Patent Publication Number: US-2003222215-A1

Title: Method for objective and accurate thickness measurement of thin films on a microscopic scale

Description:
BACKGROUND OF THE INVENTION  
       [0001] 1. Field of the Invention  
       [0002] The present invention relates to measurement techniques in which the thickness of thin films, in the range of nanometers down to atomic dimensions, have to be determined. In particular, the present invention relates to measurement techniques requiring the preparation of thin samples to obtain measurement data by radiation of small wavelengths, such as electrons, passing through the sample.  
       [0003] 2. Description of the Related Art  
       [0004] The deposition of thin films on any type of substrate has become one of the most important technologies of surface modification. The development and the production of a huge number of products requires the deposition of various coating materials and functional coatings, such as tribological, hard, high-temperature, conductive and dielectric, optical, biotechnological and decorative coatings, with a precisely adjusted thickness on various surface topologies. Since the final performance of a product may significantly be determined by the quality of the deposited thin film, precise control during manufacturing of the products is essential.  
       [0005] Furthermore, modem deposition techniques require great efforts in terms of energy and equipment so that any failure in producing a thin film of the required quality remarkably contributes to the overall cost of the product. An illustrative example in this respect is the fabrication of modem integrated circuits, wherein at various manufacturing stages, material layers have to be deposited with different composition and layer thickness on differently patterned structures. Incorrectly depositing a material layer on a 200 mm diameter wafer—a commonly used substrate size in manufacturing sophisticated integrated circuits—at a final stage of the manufacturing process may thus lead to the loss of several tens of thousands of dollars.  
       [0006] Consequently, a plurality of measurement methods have been developed for high precision measurement of thin films. Most of these methods, however, are concerned with measurements of the thickness, even down to a few atomic layers, wherein the thin film is coated on a substantially planar surface. These well-established methods are not very effective when the film whose thickness is to be measured is provided on non-planar surfaces exhibiting a curvature on the sub-millimeter scale. Moreover, the problem often arises that one or more layers have to be examined, which are enclosed by other material layers that do not allow direct inspection of the layer of interest. In particular, when the layer of interest is provided with a thickness in the nanometer range on a structure including elements in the order of some hundreds of nanometers to a few micrometers, as for example in micro-electronics or micro-mechanics, the method of choice for determining is electron-microscopy. One method, preferentially used for structures in the nanometer range down to atomic dimensions, is transmission electron microscopy (TEM) that allows resolving the structures of interest with sufficient resolution to precisely determine a layer thickness of a thin film.  
       [0007] When recording a TEM image for the purpose of measuring a layer thickness, electron-optical conditions are chosen that allow one to treat the image as a very good approximation of a two-dimensional, parallel projection of the sample volume under consideration. One major issue in determining a layer thickness from such a TEM image is the loss of the three-dimensional information when generating this two-dimensional projection. This issue is even exacerbated when the thin film is provided on non-planar structures.  
       [0008] With reference to FIGS. 1 a - 1   d  and  2   a - 2   d , the problems involved in determining a layer thickness by means of TEM will be described in more detail. In FIG. 1 a , a schematic perspective view of a portion  100  of a structure (not shown) is depicted. It should be noted that the portion  100  may be enclosed by further materials that are not shown in FIG. 1 a , so that the portion  100  may only form a small part of the total structure. The portion  100  comprises a thin film  101  having a thickness  102  that is to be determined by the TEM measurement. The thin film  101  may be enclosed by a first material  103  and a second material  104  that, at least in some properties, differ from the material comprising the thin film  101 . In TEM measurements, a section has to be prepared, the thickness of which is sufficiently small to allow the charged particles passing therethrough. In order to accurately determine the layer thickness  102 , the section with a thickness of a few hundred nanometers or less is prepared substantially perpendicularly to a longitudinal direction, indicated as  105 . The section to be made, indicated by reference  106 , is shown in dashed lines.  
