JP2006519992A - Profiling complex surface structures using scanning interferometry - Google Patents

Abstract

A method comprising comparing information derivable from a scanning interferometry signal for a first surface location of a test object with information corresponding to a plurality of models of the test object, wherein the plurality of models is a test object A method that is parameterized by a set of properties for an object. The information corresponding to the plurality of models may include information on at least one amplitude component of the converted part (for example, Fourier transform part) of the scanning interference spectral signal corresponding to each model of the test object. In the second aspect, the model is parameterized by a set of characteristics corresponding to a fixed surface height and different from the fixed surface height. In a third aspect, the comparison includes clarifying the systematic effects of the scanning interferometry signal.

Description

  The present invention relates to measuring surface topography and / or other properties of objects having complex surface structures using scanning interferometry. Complex surface structures include, for example, thin films, separate structures of different materials, or separate structures that cannot be resolved sufficiently with the optical resolution of an interference microscope. Such measurements are related to the characterization of flat panel display components, semiconductor wafer metrology, and in-situ thin film and foreign material analysis.

  This application claims priority to the following US provisional application under 35 USC 119 (e). US patent application Ser. No. 10 / 452,615 (filed Mar. 6, 2003), entitled “Profiling Complex Surface Structures Using Height Scanning Interferometry”, US Patent Application No. 10/452. , 465 (filed Mar. 6, 2003), title of invention "profiling complex surface structures using signals from height-scanning interferometry", and US patent application Ser. No. 10 / 539,437. Description (filed Jan. 26, 2004), title of invention “Surface Profiling Using Interference Pattern Matching Template”. All of which are incorporated herein by reference.

  Interferometry techniques are widely used to measure the surface profile of an object. In order to perform the measurement, the interferometer generates an interferogram by combining the measurement wavefront reflected from the target surface and the reference wavefront reflected from the reference surface. The fringe in the interferogram indicates a spatial change between the target surface and the reference surface.

  A scanning interferometer scans the optical path length difference (OPD) between the reference and measurement legs of the interferometer over a range that is comparable to or longer than the coherence length of the interference wavefront, A scanning interferometry signal is generated for each camera pixel used to measure the interferogram. If a white light source is used, a limited coherence length can be generated. This is referred to as scanning white light interferometry (SWLI). A typical scanning white light interferometry (SWLI) signal is a small number of fringes localized near the zero optical path difference (OPD) position. Signal characteristics are typically sinusoidal carrier modulation ("fringe") with a bell-shaped fringe contrast envelope. The traditional idea underlying SWLI metrology is to measure the surface profile using localized fringes.

  There are two principal trends in SWLI processing technology. The first approach is to find the position of the peak or center of the envelope. In this case, it is assumed that this position corresponds to the zero optical path difference (OPD) of the two beam interferometer. One beam of the two-beam interferometer reflects from the object surface. The second approach is to convert the signal to the frequency domain and calculate the phase change rate with respect to wavelength. In this case, it is assumed that the gradient that is basically linear is directly proportional to the object position. See, for example, US Pat. No. 1,398,113 (Peter de Grot). This latter approach is referred to as frequency domain analysis (FDA).

  Unfortunately, such assumptions may not work when applied to a test object provided with a thin film. The reason is that reflection occurs not only at the top surface but also at the underlying film / substrate interface. Recently, US Pat. No. 6,545,763 (SW Kim and GH Kim) disclosed a method for dealing with such structures. In this method, the frequency domain phase profile of the SWLI signal for the thin film structure is matched to the estimated frequency domain phase profile for various film thicknesses and surface heights. The exact film thickness and surface height were determined by performing optimization simultaneously.

  The inventor has found that there is a large amount of information in the scanning interferometry signal. This information is almost ignored in conventional processing. In the case of a complicated surface structure, for example, in the case of a thin film, the performance may be degraded by the conventional processing technology. Prior art is based on identifying the peak location in the fringe contrast envelope or calculating the slope for the frequency domain phase profile. However, according to the novel processing technique disclosed herein, surface height information and / or information about its complex surface structure can be extracted.

  For example, if it is not assumed that the surface height information is directly related to the peak in the fringe contrast envelope, in some embodiments of the present invention, the surface height changes to If the scanning interferometry signal changes, otherwise, the shape of the scanning interferometry signal is maintained. In particular, when characterizing complex surface structures, the shape of the scanning interferometry signal is useful. This is because the shape of the signal is independent of the surface height. Similarly, in the frequency domain, it is assumed in some embodiments that a linear term is introduced into the frequency domain phase profile by changing the surface height. This is assumed even if the frequency domain profile itself is not linear. However, the frequency domain amplitude profile does not change when the surface height changes. Thus, frequency domain amplitude profiles are particularly useful in characterizing complex surface structures.

  If complex surface structures are characterized, the surface height can be determined efficiently. For example, a peak can be generated at the scan coordinates corresponding to the surface height by cross-correlation between the scanning interferometry signal and a model signal having a shape corresponding to a complex surface structure. Similarly, in the frequency domain, phase effects due to complex surface structures can be subtracted from the frequency domain phase profile. The surface height can also be extracted using conventional FDA analysis.

  Examples of complex surface structures include the following. Simple thin film (in this case, for example, the variable parameter of interest may be film thickness, film refractive index, substrate refractive index, or some combination thereof); multilayer thin film; diffracted or otherwise Sharp edges and surface features that generate complex interference effects; untreated surface roughness; untreated surface features, eg sub-wavelength grooves on otherwise smooth surfaces; different materials (eg, surface May contain a combination of thin film and solid metal, in which case the library will include both surface structure types and automatically identify the thin film or solid metal by matching against the corresponding frequency domain spectrum. Optical activity such as surface structures that produce fluorescence; surface spectroscopic properties such as color and wavelength dependent reflectivity; surface polarization Presence of characteristic, distortion of the surface or deformable surface features leading to disturbance of the interference signal, vibration or movement.

  In some embodiments, the limited coherence length of the light used to generate the scanning interferometry signal is based on a white light source, more generally a broadband light source. In other embodiments, the light source may be monochromatic and the limited coherence length uses a high numerical aperture (NA) to send light to and / or receive light from the test object. Can be generated. Due to the high NA, the light ray contacts the test surface over a range of angles and various spatial frequency components are generated in the recorded signal when the OPD is scanned. In further embodiments, limited coherence can be generated from a combination of both effects.

  The cause of the limited coherence length is also the physical basis for the presence of information in the scanning interferometry signal. That is, the scanning interferometry signal includes information about a complex surface structure. The reason is that this signal is generated by contacting the test surface with many different wavelengths and / or many different angles.

  In the processing techniques described herein, information that can be derived from the scanning interference spectral signal for the first surface location of the test object (including the scanning interference spectral signal itself) corresponds to multiple models of the test object. Compare with the information you want. The multiple models are parameterized by a series of characteristics for the test object. For example, the test object can be modeled as a thin film, and the set of properties can be a set of values for the thickness of the thin film. The information being compared may include, for example, information about the frequency domain phase profile, and may further include information about the shape of the scanning interferometry data and / or information about the frequency domain amplitude profile. In addition, all the models correspond to a fixed surface height at the first surface location of the test object in order to focus the comparison on a complex surface structure rather than a surface height at the first surface location. can do. The comparison itself can be based on the calculation of a merit function that indicates the similarity between the information from the actual scanning interferometry signal and the information from each model. For example, a merit function may indicate a fit between information derivable from scanning interferometry data and a function parameterized by a set of characteristics.

  Further, in some embodiments, the set of characteristics corresponds to the characteristics of the test object at a second location that is different from the first location. Examples of such a test object include a diffractive surface structure that affects an interface signal with respect to the first surface portion. Therefore, in many cases, the complex surface structure is anything other than the surface height at the first surface location corresponding to the scanning interference spectral signal, but the complicated surface structure corresponds to the scanning interference spectral signal. It may correspond to a surface height feature that is spaced from the first surface location.

  The methods and techniques described herein can be used for in-process metrology measurements of semiconductor chips. For example, scanning interferometry can be used on a contactless surface topography measurement semiconductor wafer while chemical mechanical polishing (CMP) of a dielectric layer on the wafer. CMP is used to form a smooth surface for the dielectric layer. A smooth surface is suitable for precision photolithography. Based on the results of the interferometric topography method, process conditions for CMP (eg, pad pressure, polishing slurry composition, etc.) can be adjusted to keep the surface non-uniformity within acceptable limits.

The various aspects and features of the invention will now be summarized.
In general, in one aspect, features of the present invention compare information derivable from scanning interferometry signals for a first surface location of a test object with information corresponding to multiple models of the test object. The plurality of models is a method that is parameterized by a set of characteristics for the test object.

Embodiments of the invention may include any of the following features.
The method may further include determining an exact characteristic for the test object based on the comparison.

  The method may further include determining a relative surface height relative to the first surface location based on the comparison. Determining the relative surface height is based on a comparison that determines which model corresponds to the exact one of the properties for the test object, using a model that corresponds to the exact property. It may include calculating the relative surface height.

  For example, using a model corresponding to the exact characteristic may include correcting data from the scanning interferometry signal to reduce the effects resulting from the exact characteristic. Correcting the data includes extracting the phase effect resulting from the exact characteristic from the phase component of the transform of the scanning interferometry signal on the test object, and using the model corresponding to the exact characteristic further It may include calculating the relative surface height from the phase component of the transformation after removing the phase effect resulting from the exact characteristic.

  In another example, calculating the relative surface height using a model corresponding to the exact property is used to compare information for the test object with information for the model corresponding to the exact property. Determining the position of the peak in the correlation function.

  The method may further include comparing information derivable from the scanning interferometry signal for additional surface locations with information corresponding to the plurality of models. The method may also further include determining a surface height profile for the test object based on the comparison.

Comparing may include calculating one or more merit functions that indicate the similarity between information derivable from the scanning interferometry signal and information corresponding to each model.
Comparing may include fitting information derivable from the scanning interferometry signal into a representation for the information corresponding to the model.

  The information corresponding to the plurality of models may include information on at least one amplitude component of the converted part (for example, Fourier transform part) of the scanning interference spectral signal corresponding to each model of the test object. Similarly, information derivable from the scanning interference spectral signal may include information on at least one amplitude component of the converted scanning interference spectral signal for the test object.

Comparing may include comparing the relative strength of the at least one amplitude component for the test object with the relative strength of the at least one amplitude component for each model.
The information corresponding to the plurality of models may be a function of coordinates with respect to the converted amount. For example, the information corresponding to a plurality of models may include a converted amplitude profile for each model. Further, the comparing may include comparing the amplitude profile of the converted scanning interference spectral signal for the test object with each amplitude profile for the model.

  Comparing may further include comparing information in the phase profile of the conversion of the scanning interferometry signal for the test object with information in the phase profile of the conversion for each model. For example, the information in the phase profile may include information about the phase profile non-linearity and / or phase shift values with respect to the transformed coordinates.

  The information that can be derived and compared from the scanning interferometry signal may be a number. Alternatively, the information that can be derived and compared from the scanning interferometry signal may be a function. For example, the function may be a scan position function or a spatial frequency function.

  Information for the test object may be obtained from transforming the scanning interferometry signal for the test object into the spatial frequency domain (eg, Fourier transform). The information on the test object may include information on the converted amplitude profile and / or the converted phase profile.

  Information about the test object may relate to the shape of the scanning interferometry signal at the first location of the test object. For example, the information for the test object may relate to the fringe contrast amplitude in the shape of the scanning interferometry signal. Information about the test object may also relate to the relative spacing between zero crossings in the shape of the scanning interferometry signal. Further, information on the test object may be expressed as a function of the scanning position, and the function may be obtained from the shape of the scanning interference spectral signal.

  Comparing may include calculating a correlation function (eg, a complex correlation function) between information for the test object and information for each model. Comparing may further include determining one or more peak values in each correlation function. And the method may further comprise determining an exact characteristic for the test object based on the parameterization of the model corresponding to the maximum peak value. Alternatively or additionally, the method further includes determining a relative surface height at the first surface location of the test object based on coordinates for at least one peak value in the correlation function. Also good.

The plurality of models may correspond to a fixed surface height at the first location of the test object.
The set of characteristics may include a set of values for at least one physical parameter of the test object. For example, the test object may include a thin film layer having a thickness, and the physical parameter may be the thickness of the thin film at the first location.

  The set of properties may include a set of properties of the test object at a second surface location that is different from the first surface location. For example, it may include a structure at the second surface location that diffracts light and affects the scanning interferometry signal for the first surface location. In one example, the set of characteristics at the second surface location may include a substitution of the amplitude for the step height at the second location and the position for the second location. In another example, the series of characteristics at the second surface location may include permutation of the modulation depth relative to the grating and the offset position of the grating, and the grating may extend across the second location.

The series of properties may be a series of surface materials for the test object.
The series of properties may be a series of surface layer configurations for the test object.
The scanning interferometry signal may be generated by the scanning interferometry system and comparing may include defining systematic effects on the scanning interferometry signal that results from the scanning interferometry system. For example, the systematic effects may include information about the dispersion of phase changes upon reflection from components of the scanning interferometry system. The method may further include comparing information derivable from the scanning interferometry signal for additional surface locations with information corresponding to the plurality of models. In this case, systematic influences may be analyzed for a plurality of surface locations. The method may further include calibrating the systematic effects of the scanning interferometry system with other test objects of known characteristics.

  The generation of the scanning interferometry signal is formed by imaging the test light emerging from the test object and causing it to interfere with the reference light on the detector, and a common light source between the interference part of the test light and the interference part of the reference light The scanning interferometry signal changes the optical path length difference by extracting the test light and the reference light from a common light source (eg, a spatially extended light source) by changing the optical path length difference from the sensor to the detector. May correspond to the interference intensity measured by the detector.

The test light and the reference light may have a spectral bandwidth greater than 5% of the center frequency for the test light and the reference light.
The common light source may have a spectral coherence length, and the optical path length difference may be changed over a range longer than the spectral coherence length to generate the scanning interference spectral signal.

Depending on the optics used to send the test light onto the test object and image it on the detector, the numerical aperture for the test light may be defined as a value greater than 0.8.
The method may further include generating a scanning interferometry signal.

