Abstract:
A method for measuring overlay in semiconductor wafers includes a calibration phase in which a series of calibration samples are analyzed. Each calibration sample has an overlay that is known to be less than a predetermined limit. A difference spectrum for a pair of reflectively symmetric overlay targets is obtained for each calibration sample. The difference spectra are then combined to define a gross overlay indicator. In subsequent measurements of actual wafers, difference spectra are compared to the overlay indicator to detect cases of gross overlay.

Description:
PRIORITY CLAIM 
   The present application claims priority to U.S. Provisional Patent Application Ser. No. 60/476,567, filed Jun. 6, 2003, which is incorporated in this document by reference. 
   RELATED APPLICATION 
   The subject matter of the present application is related to the disclosure included in a concurrently filed U.S. patent application Ser. No. 10/858,587 entitled: “APPARATUS AND METHOD FOR MEASURING OVERLAY BY DIFFRACTION GRATINGS”. The disclosure of that related application is incorporated herein by reference. 

   TECHNICAL FIELD 
   This invention relates to measuring the pattern overlay alignment accuracy of a pair of patterned layers on a semiconductor wafer, possibly separated by one or more layers, made by two or more lithography steps during the manufacture of semiconductor devices. 
   BACKGROUND OF THE INVENTION 
   Manufacturing semiconductor devices involves depositing and patterning several overlaying layers. A typical semiconductor wafer might include, for example, a series of gates formed on a first layer and a series of interconnects formed on a second layer. The two layers (and their structures) are formed at different lithography steps in the manufacturing process. Alignment between the two layers is critical to ensure proper connection between the gates and their interconnects. Typically, this means that the tolerance of alignment must be less than the width of a single gate. 
   Overlay is defined as the displacement of a patterned layer from its ideal position aligned to a layer patterned earlier on the same wafer. Overlay is a two dimensional vector (Δx, Δy) in the plane of the wafer. Overlay is a vector field, i.e., the value of the vector depends on the position on the wafer. Perfect overlay and zero-overlay are used synonymously. Overlay and overlay error are used synonymously. Depending on the context, overlay may signify a vector or one of the components of the vector. 
   Overlay metrology provides the information that is necessary to correct the alignment of the stepper-scanner and thereby minimize overlay error with respect to previously patterned layers. Overlay errors, detected on a wafer after exposing and developing the photoresist, can be corrected by removing the photoresist, repeating exposure on a corrected stepper-scanner, and repeating the development of the photoresist. If the measured error is acceptable but measurable, parameters of the lithography process could be adjusted based on the overlay metrology to avoid excursions for subsequent wafers. 
   Most prior overlay metrology methods use built-in test patterns etched or otherwise formed into or on the various layers during the same plurality of lithography steps that form the patterns for circuit elements on the wafer. One typical pattern, called “box-in-box” consists of two concentric squares, formed on a lower and an upper layer, respectively. “Bar-in-bar” is a similar pattern with just the edges of the “boxes” demarcated, and broken into disjoint line segments. The outer bars are associated with one layer and the inner bars with another. Typically one is the upper pattern and the other is the lower pattern, e.g., outer bars on a lower layer, and inner bars on the top. However, with advanced processes the topographies are complex and not truly planar so the designations “upper” and “lower” are ambiguous. Typically they correspond to earlier and later in the process. The squares or bars are formed by lithographic and other processes used to make planar structures, e.g., chemical-mechanical planarization (CMP). 
   In one form of the prior art, a high performance microscope imaging system combined with image processing software estimates overlay error for the two layers. The image processing software uses the intensity of light at a multitude of pixels. Obtaining the overlay error accurately requires a high quality imaging system and means of focusing the system. One requirement for the optical system is very stable positioning of the optical system with respect to the sample. Relative vibrations blur the image and degrade the performance. Reducing vibration is a difficult requirement to meet for overlay metrology systems that are integrated into a process tool, like a lithography track. 
   As disclosed in U.S. patent application Ser. No. 2002/0158193; U.S. patent application 2003/0190793 A1; and as described in Proc. of  SPIE , Vol. 5038, February 2003, “Scatterometry-Based Overlay Metrology” by Huang et al., p. 126–137, and “A novel diffraction based spectroscopic method for overlay metrology” by Yang et al. p. 200–207 (all four incorporated in this document by reference) one approach to overcome these difficulties is to use overlay metrology targets that are made of a stack of two diffraction gratings as shown in  FIG. 1 . The grating stack  10  has one grating  20  in a lower layer and another grating  30  in an upper layer as shown in  FIG. 1 . The layers of  20  and  30  are to be aligned. There are two instances of the grating stack  10 , one for the x-component of overlay and one for the y-component. The measurement instrument is such that it does not resolve individual grating lines. It measures overall optical properties of the entire grating. Optical properties are measured as a function of wavelength, polar or azimuthal angle of incidence, polarization states of the illumination and the detected light, or any combination of these independent variables. An alternative embodiment uses two stacks of line gratings to measure x-overlay and two stacks of line gratings to measure y-overlay (four grating stacks total). Still another embodiment uses three line grating stacks in combination to simultaneously measure both x and y alignment. (See also PCT publication WO 02/25723A2, incorporated herein by reference). Scatterometry (diffraction) is proving to be an effective tool for measuring overlay. 
   A shortcoming of the prior scatterometry-based art is that, diffraction gratings cannot distinguish overlay values that differ by an integer number of periods. Let R(λ,θ,ξ) denote the specular (0-th order) reflection of the grating at wavelength λ, angle of incidence θ, and offset ξ. The offset ξ is the distance between centerlines of lower and upper grating lines as shown in  FIG. 1 . The function R(λ,θ,ξ) is periodic with respect to the offset ξ:
 