       [0009]FIG. 1 b  shows a schematic perspective view of the section  106  of FIG. 1 a  and of a corresponding TEM image  110  obtained by exposing the section  106  to an electron beam  107  that substantially perpendicularly impinges on the section  106 . Due to the different properties of the materials  103 ,  104  and the thin film  101 , the amount of electrons scattered by the various materials is different and a corresponding two-dimensional projection  108  of the section  106  is obtained on the image  110 . Thus, for an idealized thin film  101  having sharp boundaries to the neighboring materials  103  and  104 , the projection  108  of the thin film  101  will also exhibit sharp boundaries to the adjacent image portions, wherein a thickness  109  of the protection  108  precisely corresponds to the thickness  102  of the thin film  101 . Of course, any magnification caused by the electron lenses for generating the final image  110 , has to be taken into consideration when estimating the thickness  102  by means of the thickness  109  of the projection  108 . For the sake of simplicity, any magnification effects in FIG. 1 b  are not depicted.  
       [0010] According to the process illustrated in FIGS. 1 a  and  1   b , the thickness  102  of the thin film  101  may be precisely determined under the assumption that the section  106  may be prepared in an ideal manner as shown in FIGS. 1 a  and  1   b . In reality, however, preparing an appropriate section for TEM analysis requires a great deal of skill and experience of an operator, since generally a large sample, such as a semiconductor substrate, has to be cut precisely at the location where the structure to be measured is expected to be located and the cut substrate has to be thinned to the appropriate thickness in the hundred nanometer range and beyond so as to avoid undue scattering of electrons. Cutting slices of samples may be accomplished by mechanical milling and thinning the samples may be obtained by advanced ion beam milling and polishing methods. In any case, preparing the section  106  is quite complex and often produces a non-ideal section as will be explained with reference to Figures  1   c  and  1   d.    
       [0011] In FIG. 1 c , the section  106  that is to be prepared from the portion  100  is, owing to any inaccuracies during orienting the portion  100  in cutting and thinning, tilted with respect to a direction orthogonal to the longitudinal direction  105 , as indicated by an angle α.  
       [0012]FIG. 1 d  shows the section  106  with its surface oriented to the electron beam  107  in the same manner as depicted in FIG. 1 b . Consequently, the thickness of the thin film  101  appears to be larger, determined by the tilt angle α, and is now indicated as  102 ′. The electrons passing through the section  106  will encounter a varying degree of scattering along the thickness direction and will produce the projection  108  with a correspondingly enlarged thickness  109 ′. Accordingly, an operator inspecting the TEM image  110  will most likely predict a thickness for the thin film  101  that is inaccurate and thus strongly depends on the operator&#39;s skill and experience. Hence, determining a layer thickness of a thin film is extremely sensitive to variations in preparing the section and also significantly depends on the operator&#39;s skill of interpreting the TEM image.  
       [0013] This situation becomes even more exacerbated, when a thin film is coated on a structure including a curvature when the order of magnitude of the curvature is comparable to a thickness of the section. In order to more clearly demonstrate the problems with thin films provided on a curved structure, reference will now be made to FIGS. 2 a - 2   d.    