  In another aspect, the invention features a processor in a computer that can derive information from a scanning interferometry signal for a first surface location of a test object and information corresponding to a plurality of models of the test object. An apparatus including a computer readable medium having a program for comparing the plurality of models, wherein the plurality of models are parameterized by a series of characteristics for the test object.

The apparatus may include any of the features described above in connection with the method.
In another aspect, the invention features a scanning interferometry system configured to generate a scanning interferometry signal and an electronic processor coupled to the scanning interferometry system to receive the scanning interferometry signal, An apparatus comprising: an electronic processor programmed to compare information derivable from scanning interferometry signals for a first surface location of a test object with information corresponding to a plurality of models of the test object. Thus, the multiple models are devices that are parameterized by a series of characteristics for the test object.

The apparatus may include any of the features described above in connection with the method.
In general, in another aspect, features of the present invention include chemical mechanical polishing of a test object, collecting scanning interferometry data for the surface topography of the test object, and scanning interferometry data Adjusting the process conditions for chemical mechanical polishing of the test object based on the information obtained from the method. For example, the process conditions may be pad pressure and / or polishing slurry composition. In a preferred embodiment, adjusting the process conditions based on information obtained from scanning interferometry data includes information derivable from scanning interferometry signals for at least a first surface location of the test object, and the test object. The plurality of models may be parameterized by a series of characteristics for the test object. The analysis of the scanning interferometry signal may further include any of the features described above in conjunction with the first described method.

  Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. In case of conflict with publications, patent applications, patents, and other references mentioned and incorporated herein by reference, the specification, including definitions, will be reviewed.

Other features, objects, and advantages of the present invention will become apparent from the following detailed description.
Like reference numerals in different drawings refer to common elements.

FIG. 1 is a flow chart that schematically illustrates one embodiment of the present invention in which scanning interferometry data is analyzed in the spatial frequency domain.
Referring to FIG. 1, to measure data from the surface of a test object, an optical path difference (OPD) between a reference path and a measurement path is mechanically or electro-optically scanned using an interferometer. To do. The measurement path is directed to the object surface. The OPD is a function of the local height of the object surface at the start of the scan. A plurality of camera pixels correspond to different surface locations on the surface of the object, and the computer records an interference intensity signal during OPD scanning for each camera pixel. Next, after storing the interference intensity signal as a function of the OPD scan position for each of the different surface locations, the computer performs a transform (eg, a Fourier transform) to generate a frequency domain spectrum of the signal. The spectrum includes both amplitude information and phase information as a function of the spatial frequency of the signal in the scan range. For example, a suitable frequency domain analysis (FDA) for generating such a spectrum is disclosed in the following document. Commonly owned US Pat. No. 5,398,113 (Pita de Groot), title of invention “Method and apparatus for surface topography measurement by spatial frequency analysis of interferogram”. The contents of this document are incorporated herein by reference.

  In separate steps, the computer generates a theoretical prediction library for the frequency domain spectrum for various surface parameters and a model for the interferometer. These spectra may cover a range of possible thin film thicknesses, surface materials, and surface textures, for example. In a preferred embodiment, the computer generates a library spectrum for a constant surface height, eg height = zero. Thus, in such an embodiment, the information contained in the library has nothing to do with surface topography. The information included is only about the type of surface structure and the interaction of the surface structure, optical system, illumination, and detection system in generating the characteristic features of the frequency domain spectrum. As an alternative, the prediction library may be generated empirically using sample artifacts. As another alternative, the library may receive information from prior supplementary measurements of the object surface provided by other equipment, such as an ellipsometer, and any other input from the user regarding known characteristics of the object surface. It may be used to reduce the number of unknown surface parameters. Any of these approaches to library formation augmented by supplementary measurements, theoretical modeling, empirical data, or theory can be extended by interpolation to provide intermediate values as part of library formation or during library search. It may be generated in real time.

In the next step, the experimental data and the prediction library are compared using a library search that provides surface structure parameters. In the case of a film of unknown thickness, a library for a single surface type, eg SiO ₂ on Si, spans many film thicknesses where the top surface height may always be equal to zero. Another example is surface roughness. In this case, the adjustable parameter may be roughness depth and / or spatial frequency. Library searching provides a match for FDA spectral characteristics that are independent of surface height. Such characteristics are related to, for example, the average value of the amplitude spectrum (related to the overall reflectivity of the surface) or the change in amplitude as a function of spatial frequency (in a monochromatic high NA system, the scattering angle of reflected light) ).

  The analysis may also include system characterization. System characterization includes, for example, measuring one or more reference workpieces with known surface structure and surface topography to determine system wavefront error, variance, and efficiency (cannot be included in the theoretical model) And determining parameters such as

  Furthermore, the analysis may include an overall calibration. The overall calibration includes, for example, measuring one or more reference workpieces, and measuring these surface parameters, eg, film thickness determined by library search, and those determined independently, eg, by ellipsometry. Determining the correlation between values for parameters.

  Based on the comparison of the experimental data with the prediction library, the computer identifies the surface model that corresponds to the best match. The computer may then display or send the surface parameter results numerically or graphically to a user or host system for further analysis or data storage. Using the surface parameter results, the computer may then determine surface height information in addition to the characteristics specified by the library search. In some embodiments, the computer generates a corrected phase spectrum. This is done, for example, by subtracting the corresponding theoretical phase spectrum directly from the experimental phase spectrum. The computer then determines a local surface height for the one or more surface points. This is done by analyzing the corrected phase as a function of spatial frequency, for example by analyzing the coefficients generated by a linear fit. The computer then generates a complete three-dimensional image constructed from the height data and the corresponding image plane coordinates, along with a graphical or numerical display of the surface characteristics determined by the library search.

  In some cases, library searching and data collection can be performed iteratively to further improve the results. Specifically, the library search can be refined on a pixel-by-pixel basis or on a region basis. This is done by forming a refined library in relation to the local surface type. For example, if during a preliminary library search it is found that there is a thin film of approximately 1 micron on the surface, the computer generates a fine-grained library of example values close to 1 micron to further search. Make it elaborate.

  In further embodiments, the user's interest may be only the surface characteristics modeled by the prediction library, not the surface height. In this case, the step of determining the surface height is not performed. Conversely, the user's interest may be just the surface height, not the surface characteristics that are modeled in the prediction library. In this case, the computer uses a comparison between the experimental data and the prediction library to correct the experimental data for surface property effects. As a result, the surface height is determined more accurately. However, there is no need to determine the surface characteristics clearly or display the characteristics.

  This analysis may be applied to the following various surface analysis problems. The surface analysis task is, for example, a simple thin film (in this case, for example, the variable parameter of interest may be film thickness, film refractive index, substrate refractive index, or some combination thereof); Multilayer thin film; sharp edges and surface features that diffract or otherwise create complex interference effects; untreated surface roughness; untreated surface features, eg, subwavelength widths on otherwise smooth surfaces Different materials (eg, the surface may contain a combination of thin film and solid metal. In this case, the library will include both surface structure types, and the thin film or solid metal may be included in the corresponding frequency domain spectrum. May be automatically identified by matching); optical activity such as fluorescence; surface spectroscopic properties such as color and wavelength dependent reflection ; Including and data distortion associated with the data acquisition procedure; polarization dependence of the properties of the surface; surface distortion or deformable surface features leading to disturbance of the interference signal, vibration or movement. The data acquisition procedure includes, for example, a data acquisition window that does not completely contain interference intensity data.

  The interferometer may include any of the following features. A spectrally narrowband light source having a high numerical aperture (NA) objective lens; a spectrally broadband light source; a combination of a high NA objective lens and a spectrally broadband light source; Objective lenses including, for example, oil / water immersion and solid immersion formats, eg, in Michelson, Mirau, or Linnik geometry; a series of measurements at multiple wavelengths; unpolarized Light: polarized light, eg, linear, circular, or structured polarized light. For example, structured polarized light requires, for example, a polarization mask and produces different polarizations for separate segments of the illumination pupil or imaging pupil, resulting in polarization-dependent optical due to surface properties. You may clarify the effect. The interferometer may also include the overall system calibration described above.

  When comparing theoretical and experimental data, the library search may be based on any of the following: Product or difference between amplitude and / or phase data in the frequency spectrum, eg, average amplitude and average phase, average amplitude itself, and average phase itself product or difference; amplitude spectrum slope, width and / or high Interference contrast; data in frequency spectrum at DC or zero spatial frequency; nonlinearity or shape of amplitude spectrum; zero frequency cutoff of phase; nonlinearity or shape of phase spectrum; and any combination of these criteria. As used herein, amplitude and wave height are used interchangeably.

  Shown in FIG. 2 is a flow chart generally illustrating another embodiment for analysis of scanning interferometry data. The analysis is similar to that described in FIG. 1 except that the comparison between the experimental data and the prediction library is based on information in the scan coordinate domain. The experimental signal feature may be a quasi-periodic carrier vibration modulated in amplitude by an envelope function with respect to the scan coordinate. When comparing theoretical and experimental data, the library search may be based on any of the following: Average signal strength; shape of signal envelope, eg any ideal or reference shape, eg deviation from Gaussian; phase of carrier signal for envelope function; zero crossing and / or signal maximum and minimum relative spacing; maximum and The values for the minimum and their ordering; the peak value of the correlation between the library signal and the measurement signal after adjustment for the optimal relative scan position; and any combination of these criteria.

  In the following, a detailed mathematical description of the analysis is given and examples are given. First, a typical scanning interferometer will be described. Second, the determination of a mathematical model for the scanning interferometry data will be described. Third, the optical characteristics of the surface and a method for generating an accurate model of scanning interference spectral data for various surface characteristics using such information will be described. Fourth, how to obtain information about the test object by comparing experimental interference spectroscopy data with the prediction library will be described. First, thin film applications will be described, and then other complex surface structures, specifically, step heights and lattice patterns that cannot be optically sufficiently resolved will be described. First, the analysis will be focused on the analysis in the spatial frequency domain, and then the analysis in the scanning coordinate domain will be described.

  FIG. 3 shows a linic type scanning interferometer. Illumination light 102 from a light source (not shown) is partially transmitted by beam splitter 104 to form reference light 106 and partially reflected by beam splitter 104 to form measurement light 108. To do. Measurement light is focused by a measurement objective 110 onto a test sample 112 (eg, a sample comprising a thin single film or multiple films of one or more different materials). Similarly, the reference light is focused on the reference mirror 116 by the reference objective lens 114. Preferably, the measurement and reference objectives have common optical properties (eg, numerical apertures are matched). Measurement light reflected (or scattered or diffracted) from the test sample 112 propagates backward through the measurement objective 110, is transmitted by the beam splitter 104, and is detected by the imaging lens 118. The image is formed on the device 120. Similarly, the reference light reflected from the reference mirror 116 propagates backward through the reference objective lens 114, is reflected by the beam splitter 104, and is imaged on the detector 120 by the imaging lens 118, There it interferes with the measurement light.

  For ease of explanation, the measurement and reference light shown in FIG. 3 is focused to a specific point on the test sample and reference mirror, respectively, and then interferes on the corresponding point on the detector. Such light corresponds to the portion of the illumination light that propagates perpendicular to the pupil plane relative to the measurement and reference legs of the interferometer. The other part of the illumination light eventually illuminates the test sample and other points on the reference mirror. These points are then imaged at corresponding points on the detector. In FIG. 3, this is illustrated by the dashed line 122. Dashed lines 122 correspond to chief rays that emerge from different points on the test sample and are imaged at corresponding points on the detector. The chief rays intersect at the center of the pupil plane 124 of the measurement leg. The pupil plane 124 is a rear focal plane of the measurement objective lens 110. Of the light that emerges from the test sample, those that appear at an angle different from the principal ray intersect at different locations on the pupil plane 124.

  In a preferred embodiment, detector 12Q is a multi-element (ie, multi-pixel) camera that independently measures interference between measurement light and reference light corresponding to different points on the test sample and reference mirror. (Ie give spatial resolution for the interference pattern).

  A scanning stage 126 coupled to the test sample 112 scans the position of the test sample relative to the measurement objective lens 110. This is indicated by the scan coordinates in FIG. For example, the scanning stage can be based on a piezoelectric transducer (PZT). Detector 120 measures the intensity of optical interference at one or more pixels of the detector as the relative position of the test sample is being scanned, and the information is analyzed by computer 128 for analysis. Send to.

  Since the region where scanning is performed is a region where the measurement light is focused on the test sample, the optical path length of the measurement light from the light source to the detector is changed by scanning. This change depends on the angle at which the measurement light is incident on the test sample and emerges from the test sample. As a result, the optical path difference (OPD) from the light source to the detector between the interference portion of the measurement light and the reference light interference portion is increased or decreased by the scanning coordinate ζ, while the measurement light is incident on the test sample, Depending on the angle emerging from the test sample. In other embodiments of the invention, the same result is achieved by scanning the position of the reference mirror 116 relative to the reference objective lens 114 (instead of scanning the test sample 112 relative to the measurement objective lens 110). Can do.

Since there is a difference in how the OPD changes with the scanning coordinate ζ, the coherence length is limited in the interference signal measured in each pixel of the detector. For example, the interference signal (which is a function of the scan coordinate) is typically modulated by an envelope on the order of spatial coherence length λ / 2 (NA) ² . Here, λ is the nominal wavelength of the illumination light, and NA is the numerical aperture of the measurement and reference objective lenses. As described further below, the modulation of the interference signal provides angle dependent information about the reflectivity of the test sample. In order to increase the limited spatial coherence, the numerical aperture defined by the objective lens in the scanning interferometer is preferably large. For example, greater than about 0.7 (or more preferably, greater than about 0.8 or greater than about 0.9). The interference signal can also be modulated by a limited temporal coherence length associated with the spectral bandwidth of the illumination source. Depending on the configuration of the interferometer, either one of these limited coherence length effects may dominate, or both may substantially affect the overall coherence length.