 R (λ,θ,ξ)= R (λ,θ,ξ+ P )  (1)
 
   where P is the period of the grating. The scatterometry-based overlay measurements are even more ambiguous when the profiles of the grating lines are symmetric. Then, a consequence of reciprocity is:
 
 R (λ,θ,ξ)= R (λ,θ,−ξ)  (2)
 
 R (λ,θ,( P/ 2)+ξ)= R (λ,θ,( P/ 2)−ξ)  (3)
 
   There are two values of offset, separated by P/2, where the optical properties become ambiguous. Optical properties are insensitive to overlay at these points. Therefore, the largest measurement range is P/2 when the grating lines have symmetric cross-sections. When overlay exceeds this range, not only is the measurement wrong, there is no indication that the overlay is out of range.  FIG. 2  demonstrates this ambiguity for calculated data corresponding to near-normal incidence, unpolarized reflectance of a grating stack as a function of offset ξ. 
   To overcome this limitation, Huang et al., cited above, describes a means of manufacturing a grating with an asymmetric unit cell. For gratings of that type (i.e., where there is substantial asymmetry) the measurement range becomes P, a whole period. The measurement range also becomes a whole period if two grating stacks are used, the offset of each stack is biased, and the two offset biases differ by P/4 (as described in U.S. application Ser. No. 10/613,378, filed Jul. 3, 2003, incorporated in this document by reference). 
   When the periodic nature of overlay is accounted for, the overlay measured by the grating, Δx GRATING , is related to the actual overlay as follows: 
   
     
       
         
           
             
               