       [0014] In FIG. 2 a , a schematic cross-sectional view of a semiconductor structure  200  is shown. The structure  200  may comprise a substrate  220 , such as a silicon substrate, which may comprise one or more circuit elements (not shown) that in combination form an integrated circuit. A dielectric layer  221  is formed on the substrate  220  and may comprise, for example, silicon dioxide as is often used as an interlayer dielectric in integrated circuits. In the dielectric layer  221 , a via  222  is formed having dimensions in accordance with design requirements. For example, the via  222  may provide contact to any underlying circuit feature and may have a diameter of approximately 0.2 μm or even less, when sophisticated integrated circuits are considered. For the sake of convenience, a single contact region  223  is deposited and is meant to represent a contact portion of an underlying circuit feature. On the inner surfaces of the via  222 , a thin film  201  is formed having a thickness  202 . For example, the thin film  201  may represent a barrier diffusion layer comprised of, for example tantalum, titanium, titanium nitride, tantalum nitride, and the like, as is typically used in the fabrication of integrated circuits. Moreover, the via  222  is to be filled with an appropriate contact metal such as tungsten, aluminum, copper and the like. Depending on the type of integrated circuit, the via  222  may have an aspect ratio of 10 to 1 and, thus, deposition of the thin film  201  involves highly sophisticated deposition methods, wherein it is extremely important to provide the thickness profile of the thin film  201  with high precision according to design requirements. Usually, it is desired to provide the thin film  201  with a specific thickness, which may vary at the various locations in the via  222 , such as at the top region  225  and the bottom region  224 . In sophisticated integrated circuits with copper lines, the thin film layer  201  may prevent copper from diffusing into the neighboring materials, while at the same time the thin film  201  has to provide a sufficient conductivity to the underlying contact region  223  so as not to unduly degrade the performance of the complete copper plug. Thus, deposition of the thin film  201  has to be carried out within very tightly set limits. Therefore, a very accurate determination of the thickness  202  at the various locations of the via  222  is essential for appropriately adjusting deposition parameters. For the TEM analysis of the thin film  201 , a section  206  has to be prepared that includes the via  222 .  
       [0015]FIG. 2 b  shows a top view of the structure  200  as shown in FIG. 2 a . As is evident from FIG. 2 b , even if advanced sample preparation techniques are employed, a thickness  224  of the section  206  will contain a portion  225  of the thin film  201  having a curvature defining curved edge portions  226 .  
       [0016]FIG. 2 c  shows a schematic perspective view of the section  206 , wherein the curved edges  226  of the thin film  201  are visible. It should be noted, that the bottom portion  224  of the via  222  is formed on the substantially planar contact region  223  so that the bottom of the via  222  does not substantially comprise curved edges such as the edges  226  provided on the sidewalls of the via  222 .  
       [0017]FIG. 2 d  schematically shows, in an over-simplified manner, the arrangement used to obtain a TEM image of the thin film  201 . An electron source  230 , configured to provide an electron beam  207  with required characteristics to provide a TEM image  210 , is positioned to emit the electrons  207  onto the section  206 . As is evident from FIG. 2 d , although the thin film  201  has the thickness  202 , this thickness  202  does not translate into a thickness  209  of a two-dimensional projection  208  of the thin film  201 . Rather, the thickness  209  of the projection  208  represents the projection including the curvature of the thin film  201  and thus does not allow the precise determination of the actual thickness  202  on the basis of the TEM image  210 . Similar to the situation as described with reference to FIGS. 1 a - 1   d , the determination of the thickness  202  is strongly affected by the skills and experience of the corresponding operator. Moreover, the situation becomes even worse when the section  206  may not be prepared as an extremely thin sample, since then the contribution of the curvature to the entire thickness  209  of the projection  208  is increased. In particular, determining the thickness  202  at the sidewall compared to the thickness  202  at the bottom of the via  222  without a curved edge may thus yield quite different results, thereby erroneously indicating a significant non-uniformity obtained during the deposition process.  
       [0018] In view of the above-mentioned problems, it would be highly desirable to eliminate or at least reduce the influence of the quality of the section and an operator&#39;s skill and experience on the result of the TEM measurements.  
       SUMMARY OF THE INVENTION  
       [0019] Generally, the present invention is directed to a method and an apparatus in which loss of the three-dimensional information is, at least partially, compensated for by obtaining an intensity profile of a two-dimensional projection in an image generated by short wave length radiation, such as an electron beam, wherein structural characteristics, such as curved edges of thin film and/or a tilt angle in preparing the section, including the thin film of interest, are taken into account by analyzing the intensity profile on the basis of properties that are substantially independent from structural characteristics and tilt angles.  
