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Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials
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10.1063/1.2337256
/content/aip/journal/jap/100/5/10.1063/1.2337256
http://aip.metastore.ingenta.com/content/aip/journal/jap/100/5/10.1063/1.2337256

Figures

Image of FIG. 1.
FIG. 1.

Diffraction of a polarized plane wave by a layer with periodic permittivity sandwiched between a superstrate with uniform permittivity and a substrate with uniform permittivity . Owing to the symmetry properties of the planar diffraction mounting, all the wave vectors of the diffracted waves lie within the plane and each of the two principal polarizations and is preserved.

Image of FIG. 2.
FIG. 2.

Rectangular grating patterned in a homogeneous layer identified by geometrical parameters, the depth , linewidth , and period , and material parameters, the permittivity of wires and the permittivity of the medium between wires .

Image of FIG. 3.
FIG. 3.

Diffraction on the interface between media 0 and 1. In general, incident, reflected, and transmitted waves are identified with column vectors , , and , respectively, whose elements are the components of the pseudo-Fourier series of the corresponding electric fields. The diffraction response is then described by linear transforms between those vectors, for reflection and for transmission.

Image of FIG. 4.
FIG. 4.

Airy-like series as an implementation of the coupled-wave theory based on superpositioning contributions from multiple reflections and propagations inside the periodic medium 1 sandwiched between the upper medium 0 and the lower medium 2.

Image of FIG. 5.
FIG. 5.

Recursive algorithm for applying Airy-like series to a general nonrectangular profile of wires that must be sliced into sublayers. Within each of the recursive iterations, a partial multilayer consisting of the first slices is regarded as a pseudointerface whose reflection (transmission) matrices , are known. The Airy-like series is then applied to the layer to obtain the matrices , , until we reach the substrate. The slicing should be performed so that artificial roughness thereby produced is of a scale below the wavelength.

Image of FIG. 6.
FIG. 6.

Refractive index and extinction coefficient specified by an analysis of SE and energy transmittance measurement carried out on a nonpatterned reference Ta/quartz sample.

Image of FIG. 7.
FIG. 7.

Experimental (marks) and fitted four-zone null SE parameters, and , on a rectangular-relief quartz grating (sample A) in incidence of 70° for two examples of maximum Fourier harmonics, (solid curves) and (dotted curves). Obviously, the three-wave approximation (corresponding to ) is sufficient to fit the measurement almost perfectly.

Image of FIG. 8.
FIG. 8.

Experimental (marks) and fitted rotating-analyzer SE parameters, and , on a rectangular-relief Si grating (sample B) in incidence of 65.45° for two examples of maximum Fourier harmonics, (solid curves) and (dotted curves). In this case the five-wave approximation (corresponding to ) is sufficient to fit the measurement.

Image of FIG. 9.
FIG. 9.

Experimental (marks) and fitted four-zone null SE parameters, and , on a rectangular-wire-assumed Ta grating (sample C) in incidence of 70° for three examples of maximum Fourier harmonics, (dashed curves), (solid curves), and (dotted curves). Since the artificially assumed rectangular profile does not correspond to reality, none of the approximations fits the measurement correctly. Nevertheless, the nine-wave approximation (corresponding to ) can obviously be considered sufficient to simulate a rectangular-wire metallic grating with comparable dimensions.

Image of FIG. 10.
FIG. 10.

Experimental (marks) and fitted four-zone null SE parameters, and , on a paraboloidal-wire Ta grating (sample C) in incidence of 70° for three examples of maximum Fourier harmonics, (dashed curves), (solid curves), and (dotted curves), sliced into sublayers. In this case, higher-order approximations fit the measurement reasonably, especially in the longer-wavelength spectral range.

Image of FIG. 11.
FIG. 11.

Experimental (marks) and fitted four-zone null SE parameters, and , on a paraboloidal-wire Ta grating (sample C) in incidence of 70° for four examples of the number of slices, (dashed curves), (dash-dotted curves), (solid curves), and (dotted curves), in each case with the maximum Fourier harmonics . Here the quality of the fit is almost linearly improving by increasing .

Image of FIG. 12.
FIG. 12.

Convergence properties of the relative critical dimensions according to increasing [(a)–(e)] and according to increasing (f), obtained on samples A and B [(a) and (b)], on the rectangular-wire-assumed sample C (c), and on the paraboloidal-wire-assumed sample C with fixed (d), with fixed (e), and with fixed (f).

Tables

Generic image for table
Table I.

Nominal grating parameters of the samples under measurement.

Generic image for table
Table II.

Fitted critical dimensions (depth , linewidth , and period ) and errors of the fits of rectangular gratings according to increasing .

Generic image for table
Table III.

Fitted critical dimensions (depth , top linewidth , maximum linewidth , and period ) and errors of the fits of the Ta paraboloidal-wire grating according to increasing with a fixed number of slices .

Generic image for table
Table IV.

Fitted critical dimensions (depth , top linewidth , maximum linewidth , and period ) and errors of the fits of the Ta paraboloidal-wire grating according to increasing number of slices with fixed .

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/content/aip/journal/jap/100/5/10.1063/1.2337256
2006-09-08
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials
http://aip.metastore.ingenta.com/content/aip/journal/jap/100/5/10.1063/1.2337256
10.1063/1.2337256
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