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   In Equation (4), int[x] denotes the integer nearest to x. In prior art, the integer N is unknown. Therefore, Δx GRATING  represents Δx ACTUAL  only when N is zero, that is, when |Δx ACTUAL |&lt;P/4. Δx GRATING , which is also called fine-overlay measurement, has high precision but it is only accurate when N=0. Gross overlay is defined as the condition |Δx ACTUAL |≧P/4 or |Δy ACTUAL |≧P/4, that is, any one component of overlay exceeding a quarter of the period. Gratings cannot detect gross overlay until overlay gets so large that the upper and lower gratings do not overlap in part of the measurement spot. Although gross overlay is rare in well-tuned lithography processes, alignment errors larger than 100 nm, even as large as several microns occur when a new process, a new reticle, or a new projector is introduced. In these instances, there is a need to not only detect but also to measure gross overlay. 
   Although increasing the period increases the measurement range, P/2, this approach is not preferred because it reduces the sensitivity of the optical response of grating stacks to overlay. As the period is increased, if the period becomes a significant fraction of the diameter of the measurement spot, the placement of the spot on the grating affects the overlay measurement and reduces its precision. For these two reasons, increasing the period to increase the measurement range is counter-productive. Using more than one grating stack, each with a different period, reduces but does not eliminate ambiguity. 
   SUMMARY OF THE INVENTION 
   The present invention provides a method and apparatus for analyzing overlay in semiconductor wafers. For a typical embodiment, a semiconductor wafer is patterned to include at least one pair of overlay targets for each direction for which overlay is to be measured. For the most common case, where overlay is measured in both X and Y directions, this means that a total of at least two pairs are used (one for X and one for Y directions). The overlay targets within a pair are configured to be reflectively symmetric when overlay is zero. This means that the overlay targets within a pair appear to be identical to an optical instrument (an instrument that measures the reflection of light as a function of wavelength, polar angle of incidence, azimuthal angle of incidence, polarization state, or any combination of these independent parameters) at zero overlay. When overlay is nonzero, the reflection symmetry within a pair is broken and the optical properties differ. For small values of overlay (typically up to 100 nm), the difference is directly proportional to overlay. 
   A calibration process is used to calculate a gross overlay indicator. For this process, a series of calibration samples are used. Each calibration sample includes the same overlay targets that are used for measuring overlay in actual wafers. Each calibration sample is also selected to have an overlay that is less than a predetermined limit. The calibration samples are inspected and a difference spectrum is constructed for each. Each difference spectrum represents the difference in reflectivity for the two halves of an overlay target pair. Typically, this is measured as a function of wavelength. So each difference spectrum will contain individual measurements obtained at a series of different wavelengths. The difference spectra of the calibration samples are combined to form a matrix S. The principal directions of S are obtained by finding the largest n nonzero eigen-values and the corresponding normalized eigen-vectors of matrix S. The principal directions are used to form a matrix Q. The gross overlay indicator is a function of Q. 
   During subsequent measurements on actual wafers, difference spectra are once again obtained by measuring pairs of overlay targets at multiple wavelengths. The difference spectra are then compared to the gross overlay indicator to determine if gross overlay is present. If not, then overlay may be computed as the difference spectra and a linear differential estimator. 
   A wide range of different instruments may be used to measure the reflectivity of the overlay targets used in the calibration samples and the wafers to be measured. One particularly suitable technique is to use an optical microscope in which the illumination and collection optics are configured so that the gratings of the overlay targets are resolved to have substantially uniform colors without resolving their unit cells. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows in cross-section the stack of two gratings that are used in the diffraction-based overlay metrology in the prior art. 
       FIG. 2  shows calculated near-normal specular reflectance of the grating stack in  FIG. 1  as a function of offset ξ for four different wavelengths. 
       FIG. 3A  shows an overlay target as provided by an embodiment of the present invention. 
       FIG. 3B  shows a cross section of the overlay target of  FIG. 3A . 
       FIG. 4  shows alternate locations for a gross overlay feature within the overlay target of  FIG. 3A . 
       FIG. 5  shows alternate configurations for a measurement spot for the overlay target of  FIG. 3A . 
       FIG. 6  shows a first embodiment of an overlay target that includes two pitches to support simultaneous measurement of gross and fine overlay. 
       FIG. 7  shows a second embodiment of an overlay target that includes two pitches to support simultaneous measurement of gross and fine overlay. 
       FIG. 8  shows an embodiment of an aperiodic overlay target. 
       FIG. 9  shows two pairs of reflectively symmetric overlay targets enabling the simultaneous measurement of overlay in X and Y directions. 
       FIG. 10  is a cross-section of an embodiment of a reflectively symmetric overlay target. 
       FIG. 11A  is a plot showing estimated overlay as a function of actual overlay for the overlay target of  FIG. 7 . 
       FIG. 11B  is a plot showing a gross overlay indicator as a function of actual overlay for the overlay target of  FIG. 7 . 