Another example of the scanning interferometer is the Mirau interferometer shown in FIG.
Referring to FIG. 4, illumination light 206 is supplied to beam splitter 208 by light source module 205. Beam splitter 208 sends illumination light 206 to Mirau interferometry objective assembly 210. The assembly 210 includes an objective lens 211, a reference plane 212 (having a reflective coating on its small central portion to define a reference mirror 215), and a beam splitter 213. In operation, illumination light is focused through the reference plane 212 toward the test sample 220 by the objective lens 211. The beam splitter 213 reflects the first portion of the focus light to the reference mirror 215 to define the reference light 222, and the second portion of the focus light is transmitted to the test sample 220 so that the measurement light 224 is It is formed. The beam splitter 213 then recombines the measurement light reflected (or scattered) from the test sample 220 with the reference light reflected from the reference mirror 215. The objective lens 211 and the imaging lens 230 form an image of the combined light and cause it to interfere on the detector (for example, a multi-pixel camera) 240. As with the system of FIG. 3, the measurement signal from the detector is sent to a computer (not shown).

  In the scan in the embodiment of FIG. 4, a piezoelectric transducer (PZT) 260 is coupled to the Mirau interferometry objective lens assembly 210. The piezoelectric transducer (PZT) 260 scans the entire assembly 210 with respect to the test sample 220 along the optical axis of the objective lens 211 to obtain scanning interference spectral data I (ζ, h) at each pixel of the camera. It is configured to be. Alternatively, the PZT may be coupled to the test sample rather than the assembly 210 to provide relative motion between them. This is illustrated by the PZT actuator 270. In still other embodiments, scanning may be performed by moving one or both of the reference mirror 215 and the beam splitter 213 relative to the objective lens 211 along the optical axis of the objective lens 211.

  The light source module 205 includes a spatially expanded light source 201, a telescope (formed by the lenses 202 and 203), a diaphragm disposed in the front focal plane of the lens 202 (coincident with the rear focal plane of the lens 203). 204. With this arrangement, a spatially expanded light source is imaged onto the pupil plane 245 of the Mirau interferometry objective lens assembly 210. This is an example of Koehler imaging. The size of the aperture controls the size of the illumination field on the test sample 220. In other embodiments, the light source module may include an arrangement in which a spatially expanded light source is imaged directly onto the test sample. This is known as critical imaging. Either type of light source module may be used with the linic scanning interferometry system of FIG.

  In a further embodiment of the invention, a scanning interferometry system may be used to determine angle-dependent scatter or diffraction information for a test sample, ie for light wave scatterometry. For example, using a scanning interferometry system, the test sample is illuminated by a test incidence over only a very narrow range of incidence angles (eg, substantially normal incidence or otherwise collimated). Also good. The incident is then scattered or diffracted by the test sample. The light emerging from the sample is imaged on the camera and interferes with the reference light. This is as described above. The spatial frequency of each component in the scanning interference spectroscopic signal varies depending on the angle of the test light emerging from the test sample. Therefore, performing a Fourier analysis after a vertical scan (ie, scanning along the optical axis of the objective lens) does not directly access or image the back focal plane of the objective lens. In addition, measurements of diffracted light and / or scattered light can be made as a function of appearance angle. In order to obtain substantially normal incidence illumination, for example, the configuration of the light source module is such that the point light source is imaged on the pupil plane, otherwise the illumination light is set to the numerical aperture of the measuring objective lens. This can be done to reduce the degree of filling. Lightwave scatterometry techniques are for analysis of discrete structures within a sample surface, such as grating lines, edges, or general surface roughness, which can diffract and / or scatter light to higher angles. Can be useful.

  It is assumed in most of the analysis herein that the polarization state of the light in the pupil plane is random, that is, approximately equal amounts of s-polarized light (perpendicular to the incident plane) and p (incident plane). (Both orthogonally polarized). Alternative polarizations are possible, such as pure s-polarized light. For example, a radial polarizer in the pupil plane (for example, in the back focal plane of the object to be measured in the case of a linic interferometer and in the back focal plane of a common objective lens in the Mirau interferometer) You may implement | achieve by arrange | positioning. Other possible polarizations include radial p-polarization, circular polarization, and modulated (eg, two states, one following the other) polarization for polarization measurements. In other words, the analysis of the optical properties of the test samples can be done not only for their angle or wavelength dependence, but also for their polarization dependence or for selected polarizations. Such information may also be used to improve the characterization accuracy of the thin film structure.

  In order to implement such ellipsometric measurements, the scanning interferometry system may include a fixed or variable polarizer in the pupil plane. Referring again to FIG. 4, for example, the Mirau-type interferometry system includes polarization optics 280 in the pupil plane to select the desired polarization for light incident on and emerging from the test sample. It has become. Further, the polarization optics may be reconfigurable to change the selected polarization. The polarization optics may include one or more elements including a polarizer, a waveplate, an apodization aperture, and / or a modulation element for selecting a predetermined polarization. Furthermore, the polarization optics may be fixed, structured, or reconfigurable to generate data similar to ellipsometer data. For example, after a first measurement using a radially polarized pupil for s-polarization, a radially polarized pupil for p-polarization may be used. In other examples, an apodized pupil plane may be used with linearly polarized light, such as a slit or wedge. The slit or wedge is a screen that can be rotated or reconfigurable in the pupil plane to send any desired linear polarization state to the object, such as a liquid crystal display.

  In addition, polarization optics may provide variable polarization across the entire pupil plane (eg, by including multiple polarizers or spatial modulators). In this way, the polarization state can be “tagged” according to the spatial frequency. This is done, for example, by providing a different polarization from a shallow angle for a high angle of incidence.

  In still other embodiments, selectable polarization may be combined with phase shift as a function of polarization. For example, polarization optics may include a linear polarizer placed in the pupil plane, after which two wave plates (eg, 1/8 wave plates) may be placed in opposing quadrants of the pupil plane. Good. As a result of the linear polarization, the polarization angle covers the entire range with respect to the entrance surface of the objective lens. For example, if the wave plates are aligned so that the phase shift of the dominant s-polarized light is fixed, both radial s-polarized light and p-polarized light are present at the same time, but the phase is only π, for example, from each other Due to the shift, the interferometer effectively detects the difference between these two polarization states as a fundamental signal.

In further embodiments, the polarization optics may be located elsewhere in the device. For example, linear polarization can be realized anywhere in the system.
Next, a physical model for the scanning interference spectral signal will be described.

Suppose that the object surface has a height feature h and that it is desired to profile the height feature h over the region indicated by the lateral coordinates x, y. By means of the stage, a smooth continuous scan ζ is realized for the interference objective lens or the object itself as shown. During the scan, intensity data I _{ζ, h} for each image point or camera pixel is recorded by the computer in successive camera frames. The main dependence of the intensity I _{ζ, h} on the scanning position and the surface height is indicated by a subscript. This subscript is a notation used throughout.

  A suitable physical model of optics can be very elaborate. This reduces the partial coherence of the light source, polarization mixing within the interferometer, the imaging properties of high NA objectives, and the interaction of electric field vectors at high angles of incidence and in the presence of discontinuous surface features. Made by considering. For convenience, the model is simplified by assuming random polarization and diffuse, low coherence extended light sources. The interference signal modeling makes it easy to examine the effects of all the light bundles (shown in FIG. 5) that are reflected from the object surface at the incident angle Ψ after passing through the pupil plane of the objective lens.

  The effect of interference on a single beam bundle through the optical system is proportional to:

ここでＺ_β，ｋは、有効な対象物強度反射率であり、たとえばビーム・スプリッタの効果を含む。Ｒ_β，ｋは、有効な基準反射率であり、ビーム・スプリッタおよび基準ミラーの両方を含む。周囲媒体の屈折率はｎ_０であり、入射角Ψに対する方向余弦は、以下の通りである。

Here, Z _{β, k} is an effective object intensity reflectance, and includes, for example, the effect of a beam splitter. R _{β, k} is an effective reference reflectivity and includes both a beam splitter and a reference mirror. The refractive index of the surrounding medium is n ₀ , and the direction cosine with respect to the incident angle Ψ is as follows.

また光源照明に対する波数は、以下の通りである。

The wave numbers for the light source illumination are as follows.

位相に対する記号の取り決めによって、表面高さの増加を、位相の正の変化に対応させる。位相項は、干渉計内の対象物経路に対する影響ω_β，ｋ、たとえば対象物表面からの薄膜効果と、基準経路に対する影響ν_β，ｋ、たとえば基準ミラーおよび対物レンズ内の他のオプティクスとを有する。

The symbol convention for phase allows the increase in surface height to correspond to a positive change in phase. The phase term determines the effect ω _{β, k} on the object path in the interferometer, eg the thin film effect from the object surface, and the effect ν _{β, k} on the reference path, eg other optics in the reference mirror and objective lens Have.

  The total interference signal integrated on the pupil plane is proportional to:

ここで、Ｕ_βは瞳平面の光分散であり、Ｖ_ｋは光学的なスペクトル分散である。方程式（４）における重み付け係数βは、投影角度に起因するｃｏｓ（Ψ）項と、瞳平面内の幅ｄΨの環の直径に対するｓｉｎ（Ψ）項とから得られる。

Here, U _β is the light dispersion in the pupil plane, and V _k is the optical spectral dispersion. The weighting factor β in equation (4) is obtained from the cos (ψ) term due to the projection angle and the sin (ψ) term for the diameter of the ring of width dψ in the pupil plane.

ここで、図５に図示したように、対物レンズはアッベの正弦条件に従うと仮定する。このような比較的簡単な重み付けは、ランダムに偏光された、空間的にインコヒーレントな照明（すべての光線束が互いに無関係である）の場合に可能である。最後に、すべての入射角に亘る積分範囲は、０≦ｐ≦１を意味し、すべての波数に亘るスペクトル積分は、０≦ｋ≦∞である。

Here, as shown in FIG. 5, it is assumed that the objective lens obeys Abbe's sine condition. Such a relatively simple weighting is possible in the case of randomly polarized, spatially incoherent illumination (all beam bundles are independent of each other). Finally, the integration range over all incident angles means 0 ≦ p ≦ 1, and the spectral integration over all wave numbers is 0 ≦ k ≦ ∞.

In the frequency domain analysis (FDA), first, Fourier transform of the interference intensity signal I _{ζ, h} is calculated. Use non-standardized Fourier integrals for literal (non-numeric) analysis.

ここで、Ｋは空間周波数であり、たとえば、単位は周期／ミクロンである。周波数ドメイン値ｑ_Ｋ，ｈの単位は、波数の逆であり、たとえばμｍである。これから、パワー・スペクトルと、

Here, K is a spatial frequency. For example, the unit is a period / micron. The unit of the frequency domain value q _{K, h} is the inverse of the wave number, for example, μm. From now on, the power spectrum,

位相スペクトルとが得られる。

And a phase spectrum.

φ”_Ｋ，ｈに対するダブル・プライムの意味は、フリンジのオーダに２倍の不確かさがあるということである。すなわち画素ごとのものと、走査の出発点に対する全体的なものとの両方である。そのため、従来のＦＤＡでは、パワー・スペクトルＱ_Ｋ，ｈによって重み付けされた位相スペクトルφ”_Ｋ，ｈに対する線形フィットによって表面トポグラフィを決定することに、直接進む。フィットによって、各画素に対して、勾配と、

The meaning of double prime for φ ″ _{K, h} is that there is twice the uncertainty in the fringe order, both pixel-by-pixel and global to the starting point of the scan. Therefore, conventional FDA goes directly to determining the surface topography by a linear fit to the phase spectrum φ ″ _{K, h} weighted by the power spectrum Q _{K, h} . By fitting, for each pixel, a gradient,

切片とが得られる。

Sections are obtained.

なお、切片または「位相ずれ」Ａ”は、高さｈとは無関係であるが、位相データのフリンジのオーダの不確かさから引き継いだダブル・プライムを備える。勾配σには、この不確かさはない。切片Ａ”および傾斜σ_ｈから、特定の平均または公称上の空間周波数Ｋ０に対して、「コヒーレンス・プロファイル」と、

Note that the intercept or “phase shift” A ″ is independent of the height h, but comprises a double prime inherited from the fringe order uncertainty of the phase data. The slope σ does not have this uncertainty. From the intercept A ″ and the slope σ _h , for a specific average or nominal spatial frequency K 0, a “coherence profile”;

「位相プロファイル」とを規定することができる。

A “phase profile” can be defined.

単純で理想的な場合、すなわち誘電体表面が完全に均一、均質で、薄膜および異なる材料の影響がなく、ならびに光学システムが分散に対して完全に平衡状態である場合には、位相およびコヒーレンス・プロファイルは、表面高さに直線的に比例する。

In simple and ideal cases, i.e. where the dielectric surface is perfectly uniform and homogeneous, without the influence of thin films and different materials, and the optical system is perfectly balanced with respect to dispersion, the phase and coherence The profile is linearly proportional to the surface height.

２つの高さ計算のうち、位相に基づく高さ値ｈ”_θの方が正確である。しかし、この値は、単色の干渉分光法のフリンジのオーダの特性に不確かさがある。高分解能を得るために、コヒーレンスに基づく明瞭であるが正確さに劣る値ｈ_Θを用いることで、この不確かさを取り除き、最終的な値ｈ_θを得る。

Of the two height calculations, the phase-based height value h ″ _θ is more accurate. However, this value is uncertain in the fringe order characteristics of monochromatic interferometry. In order to obtain this, the uncertainties are removed by using a clear but inaccurate value h _Θ based on coherence, and a final value h _θ is obtained.

The conventional FDA assumes that the interference phase φ ″ _{K, h} is still a substantially linear function of the spatial frequency even in a less ideal situation. However, in this embodiment, the main parameter of the surface structure is assumed. For example, film thickness determination is made by comparing experimental data with theoretical predictions, which can include a highly nonlinear phase spectrum and the accompanying modulation of the power spectrum.

  To do this, the equation (6), which is the definition of the Fourier transform, is combined with the interference signal equation (4) into the following equation for the predicted FDA spectrum:

計算の効率を向上させるために、方程式（１５）における３重積分の部分的な文字上の評価を行なうことができる。

In order to improve the computational efficiency, a partial character evaluation of the triple integral in equation (15) can be performed.