       [0020] According to one illustrative embodiment of the present invention, a method of determining the thickness of a thin film comprises preparing a cross-sectional specimen of the film and irradiating the film with a radiation beam substantially perpendicularly to a thickness direction of the film so as to provide a digital image of the specimen. The method further includes extracting an intensity profile from the digital image, substantially parallel to the thickness direction, and analyzing the intensity profile of the digital image to determine the thickness of the film. In a further embodiment, the thin film is a curved thin film.  
       [0021] In a further illustrative embodiment of the present invention, a method of determining the thickness of a material layer formed in a substrate comprises preparing a section of the substrate, exposing a layer indicative of a layer thickness and obtaining a digital image of at least a portion of the section from radiation passing through the section. The method further includes extracting an intensity profile from the image substantially perpendicular to a thickness direction of the layer, and estimating the layer thickness on the basis of at least one predefined characteristic of the intensity profile.  
       [0022] Pursuant to a further illustrative embodiment of the present invention, an apparatus for determining the thickness of a curved thin film comprises a radiation source configured to irradiate a specimen of the curved film and a particle detector configured to detect radiation passing through the specimen to provide a digital image of the specimen. The apparatus further comprises an extraction unit configured to extract an intensity profile from the digital image and an analyzer for analyzing the intensity profile of the digital image.  
       [0023] According to still another illustrative embodiment of the present invention, an apparatus for determining the thickness of a material area formed in a substrate comprises a radiation source configured to emit a, beam of radiation of predefined characteristics and a detector configured and arranged to detect radiation passed through a section placed between the radiation source and the detector. Moreover, an extraction unit is provided that is configured to extract an intensity profile from a digital image along a predefined direction in the digital image. Additionally, a calculation unit is configured to determine a layer thickness of the material layer on the basis of at least one predefined characteristic of the intensity profile. 
     
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
     [0024] The invention may be understood by reference to. the following description taken in conjunction with the accompanying drawings, in which like reference numerals identify like elements, and in which:  
     [0025]FIGS. 1 a - 1   d  show schematic perspective views of a structure including a thin film for which a TEM image is to be gathered;  
     [0026]FIGS. 2 a - 2   d  schematically show cross-sectional views and perspective views of a typical application in determining the thickness of a thin film, wherein the thin film is coated on a structured surface;  
     [0027]FIG. 3 a  schematically depicts an apparatus for determining a layer thickness according to one illustrative embodiment of the present invention;  
     [0028]FIG. 3 b  schematically shows a further embodiment of an apparatus that allows precise measurements of thin films;  
     [0029]FIG. 4 a  schematically depicts a perspective view of a curved film and the projection thereof;  
     [0030]FIG. 4 b  shows the structure of FIG. 4 a  with an area for extracting an intensity profile; and  
     [0031]FIG. 4 c  depicts an intensity profile obtained from the structure depicted in FIGS. 4 a  and  4   b  in accordance with one illustrative embodiment of the present invention. 
    
    
     [0032] While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.  
     DETAILED DESCRIPTION OF THE INVENTION  
     [0033] Illustrative embodiments of the invention are described below. In the interest of clarity, not all features of an actual implementation are described in this specification. It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which will vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure.  
     [0034] As previously noted, the present invention is based on the inventors&#39; finding that the loss of the third dimension in producing a transmission image of a thin sample including a thin film, the thickness of which has to be determined, may be compensated for by extracting an intensity profile of the projected image of the thin film and analyzing the intensity profile. The analysis may be based upon typical characteristics of the intensity profile that are substantially independent from properties of the sample, such as sample thickness, radius of curvature of the thin film in a thickness direction of the thin film, and a tilt angle introduced during the preparation of the sample. Such sample-independent characteristics and criteria may be, for example, any extrema of the profile curve, appropriately set threshold values in predefined regions of the profile curve, and the like. The interaction of moderate energy radiation and charged particles with matter is well-understood and therefore suitable criteria for estimating profile curves may be obtained by carrying out simulation calculations regarding the sample to be measured. Moreover, the results of the simulations may be used to establish reference data or sets of reference data in which variations of parameters, such as sample thickness and/or layer thickness of a thin film to be measured, and the like, are taken account of, so that the reference data may be compared to the measurement data to determine the layer thickness. Hence, since such characteristics and/or criteria and/or reference data may be determined in an objective manner, influences of the sample preparation methods used and an operator&#39;s influence on estimating a transmission image may be substantially reduced or eliminated.  