       FIG. 12A  is a plot showing estimated overlay as a function of actual overlay for the overlay target of  FIG. 3A . 
       FIG. 12B  is a plot showing a gross overlay indicator as a function of actual overlay for the overlay target of  FIG. 3A . 
       FIG. 13A  is a map showing the difference spectrum ΔR as a function of wavelength and overlay for the overlay target of  FIG. 7 . 
       FIG. 13B  is a map showing the orthogonal projection (I−QQ T )ΔR as a function of wavelength and overlay for the overlay target of  FIG. 7 . 
       FIGS. 14A through 14C  are plots showing gross overlay indicators as a function of actual overlay for n equal to 2, 3 and 4 where n is the number of principal directions used to compute the gross overlay indicator. 
       FIG. 15  shows an arrangement that includes multiple overlay targets as provided by an embodiment of the present invention. 
       FIG. 16A  is a plot showing estimated overlay as a function of actual overlay for the overlay target of  FIG. 15 . 
       FIG. 16B  is a plot showing a gross overlay indicator as a function of actual overlay for the overlay target of  FIG. 15 . 
       FIG. 17  is a block diagram of a metrology tool suitable for measuring overlay using the targets and methods provided by the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The present invention provides one or more overlay targets along with a method for using the overlay targets to analyze overlay in semiconductor wafers. Together, the targets and analysis method increase the range of overlay that can be measured without sacrificing overlay measurement sensitivity. The following sections describe the targets first and then focus on the analysis method. 
   1. Description of the Overlay Target 
   In  FIG. 3A , a first embodiment of an overlay target as provided by the present invention is shown and generally designated  300 . Target  300  is shown in the plan view (viewed in a direction that is perpendicular to the plane of the wafer) and includes a fine overlay pattern  302  and a gross overlay pattern  304 . The gross overlay pattern  304  is used to measure or detect gross overlay and occupies a small fraction of the entire target  300 . 
     FIG. 3B  shows a cross section of target  300  taken along the line A—A in  FIG. 3A . As shown by the cross-section, target  300  includes an upper grating  306  and a lower grating  308  that are formed in the two layers that are to be aligned. For lithography applications, upper grating  306  is typically part of the resist mask. Lower grating  308  is previously formed in a patterned layer, for example, in an isolation trench, gate, metal contact (via) or metal interconnection level. One or more intermediate layers  310  may be positioned between upper grating  306  and lower grating  308 . The number of intermediate layers  310  and their compositions are application dependent. 
   Gross overlay pattern  304  can be implemented as a grating stack with a different pitch from the fine pitch P as shown in  FIG. 3B . The line-to-space ratios in fine overlay pattern  302  and gross overlay pattern  304  may be different. The line-to-space ratios in upper grating  306  and lower grating  308  may also be different. Gross overlay pattern  304  can also be implemented as an aperiodic pattern without any periodicity in upper grating  306  or lower grating  308 . By choosing a proper pitch and offset bias for gross overlay pattern  304 , the higher pitch in gross overlay pattern  304  does not reduce the range in which a gross overlay is detected but improves the discrimination of both gross and fine overlay. 
   For the implementation shown in  FIGS. 3A and 3B , gross overlay pattern  304  breaks the periodicity of the fine grating pattern  302 .  FIG. 4  shows an alternate configuration where gross overlay pattern  304  is distributed at the corners of fine grating pattern  302 . 
   Optical response of the overlay target is measured as a function of wavelength, polar angle of incidence, azimuthal angle of incidence, polarization state, or any combination of these independent variables. As shown in  FIG. 5 , the measurement spot can be smaller ( 502   a ) or larger ( 502   b ) than the overlay target  300 . In either case, the gross overlay pattern  304  occupies a small fraction, such as 4%, of the area of the measurement spot. The more uniform the measurement spot, the higher is the tolerance to errors in positioning the spot. The overlay target in  FIG. 4  has the gross overlay pattern  304  on the periphery of the target, and is not preferred for cases where the measurement spot is smaller than overlay target  300  (e.g.,  502   a ). 
   In  FIG. 6 , a second embodiment of an overlay target as provided by the present invention is shown and generally designated  600 . Target  600  includes an upper grating  602  and a lower grating  604  that are formed in the two patterned layers that are to be aligned. For lithography applications, upper grating  602  is typically part of the resist mask. Lower grating  604  is a previously formed patterned layer, for example, in an isolation trench, gate, metal contact (via) or metal interconnection level. One or more layers  606  may be positioned between gratings  602  and  604 . The number of these intermediate layers  606  and their compositions are application dependent. 
   The lines in gratings  602  and  604  are configured to establish a repeating pattern. As shown in  FIG. 6 , the repeating pattern is composed of a series of gross periods. Each gross period is further composed of a series of fine periods (labeled P 1  through P N  in  FIG. 6 ). Each gross period is identical meaning that it includes the same series of fine periods. Gross periods are used to measure gross (micron-level) overlay while fine periods are used to measure fine overlay. For the particular example of  FIG. 6 , the gross period is 2455.38 nm. It is composed of a 615.38 nm fine period followed by an 800 nm period that is followed, in turn by a 1040 nm period. The fine periods are a geometric progression: the first is equal to 
           800   C         
nm, the second is 800 nm and the third is 800*C nm (where C is 1.3).
 
