# Correlation and convolution filtering and image processing for pitch evaluation of 2D micro- and nano-scale gratings and lattices.

TL;DR: It is elucidated that the pitch average, uniformity, rotation angle, and orthogonal angle can be calculated using the PD method and this has been applied to the pitch evaluation of several 2D gratings and lattices, and the results are compared with the results of using the center-of-gravity and Fourier-transform-based method.

Abstract: We have mathematically explicated and experimentally demonstrated how a correlation and convolution filter can dramatically suppress the noise that coexists with the scanned topographic signals of two-dimensional (2D) gratings and lattices with 2D perspectives. To realize pitch evaluation, the true peaks' coordinates have been precisely acquired after detecting the local maxima from the filtered signal, followed by image processing. The combination of 2D filtering, local-maxima detecting, and image processing make up the pitch detection (PD) method. It is elucidated that the pitch average, uniformity, rotation angle, and orthogonal angle can be calculated using the PD method. This has been applied to the pitch evaluation of several 2D gratings and lattices, and the results are compared with the results of using the center-of-gravity (CG) and Fourier-transform-based (FT) method. The differences of pitch averages which are produced using the PD, CG, and FT methods are within 1.5 pixels. Moreover, the PD method has also been applied to detect the dense peaks of Si (111) 7×7 surface and the highly oriented pyrolytic graphite (HOPG) basal plane.

## Summary (3 min read)

### 1. INTRODUCTION

- The pitch described in this paper is the distance between adjacent similar structural features of one-dimensional (1D) and twodimensional (2D) gratings and lattices on surfaces.
- In nanometer metrology and measurement, the International Organization for Standardization (ISO) stipulates 1D and 2D gratings and lattices in several documents to calibrate diverse microscopes and instruments after metrologically verifying the pitch-related parameters, such as pitch average, pitch uniformity, etc.
- It has always been cumbersome to evaluate the pitches of 2D gratings and lattices based on CG- and FT-methods.
- When raster-scanning only a cluster of 2D grating or lattice features into image, the authors will find that the grating and lattice structures are always orientating an unknown θ angle around z-axis relative to the xoy plane.
- It can achieve the equivalent impact and high credibility for analyzing the images and topographic signals containing repeated 3D structural features.

### A. Topographic and coexisted signals

- 2) The Xand Y-axes are parallel to the direction of the X-pitch and Y-pitch, PX and PY, respectively; 3) Px=PXcosϕ x and Py=PYcosϕ y mean the projectionsof PXand PY in the xoy plane.
- The diminishing effect on the accurate pitch evaluation can be leveled by rotationtransforming ϕx and ϕy angles around the x- and y-axis respectively, so that U(X,Y) ≈0 in XYZ system, which means the drift signal theoretically does not exist in XYZ system.
- RTF(x,y), RTf(x,y), RTU(x,y) and RTW(x,y) are named as filtered signal, filtered topographic signal, filtered nonlinear drift signal and noise residue signal, respectively.
- Compared with equation (4), equation (9) interprets that RTU(x,y) still is a nonlinear drift signal.

### 3. 2D LATTICES

- The 3D convex type features have parallelogram (rectangle, square, diamond, etc.) or circle bottoms and the 3D concave type features have parallelogram or circle tops.
- Lattices are fabricated such that the features are arranged in square, rectangular, hexagonal and oblique array.
- Mathematically, they are described by different analytic functions inside and zero outside the 3D features.

### A. Topographic signals

- For a Px - and Py -pitch lattice with any 3D feature in square, rectangular, hexagonal, and oblique array, the raster-scanned topographic signal (with Px- and Py-periods) is defined as f(x,y) inside the 2D waveforms and zero outside.
- Py/2} so that the 3D feature waveforms at the origin lies entirely within the primitive unit bottom or top, where x and y are two independent real variables in the whole feature array.
- The topographic signal of two exemplar square lattices with 3D central-symmetric features in parallelogram holes and hills is shown in Fig. 3(a) and (b), respectively.
- Since the amplitude AIJ decreases sharply with I and J increasing [25], the sinusoidal waveform in fundamental period (I,J=1) dominates equation (11).
- The actual raster-scan ranges are 90µm×90µm and 50µm×50µm, respectively.