     [0035] With reference to FIGS. 3 a  and  3   b , illustrative embodiments of apparatus allowing objective and precise thickness measurements will now be described. In FIG. 3 a , an apparatus  300  comprises a radiation source  330  that is configured to emit a beam of radiation  307  of required characteristics. For instance, the radiation source  330  may be an electron source as used in a standard transmission electron microscope. It should be noted, however, that the principles of the present invention may be readily applied to any radiation source emitting a radiation with a wavelength that is sufficient to precisely resolve the structures to be investigated. Thus, the radiation source  330  may represent an x-ray source, an ion beam source and the like. The apparatus  300  further comprises any of a variety of known means for receiving, positioning and holding in place a sample, such as the section already described with reference to FIGS. 1 and 2. So as to not obscure the present invention, such means are not expressly shown in the attached drawings. For the sake of simplicity, this means, as well as the sample, will be commonly indicated by reference number  306 . In one embodiment, a standard TEM apparatus may be used as the radiation source  330  and the means  306 .  
     [0036] The apparatus  300  further comprises a screen  331  configured and arranged to receive any radiation that has passed the sample  306 . For instance, the screen  331  may be adapted to produce light of appropriate wavelength upon incidence of a portion of the radiation  307 . Moreover, an image generating means  332  is provided and arranged so as to receive the light generated by the screen  331  and to generate an image corresponding to the radiation incident on and converted by the screen  331 . For example, the image generating means  332  may be a digital camera that produces an image, which may readily be stored and subjected to further electronic processing. In other embodiments, the image generating means  332  may be a standard analog device coupled to a scanner device that allows digitizing an analog image obtained from the image generating means  332 . An extraction unit  333  is configured to receive an image from the image generating means  332  or any other appropriate device that allows the generation of a digital image representing the distribution of radiation that has arrived on the screen  331 . The extraction unit may be directly coupled to the image generating means  332  or may be a stand-alone device. The extraction unit  333  is configured to obtain one or more intensity profiles of a predefined portion of the digital image supplied to the extraction unit  333 . In one embodiment, the extraction unit  333  may have implemented an image processing unit that allows analysis of the information contained in the digital image on a pixel basis. Thus, the extraction unit  333  may be adapted to select a certain region of interest of the digital image and to provide the contents representing the selected region to a calculation unit  334  that is adapted to perform any required manipulation on the pixel content supplied by the extraction unit  333 . The extraction unit  333  and the calculation unit  334  may be implemented in a common control unit, such as a computer device, wherein the computer may communicate with the image generating means  332 , or the computer may receive image data by an operator, and the like. For example, the calculation unit  334  may be adapted to determine gray scales on a pixel basis and compare the gray scales to predefined reference values so as to extract information regarding the intensity distribution in the region of interest, i.e., of one or more intensity profiles provided by the extraction unit  333 . Such information may include extrema of the intensity profile, any plateaus in the intensity profile and the like.  
     [0037] In another embodiment, the calculation unit  334  may have a required computational power and resources including an appropriate instruction set to provide for an advanced image processing of the digital image.  