   In general, it should be noted that the number of fine periods included in a gross period, as well as their sizes and line-to-space ratio is subject to endless variation. The important feature is that not all fine periods in one gross period are equal. For this particular example, each fine period has a line-to-space ratio of 1:1. For other implementations, different fine periods may be constructed to have different line-to-space ratios. Gratings  602  and  604  may also be constructed using different line-to-space ratios. 
   For the example of  FIG. 6 , the gross periods in gratings  602  and  604  are the same so that there are no Moire patterns across overlay target  600 . Consequently, placement of the measurement spot on overlay target  600  is not critical for measurement spots smaller than the gross period. For scatterometry measurement, a typically measurement spot is on the order of 20 to 40 μm in diameter. The tolerance for positioning of the measurement spot within overlay target  600  is typically several micrometers. 
   In  FIG. 7 , a third embodiment of an overlay target as provided by the present invention is shown and generally designated  700 . Target  700  includes an upper grating  702  and a lower grating  704  that are formed in the two patterned layers that are to be aligned. One or more intermediate layers  706  may be positioned between upper grating  702  and lower grating  704 . Both upper grating  702  and lower grating  704  have a constant fine pitch. Depending on the implementation, the fine pitch may or may not be the same on both grating. Each gross period includes N fine periods. One of the N fine periods in every gross period has a line or space missing. In the example of  FIG. 7 , the fine period is 800 nm, the gross period is 4 μm and N=5. 
   Many other arrangements that are similar to those shown in  FIGS. 6 and 7  are possible. The key feature of the targets is that there is a fine-scale structure to increase the sensitivity to overlay, and there is a gross-scale structure to measure or at least detect gross (on the order of a micrometer) overlay. The gross period is much smaller than the measurement spot of the optical instrument so that the target appears uniform (laterally homogeneous) to the optical instrument. This increases the tolerance to the position of the measurement spot relative to the target. 
     FIG. 8  shows an overlay target  800  that has no gross-periodicity. In the pseudo-random implementation, the width of every line and space in layer  802  and  804  is randomly selected. In practice, the width of lines and spaces are randomly selected from a finite list since there is a minimum grid size for writing reticles. 
   2. Description of Overlay Measurement and Gross Overlay Detection 
   As shown in  FIG. 9 , four copies of an overlay target labeled XA, XB, YA, and YB are placed in close proximity of each other, preferably in a scribe line  902  between the dies on the wafer. Overlay targets XA, XB, YA, and YB may be selected from any of the types described above (i.e., overlay target  300 ,  600 ,  700  or  800 ). Overlay targets XA, XB, YA, and YB are measured with the optical instrument, one at a time or simultaneously. Targets XA and XB are used to obtain the overlay component Δx, and targets YA and YB are used to obtain another overlay component Δy. 
   The overlay targets XA and XB have reflective symmetry with respect to the x-axis when the x-component of overlay is zero. The lines of the fine gratings in XA and XB are parallel to the y-axis. Similarly, targets YA and YB have reflection symmetry with respect to the y-axis when the y-component of overlay is zero. The lines of the fine gratings in YA and YB are parallel to the x-axis. 
   For line gratings, reflection symmetry is equivalent to making a copy of a first target, rotating the copy by 180° in the plane of the wafer, and translating the copy in the plane of the wafer. The two overlay targets XA and XB look identical to an optical instrument (that measures the reflection of light as a function of wavelength, polar angle of incidence, azimuthal angle of incidence, polarization state, or any combination of these independent parameters) at zero overlay. When overlay is nonzero, the reflection symmetry is broken and the optical properties of target XA and XB differ. For small values of overlay (typically up to 100 nm), the difference is directly proportional to overlay. 
   Obtaining overlay from two targets (e.g., XA and XB) that only differ in the amount of offset between their upper and lower gratings increases the accuracy and precision of overlay that can be determined from optical response of the targets.  FIG. 10  shows a cross section of two targets (XA and XB) configured in this fashion. The distance  1002  between the centerlines of the upper and lower grating lines in target XA is equal to the distance  1004  between the upper and lower grating lines in target XB at perfect alignment (zero overlay). When overlay Δx is nonzero, distance  1002  increases by Δx while distance  1004  decreases by Δx. Overlay targets XA and XB are identical in all respects other than the offset between their upper and lower gratings. 
   Reflection symmetry of target XA and XB leads to the property:
 
