Browse Prior Art Database

Algorithm for Line Width Extraction from Cosine Transform

IP.com Disclosure Number: IPCOM000088457D
Original Publication Date: 1977-Jun-01
Included in the Prior Art Database: 2005-Mar-04
Document File: 5 page(s) / 87K

Publishing Venue

IBM

Related People

Ananthakrishnan, RB: AUTHOR [+4]

Abstract

One width and overlay system (which can be employed with that of U. S. Patent 3,957,376) does not provide consistent results in measuring the width of lines with gradients using algorithms involving the Fourier transform of the diffraction pattern of the line.

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Algorithm for Line Width Extraction from Cosine Transform

One width and overlay system (which can be employed with that of U. S. Patent 3,957,376) does not provide consistent results in measuring the width of lines with gradients using algorithms involving the Fourier transform of the diffraction pattern of the line.

One conventional algorithm, the "Burke" algorithm, smooths and differentiates one side of the diffraction pattern, takes the modulus of the Fourier transform of the smoothed, differentiated pattern, and assigns the component with the highest amplitude in the modulus to the line width.

Analysis of the system has shown that the Fourier transform has terms representing b, a-b, a, and a + b (Fig. 1). The coefficients of these terms are functions of the amplitude and phase of the complex reflectances of the different parts of the pattern on the wafers, and it is not generally true that the line-width term has the greatest amplitude, as previously assumed. In fact, for example, over the range of nominal tolerances in the film thicknesses (of photoresist and the underlying SiO(2) film), the coefficient of the line-width term, a, is predicted to be zero over a wide range. In particular, this coefficient is predicted to be zero at several points within the operating range of tolerances. In practice, the Burke algorithm has been observed to give unreliable results, and it has displayed the predictable behavior of switching between peaks in the transform, as the amplitudes of the peaks change as a result of film thickness variations. The peaks that are chosen in error are usually those representing the line-width-plus- gradient (a+b) or the line-width-minus-gradient (a-b) peaks. Consequently, the Burke algorithm is characterized as having an uncertainty approximately equal to the gradient width.

A more rigorous analysis of the Fourier transform is diagrammed in Fig. 2. The diffraction data is multiplied by a quadratic function of the diode number, the DC component is removed, and the cosine transform of the diffraction pattern is found. On the assumption that the diffraction pattern is symmetrical, the sine transform is expected to be zero and is discarded so the cosine transform is the whole transform. Fig. 3 is a representation of a cosine transform amplitude versus line-width spectrum. The component b (gradient width) has an amplitude -Rho cos Phi, the component (a-b), line-width-minus gradient, has an amplitude of -1/2, the component a, line-width, has an amplitude +Rho cos Phi, and the component (a+b), line-width plus gradient, has an amplitude -Rho/2//2. The value Rho is the ratio of reflectances inside and outside the line, and Phi is the phase angle between reflectances inside and outside the line. The values of Rho and Phi vary significantly over the expected film thickness tolerance variations. Consequently, the amplitude of the line-width component varies between + Rho and - Rho.

The line-width is, at present, found b...