The lexical analysis of equation (15) begins by changing the order of integration, and first the values of the individual interference signals g _{β, k, ζ, h} are over all scanning positions ζ, Determined with β and k fixed.

余弦項ｇ_{β，ｋ，ζ，ｈ}を、以下の式を用いて通常の方法で展開した後、

After expanding the cosine terms g _{β, k, ζ, h} in the usual way using the following formula:

ζについての内部の積分を以下のように求める。

The internal integral for ζ is determined as follows:

ここで、以下の式を用いた。

Here, the following formula was used.

δ関数は、引数の逆の物理単位、この場合には逆の波数を有する。

The δ function has the opposite physical unit of the argument, in this case the opposite wavenumber.

These δ functions confirm the equivalence between the spatial frequency K and the product 2βkn ₀ .
The theoretical variation of the variable for the next integral is thus

ここで、κ＾は、空間周波数Ｋと同じ意味であるが、積分の自由変数として使用される。方程式（１８）は、以下のように書くことができる。

Here, κ ^ has the same meaning as the spatial frequency K, but is used as a free variable of integration. Equation (18) can be written as:

ここで、

here,

なお、変数を変えることによって、方程式（２３）におけるＲ、Ｚ、ν、ω項に対するβ依存性は、κ＾およびｋに対する依存性となる。

Note that by changing the variables, the β dependence on the R, Z, ν, and ω terms in equation (23) becomes dependence on κ ^ and k.

  As the next step, the following will be described first.

ここで、Ｈは、以下にように定義される無単位のヘビサイドのステップ関数である。

Here, H is a unitless snake-side step function defined as follows.

ｆは、Ｋおよびｋの任意の関数である。方程式（２５）〜（２７）を用いて、方程式（２３）は以下のようになる。

f is an arbitrary function of K and k. Using equations (25)-(27), equation (23) becomes:

ここで以下の式を用いると、

Here, using the following formula,

最終的に以下のような結果となる。

The final result is as follows.

方程式（３３）は、積分の数が少ないので、当初の式（１５）の三重積分と比べて著しく能率的な計算となる。

Since the number of integrals is small in the equation (33), the calculation is significantly more efficient than the triple integral of the original equation (15).

There are some extreme cases that are interesting to solve analytically. For example, when the influence of the phase (ν _{K, k} −ω _{K, k} ) = 0 and the reflectances R and Z are independent of the incident angle and the wavelength, the equation (33) is as follows: It will be easy.

したがって、方程式（２４）において定義される重み付け係数Γ_Ｋ，ｋを伴う積分のみを取り扱えばよい。この理想的なケースでは、方程式（３４）に対してさらに２つの極限的なケースを評価することが簡単になる。すなわち、高ＮＡ対物レンズを有する準単色の照明、および低ＮＡを有する広帯域の照明である。

Therefore, only the integration with the weighting coefficients Γ _{K, k} defined in equation (24) need be handled. In this ideal case, it becomes easier to evaluate two more extreme cases for equation (34). That is, quasi-monochromatic illumination with a high NA objective lens and broadband illumination with a low NA.

In the case of a quasi-monochromatic light source having a narrow spectral bandwidth _kΔ , the spectrum is normalized as follows.

ここで、ｋ_０は、公称上の光源波数である。方程式（３４）における積分は、以下のような形となる。

Here, k ₀ is a light source wavenumber nominal. The integral in equation (34) takes the form:

Ｕ_Ｋ，ｋは、小さい帯域幅ｋ_Δ上で本質的に一定であると仮定すると、以下の式が得られる。

Assuming U _{K, k} is essentially constant over a small bandwidth k _Δ , the following equation is obtained:

ここで、積分の値を求める際に、以下の関係を用いた。

Here, the following relationship was used when calculating the value of integration.

これは、狭帯域幅ｋ_Δ≪ｋ０に対しては有効である。特に、スペクトルの正の非ゼロ部分は、以下のようになる。

This is effective for a narrow bandwidth k _Δ << k0. In particular, the positive non-zero part of the spectrum is

結果として、光源のスペクトル帯域幅が狭く、反射率Ｒ、Ｚが一定で、位相の影響ωがないこの特別な場合には、以下のようになる。

As a result, in this special case where the spectral bandwidth of the light source is narrow, the reflectivities R and Z are constant, and there is no phase effect ω:

この特別な場合には、位相は、表面高さに直線的に比例し、従来のＦＤＡと一致する。また空間周波数は、方向余弦に直接対応する。

In this special case, the phase is linearly proportional to the surface height and is consistent with conventional FDA. The spatial frequency directly corresponds to the direction cosine.

したがって、ＦＤＡスペクトルの空間周波数座標と入射角との間には、１対１の関係がある。さらに、フーリエ振幅√Ｑ_ＫにおけるＫ重み付けは、方程式（４１）から計算されることに注意されたい。このことは、図６（ａ）の例示スペクトルにおいて明らかである。同図では、垂直入射から、対物レンズＮＡによって課される方向余弦限界までの範囲に亘って、瞳平面を完全に均一に充填する場合に対する理論的な予測を示す。

Therefore, there is a one-to-one relationship between the spatial frequency coordinates of the FDA spectrum and the incident angle. Furthermore, note that the K weighting at the Fourier amplitude √Q _K is calculated from equation (41). This is evident in the exemplary spectrum of FIG. The figure shows a theoretical prediction for the case where the pupil plane is filled completely uniformly over the range from normal incidence to the direction cosine limit imposed by the objective lens NA.

第２の例として、垂直入射付近の方向余弦の狭い範囲β_Δに限定された均一な照明を伴う広帯域の照明の場合を考える。したがって、規格化された瞳平面分散は以下のようになる。

As a second example, consider the case of broadband illumination with uniform illumination limited to a narrow range β _Δ of the direction cosine near normal incidence. Therefore, the normalized pupil plane dispersion is as follows:

変数を変えた後、

After changing the variable

方程式（３４）における定積分は、この場合には以下の形となる。

The definite integral in equation (34) takes the following form in this case:

これは以下の値となる。

This is the following value:

ここで、以下の関係を用いた。

Here, the following relationship was used.

スペクトルの正の非ゼロ部分は、この広帯域の光源照明用であり、したがって、垂直入射付近は、以下のようになる。

The positive non-zero part of the spectrum is for this broadband light source illumination, so near normal incidence is:

これは、フーリエ振幅（√Ｑ_Ｋ）が光源スペクトル分散Ｖ_{Ｋ／２ｎ０}に比例するという熟知結果に密接に対応する。このことはたとえば、図６（ｂ）において、公称上または平均の波長ｋ_０の中心に置かれたガウシアン・スペクトルに対して示されている。なお方程式（５２）は、以下の線形的な位相の展開が、

This closely corresponds to the familiar result that the Fourier amplitude (√Q _K ) is proportional to the light source spectral dispersion V _{K / 2n0} . This is for example, in FIG. 6 (b), are shown for Gaussian spectra placed nominally or average of the center wavelength k _0. Equation (52) has the following linear phase expansion:

従来のＦＤＡと一致するという仮定にも適合する。

It also meets the assumption that it is consistent with conventional FDA.

Since the Fourier amplitude √Q _{K, h} = | q _{K, h} | and the phase φ ″ _{K, h} = arg (q _{K, h} ) are obtained from the Fourier transform of the interference intensity I _{ζ, h} , Return to the interference signal domain.

ここで、もう一度、κ＾を空間周波数に対して用いて、これが方程式（５４）における積分の自由変数であることを強調した。したがって、強度信号を計算する１つの方法は、方程式（３３）によってフーリエ成分ｑ_Ｋ，ｈを生成し、方程式（５４）を用いてＩ_ζ，ｈに変換することである。

Here again, κ ^ is used for the spatial frequency to emphasize that this is a free variable of integration in equation (54). Thus, one way to calculate the intensity signal is to generate a Fourier component q _{K, h} by equation (33) and convert it to I _{ζ, h} using equation (54).

  In this model, it is assumed that the light source light is randomly polarized. However, this does not mean that the polarization effect should be ignored. Rather, the above calculations assume an incoherent superposition that is the result of being equally weighted from two orthogonal polarization states s and p defined by the entrance plane of the illumination. For polarization, superscript notation is used.

したがって、このβ、ｋにおける未偏光光に対する平均の位相角度は、以下のようになる。

Therefore, the average phase angle with respect to unpolarized light at β and k is as follows.

なお、２つの偏光の影響に対して振幅が同一である場合を除いて、ほとんどの場合、以下のようになる。

In most cases, except when the amplitude is the same for the two polarization effects,

また、ｑ^ｓ _β，ｋおよびｑ^ｐ _β，ｋが、複素平面において完全に平行である場合を除いて、以下のようになる。

Further, except for the case where q ^s _{β, k} and q ^p _{β, k} are completely parallel in the complex plane, they are as follows.

同じ所見を、システムおよび対象物反射率であるＲ^ｓ _β，ｋ、Ｒ^ｐ _β，ｋ、およびＺ^ｓ _β，ｋ、Ｚ^ｐ _β，ｋに、それぞれ適用する。これらは、位相が同一である場合を除いて、直接足し合わせることはできない。

The same observations apply to the system and object reflectivity R ^s _{β, k} , R ^p _{β, k} and Z ^s _{β, k} , Z ^p _{β, k} respectively. They cannot be added directly, except when the phases are the same.

  If the polarization effect is properly taken into account in the calculation of the object surface reflectivity, the modeling is fairly simple and quite adaptable, and can handle the more interesting cases of polarized light.

The next step is to convert it into a separate numeric expression, taking into account software development. The relationship between the interference signal I _{ζ, h} and the Fourier spectrum q _{K, h} is redefined using a separate Fourier transform as follows:

ここで、ｑ~_Ｋ^，ｈは、ｑ_Ｋ^，ｈの複素共役であり、干渉信号にはＮ個の別個のサンプルが存在する。方程式（６０）および後述の式において、導出時には重要であったが空間周波数Ｋに対する代わりとしてはもはや必要ではない自由変数Ｋを用いることは、やめている。したがって、予測される正の周波数ＦＤＡ複素スペクトルは、以下のようになる。

Here, q ~ _{K ^, h} is a complex conjugate of qK _{^, h} , and there are N separate samples in the interference signal. In equation (60) and the equations described below, the use of free variable K, which was important at the time of derivation but is no longer necessary as a substitute for spatial frequency K, is ceased. Thus, the predicted positive frequency FDA complex spectrum is:

ここで、規格化された、高さに無関係な係数は、以下のようになる。

Here, the normalized coefficient independent of the height is as follows.

ここで、積分範囲に対する規格化は、以下のようになる。

Here, normalization with respect to the integration range is as follows.

方程式（６２）におけるヘビサイドのステップ関数Ｈによって、足し合わせに対する不必要な影響が防止されている。重み付け因子Γ_Ｋ，ｋは、方程式（２４）において定義される通りである。

The snake side step function H in equation (62) prevents unwanted effects on summing. The weighting factor Γ _{K, k} is as defined in equation (24).

To compare the experiment with theory, an equation (61) is used to generate an experimental FDA spectrum, and equation (62) is used to convert back to the space domain to obtain the theory of I _{ζ, h} . Predictive. This is most efficiently done when fast Fourier transform (FFT) is used. The range of the K value is determined by the characteristics of the FFT. If N separate samples for I _{ζ, h} are spaced apart by an increment ζ _step , start from zero and increase to N / 2 periods per data trace (N / 2) +1 positive There is a spatial frequency. They are spaced apart by the following increments:

周波数ドメインにおける位相アンラッピングを容易にするために、走査に対するゼロ位置の調整を試みて、その位置が信号ピーク付近にくるようにすることで、周波数ドメインにおける位相勾配が小さくなる。ＦＦＴでは、走査における最初のデータ点がゼロにあることが常に仮定されているため、信号を適切にオフセットしなければならない。

In order to facilitate phase unwrapping in the frequency domain, attempting to adjust the zero position relative to the scan so that the position is near the signal peak reduces the phase gradient in the frequency domain. In FFT, it is always assumed that the first data point in the scan is at zero, so the signal must be offset appropriately.

Focus on modeling of sample surfaces with thin films.
Shown in FIG. 7 are two surface types with and without thin films. In both cases, the effective amplitude reflectivity Z _{β, k} is defined according to:

ここで、Ｚ_β，ｋは強度反射率であり、ω_β，ｋは反射時の位相変化である。添え字β、ｋは、照明の方向余弦への依存性を強調するものである。

Here, Z _{β, k} is the intensity reflectance, and ω _{β, k} is the phase change during reflection. The subscripts β and k emphasize the dependence of the illumination on the direction cosine.

ここで、Ψ_０は入射角であり、波数については以下のようになる。

Here, Ψ ₀ is an incident angle, and the wave number is as follows.

ここで、λは光源の波長である。添え字βは、第１の入射方向余弦β_０を指すものと理解される。

Here, λ is the wavelength of the light source. The subscript β is understood to refer to the first incident direction cosine β ₀ .

The surface is characterized, in part, by its refractive index. The refractive index of the surrounding medium (usually air) is n _0. In the case of a simple surface in FIG. 7 (a), only one refractive index n _1. In the case of the thin film FIG. 7B, there are two surface refractive indexes. That is, n ₁ for a transparent or partially transparent film and n ₂ for a substrate. Most commonly, these refractive indices are complex numbers characterized by real and imaginary parts. For example, a typical refractive index is n ₁ = 3.18 + 4.41i at λ = 550 nm, for example in the case of chromium. Here, the convention that the imaginary part is defined as positive is adopted.

The refractive index of the material depends on the wavelength. The dispersion of the refractive index n ₀ with respect to air is not very large, but is important for many sample surfaces, especially metals. At small wavelength changes near the nominal k0, most materials depend almost linearly on wavenumber. So we can write:

ここで、ｖ^（０） _１、ｖ^（１） _１はそれぞれ、屈折率ｎ_１に対して公称上の波数ｋ０における、切片および勾配である。

Here, v ⁽⁰⁾ ₁ and v ⁽¹⁾ ₁ are an intercept and a gradient at a nominal wavenumber k0 with respect to the refractive index n ₁ , respectively.