 R   B (λ,Δ x )= R   A (λ,−Δ x )  (1)
 
   R A (λ,Δx) and R B (λ,Δx) are optical responses of target XA and XB, respectively, for overlay Δx. Although we only explicitly show the dependence of R A  and R B  on wavelength λ, R A  and R B  can be a function of wavelength, polar angle of incidence, azimuthal angle of incidence, polarization state, or any combination of these independent parameters. The difference of the two spectra R A  and R B  is: 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   In the last equation, the first term in the Taylor series expansion with respect to a small overlay Δx is retained. Higher order terms are neglected. This approximation holds for |Δx|&lt;100 nm for practical grating geometries. The left-hand side of Equation (2) is a measurable quantity that depends linearly on the parameter of interest, namely, the overlay Δx. Estimation of Δx from equation (2) takes the form of a linear operator on the difference spectrum: 
   
     
       
         
           
             
               
                 
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   The summation is over the discrete values of wavelength at which measurements are taken. ΔR MEAS (λ) in Equation (3) is the difference of the reflection spectra measured for XA and XB. LDE(λ) is the so-called linear differential estimator. LDE(λ) is a spectrum that can be calculated by simulating diffraction of electromagnetic waves, or it can be obtained from measurements of targets with known overlay values. Equations (1) to (3) are prior art and they are repeated here for clarity.  FIG. 11   a  shows the estimate Δ{circumflex over (x)} of overlay obtained according to Equation (3) for the target described in  FIG. 7 . The estimate is valid in a linear range  1102  shown in  FIG. 11   a . Outside the linear range the estimate (3) is not valid. Modern lithography processes rarely make overlay excursions larger than 100 nm. In the event that overlay is outside the linear working range  1102 , that condition must be flagged. The linear differential estimator (3) of the prior art provides no such warning. 
   The targets shown in  FIGS. 3  though  8  have features that break the periodicity at the fine pitch. When the overlay is one or two times the fine period, the difference spectrum ΔR MEAS (λ) is different than in Equation (2). This condition is detected by the following algorithm: 
   Step  1 : Form the symmetric, positive semi-definite matrix S by adding rank-1 matrices ΔR(ΔR) T . 
   
     
       
         
           
             
               
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   The entries of the column vector ΔR are the measurements of the difference spectra at discreet wavelengths: 
                   Δ   ⁢           ⁢   R     =     [           Δ   ⁢           ⁢       R   MEAS     ⁡     (     λ   1     )                   Δ   ⁢           ⁢       R   MEAS     ⁡     (     λ   2     )                 ⋮             Δ   ⁢           ⁢       R   MEAS     ⁡     (     λ   M     )               ]             (   5   )               
(ΔR) T , which is a row vector, is the transpose of ΔR. The summation in Equation (4) is over multiple training measurements where targets are known to have overlay values smaller than a predetermined threshold δ such as 100 nm. The training targets may have multiple values of offset smaller than δ, and representative variations of other process parameters such as layer thicknesses and line widths.
 
   Step  2 : Find the largest n nonzero eigen-values and the corresponding normalized eigen-vectors q 1 , . . . , q n  of matrix S. The eigen-vectors of S are called principle directions. The integer number n is a parameter of the algorithm. Its value is typically between 1 and 4. The matrix:
 
 Q=[q   1    q   2    . . . q   n ]  (6)
 
has orthonormal columns. Store matrix Q for later use. This is the end of training.
 
   Step  3 : A gross overlay indicator (GOI) is obtained by the following operation during the measurement of overlay. 
                       GOI   =       ⁢            (     I   -     QQ   T       )     ⁢   Δ   ⁢           ⁢   R                      =       ⁢            Δ   ⁢           ⁢   R     -       ∑     j   =   1     n     ⁢           ⁢       q   j     ⁡     (       q   j   T     ⁢   Δ   ⁢           ⁢   R     )                              (   7   )               
GOI is a nonnegative scalar quantity. The double vertical bars in (7) indicate the Euclidian-norm of the vector. The vector (I−QQ T )ΔR is the component of ΔR that is orthogonal to all principle directions q 1 , . . . , q n . The projection matrix P=(I−QQ T ) is the orthogonal projector of the linear space spanned by q 1 , . . . , q n . Raise an alarm if GOI is larger than a previously determined threshold:
 Gross overlay is detected if  GOI &gt;threshold  (8) 
The value of the threshold is determined during training such that the alarm is not raised for any of the training cases.
 
