in Table 1 are mainly caused by the variation of the grating parameters over the target area, and partly by the user influence occurring during the cursor placement in a measurement. Additionally, both AFM and SEM may feature systematic calibration uncertainties that are not considered in Table 1. Although the good agreement between the measured height values suggests that the systematic errors are insignificant, we will still discuss the AFM and SEM calibration at the end of this section.
In our simulations, the width of the grating line is defined by the silicon mid-cd (at half of the grating line’s height) without the oxide layer, as shown in Fig. 7. With the mean SEM values from Table 1 and a constant oxide thickness of 3 nm, a silicon mid-cd of approximately 62 nm is obtained. For the computation of the library, we vary the silicon mid-cd by 20 nm around this value, that is, between 52 and 72 nm in steps of 1 nm. The line’s height is varied between 60 and 80 nm, also in steps of 1 nm, and the sidewall angle between 73 and 86° in steps of 1°. The top and bottom roundings, as well as the line edge roughness, are not considered here. The library is then compared to the measurement, and for each combination
$ \overrightarrow{p} $ of parameter values, the differences between the measurement and simulation are quantified by calculating the RMS:$${\rm{RMS}}\left( {\vec p} \right) = \frac{1}{{15 \cdot L}}\mathop \sum \nolimits_{l = 1}^L \underbrace {\mathop \sum \nolimits_{i = 1}^4 \mathop \sum \nolimits_{j = 1}^4 \sqrt {\frac{{\displaystyle\mathop \sum \nolimits_{k = 1}^K {{\left[ {m_{ij}^{{\rm{meas}}}\left( {k,l} \right) - m_{ij}^{{\rm{sim}}}\left( {k,l,\vec p} \right)} \right]}^2}}}{K}} }_{\left( {\neg \;i = j = 1} \right)}$$ (13) In Eq. 13,
$ {m}_{ij}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}} $ ($ {m}_{ij}^{\mathrm{s}\mathrm{i}\mathrm{m}} $ ) is the measured (simulated) Mueller-matrix element$ {m}_{ij} $ at the wavelength index$ l $ and angle index$ k $ . L is the total number of wavelengths, and K is the total number of angles, that is, the number of NA coordinates$ \left({NA}_{x},{NA}_{y}\right) $ within the accessible maximum NA of 0.8. The Mueller-matrix element$ {m}_{11} $ is omitted because it is used to normalise all elements and is, hence, always equal to one.For the reconstruction of the silicon line grating, we consider L = 14 wavelengths and K = 1257 NA coordinates. The value of K corresponds to a 41 × 41 grid in the Fourier plane. Our measurements offer a considerably higher angular resolution, but we still limited the number of pixels to reduce the computational effort required for library generation. Within the analysed parameter ranges, the largest RMS value is 26.2%, whereas the smallest is 8.2%. The minimum RMS value defines the best match, and hence, the optimum parameter set
$ \overrightarrow{p} $ , whose values represent the theoretical grating profile coming closest to the real one. The optimum parameter values are listed in the fourth column of Table 1. Note that to simplify the comparison, the silicon mid-cd was re-converted to the total top-cd, including the oxide layer. The corresponding simulation results are depicted in Figs. 9b, 10b, and the remaining differences between the measurement and simulation can be assessed from Figs. 9c, 10c. All results are discussed in Section 6. -
The SEM was calibrated on a waffle-pattern diffraction grating featuring 2160 lines per mm, which corresponds to a pitch of approximately 463 nm (magnification calibration target no. 607 from Ted Pella, Redding, CA, USA). The line width uncertainty amounts to ± 10 nm, but the corresponding calibration error decreases with the feature size of the device under test.
The calibration of the AFM was verified using fused-silica reference-artefact structures on a silicon substrate (target HS-100MG from BudgetSensors, Sofia, Bulgaria). To reduce sample-induced measurement errors, the sample was overcoated with chromium (75 nm). The sample features binary step structures with a nominal height of 113 nm ± 3%, as calibrated by the vendor. The lateral dimensions vary from 5 to 30 µm. With the overcoating, the sample is well suited for AFM probes as well as optical probes. The sample was calibrated on the nanopositioning and nanomeasuring machine NPMM-200 (ref. 28) using an optical fixed-focus probe. The NPMM-200 employs six interferometers with a five-axis stage control to implement an extended Abbe principle in three axes, allowing for high-accuracy height measurements down to sub-0.1-nm repeatability even for large step heights. The measured height of the sample is 113.70 nm and the standard deviation of 10 measurements amounts to 0.15 nm. The measurement uncertainty is dominated by the sample nanotopography and is estimated from measurements at different positions to be better than ± 1 nm. In summary, the agreement between the vendor specification and NPMM-200 measurement results is excellent.
In addition to this successful cross-check with the NPMM-200, it should be emphasised again that the height values measured with the SEM and AFM coincide well (see Table 1). Therefore, we are convinced that, at least for our specific application, the systematic calibration error per technique is small enough to enable reliable quantitative comparisons between the two techniques, and even more important, with our scatterometric results.
Model-based characterisation of complex periodic nanostructures by white-light Mueller-matrix Fourier scatterometry
- Light: Advanced Manufacturing 2, Article number: (2021)
- Received: 22 January 2021
- Revised: 26 May 2021
- Accepted: 26 May 2021 Published online: 23 June 2021
doi: https://doi.org/10.37188/lam.2021.018
Abstract: Optical scatterometry is one of the most important metrology techniques for process monitoring in high-volume semiconductor manufacturing. By comparing measured signatures to modelled ones, scatterometry indirectly retrieves the dimensions of nanostructures and, hence, solves an inverse problem. However, the increasing design complexity of modern semiconductor devices makes modelling of the structures ever more difficult and requires a multitude of parameters. Such large parameter spaces typically cause ambiguities in the reconstruction process, thereby complicating the solution of the inherently ill-posed inverse problem further. An effective means of regularisation consists of systematically maximising the information content provided by the optical sensor. With this in mind, we combined the classical techniques of white-light interferometry, Mueller polarimetry, and Fourier scatterometry into one apparatus, allowing for the acquisition of fully angle- and wavelength-resolved Mueller matrices. The large amount of uncorrelated measurement data improve the robustness of the reconstruction in the case of complex multi-parameter problems by increasing the overall sensitivity and reducing cross-correlations. In this study, we discuss the sensor concept and introduce the measurement strategy, calibration routine, and numerical post-processing steps. We verify the practical feasibility of our method by reconstructing the profile parameters of a sub-wavelength silicon line grating. All necessary simulations are based on the rigorous coupled-wave analysis method. Additional measurements performed using a scanning electron microscope and an atomic force microscope confirm the accuracy of the reconstruction results, and hence, the real-world applicability of the proposed sensor concept.
Research Summary
Scatterometry: Novel sensor facilitates the model-based reconstruction of complex nanostructures
With the structure dimensions on modern semiconductor chips having reached the nanometre scale and circuit designs becoming increasingly complex, process monitoring is facing veritable challenges. Optical scatterometry retrieves the parameter values of a nanostructure indirectly by comparing between measured and modelled signatures. However, modelling complex structures requires large parameter spaces, which are known to cause large measurement uncertainties. Maria Laura Gödecke from Germany’s University of Stuttgart and colleagues now report the development of a novel scatterometric sensor that makes full use of the rich polarisation information contained in the angle- and wavelength-resolved Mueller matrix of a nanostructure. Compared to other scatterometric sensors, the information gain improves the sensitivity and robustness of the method. The team verified the practical feasibility of the proposed concept by reconstructing the profile parameters of a sub-wavelength silicon line grating.
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