     [0038]FIG. 3 b  schematically shows a variation of the apparatus of FIG. 3 a  according to a further illustrative embodiment of the present invention. In FIG. 3 b , parts that are identical to those described in FIG. 3 a  are denoted by the same reference numerals and a corresponding description of these parts is omitted. In FIG. 3 b , the apparatus  300  comprises the radiation source  330  adapted to emit the beam of radiation  307  with the required characteristics. Other than in the embodiment shown in FIG. 3 a , a positioning system  335  is provided and is mechanically coupled to the radiation source  330 . The positioning system  335  is configured to move the radiation source  330  in at least one direction, as indicated by arrow  336 , by correspondingly moving the radiation source  330  to thereby enable the beam  307 , exhibiting a relatively small radiation spot at the location of the sample  306 , to be scanned over the sample  306 . In other embodiments, additionally or alternatively, the sample  306  may be supported by a corresponding sample positioning system (not shown) that allows moving the sample  306  relative to the radiation source  330 . The apparatus  300  further comprises a beam optical system  337  that is configured to direct the radiation  307  emitted by the radiation source  330  and passed through the sample  306  onto a detector  338  that has a sufficient spatial resolution for the measurements to be performed. An output  339  of the detector  338  may be configured to supply digital information to the extraction unit  333 .  
     [0039] Thus, the embodiments of FIG. 3 a  differ from the embodiments of FIG. 3 b  in that the radiation transmitted through the sample  306  may directly be converted into a digital image without requiring the screen  331  as shown in FIG. 3 a . Moreover, the apparatus  300  of FIG. 3 b  may be operated in a scan mode so that the apparatus of FIG. 3 b  allows one to select a region of interest by correspondingly positioning the radiation source  330  and/or the sample  306 .  
     [0040] The operation of the apparatus  300  shown in FIGS. 3 a  and  3   b  will now be described with reference to FIGS. 4 a - 4   c  irrespective of the mode of irradiating the sample  306 . In FIG. 4 a , a schematic perspective view of a portion of the sample  306  is shown. The sample may include a via, such as the via  222 , as shown in FIGS. 2 a - 2   d . Thus, the sample  306  comprises a thin film  301  having curved edges  326 , wherein a thickness of the thin film resting on a curved surface is to be determined. Regarding the preparation of the sample  306 , the same criteria apply as already explained with reference to FIGS. 1 and 2. Upon illumination with the beam  307 , for example comprised of electrons, a portion of the radiation is absorbed in accordance with the properties of the material forming the thin film  301 . Since a neighboring material  303  or  304  differs in at least one property from the material of the thin film  301 , a two-dimensional projection  308  is obtained, the thickness  309  of which is, however, affected by the magnitude of the curvature of the curved edges  326  as is previously explained with reference to FIGS. 2 a - 2   d . Thus, the digital image  310  including the projection  308  and generated by the screen  331  in combination with the image generating means  332 , when the apparatus  300  of FIG. 3 a  is considered, or that is directly generated by the detector  338 , when the apparatus  300  of FIG. 3 b  is considered, does not allow a precise determination of an actual thickness  302  of the thin film  301  for the same reasons as already pointed out earlier.  
     [0041] In FIG. 4 b , by means of the extraction unit  333  a region of interest  311  of the digital image  310  is selected that includes partially the projection  308 . The region of interest  311  may be selected according to requirements, such as desired position, characteristics of the thin film  301 , contrast of the projection  308  and the like. The region of interest  311  is selected to at least include a transition to the neighboring regions  303  and  304 . In one embodiment, the region of interest  311  may represent a single pixel line of the digital image  310 , taken along a direction that is substantially perpendicular to a length direction  312  defined by the thin film  301 . In another embodiment, as shown in FIG. 4 b , the region of interest  311  extends along the direction  312  and thus may include a plurality of sections of the projection  308 . The corresponding plurality of sections, each representing a single intensity profile, may then be summed and weighted to establish an averaged intensity profile of the region of interest  311 . In this way, any fluctuations between individual pixel lines representing a section of the projection  308  may be smoothed. In one embodiment, averaging a plurality of intensity profiles may automatically be performed once the region of interest  311  is selected by an operator.  