  The most common use of refractive index is Snell's law. Referring to FIG. 7B, the refractive beam angle inside the film is as follows.

ここで、Ψ_０は、屈折率ｎ_０の媒体内の角度であり、これが屈折率ｎ_１の媒体の最上面に入射する。また、Ψ_{１，β，ｋ}は、屈折角である。これらの角度は、屈折率が複素数である場合には、複素数値を取ることができ、部分的にエバネッセントな伝搬を示す。

Here, Ψ ₀ is an angle in the medium having the refractive index n ₀ , and this is incident on the uppermost surface of the medium having the refractive index n ₁ . Moreover, Ψ _{1, β, k} is a refraction angle. These angles can be complex values when the index of refraction is complex, and show partially evanescent propagation.

  The complex amplitude reflectivity at the boundary between the two media depends on the polarization, wavelength, incident angle, and refractive index. The s- and p-polarized reflectivities of the top surface of the film of Fig. 7 (b) are given by the Fresnel equation as follows:

β、ｋに対する依存性は、角度Ψ_０、Ψ_{１，β，ｋ}から生じる。出口角Ψ_{１，β，ｋ}によって、屈折率ｎ_１，ｋを介したｋ依存関係が導入される。同様に、基板−膜界面の反射率は、以下の通りである。

The dependence on _{β, k} arises from the angles Ψ ₀ , Ψ _{1, β, k} . The exit angle Ψ _{1, β, k} introduces a k dependence via the refractive index n _{1, k} . Similarly, the reflectance at the substrate-film interface is as follows.

なお、フレネル方程式において、入射角および屈折角が同じ場合には、両方の偏光に対する反射率はゼロになる。

In the Fresnel equation, when the incident angle and the refraction angle are the same, the reflectivity for both polarized lights is zero.

  In the case of a simple surface (no thin film), the reflectance of the sample surface is the same as that of the top surface.

その結果、表面反射によって引き起こされる反射時の位相変化（ＰＣＯＲ）は、以下のようになる。

As a result, the phase change (PCOR) during reflection caused by surface reflection is as follows.

なお、境界条件を満たすために、ｓ−偏光は、反射時に「反転する」（＝誘電体の場合のπ位相シフト）が、ｐ−偏光は反転しない。正確に垂直入射の場合には偏光状態間の区別は意味がなくなる。すなわち、どんな場合でもフレネル方程式においてゼロ除算となるため、この極限的なケースを取り扱う式は異なるものとなる。

In order to satisfy the boundary condition, s-polarized light is “inverted” upon reflection (= π phase shift in the case of a dielectric), but p-polarized light is not inverted. In the case of exactly normal incidence, the distinction between polarization states is meaningless. In other words, since the Fresnel equation is divided by zero in any case, the formula for handling this extreme case is different.

When the plus sign convention is used for the complex part of the refractive index, the greater the absorption (complex part), the greater the PCORωβ _{, k} . In other words, increasing the absorption coefficient is equivalent to reducing the effective surface height. This can be understood intuitively. That is, absorption is envisioned as the light beam penetrates into the material before reflection, rather than the precise reflection and transmission that occurs exactly at the boundary. The normal surface, i.e. the increase in surface height, corresponds to a positive change in the phase difference between the reference surface and the measurement surface, subtracting the positive surface PCOR from the interferometer phase.

  A thin film is a special case of parallel plate reflection. The light is partially reflected as it passes through the top surface (see FIG. 7) and travels to the substrate surface. Second reflection occurs on the substrate surface. This reflection is delayed in phase with respect to the first reflection. But that doesn't end there. The light reflected from the substrate is partially reflected again when it passes back through the top surface. The result is a further reflected beam that faces down again to the substrate. This continues in principle forever, and each further reflection is slightly weaker than the previous one. Assuming all these multiple reflections persist and affect the final surface reflectance, the infinite set has the following values:

説明の備考として、β_{１，β，ｋ}のβ依存関係が、屈折率ｎ_０の周囲媒体中の入射方向余弦β_０への依存関係を指すことを、思い起こされたい。同じ方程式（７７）が、対応する単一の表面反射率を有する両方の偏光状態に適用される。

As a remark of explanation, recall that the β dependence of β _{1, β, k} refers to the dependence of the refractive index n ₀ on the incident cosine β _{0 in} the surrounding medium. The same equation (77) applies to both polarization states with a corresponding single surface reflectance.

  Examining these equations shows why conventional FDA processing fails in the presence of thin films. In conventional FDA, the surface height is determined by a linear fit to the Fourier phase spectrum weighted by the Fourier power spectrum, and a broad band (white) light is used to form the spread of the Fourier spatial frequency. . The idea is that the phase expansion is brought about by the expected linear phase dependence on the surface height. Any other constant offset or linear factor (eg, “dispersion”) associated with the surface characteristics is removed by characterization of the system or by simply ignoring phase effects that do not vary with field position.

  This works perfectly accurately for simple surfaces. In the case of unpolarized light, and perhaps circularly polarized light, the wavelength dependence of PCOR is approximately linear with wavenumber and constant for a given material. However, if a thin film is present, conventional analysis will not work. The phase is non-linear and the phase gradient is sensitive to film thickness and may vary across the field of view. Therefore, in this analysis, the determination of the main parameters of the surface structure, such as the film thickness, is made by comparing experimental data with theoretical predictions, for example how the thin film modulates the reflectivity of the surface. Knowledge is used.

Next, it will be described how surface structure parameters such as film thickness and phase change (PCOR) during reflection are obtained by comparing experimental data with a library of theoretical predictions. For films of unknown thickness, a library for a single surface type, eg, SiO ₂ on Si, spans many possible film thicknesses. In the frequency domain embodiment, the idea is to search this library for a match to the FDA spectral characteristics unrelated to the surface topography, for example a unique structure for the amplitude spectrum due to thin film interference effects. The library spectrum is then used in the computer to correct the FDA data to allow an accurate surface topography map.

In one embodiment, the library contains exemplary FDA spectra for surface structures. Each spectrum provides a series of complex coefficients ρ _K that represent the Fourier coefficients as a function of the spatial frequency K. These spectra are Fourier transforms of the intensity data I _{ζ, h} acquired during the optical path length scan ζ of the interferometer. The spatial frequency K is proportional to the angular wave number k = 2π / λ for the segment of the light source light spectrum, the refractive index n ₀ of the surrounding medium, and the direction cosine β = cos (ψ). Here, Ψ is an incident angle with respect to a light beam transmitted to the surface of the object.

予測ライブラリに対するρ_Ｋ係数は、ＦＤＡスペクトルの外観に影響を与える可能性がある表面の光学特性（表面高さを除く）を含む。

The ρ _K coefficient for the prediction library includes surface optical properties (excluding surface height) that can affect the appearance of the FDA spectrum.

In order to predict the FDA spectrum, it is necessary to perform an integration representing the incoherent sum of the light bundles over the range of the incident angle Ψ and the angular wave number k with respect to the light source light. As described above, the numerical integration can be a single sum over N angular wave numbers K (weighted by factors Γ _{K, k} ) that make the calculation efficient.

重み付け因子は、以下の通りである。

The weighting factors are as follows.

ここで、Ｖ_Ｋは、光源スペクトルであり、Ｕ_Ｋ、ｋは、瞳平面の光分散である。対応する規格化γは、重み付け因子のすべての空間周波数に亘る合計である。

Here, V _K is the light source spectrum, and U _{K, k} is the light dispersion in the pupil plane. The corresponding normalization γ is the sum of all weighting factors over all spatial frequencies.

ここで、γは、簡潔に規定されるべき規格化であり、Ｈは、ヘビサイドのステップ関数である。

Here, γ is a normalization to be succinctly defined, and H is a snake side step function.

The surface structure of the object, in particular the unique properties of the thin film, becomes part of the spectrum ρ _K through the object-path phase ω _{K, k} and the reflectivity Z _{K, k} . This has already been described in detail. Equally important are the reference-path phase ν _{K, k} and the reflectivity R _{K, k} . These depend on the scanning interferometer itself. Such factors can be determined by theoretically modeling a scanning interferometer or by calibrating it with a test sample of known characteristics. This will be further described later.

A typical prediction library for a thin film is a series of spectra ρ _K indexed by film thickness L. The range of spectra stored is only a narrow spatial frequency region of interest (ROI), typically 15 or 16 values for 256 frames of intensity data acquisition. The remaining value outside this ROI is zero. The ROI range is derived from the definition of spatial frequency.

１００Ｘのミラウ対物レンズおよび狭帯域幅、５００ｎｍ光源に基づく走査型干渉計に対する空間周波数の典型的な範囲は、２．７μｍ^−１〜４．０μｍ^−１である。計算を能率的に行なうために、数式（８０）〜（８３）を用いて各画素に対して複数回再計算することを伴う解析的な検索ルーチンではなく、サンプル・スペクトル間の０．５〜５ｎｍによってインデックスされる高密度なルック・アップ・テーブルを用いることができる。

100X of Mirau objective and narrow bandwidth, the typical range of spatial frequency for scanning interferometer-based 500nm light source is 2.7μm ^-1 ~4.0μm ^-1. In order to perform the calculation efficiently, it is not an analytical search routine involving recalculation for each pixel by using the equations (80) to (83), but 0.5 to 0.5 between the sample spectra. A dense look-up table indexed by 5 nm can be used.

  Library searching involves the following steps: (1) Select a predicted FDA spectrum from a library corresponding to a particular surface type. (2) Calculate how closely this spectrum matches the experimental data using the merit function. And (3) iterate through some or all library data sets to determine from which theoretical spectrum the best match is obtained. What we are looking for are “discriminating characteristics” in the frequency domain that are uniquely related to the surface characteristics. Surface properties are, for example, thin films, foreign materials, step structures, roughness, and their interaction with the interferometer optical system. Thus, such a comparison clearly removes the linear rate of phase change with respect to spatial frequency. This is a property of the FDA spectrum that varies directly with surface topography and is therefore not relevant for library searching.

  In the comparative spectrum, it is beneficial to separate the phase effect and the amplitude effect on the calculation. Therefore, with respect to the theory:

ここで、ｃｏｎｎｅｃｔ_Ｋは、φ_Ｋ，ｈの空間周波数依存性における２−πステップを取り除く関数である。実験データに対しては、以下のようになる。

Here, connect _K is a function that removes the 2-π step in the spatial frequency dependence of φK _{, h} . The experimental data is as follows.

φ”^ｅｘ _Ｋに対するダブル・プライムは、走査における出発点に対する画素間および全体の両方からのフリンジのオーダにおける不確かさを示す。実験データは必然的に、局所的な表面高さに関係する勾配項を含む。これが、ｐ符号の代わりにｑ符号を用いる理由である。

The double prime for φ ″ ^ex _K shows the uncertainty in the order of fringes from both the inter-pixel and the whole relative to the starting point in the scan. The experimental data necessarily inevitably has a gradient term related to the local surface height. This is why the q code is used instead of the p code.

  The phase difference can be calculated for a particular set of trial surface parameters.

トライアルなパラメータは正確であると仮定して、位相差ζ”_Ｋ，ｈは、補正されたＦＤＡ位相である。理論が実験に良好にマッチングすることにより、原理的には切片がゼロ（すなわちゼロ位相ずれ）の空間周波数Ｋの単純な線形関数である位相ζ”_Ｋ，ｈが得られる。したがって、先を見越すと、成功裏に補正された位相ζ”_Ｋ，ｈは、最終的には従来のＦＤＡ解析へと下流に供給されるものである。従来のＦＤＡ解析では、周波数空間における位相の勾配は、表面高さに正比例していると仮定している。

Assuming that the trial parameters are correct, the phase difference ζ ″ _{K, h} is the corrected FDA phase. In principle, the intercept is zero (ie, zero because the theory matches well with the experiment. A phase ζ ″ _{K, h} which is a simple linear function of the spatial frequency K of (phase shift) is obtained. Therefore, in anticipation, the successfully corrected phase ζ ″ _{K, h} is ultimately supplied downstream to the conventional FDA analysis. In the conventional FDA analysis, the phase in the frequency space is Is assumed to be directly proportional to the surface height.

Based on the findings in the previous paragraph, the corrected phase ζ ″ _{K, h} has two features of interest that allow the evaluation of matching theory against experiments independent of surface height. The first is the phase shift A ″ or the intercept value ζ ″ _{K = 0, h} of _{K = 0} , which is obtained by a linear fit. The second is the remaining nonlinearity with respect to the wavenumber after the linear fit. The corresponding merit function is, for example:

ここで、σ_ｈは、補正された位相ζ”_Ｋ，ｈに対する（振幅が重み付けされた）線形フィットの勾配である。方程式（９１）におけるｒｏｕｎｄ（）関数によって、位相ずれＡ”が範囲±πに制限される。

Where σ _h is the slope of the linear fit (amplitude weighted) with respect to the corrected phase ζ ″ _{K, h} . The round () function in equation (91) causes the phase shift A ″ to be in the range ± π. Limited to

A library search can proceed using phase information alone, ie, minimizing one or both of the merit function values χ _φ and / or χ _φnon , but important and useful discriminating properties are Fourier transforms. It also exists in amplitude. The amplitude is particularly interesting in that it is inherently independent of the surface height. Therefore, for example, the following amplitude merit function can be defined almost the same as the phase merit.

ここで、Ωは経験的なスケーリング因子である。

Where Ω is an empirical scaling factor.

メリットχ_Ｐは、対象物表面の全反射率に最も密接に関係しており、空間周波数依存性とは無関係である。一方で、χ_Ｐｎｏｎは、理論および実験的な振幅プロットが、形状においてどの程度良好にマッチングするかを表現する。

The merit χ _P is most closely related to the total reflectance of the object surface, and is independent of the spatial frequency dependence. On the other hand, χ _Pnon represents how well the theoretical and experimental amplitude plots match in shape.

The amplitude merit function χ _P and / or χ _Pnon is separate from or even in place of the phase merit χ _φ and / or χ _φnon . Therefore, a general library search merit function is as follows.