   The gross overlay indicator GOI is shown as a function of overlay in  FIG. 11   b  for the target described in  FIG. 7 . GOI is below a threshold  1104  in the linear working range  1102  where the linear differential estimation (3) is valid. For larger overlay values, GOI is above threshold  1104  indicating that estimation (3) should not be used. As overlay approaches (Gross Period)/2, GOI falls below threshold  1104  and no is longer an accurate indication of gross overlay. This happens for a 2 μm overlay for the target in  FIG. 5 . Such a large overlay can be detected by visual inspection using a visible light microscope. 
   The aperiodic target shown in  FIGS. 3–5  has no periodicity. Therefore, the GOI will not return to zero for any large value of overlay. The GOI is shown as a function of overlay in  FIG. 12   b  for the target described in  FIG. 3   a . The gross overlay pattern occupies 4% the area of the measurement spot. In this preferred overlay target, there is no fail of GOI (i.e., no false negative region) as that shown in  FIG. 11   b . For overlay values larger than the measurement range  1202 , GOI is always above the threshold  1204 .  FIG. 12   a  shows the estimate Δ{circumflex over (x)} of overlay obtained according to Equation (3) for the target described in  FIG. 3   a.    
   This is why the gross overlay algorithm works: if the Taylor approximation in Equation (2) were exact, the matrix S would have only one nonzero eigen-value and all measured ΔR would be proportional to the corresponding principle direction, q 1 . Then, GOI=∥ΔR−q 1 (q 1   T ΔR)∥=0 would be zero in the absence of noise. GOI would be zero in the linear regime where Equation (2) holds, and positive where ΔR is no longer linearly related to overlay. In reality, Equation (2) is not exact but GOI still discriminates between the linear regime where Equation (2) is a good approximation, especially if more than one principle direction is used (n&gt;1). 
   The map in  FIG. 13   a  shows ΔR as a function of wavelength and overlay for the target in  FIG. 7 . The map appears almost periodic with respect to overlay with a period equal to the fine period (800 nm). ΔR deviates sufficiently from a periodic function. This is brought out in  FIG. 13   b  where the orthogonal projection (I−QQ T ) ΔR is plotted as a color map as a function of wavelength and overlay. The resulting map is close to zero in the linear working range  1302 . The gross overlay shown in  FIG. 10   b  is obtained from the map in  FIG. 13   b  by the operation ∥(I−QQ T )ΔR∥. This amounts to squaring the map in  FIG. 13   b , summing the squares along the wavelength dimension, and taking the square root of the sum. The GOI in  FIG. 11   b  is close to zero in the working linear range  1102  because the map of (I−QQ T )ΔR is close to zero in the linear working range  1302  in  FIG. 13   b .  FIGS. 14   a  and  14   b  show the effect of taking more principle directions. The GOI is closer to zero in the working linear range when more principle directions are taken.  FIGS. 11   a  and  11   b  also show that very few principle directions are needed. 
     FIG. 14   c  shows two curves. The first is the gross overlay indicator GOI in the linear working range. No measurement noise was included in calculating GOI. The second curve shows GOI under typical noise of an actual near-normal spectroscopic reflectometer. As may be appreciated by inspection, discrimination of gross overlay from linear working range is robust under noise. 
   3. Description of Methods Using Multiple Targets for Overlay Metrology 
   The methods described in the previous section uses 4 overlay targets XA, XB, YA, and YB (as shown in  FIG. 9 ) to measure the X- and Y-components of overlay. The overlay can be obtained using Equation (3). The spectrum LDE(λ) in Equation (3) can be calculated by simulating diffraction of electromagnetic waves, or it can be obtained from measurements of targets with known overlay values. The overlay targets XA and XB have reflection symmetry with respect to the x-axis when the x-component of overlay is zero. One way to implement XA and XB is to employ offset biases between the upper and lower gratings in overlay targets XA and XB. The offset biases in XA and XB have the same magnitude but opposite directions along the x-axis as shown in  FIG. 10 . 
   Consider a set of overlay targets  302 X 1 ,  302 X 2 , . . . , and  302 Xn placed in close proximity as shown in  FIG. 15 . The overlay targets in  300  are lined up in  FIG. 15 , but they can actually be arranged in various ways. By choosing offset biases ΔX 1 , ΔX 2 , . . . , and ΔXn for each target respectively, the total offsets in each target are ΔX 1 +Δx, ΔX 2 +Δx, . . . , and ΔXn+Δx where Δx is the overlay (alignment error). As long as these targets are close enough to one another (not necessarily adjacent to one another) on the wafer, the process conditions are the same for all targets. Therefore, the pattern profiles and layer thicknesses are essentially the same regardless of the different lateral offsets in each target. The optical response in each target R 1 , R 2 , . . . , and R n  depends only on the total offset in each target and the optical instrument parameters:
 
 R   i   =R (λ,Δ v )  (4)
 
where λ is the incident wavelength and Δv=ΔX i +Δx is the total offset for the target Xi.
 
   The optical instrument parameters can include wavelength, polar angle of incidence, azimuthal angle of incidence, polarization state, or any combination of these independent parameters. For targets with reflection symmetry to the lateral offset:
 
 R (λ,Δ v )= R (λ,−Δ v )  (5)
 