### 4. AUTOMATIC PEAK DETECTION

- First, two M×N zero matrices BR and BC are constructed.
- Subsequently, the new values are assigned to the corresponding positionsin BR and BC, respectively: (1) The local maxima of RTF(m,n) are detected row by row.
- The local maxima (grey-scale=1), which have disappeared in the binary image, can produce good visual effect in the ternary image to associate the peaks with the original and filtered images.
- The peaks detection to the raster-scanned signal of the 2D sinusoidal grating in Fig. 1(a) is shown by the ternary image in Fig.1 (c).
- It is made possible to use atoms positions and unit cells to detect the directional drift of the sample, i.e., the motion of the scanner in an STM.

### 5. PITCH EVALUATION

- The ternary image of the 2D square holes in Fig. 7 (a) has nine LSMLs along Px-direction and ten LSMLs along Py-direction.
- If the pitches in a narrow window are evaluated as P1, P2, ⋅⋅⋅, PL, (nm), the pitch average P and uniformity δ can be automatically calculated using statistical mathematics.

### 6. PITCH EVALUATION RESULTS

- Before CC-filtering, three raster-scanned signals are leveled using the coordinate rotation-transformation to diminish the drift component U(x,y), so as to make Px ≈PX and Py ≈PY.
- After leveling, CC-filtering and peak detecting, the fitted LSML coefficients ar and ac of 2D gratings and lattices can be acquired.
- The pitch averages xP -PD and yP -PD, uniformity δx and δy, averages of rotation angles cθ (deg) and rθ (deg), orthogonal angles oθ , etc. are listed in table 1, where 2D holes, 2D CCD and Hutley represent 2D square holes in square array, 2D CCD array panel and 2D sinusoidal grating, respectively.
- It implicates that the three different AFMs that were used for raster-scanning three 2D gratings and lattices have unequal scale factor Cx and Cy, cross-talking between x- and y-scanners, and other geometrical errors described in [17].

### 7. COMPARISION OF PITCH EVALUATION METHOS

- As two series of LSMLs (i.e., r1, r2, ⋅⋅⋅, rM and c1, c2, ⋅⋅⋅, cN) along the Pxand Py-directions can be fitted to the peaks coordinates in the ternary image based on PD-method, two groups of 1D topographic signal sequences of a 2D grating or lattice can be extracted along the series of LSMLs.
- The intercomparison of three pitch evaluation methods is realized about 2D gratings and lattices.
- Any LSML does not completely cross through all the peaks detected by the PD-method within the corresponding narrow window.
- Nevertheless, the PD-method truly deals with the 2D topographic signals of 2D gratings and lattices.

### 8. CONCLUSION

- Mathematic analysis with 2D perspective has explicated that a half 2D sinusoidal waveform template can be used as a 2D correlation and convolution (CC) filter.
- After CC filtering, the peaks can be acquired based on local-maxima detecting, followed by image processing.
- The PD-method will not be influenced by the unknown angles of 2D gratings and lattices rotating in-plane relatively to the stage of measuring instruments.
- The 2D nonlinear drafting signal U(x,y) which are simultaneously generated in the raster-scan process will not interfere the CC filtering, whether or not it is leveled using coordinate rotation-transformation.
- The CC filtering allows conveniently and reliably evaluating the local pitches, the average and uniformity of the pitches, rotation angle, orthogonal angle between two pitches of 2D gratings and lattices.

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### "Correlation and convolution filteri..." refers methods in this paper

...Based on the algorithm to find the local maxima in a sequence signals by applying quadratic/parabolic interpolation of three adjacent samples [23, 24], the following two steps are taken to detect the local maxima from row vectors and column vectors, respectively....

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### "Correlation and convolution filteri..." refers background in this paper

...ANNEX C: EXPRESSION OF 2D SIGNAL OF LATTICES To expand the topographic signal of a lattice expressed by equation (12) as the real form [25]:...

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...Since the amplitude AIJ decreases sharply with I and J increasing [25], the sinusoidal waveform in fundamental period (I,J=1) dominates equation (11)....

[...]