ここで、ｗは重み付け因子である。原理的には、種々のパラメータに対する標準偏差を知ることで、方程式（９６）における重みを決定することができる。より経験的なアプローチは、実際のデータおよびシミュレートされたデータに対して種々の重みを試して、それらがどの程度良好に機能するかを調べることである。以下の例では、すべてのメリットの影響に対して、等しい重みｗ_φ＝ｗ_φｎｏｎ＝ｗ_Ｐ＝ｗ_Ｐｎｏｎ＝１を選択する。

Here, w is a weighting factor. In principle, knowing the standard deviation for various parameters, the weight in equation (96) can be determined. A more empirical approach is to try different weights on the actual and simulated data to see how well they work. In the following example, an equal weight w _φ = w _φnon = w _P = w _Pnon = 1 is selected for all merit effects.

In the examples of FIGS. 8 to 13, the merit function search procedure is shown for six film thicknesses of SiO ₂ on Si: 0, 50, 100, 300, 600, and 1200 nm, respectively. The single library for all examples contains the range 0-1500 nm with 2 nm spacing. The data is a simulation and there is no noise. As with all examples described herein, the scanning step is 40 nm, the source wavelength is 498 nm, and the source Gaussian FWHM is 30 nm (pseudo-monochromatic).

The most interesting aspect of these simulated searches is the behavior of the four merit functions. It is generally accepted that the inclusion of these four functions reduces the ambiguity in the final merit value and there is a strong periodicity as a function of film thickness for each merit value. Another common finding is that the benefits based on non-linearity are most effective at 300 nm and above, both in phase and amplitude, but the phase shift and average amplitude are dominant at film thicknesses below 300 nm. ,That's what it means. This shows that χ _φ , χ _P merit functions are particularly useful for true thin films, where system characterization is important and directly combines to result in phase shifts and amplitudes That is.

Once the film thickness is determined (or other usage for the material or algorithm is specified), the FDA process proceeds in the usual way. However, instead of the original experimental phase data, the corrected FDA phase ζ ″ _{K, h} is used. In principle, if modeling is successful, ζ ″ _{K, h} has no nonlinearity. The phase shift should be zero. The next step is therefore a linear fit to the phase spectrum ζ ″ _{K, h} . It appears that using the amplitude spectrum P _K instead of the square of the amplitude is more effective for high NA FDA. What is obtained for each pixel by the fit is the gradient,

切片（位相ずれ）とである。

It is an intercept (phase shift).

なお、位相ずれＡ”は、位相データのフリンジのオーダの不確かさから引き継いだダブル・プライムを有する。傾斜σ_ｈには、この不確かさはない。切片Ａ”および傾斜σ_ｈから、特定の平均または公称上の空間周波数Ｋ０に対して、以下のように「コヒーレンス・プロファイル」と、

Note that the phase shift A ″ has a double prime inherited from the fringe order uncertainty of the phase data. The slope σ _h does not have this uncertainty. From the intercept A ″ and the slope σ _h , a specific average Or, for the nominal spatial frequency K0, “coherence profile” as follows:

「位相プロファイル」とを定義する。

Define a “phase profile”.

次に、位相θ”_ｈにおける画素間のフリンジのオーダの不確かさを取り除く。

Next, the uncertainty of the fringe order between pixels in the phase θ ″ _h is removed.

ここで、α’は、画素間２πステップがない本来の位相ずれＡ”に対する近似である。

Here, α ′ is an approximation to the original phase shift A ″ with no 2π step between pixels.

  Finally, the height profile is obtained from:

なお、位相オフセットγを差し引く必要はない。その理由は、補正された位相ζ_Ｋ，ｈを生成する際に、差し引くことはすでに行なわれているからである。

It is not necessary to subtract the phase offset γ. This is because the subtraction has already been performed when the corrected phase ζ _{K, h} is generated.

  The first example of surface topography measurement (FIG. 14) is a pure simulation. The surface topography is zero everywhere. However, the underlying film layer increases from 0 to 1500 nm in 10 nm increments. Using the same prediction library as in FIGS. 8-13, this test demonstrates that the film thickness is clearly determined over the entire range of the prediction library, albeit with perfect noise-free data. Yes.

Although the next example (FIG. 15) is also a simulation, noise is added. The random additive noise is Gaussian, the standard deviation is 2 bits, and the average is 128 intensity bits. This seems to be typical of actual data. Although there is a significant difference in reflectance between the SiO ₂ and Si (4% ~45%), the result is to go unequivocally satisfactory.

Next, system characterization will be described.
Data collected during the system characterization procedure is used to define the phase offset γ _sys and the linear variance τ _sys . To include system characterization data, the Fourier transformed experimental data is corrected using the following before library search and before any other inter-pixel based FDA processing.

ここで、Ｋ０は公称上の空間周波数であり、ＦＤＡデータ・セットに対する公称上のスペクトルの周波数を表わす。これは、たとえば、ＲＯＩの中心点を位置決めすることによって特定される。なお、理論的なライブラリは変わってはいない。スケーリング係数Ｍ（ギリシャ語大文字の「Ｍ」）は、対象物の表面反射率をライブラリ検索におけるパラメータとして使用することを可能にする新しいシステム特徴付けである。

Where K0 is the nominal spatial frequency and represents the nominal spectral frequency for the FDA data set. This is specified, for example, by positioning the center point of the ROI. The theoretical library has not changed. The scaling factor M (Greek capital letter “M”) is a new system characterization that allows the surface reflectance of an object to be used as a parameter in a library search.

The phase offset γ _sys and system phase shift A _sys as a function of field position can be stored as a function of field position, and the true system variance can be calculated according to the following equation:

振幅係数Ｍも、フィールド依存性である。

The amplitude coefficient M is also field dependent.

  The generation of system characterization data proceeds in a similar manner as described above for the object sample. The features are transferred to a known workpiece and measured to determine the system characterization. This decision is made by looking at how the results differ from what is expected for a complete system. Specifically, using a known sample whose exact library entry is predetermined, the phase shift A "as in equation (98) and the final height h as in equation (102). Next, assuming a perfectly flat workpiece, the system phase offset and

システム位相ずれとを、計算する。

The system phase shift is calculated.

ここで、ｃｏｎｎｅｃｔ_ｘｙ（）は、画素間位相アンラッピングである。振幅マップは、以下の通りである。

Here, connect _xy () is inter-pixel phase unwrapping. The amplitude map is as follows.

種々の実施形態においては、複数のシステム特徴付けを平均化することができる。これはおそらく、ある範囲のサンプル形式に亘る最終的な応用例（たとえばＳｉ上のＳｉＯ_２）と同様の表面構造を有する加工品を用いることによって、行なわれる。

In various embodiments, multiple system characterizations can be averaged. This is probably done by using a workpiece with a surface structure similar to the final application (eg SiO ₂ on Si) over a range of sample formats.

  Although most of the foregoing description and simulation has focused on thin film surface structures, the analysis is applicable to other types of complex surface structures. In the following, it will be shown how the scanning interferometry data can be analyzed to reveal surface structures that are smaller than the optical resolution of the scanning interferometer microscope. The optical resolution is ultimately limited by the wavelength of the light source and the NA of the collection optics.

  FIG. 16a shows a height profile determined from actual scanning interference spectroscopy data. In this case, with a grating of 2400 lines / mm (lpmm), the peak-valley (PV) modulation depth is 120 nm, and the light source used has a nominal wavelength of 500 nm. The upper profile of FIG. 16a shows the height profile determined using conventional FDA analysis. The PV modulation depth shown by the conventional analysis is only about 10 nm, and the actual modulation depth is extremely underestimated. The reason for this inaccuracy is that the grating features exist at the limits of the optical resolution of the 500 nm instrument. This is true even if the pixel resolution of the camera in the device is high enough to accurately resolve the grid.

  One way to consider this effect is that the scanning interferometry signal for the first camera pixel that generally corresponds to the first surface location may also include the effects from adjacent surface locations, and this occurs because The surface feature of the further surface location is when it is sufficiently sharp with respect to the light wavelength to diffract the light into the first pixel. These surface height features from adjacent surface locations impair conventional analysis of the scanning interferometry signal corresponding to the first surface location.

  At the same time, however, this means that the scanning interferometry signal corresponding to the first surface location contains information about nearby complex surface features. This is shown in FIG. In the figure, scanning interference spectral signals from pixels corresponding to various locations around this step height feature are shown. In the signal in (a), the step height is on the right side of the pixel and is higher. In the signal in (b), the step passes directly through the pixel. In the signal in (c), the step is on the left side of the pixel and is lower. One of the discriminating characteristics immediately apparent in the signal is that the fringe contrast in (b) is lower than in (a) and (c). For example, if the step height is equal to a quarter of the wavelength and the pixel location corresponds exactly to the position of the step height, the fringe contrast in (b) is completely extinguished. The reason is that the interference from the two sides of the step cancels each other exactly. There is a lot of information in the signals shown in (a) and (c). For example, FIG. 18 shows nonlinear distortion in the frequency domain phase spectrum for signals (a) and (c) of FIG. 17, respectively. This distortion is due to the nearby step height. These spectra are shown in FIG. 18 as (a) and (b), respectively. In the absence of a step height, the frequency domain phase spectrum is linear. That is, the non-linear feature in the frequency domain phase spectrum for the pixel corresponding to the surface location adjacent to the step height includes information about the step height.

  In order to more accurately measure the surface profile of the test surface when there are surface features that cannot be sufficiently resolved in this way, the library search technique described above for thin films can be used. For example, for a test surface with a grid that cannot be sufficiently resolved, a series of model FDA spectra are generated for different values of PV modulation depth and offset position. As with the thin film example, the surface height for the model spectrum remains fixed. The analysis is then continued as in the thin film example described above, except that the model spectrum is not parameterized by the thin film thickness but by the modulation depth and offset position. Matching can then be determined using a comparison between the identification characteristics of the FDA spectrum for the actual test surface and the identification characteristics of the various model spectra. Based on the matching, the distortion in the actual FDA spectrum for each pixel caused by the presence of the grid is removed so that the surface height for each pixel can be determined using conventional processing. Figures 16b and 19b show the results of such an analysis using the same merit function as described above for the thin film.

  FIG. 16b shows the height profile determined using library search analysis for the 2400 line / mm grid described above with reference to FIG. 16a. The same data was used in FIGS. 16a and 16b. However, by library search analysis, the PV modulation depth for the grating was determined to be 100 nm. This is much closer to the actual modulation depth of 120 nm than the 10 nm result determined by the conventional FDA process of FIG. 16a. Shown in FIGS. 19a and 19b is a similar analysis for a simulation with a discrete step height and assuming a nominal 500 nm light source. FIG. 19a shows a comparison between a height profile (solid line) determined using conventional FDA processing and an actual height profile (broken line) for the simulation. FIG. 19b shows a comparison between the height profile (solid line) determined using the library search method and the actual height profile (broken line) for the simulation. The parameters for the model spectrum in the library search were location and step height amplitude. As illustrated, the library search analysis improves the lateral resolution from about 0.5 microns to about 0.3 microns.

  In the detailed analysis described above, a comparison between information in actual data and information corresponding to various models was made in the frequency domain. In other embodiments, the comparison can be made in the scan coordinate domain. For example, a change in the absolute position of the fringe contrast envelope generally indicates a change in surface height at the first surface location corresponding to the signal of interest, but the shape of the signal (regardless of its absolute position) is complex. Information on the surface structure at the first surface location, such as the underlying layer at the first surface location and / or the adjacent location.

  One simple case is to consider the amplitude of the fringe contrast envelope itself. For example, if the thickness of the thin film is very small compared to the wavelength range generated by the light source, the interference effect generated by the thin film becomes independent of the wavelength. In this case, the amplitude of the fringe contrast envelope is directly modulated by the thin film thickness. Thus, in general, the fringe contrast amplitude can be compared with the fringe contrast amplitude for models corresponding to various thin film thicknesses to identify a match for a particular thin film thickness (from the interferometer itself). Systematic impacts of

  Another simple case is to look at the relative spacing of the fringe zero crossing below the fringe contrast envelope. When a simple surface structure is illuminated with symmetric frequency dispersion, the relative spacing between the various zero crossings should nominally be the same. Thus, the change in relative spacing indicates a complex surface structure (when considering systematic effects from the interferometer itself). The change in relative spacing can also be compared to models for various complex surface structures to identify a match for a particular surface structure.

  In other cases, look at the correlation between the scan domain signal and the scan domain signal corresponding to the various models of the test surface. Matching generally corresponds to the correlation with the highest peak value. The highest peak value indicates a model in which the shape of the scan domain signal is most similar to the shape of the actual signal. Such an analysis is generally independent of the surface height. The reason is that the difference between the actual sample surface height and the surface height of each model only shifts the peak location in the correlation function, and generally does not affect the peak value itself. is there. On the other hand, once an accurate model is identified, the peak location in the correlation function of the accurate model provides the surface height for the test sample without the need for further analysis (eg, conventional FDA).

  Similar to analysis in the spatial frequency domain, analysis in the scan coordinate domain can be used for many different types of complex surfaces. Such surfaces include not only thin films, but also other complex surface structures such as the surface height features that cannot be fully resolved as described above.

The scanning coordinate library search analysis with correlation between the signal for the test sample and the corresponding signal for the various models of the test sample will now be described in detail.
This approach excludes any assumptions about the interference pattern except as described below. That is, in a data set corresponding to surface locations having the same complex surface characteristics, all pixels contain the same basic localized interference pattern, which is simply a location for each pixel. Is only shifted (or rescaled). It doesn't matter what the signal really looks like, whether it is a Gaussian envelope or a linear phase behavior in the frequency domain. The idea is to generate a sample signal or template that represents this local interference pattern for various models of complex surface structures for the test object, and for each pixel the local interference pattern is the actual local pattern. Find a model that best matches the shape of the interference pattern and, for that model, determine the scan position in the data set that gives the best match between the interference pattern template and the observation signal that gives the surface height. To find. Several techniques are available for pattern matching. One approach is to mathematically correlate each template with the data. For each model, two profiles are reproduced by using a complex (ie real part plus imaginary part) template function. One is closely related to the signal envelope and the other is related to the phase of the underlying carrier signal.