   With sufficient number of targets, the spectrum LDE(λ) in Equation (3) can be obtained immediately from the measurements on these targets without any prior diffraction simulation or pre-measurements. Other interpolation methods can also be used to obtain overlay directly from measurements (e.g., see U.S. Provisional Application No. 60/519,345 and “The Color-Box Alignment Vernier: A Sensitive Lithographic Alignment Vernier Read At Low Magnification,” Peter Heimann,  Optical Engineering , July 1990, Vol. 29, No. 7, p. 828–836 (both incorporated in this document by reference). More terms in the Taylor expansion in Equation (2) can be included in the calculation if necessary as long as there are more measured targets than unknown quantities.  FIG. 16   a  shows overlay estimation using 9 targets (as that shown in  FIG. 5   a  or  FIG. 5   b ) for each overlay component. 
   The gross overlay detect method described in the previous section can be directly applied to the case with multiple overlay targets. A simple way to implement it is to make two of the targets have the reflective symmetry described in Equation (1). The reflective difference of these two targets can be used to calculate the gross overlay indicator.  FIG. 16   b  illustrates the GOI calculated from 2 of the 9 overlay targets using the Karhunen-Loeve expansion method described in the previous section. By proper choosing the offset biases ΔX i &#39;s, more than one pair of reflection symmetry targets can be formed in a set of multiple overlay targets. A GOI with higher discrimination can be derived from these pairs of targets. 
   The multiple overlay targets  300  can be measured one-by-one in a series by any optical instrument suitable for scatterometry, such as an ellipsometer or reflectometer. The spectrum can be measured over different wavelengths, polar angles of incidence, azimuthal angles of incidence, or polarization states, et cetera. These multiple targets can also be measured simultaneously using an apparatus described in U.S. Provisional Application No. 60/519,345. For instance, imaging spectrometer  1700  in  FIG. 17  can be used for the measurement purpose. It resolves the reflection from the wafer in the x-y plane of the wafer and in wavelength. In other words, the target is imaged at several wavelengths, either sequentially or simultaneously. The apparatus shown in  FIG. 15  acquires images at different wavelengths sequentially. 
   Light from a broadband source  1702  is collected and collimated by optics  1704  before being directed by a beam splitter  1706  to an objective  1708 . Objective  1708  then focuses the collimated illumination on a sample  1710 . An aperture  1712  controls the numerical aperture of illumination (“illumination NA”) and is preferably at a focal plane of objective  1708 . A filter  1714  is positioned to select the wavelength at which the sample  1710  is illuminated. Preferably, filter  1714  is positioned in the collimated beam, downstream of optics  1704  and is preferably a band-pass interference filter with a bandwidth on the order of 10 nm. Other filters such as long-pass filters are also possible. Filter  1714  is mounted on a filter wheel that supports a multitude of filters with different pass bands. A motor  1716  rotates the filter wheel under the control of controller-processor  1718 . 
   Sample  1710  includes an overlay target of the type described above. The overlay target is fully illuminated and the size of the illuminated spot on sample  1710  is controlled by a field stop  1720  in the illumination optics at a plane that is conjugate to the wafer. Aperture  1712  determines the collection NA for reflected light collected by objective  1708 , and is preferably at a focal plane of objective  1708 . Light collected by objective  1708  is imaged onto a detector array  1722  by an imager  1724 . Optical elements  1704 ,  1708  and  1724  are schematically shown as single lenses in  FIG. 17 , but in practice they are compound reflective or refractive elements. The output of detector array  1722  representing the image of the sample  1710  is digitized by electronics  1726  and transmitted to controller-processor  1718 . An algorithm that runs on  1718  processes images of the sample  1710  and returns overlay (Δx, Δy) and gross overlay indicator GOI for the x- and y-overlay. For cases with extremely large overlay error in one direction, the GOI for the other direction may also exceed threshold  162  shown in  FIG. 14 . 
   Apertures  1712  and  1720  control the imaging resolution of spectrometer  1700 . The resolution is selected so that overlay targets  302 X 1 , etc. (shown in  FIG. 13 ) are resolved but their unit cells are not. The overlay targets are typically 10 μm×10 μm, and their units cells are typically sub-micron. Microscope objectives with NA larger than 0.5 can resolve 10 μm×10 μm features at visible wavelengths. It is preferable to limit the illumination NA with aperture  1720  to less than 0.1, for two reasons. The first reason is that the spectral (color) contrast between the overlay targets of different offsets is maximized by reducing the sum of the illumination and detection numerical apertures, because the diffracted colors of a grating depends on the angle of illumination and detection. The second benefit of a small illumination aperture is reduced cross talk between adjacent gratings stacks within the overlay target being imaged. The image of a grating stack has a diffraction tail in the image plane that extends beyond the bounds of the grating stack. The diffraction tails fall off faster when illumination NA is smaller than the collection NA. 
   Beam splitter  1706  splits the filtered illumination output of filter  1714  into a test beam and a monitor beam. A photodetector  1728  measures the intensity of the monitor beam, which is indicative of the intensity of the test beam that illuminates the targets. Detector array  1722  and photo-detector  1728  preferably collect photons over the same time interval, where sequential collection is possible. Alternatively, detector  1728  could be an array into which the monitor beam is imaged (imaging optics not shown in  FIG. 17 ). In yet another alternative, detector  1728  and array  1722  could be part of the same physical detector array, with mirrors and optics, not shown. The exposure (integration) time is preferably different for each setting of the filter. At the wavelengths where the light source is weaker or the detectors have smaller quantum efficiency, the integration time can be longer.