     [0042]FIG. 4 c  shows a diagram depicting a typical intensity profile  313  taken along a direction substantially perpendicular to the longitudinal direction  312 , which will also be referred to as x direction. In FIG. 4 c , the intensity, i.e., the gray scale of the pixels, is depicted on the vertical axis whereas the position in x is depicted in the horizontal direction. The intensity profile  313  extracted by the extraction unit  333  may then be subjected to further analysis by calculation unit  334 , since the shape of the intensity profile  313  is strongly affected by the characteristics of the sample  306 , such as the thickness thereof, the characteristics of the materials comprising the regions  303 ,  304  and the thin film  301 . For example, if the electron scattering capability of the regions  303  and  304  is quite similar to that of the thin film  301 , a minimum as depicted in FIG. 4 c  will be significantly less accentuated and, thus, estimation of the thickness  301  requires further analysis. To this end, the interaction of the beam  307 , for example comprised of electrons, with the materials included in the sample  306  may be calculated by means of well-established routines that exactly describe the interaction of matter with electromagnetic radiation and charged particles. In these calculations, the thickness of the sample  306  may be varied to take account of any impreciseness in preparing the sample  306 . For example, a plurality of thicknesses of the sample may be assumed and the corresponding “responses,” for instance in the form of contrast differences between the regions  303 ,  304  and  301 , of the (simulated) sample  306  may be calculated. The results of the simulation may then be used to establish a corresponding set of reference data that may be compared to actual measurement data, or, in other embodiments, the results may be used to determine criteria as to how to determine the precise location of a transition between two adjacent regions in the sample  306 . For instance, threshold values ×1 and ×2 may be determined in the transition regions of adjacent materials, that is, in the falling edge and the rising edge of the intensity profile  313 , which specify the actual thickness  302 .  
     [0043] Alternatively or additionally, the magnitude of the curvature of the curved edges  326  and/or the thickness of the (simulated) thin film  306  may be varied to establish a set of possible “responses” of the thin film  301  to the incident beam  307 . The corresponding set of reference data may then be compared to the actual measurement results so as to determine the actual thickness  301  on the basis of the result of the comparison.  
     [0044] In one embodiment, the direction of the simulated incident beam  307  is varied for a plurality of different thicknesses  302  of the thin film  301  and a plurality of different thicknesses of the sample  306 . Thus, corresponding reference intensity profiles may be obtained, in which a tilt angle possibly introduced during the preparation of the (actual) sample  306  is compensated for by varying the (simulated) angle of incidence of the beam  307 . The reference data may then be compared to the measurement data to extract the thickness  302 . These reference data may be obtained at any appropriate time and may be stored in a library to be available for subsequent measurements.  
     [0045] It is to be noted that extracting an intensity profile from a digital image of a sample is also advantageous in precisely determining the layer thickness of a thin film coated on a substantially planar surface, as is shown FIGS. 1 a - 1   d , or the bottom region  224  of the via  222 , as shown in FIGS. 2 a - 2   d . Thus, any imperfections in preparing a sample including these “planar” features, i.e., introducing a tilt angle in cutting the sample, that may conventionally result in an inaccurate determination of the thickness may effectively be compensated by obtaining an intensity profile and analyzing the intensity profile in the above explained manner. For example, by precisely obtaining the actual thickness, such as the thickness  102  of the thin film  101  in FIGS. 1 a - 1   d , from the thickness  109 ′ (FIG. 1 d ), the tilt angle α (FIG. 1 c ) may be determined. The knowledge regarding the tilt angle α may be advantageous in further analyzing the sample of interest or in estimating the quality of the sample preparation technique.  
     [0046] The particular embodiments disclosed above are illustrative only, as the invention may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein. For example, the process steps set forth above may be performed in a different order. Furthermore, no limitations are intended to the details of construction or design herein shown, other than as described in the claims below. It is therefore evident that the particular embodiments disclosed above may be altered or modified and all such variations are considered within the scope and spirit of the invention. Accordingly, the protection sought herein is as set forth in the claims below.