  In one embodiment, for example, the analysis for each pixel includes: (1) select a test template from a library of tunable parameters, eg calculated or recorded for a specific value of film thickness, (2) using the selected test template and correlation technique Find the local surface height (examples will be described later), (3) record the peak merit function value for the test template selected based on the correlation technique, (4) all of the templates in the library or Repeat steps 1-3 for the subset, (5) determine which test template gives the best match (= highest peak merit function value), (6) tunable for best matched template Record values for various parameters (eg thin film thickness), (7) Data Call the height values that gave peak matching position in the race.

  Next, a suitable correlation technique based on complex correlation will be described. A template interference pattern is generated for each model of the test surface.

ここで、屈折率ｊは、テンプレート・パターンに対する特定のモデルを示す。関数ｍ^ｊ _ｔｅｍｐ（ζ）およびφ^ｊ _ｔｅｍｐ（ζ）によって、複雑な表面構造が特徴付けられる。しかしこれらの関数は、信号に対応する箇所における表面高さには無関係である。表面高さはゼロに設定される。好ましい実施形態においては、関数ｍ^ｊ _ｔｅｍｐ（ζ）およびφ^ｊ _ｔｅｍｐ（ζ）によって、干渉計からの系統的な影響も明らかになる。そして、テンプレート・パターンに対する複素数表現を用いる。

Here, the refractive index j indicates a specific model for the template pattern. The functions m ^j _temp (ζ) and φ ^j _temp (ζ) characterize complex surface structures. However, these functions are independent of the surface height at the location corresponding to the signal. The surface height is set to zero. In the preferred embodiment, the functions m ^j _temp (ζ) and φ ^j _temp (ζ) also reveal the systematic effects from the interferometer. A complex number representation for the template pattern is used.

さらにウィンドウ関数を用いて、複素テンプレート関数の特定の部分を選択する。

Further, a specific part of the complex template function is selected using a window function.

たとえば、適切なウィンドウは、以下のようであってもよい。

For example, a suitable window may be as follows:

ここで、ウィンドウ幅Δζは、手入力で設定することができる。

Here, the window width Δζ can be set manually.

Since there is an interference pattern template I ^ ^j _pat , it can be used to compare with the actual data set. In preparing this, it is convenient to generate a complex signal I ^ _ex starting from the actual experimental data set.

この信号のフーリエ変換は、以下の通りである。

The Fourier transform of this signal is as follows.

ここで、

here,

そして、スペクトルの正の周波数部分から部分的なスペクトルを構築する。

A partial spectrum is then constructed from the positive frequency portion of the spectrum.

そして逆変換は、以下の通りである。

The inverse transformation is as follows.

ここで、この複素関数Ｉ＾_ｅｘの実部は、当初の実験データＩ_ｅｘである。さらに、位相および包絡線は、簡単な操作によって分離することができる。たとえば、複素関数Ｉ＾_ｅｘの振幅を用いて、信号強さＡＣ_ｅｘ（ｘ）および包絡線ｍ_ｅｘの積を利用することができる。

Here, the real part of this complex function I ^ _ex is the original experimental data _Iex . Furthermore, the phase and envelope can be separated by a simple operation. For example, the product of the signal strength AC _ex (x) and the envelope m _ex can be utilized using the amplitude of the complex function I ^ _ex .

技術の基礎となる理論にしたがって、ｍ_ｅｘの少なくとも有意義な部分が、正確なモデルに対するｍ^ｊ _ｔｅｍｐと同じ一般的な形状を有し、唯一の違いは、線形のオフセットｈ_ｅｘおよびスケーリング因子ＡＣ_ｅｘ（ｘ）であると予想される。また実験および干渉パターン・テンプレート位相オフセットφ_ｅｘ、φ^ｊ _ｐａｔ間の差が、正確なモデルに対する高さｈ_ｅｘに直線的に比例していると予想される。

According to the underlying theory of the technology, at least a significant part of m _ex has the same general shape as m ^j _temp for the exact model, the only difference being the linear offset h _ex and the scaling factor AC _ex Expected to be (x). It is also expected that the difference between the experimental and interference pattern template phase offsets φ _ex , φ ^j _pat is linearly proportional to the height h _ex for the exact model.

The challenge for the time being is to place a specific signal pattern represented by the interference pattern template I ^ ^j _pat in the experimental data I ^ _ex to determine how good a matching exists for each of the different models j It is to be. In the following, it is described that the refractive index j is omitted and the matching analysis proceeds for each model.

The first step is to find the scan position ζ _best where the shapes of the envelopes m _ex , m _pat and φ _ex , φ _pat are best matched. A feasible approach is a merit function based on a normalized correlation between the interference pattern template and the signal in the segment of the scan defined by the window w.

ここで、

here,

は、複素相関関数であり、および

Is the complex correlation function, and

は、メリット関数Πを信号強さとは無関係にする規格化である。テンプレートの複素共役Ｉ＾^＊ _ｐａｔを用いることによって、同位相の線形の位相項Ｋ_０ζが打ち消され、φ_ｅｘ、φ_ｐｌａｔがマッチングするときにΠが最大になる。相関関係の絶対値によって、残りのわずかな複素位相も取り除かれる。

Is a standardization that makes the merit function Π independent of signal strength. By using the complex conjugate I ^ ^* _pat of the template, the linear phase term K ₀ ζ of the same phase is canceled, and Π is maximized when φ _ex and φ _plate match. The absolute value of the correlation also removes the remaining few complex phases.

  Carefully add a minimum value to the denominator so that ζ (ζ) does not yield false high values or encounter singularities at low signal levels. For example:

ここで、ｍａｘ（）関数は、全走査長さζにおける信号強さ｜Ｉ＾_ｅｘ｜の最大値を戻す。ＭｉｎＤｅｎｏｍは、メリット関数検索において有効であると考えられる最小の相対信号強さである。ＭｉｎＤｅｎｏｍの値は、５％または他の何らかの小さい値においてハード・コードすることもできるし、調整可能なパラメータのままにしておくこともできる。

Here, the max () function returns the maximum value of the signal strength | I ^ _ex | for the entire scanning length ζ. MinDenom is the minimum relative signal strength that is considered to be effective in the merit function search. The value of MinDenom can be hard coded at 5% or some other small value, or it can be left as an adjustable parameter.

  The correlation integral I ^ can also be performed in the frequency domain using the correlation theorem.

ここで、Ｉは以下の関係を用いた。

Here, I used the following relationship.

ここで、

here,

Πを通して検索してピーク値を見出すことによって、最良のマッチング位置ζ_ｂｅｓｔが得られる。Πの値は、マッチングの品質の測定値であり、ゼロから１の範囲である。なお１は、完全なマッチングに対応する。メリット関数のピーク値を、種々のモデルのそれぞれに対して計算して、どのモデルが最良のマッチングであるかを決定する。そしてそのモデルに対する最良のマッチング位置ζ_ｂｅｓｔによって、表面高さが与えられる。

The best matching position ζ _best is obtained by searching through Π to find the peak value. The value of Π is a measure of the quality of matching and ranges from zero to one. Note that 1 corresponds to perfect matching. The peak value of the merit function is calculated for each of the various models to determine which model is the best match. The surface height is then given by the _best matching position ζ _best for that model.

Examples of the present technology are illustrated in FIGS. FIG. 20 shows an actual scanning interference spectral signal of a base Si substrate without a thin film. Shown in FIGS. 21 and 22 are interference template patterns for a bare Si substrate and a thin film structure with 1 micron SiO ₂ formed on Si, respectively. Shown in FIGS. 23 and 24 are merit functions as a function of scan position for the template functions of FIGS. 21 and 22, respectively. The merit function shows that the matching of the interference template pattern to the bare substrate (peak value 0.92) is much better than the matching to the thin film template pattern (peak value 0.76). . Therefore, the merit function indicates that the test sample is a bare substrate. Furthermore, the position of the peak in the merit function for the correct template pattern gives the relative surface height position for the test sample.

  The methods and systems described above can be particularly useful in semiconductor applications. Further embodiments of the present invention include applying any of the measurement techniques described above to address any of the semiconductor applications described below, and a system that performs both the measurement techniques and the semiconductor applications.

  Currently, the quantitative measurement of surface topography is of considerable interest in the semiconductor industry. Due to the small size of typical chip features, the equipment used to make these measurements typically must have a high spatial resolution both parallel and perpendicular to the chip surface. Engineers and scientists use surface topography measurement systems to control processes and detect defects that occur during manufacturing, particularly defects that result from processes such as etching, polishing, cleaning, and patterning.

  In order for process control and defect detection to be particularly useful, surface topography measurement systems have a minimum horizontal resolution that is comparable to the horizontal size of typical surface features and the vertical resolution is acceptable. The surface step height must be comparable. This usually requires that the lateral resolution is less than a micron and the vertical resolution is less than 1 nanometer. Such a system also makes its measurements without touching the chip surface, and without otherwise exerting potentially damaging forces on the surface to avoid surface modification or defect introduction. It is preferable. In addition, it is well known that the effects of many processes used in chip fabrication are strongly dependent on local factors such as pattern density and edge approximation, so that surface topography measurement systems can achieve high measurement throughput, It is also important to have the ability to sample densely over a large area in a region that may contain one or many surface features.

  It is becoming common among chip manufacturers to use the so-called “dual damascene copper” process to create electrical interconnects between various parts of a chip. This is an example of a process that can be effectively characterized using a suitable surface topography system. The dual damascene process can be considered to have five parts. (1) Interlayer dielectric (ILD) deposition in which a layer of dielectric material (eg, polymer or glass) is deposited on the surface of a wafer (containing a plurality of separate chips). (2) Chemical mechanical polishing (CMP) that polishes the dielectric layer to form a smooth surface suitable for precision photolithography. (3) Lithographic patterning and reaction in which a complex network is formed that includes narrow trenches that run parallel to the wafer surface and small vias that run from the bottom of the trench to the lower (predefined) electrically conductive layer A combination of reactive ion etching steps. (4) A combination of metal deposition steps resulting in trenches and vias that are overfilled with copper. (5) A final chemical mechanical polishing (CMP) step that removes excess copper and leaves a network of copper filled trenches (and / or vias) surrounded by dielectric material.

  Typically, the copper thickness (ie, trench depth) in the trench region and the surrounding dielectric thickness is in the range of 0.2 to 0.5 microns. The resulting trench width may be in the range of 100 to 100,000 nanometers. Also, the copper regions in each chip may form a regular pattern, such as an array of parallel lines, in certain regions and no obvious pattern in other regions. Similarly, in some areas, the surface may be densely covered with copper areas, and in other areas the copper areas may be sparse. It is important to understand the following: That is, the polishing rate, and hence the thickness of the remaining copper (and dielectric) after polishing, as well as the polishing conditions (eg, pad pressure and polishing slurry composition) as well as local details of the copper and surrounding dielectric regions. Depends on placement (ie, orientation, proximity, shape) in a strong and complex manner.

  This “position dependent polishing rate” is known to produce variable surface topography at many lateral length scales. For example, a chip located near the edge of the wafer on the assembly is polished more rapidly than a chip located near the center, so that the copper area formed is thinner than desired near the edge and desired at the center. May mean thicker. This is an example of "wafer scale" process non-uniformity, i.e. occurs on a length scale comparable to the wafer diameter. Further, it is known that the polishing rate in the region where the high-density copper trench is present is higher than that in the nearby region where the copper line density is low. This causes a phenomenon known as “CMP induced corrosion” in the high copper density region. This is an example of "chip scale" process non-uniformity, i.e. occurs on a length scale comparable to the length dimension of a single chip (which may be much smaller). Within a single copper-filled trench region (which tends to polish at a higher rate than the surrounding dielectric material), another type of chip-scale non-uniformity (known as “desching”) is present. Occur. It is dangerous if the trench is larger than a few microns in width dishing, and the electrical resistance of the affected line may later become excessive. This causes a chip failure.

  CMP-induced wafer and chip-scale process non-uniformity is inherently difficult to predict. They also change over time as the state in the CMP processing system evolves. To ensure that process conditions are effectively monitored and adjusted appropriately to ensure that any non-uniformity remains within acceptable limits, the process engineer must perform many non-contact surface topography measurements on the chip and It is important to do it frequently in a wide area. This is possible using the embodiment of the interference spectroscopy technique described above.

  Any of the computer analysis methods described above can be implemented in hardware or software or a combination of both. The method can be implemented in a computer program using standard programming techniques in accordance with the methods and figures described herein. Program code is applied to input data to execute the functions described herein and generate output information. The output information is applied to one or more output devices such as a display monitor. Each program may be implemented in a high level procedural or object oriented programming language to communicate with a computer system. However, the program can be implemented in assembly or machine language as required. In any case, the language can be a compiled language or an interpreted language. Furthermore, the program can be executed on a dedicated integrated circuit preprogrammed for that purpose.

  Each such computer program is preferably stored on a storage medium or device (eg, ROM or magnetic disk) readable by a general or special purpose programmable computer such that the storage medium or device is a computer. Configure and operate the computer to perform the procedures described herein. The computer program can also reside in cache or main memory during program execution. The analysis method can be realized as a computer-readable storage medium and can be configured by a computer program. By configuring the storage medium as such, the functions described herein are performed by operating the computer in a specific and predetermined manner.

  While many embodiments of the invention have been described, it will be appreciated that various changes may be made without departing from the spirit and scope of the invention.

It is a flow chart of interferometry. 2 is a flow chart showing a modification of the interferometry method of FIG. It is the schematic of a linic type scanning interferometer. It is the schematic of a Mirau type scanning interferometer. 2 is a diagram illustrating illumination of a test sample through an objective lens. It is a figure which shows the theoretical Fourier amplitude spectrum with respect to the scanning interference spectroscopy data in two limits. It is a figure which shows two surface types with / without a thin film. It shows a merit function search procedure for a simulation of a SiO ₂ film on a Si substrate with a thin film thickness of 0. Film thickness is a diagram showing a merit function search procedure for simulation of a SiO ₂ film on the Si substrate of 50nm. Film thickness is a diagram showing a merit function search procedure for simulation of a SiO ₂ film on the Si substrate of 100 nm. Film thickness is a diagram showing a merit function search procedure for simulation of a SiO ₂ film on the Si substrate of 300 nm. Film thickness is a diagram showing a merit function search procedure for simulation of a SiO ₂ film on the Si substrate of 600 nm. Film thickness is a diagram showing a merit function search procedure for simulation of a SiO ₂ film on 1200nm of the Si substrate. FIG. 6 shows the surface and substrate profile determined for a simulation of a SiO ₂ thin film on Si with the top surface always zero and the film thickness varying uniformly in increments of 10 nm per pixel from 0 to 1500 nm. FIG. 15 is a diagram showing the surface and substrate profiles determined for the simulation of FIG. 14 and the same simulation except that random noise was added (2-bit root mean square from an average value of 128 intensity bits). A library search method described herein for a surface height profile determined using conventional FDA analysis (FIG. 16a) and a grid of 2400 lines per mm with an actual peak / valley modulation depth of 120 nm ( FIG. 16b) is a diagram. FIG. 6 is a diagram illustrating distortion caused by a step height that cannot be sufficiently resolved when scanning interference signals for pixels corresponding to various surface locations near the step height. FIG. 18 shows non-linear distortion in the frequency domain phase spectrum for pixels corresponding to surface locations for left (FIG. 18a) and right (FIG. 18b) step heights that cannot be sufficiently resolved in FIG. FIG. 19 shows a surface height profile determined using conventional FDA analysis (FIG. 19a) and using the library search method described herein (FIG. 19b) for step heights that cannot be fully resolved. . It is a figure which shows the actual scanning interference spectroscopy signal of the Si substrate used as a base without a thin film. It is a figure which shows the interference template pattern with respect to a bare Si substrate. FIG. 6 shows an interference template pattern for a thin film structure with 1 micron SiO ₂ formed on Si. It is a figure which shows the merit function as a function of the scanning position with respect to the template function in FIG. It is a figure which shows the merit function as a function of the scanning position with respect to the template function in FIG.

Claims (77)

A method,
Comparing information derivable from a scanning interferometry signal for a first surface location of the test object with information corresponding to a plurality of models of the test object;
The plurality of models are parameterized by a set of characteristics for a test object;
The information corresponding to the plurality of models includes information on at least one amplitude component of a conversion of the scanning interference spectral signal corresponding to each model of the test object.   The method of claim 1, further comprising determining an exact characteristic for a test object based on the comparison.   The method of claim 1, further comprising determining a relative surface height for the first surface location based on the comparison.   Determining the relative surface height is based on the comparison to determine which model corresponds to the correct one of the properties for the test object, the model corresponding to the exact property 4. The method of claim 3, comprising calculating a relative surface height using.   5. The method of claim 4, wherein using a model corresponding to the exact characteristic includes correcting data from a scanning interferometry signal to reduce the effects resulting from the accurate characteristic.   Correcting the data includes extracting the phase effect resulting from the accurate characteristic from the phase component of the scanning interference spectral signal converted to the test object, and using a model corresponding to the accurate characteristic is 6. The method of claim 5, further comprising calculating a relative surface height from the phase component of the transform after removing the phase effect resulting from the exact characteristic.   The method of claim 1, further comprising comparing information derivable from scanning interferometry signals for additional surface locations with information corresponding to the plurality of models.   8. The method of claim 7, further comprising determining a surface height profile for a test object based on the comparison.   The comparison of claim 1, wherein the comparing includes calculating one or more merit functions indicating similarity between information derivable from scanning interferometry signals and information corresponding to each model. Method.   The method of claim 1, wherein the comparing comprises fitting information derivable from scanning interferometry signals into a representation for information corresponding to the model.   The method according to claim 1, wherein the information derivable from the scanning interference spectral signal includes information on at least one amplitude component of the conversion of the scanning interference spectral signal with respect to the test object.   The method of claim 11, wherein the comparing includes comparing a relative strength of at least one amplitude component for a test object with a relative strength of at least one amplitude component for each model.   The method according to claim 1, wherein the information corresponding to the plurality of models is a function of coordinates with respect to a transformation.   The method according to claim 13, wherein the information corresponding to the plurality of models includes a transformed amplitude profile for each model.   The method according to claim 14, wherein the comparing includes comparing the amplitude profile of the transform of the scanning interferometry signal with respect to the test object and each amplitude profile with respect to the plurality of models.   The method of claim 14, wherein the comparing further comprises comparing information in the phase profile of the transform of the scanning interferometry signal for the test object with information in the phase profile of the transform for each model. .   The method of claim 16, wherein the information in the phase profile includes a phase profile non-linearity with respect to transformed coordinates.   The method of claim 16, wherein the information in the phase profile includes a test object and a phase shift value for each model.   The method of claim 1, wherein the plurality of models correspond to a fixed surface height at a first location of a test object.   The method of claim 1, wherein the set of characteristics includes a set of values for at least one physical parameter of the test object.   21. The method of claim 20, wherein the test object includes a thin film layer having a thickness, and the physical parameter is a thickness of the thin film at a first location.   The method of claim 1, wherein the set of characteristics includes a set of characteristics of a test object at a second surface location that is different from the first surface location.   23. The method of claim 22, wherein the test object includes a structure at a second surface location that diffracts light and affects a scanning interferometry signal for the first surface location.   23. The method of claim 22, wherein the set of characteristics at the second surface location includes a substitution of an amplitude for a step height at the second location and a position relative to the second location.   23. The method of claim 22, wherein the set of characteristics at the second surface location includes a permutation of a modulation depth relative to a grating and an offset position of the grating, the grating extending across the second location.   The method of claim 1, wherein the set of properties is a set of surface materials for a test object.   The method of claim 1, wherein the set of characteristics is a set of surface layer configurations for a test object.   The method according to claim 1, wherein the transformation is a Fourier transformation.   The scanning interferometry signal is generated by a scanning interferometry system, and the comparing includes defining systematic effects on the scanning interferometry signal resulting from the scanning interferometry system. the method of.   30. The method of claim 29, further comprising calibrating systematic effects of the scanning interferometry system with other test objects having known characteristics.   The test light emerging from the test object is imaged and interfered with the reference light on the detector, and the optical path length difference from the common light source to the detector between the interference part of the test light and the interference part of the reference light And the test light and the reference light are extracted from a common light source, and the scanning interference spectral signal is measured by the detector when the optical path length difference is changed. The method of claim 1, wherein the method corresponds to an interference intensity.   32. The method of claim 31, further comprising generating the scanning interferometry signal.   32. The method of claim 31, wherein the test light and the reference light have a spectral bandwidth greater than 5% of a center frequency for the test light and reference light.   32. The method of claim 31, wherein the common light source has a spectral coherence length and the optical path length difference varies over a range longer than the spectral coherence length to generate the scanning interferometry signal. .   32. The numerical aperture for the test light is defined as a value greater than 0.8 by optics used to send the test light onto a test object and image it on a detector. Method.   The method of claim 32, wherein the common light source is a spatially extended light source. A device,
A computer readable medium having a program for causing a processor in a computer to compare information derivable from a scanning interferometry signal for a first surface location of a test object with information corresponding to a plurality of models of the test object. Prepared,
The plurality of models are parameterized by a set of characteristics for a test object;
The information corresponding to the plurality of models includes information on at least one amplitude component of a transform of the scanning interference spectral signal corresponding to one of the models of the test object. A device,
A scanning interferometry system configured to generate a scanning interferometry signal;
An electronic processor connected to the scanning interferometry system and receiving the scanning interferometry signal, information derivable from the scanning interferometry signal for the first surface location of the test object, and a plurality of test objects An electronic processor programmed to compare information corresponding to the model of
The plurality of models are parameterized by a set of characteristics for a test object;
The information corresponding to the plurality of models includes information on at least one amplitude component of a transform of the scanning interference spectral signal corresponding to one of the models of the test object. A method,
Comparing information derivable from the scanning interferometry signal for the first surface location of the test object with information corresponding to a plurality of models of the test object;
The plurality of models correspond to a fixed surface height at a first location of the test object and are parameterized by a series of characteristics for the test object that are different from the fixed surface height. ,Method.   40. The method of claim 39, further comprising determining an exact characteristic for a test object based on the comparison.   40. The method of claim 39, further comprising determining a relative surface height for the first surface location based on the comparison.   Determining the relative surface height is based on the comparison to determine which model corresponds to the correct one of the properties for the test object, the model corresponding to the exact property 42. The method of claim 41, comprising calculating a relative surface height using.   43. The method of claim 42, wherein using a model corresponding to the exact characteristic includes correcting data from the scanning interferometry signal to reduce effects resulting from the accurate characteristic.   Correcting the data includes extracting a phase effect resulting from the accurate characteristic from a phase component corresponding to the conversion of a scanning interference spectroscopic signal for a test object, and using a model corresponding to the accurate characteristic. 44. The method of claim 43, further comprising calculating a relative surface height from the phase component of the transform after removing the phase effect resulting from the exact characteristic.   Calculating the relative surface height using the model corresponding to the exact characteristic is in the correlation function used to compare the information for the test object with the information for the model corresponding to the exact characteristic. 43. The method of claim 42, comprising determining a peak location.   40. The method of claim 39, further comprising comparing information derivable from scanning interferometry signals for additional surface locations with information corresponding to the plurality of models.   47. The method of claim 46, further comprising determining a surface height profile for the test object based on the comparison.   40. The comparing of claim 39, wherein the comparing includes calculating one or more merit functions that indicate similarity between information derivable from scanning interferometry signals and information corresponding to each model. Method.   40. The method of claim 39, wherein the comparing comprises fitting information derivable from a scanning interferometry signal to a representation for information corresponding to a model.   40. The method of claim 39, wherein the information that can be derived from the scanning interferometry signal and that is being compared is a number.   40. The method of claim 39, wherein the information derivable from the scanning interferometry signal and the information being compared is a function.   52. The method of claim 51, wherein the function is a function of spatial frequency.   52. The method of claim 51, wherein the function is a function of scan position.   40. The method of claim 39, wherein the information for the test object is obtained from transforming a scanning interferometry signal for the test object into a spatial frequency domain.   55. The method of claim 54, wherein the transform is a Fourier transform.   55. The method of claim 54, wherein the information for the test object includes information about an amplitude profile for the transform.   55. The method of claim 54, wherein the information for the test object includes information about a phase profile for the transform.   40. The method of claim 39, wherein the information for the test object relates to a shape of a scanning interferometry signal at a first location of the test object.   59. The method of claim 58, wherein the information for the test object relates to a fringe contrast amplitude in the shape of a scanning interferometry signal.   59. The method of claim 58, wherein the information for the test object relates to a relative spacing between zero crossings in the shape of a scanning interferometry signal.   59. The method of claim 58, wherein the information for the test object is expressed as a function of scanning position, and the function is obtained from the shape of a scanning interferometry signal.   40. The method of claim 39, wherein the comparing comprises calculating a correlation function between information for the test object and information for each model.   64. The method of claim 62, wherein the correlation function is a complex function.   64. The method of claim 62, wherein the comparing further comprises determining one or more peak values in each correlation function.   65. The method of claim 64, further comprising determining an exact characteristic for the test object based on a parameterization of the model corresponding to the maximum peak value.   65. The method of claim 64, further comprising determining a relative surface height at a first surface location of the test object based on coordinates for at least one peak value in the correlation function.   40. The method of claim 39, wherein a scanning interferometry signal is generated by a scanning interferometry system and the comparing comprises defining a systematic effect on a scanning interferometry signal resulting from the scanning interferometry system. .   68. The method of claim 67, further comprising calibrating systematic effects of the scanning interferometry system with other test objects of known characteristics. A device,
A computer-readable medium having a program for causing a processor in a computer to compare information derivable from a scanning interferometry signal with respect to a first surface portion of the test object and information corresponding to a plurality of models of the test object With
The plurality of models corresponds to a fixed surface height at a first location of the test object and is parameterized by a set of characteristics for the test object that are different from the fixed surface height; apparatus. A device,
A scanning interferometry system configured to generate a scanning interferometry signal;
An electronic processor connected to the scanning interferometry system and receiving a scanning interferometry signal, information derivable from the scanning interferometry signal for the first surface location of the test object, and a plurality of models of the test object An electronic processor programmed to compare information corresponding to
The apparatus wherein the plurality of models are parameterized by a series of characteristics for a test object that correspond to a fixed surface height at a first location of the test object and that are different from the fixed surface height. A method,
Comparing information derivable from the scanning interferometry signal for the first surface location of the test object with information corresponding to a plurality of models of the test object;
The plurality of models are parameterized by a set of characteristics for a test object;
The comparing includes defining a systematic effect on the scanning interferometry signal resulting from the scanning interferometry system used to generate the scanning interferometry signal.   72. The method of claim 71, wherein the systematic influence includes information about dispersion in phase change upon reflection from a component of the scanning interferometry system.   72. The method of claim 71, further comprising comparing information derivable from scanning interferometry signals for additional surface locations with information corresponding to the plurality of models.   74. The method of claim 73, wherein the systematic influence is analyzed for a plurality of surface locations.   72. The method of claim 71, further comprising calibrating systematic effects of the scanning interferometry system with other test objects of known characteristics. A device,
A computer readable medium having a program for causing a processor in a computer to compare information derivable from a scanning interferometry signal for a first surface location of a test object with information corresponding to a plurality of models of the test object. Prepared,
The plurality of models are parameterized by a set of characteristics for a test object;
The program further causes the processor to define systematic effects on a scanning interferometry signal that results from a scanning interferometry system used to generate a scanning interferometry signal during the comparison. A device,
A scanning interferometry system configured to generate a scanning interferometry signal;
An electronic processor connected to the scanning interferometry system for receiving the scanning interferometry signal, information derivable from the scanning interferometry signal for the first surface location of the test object, and a plurality of the test object An electronic processor programmed to compare information corresponding to the model;
The plurality of models are parameterized by a set of characteristics for a test object;
The electronic processor is further programmed to account for systematic effects on the scanning interferometry signal that results from the scanning interferometry system used to generate the scanning interferometry signal during the comparison.

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