Abstract:
The present invention is directed towards a method for determining deformation parameters that a patterned device would undergo to minimize dimensional variations between a recorded pattern thereon and a reference pattern, the method including, inter alia, comparing spatial variation between features of the recorded pattern with respect to corresponding features of the reference pattern; and determining deformation forces to apply to the patterned device to attenuate the dimensional variations, with the forces having predetermined constraints, wherein a summation of a magnitude of the forces is substantially zero and a summation of moment of the forces is substantially zero.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     The present application claims priority to U.S. Provisional Application No. 60/788,811, filed on Apr. 3, 2006, entitled “Solutions for Force Combinations for Template Deformation using Nullspace Method and Optimization Techniques” and U.S. Provisional Application No. 60/788,812, filed on Apr. 3, 2006, entitled “Template Geometric Design for Max and Match Nano Imprint Lithography Processes”; and is a Continuation-in-Part of U.S. patent application Ser. No. 11/143,076, filed on Jun. 2, 2005, entitled “System and Method for Improvement of Alignment and Overlay for Microlithography” which claims priority to U.S. Provisional Application No. 60/576,570, filed on Jun. 3, 2004 entitled “System and Method for Improvement of Alignment and Overlay for Microlithography”, all of which are incorporated herein by reference. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT  
       [0002]     The United States government has a paid-up license in this invention and the right in limited circumstance to require the patent owner to license others on reasonable terms as provided by the terms of 70NANB4H3012 awarded by National Institute of Standards (NIST) ATP Award and N66001-06-C-2003 awarded by Nanoimprint Lithography Manufacturing Scale (NIMS) Award. 
     
    
     BACKGROUND INFORMATION  
       [0003]     Nano-fabrication involves the fabrication of very small structures, e.g., having features on the order of nanometers or smaller. One area in which nano-fabrication has had a sizeable impact is in the processing of integrated circuits. As the semiconductor processing industry continues to strive for larger production yields while increasing the circuits per unit area formed on a substrate, nano-fabrication becomes increasingly important. Nano-fabrication provides greater process control while allowing increased reduction of the minimum feature dimension of the structures formed. Other areas of development in which nano-fabrication has been employed include biotechnology, optical technology, mechanical systems and the like.  
         [0004]     An exemplary nano-fabrication technique is commonly referred to as imprint lithography. Exemplary imprint lithography processes are described in detail in numerous publications, such as United States patent application publication 2004/0065976 filed as U.S. patent application Ser. No. 10/264,960, entitled, “Method and a Mold to Arrange Features on a Substrate to Replicate Features having Minimal Dimensional Variability”; United States patent application publication 2004/0065252 filed as U.S. patent application Ser. No. 10/264,926, entitled “Method of Forming a Layer on a Substrate to Facilitate Fabrication of Metrology Standards”; and U.S. Pat. No. 6,936,194, entitled “Functional Patterning Material for Imprint Lithography Processes,” all of which are assigned to the assignee of the present invention.  
         [0005]     The imprint lithography technique disclosed in each of the aforementioned United States patent application publications and United States patent includes formation of a relief pattern in a polymerizable layer and transferring a pattern corresponding to the relief pattern into an underlying substrate. The substrate may be positioned upon a stage to obtain a desired position to facilitate patterning thereof. To that end, a mold is employed spaced-apart from the substrate with a formable liquid present between the mold and the substrate. The liquid is solidified to form a patterned layer that has a pattern recorded therein that is conforming to a shape of the surface of the mold in contact with the liquid. The mold is then separated from the patterned layer such that the mold and the substrate are spaced-apart. The substrate and the patterned layer are then subjected to processes to transfer, into the substrate, a relief image that corresponds to the pattern in the patterned layer. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0006]      FIG. 1  is a simplified side view of a lithographic system having a template spaced-apart from a substrate;  
         [0007]      FIG. 2  is a simplified side view of the substrate shown in  FIG. 1 , having a patterned layer positioned thereon;  
         [0008]      FIG. 3  is a simplified plan view of a holder for the template, both shown in  FIG. 1 , in accordance with the present invention;  
         [0009]      FIG. 4  is a simplified plan view of the template shown in  FIG. 1 , having a plurality of alignment marks;  
         [0010]      FIG. 5  is a simplified plan view showing distortion vectors determined in accordance with the present invention;  
         [0011]      FIG. 6  is a top down view of the template shown in  FIG. 1 ;  
         [0012]      FIG. 7  is a side view of the template shown in  FIG. 1 ; and  
         [0013]      FIG. 8  is a exploded view of a portion of the template shown in  FIG. 7 . 
     
    
     DETAILED DESCRIPTION  
       [0014]     Referring to  FIG. 1 , a system  10  to form a relief pattern on a substrate  12  is shown. Substrate  12  may be coupled to a substrate chuck  14 . Substrate chuck  14  may be any chuck including, but not limited to, vacuum, pin-type, groove-type, or electromagnetic, as described in U.S. Pat. No. 6,873,087 entitled “High-Precision Orientation Alignment and Gap Control Stages for Imprint Lithography Processes,” which is incorporated herein by reference. Substrate  12  and substrate chuck  14  may be supported upon a stage  16 . Further, stage  16 , substrate  12 , and substrate chuck  14  may be positioned on a base (not shown). Stage  16  may provide motion about the x and y axes.  
         [0015]     Spaced-apart from substrate  12  is a template  18  having first and second opposed sides  20  and  22 . Positioned on first side  20  of template  18  is a mesa  24  extending therefrom towards substrate  12  with a patterning surface  26  thereon. Further, mesa  24  may be referred to as a mold  24 . Mesa  24  may also be referred to as a nanoimprint mold  24 . In a further embodiment, template  18  may be substantially absent of mold  24 . Template  18  and/or mold  24  may be formed from such materials including but not limited to, fused-silica, quartz, silicon, organic polymers, siloxane polymers, borosilicate glass, fluorocarbon polymers, metal, and hardened sapphire. In a further embodiment, template  18  and mold  24  may be commonly referred to as patterned device  28 . As shown, patterning surface  24  comprises features defined by a plurality of spaced-apart recesses  30  and protrusions  32 . However, in a further embodiment, patterning surface  24  may be substantially smooth and/or planar. Patterning surface  24  may define an original pattern that forms the basis of a pattern to be formed on substrate  12 .  
         [0016]     Template  18  may be coupled to a template chuck (not shown), the template chuck (not shown) being any chuck including, but not limited to, vacuum, pin-type, groove-type, or electromagnetic, as described in U.S. Pat. No. 6,873,087 entitled “High-Precision Orientation Alignment and Gap Control Stages for Imprint Lithography Processes.” Template  18  may be coupled to an imprint head  34  to facilitate movement of template  18  and mold  26 . In a further embodiment, the template chuck (not shown) may be coupled to imprint head  34  to facilitate movement of template  18  and mold  26 .  
         [0017]     System  10  further comprises a fluid dispense system  36 . Fluid dispense system  36  may be in fluid communication with substrate  12  so as to deposit a polymeric material  38  thereon. System  10  may comprise any number of fluid dispensers and fluid dispense system  36  may comprise a plurality of dispensing units therein. Polymeric material  38  may be positioned upon substrate  12  using any known technique, e.g., drop dispense, spin-coating, dip coating, chemical vapor deposition (CVD), physical vapor deposition (PVD), thin film deposition, thick film deposition, and the like. As shown, polymeric material  38  may be deposited upon substrate  12  as a plurality of spaced-apart droplets  40 . Typically, polymeric material  38  is disposed upon substrate  12  before the desired volume is defined between mold  24  and substrate  12 . However, polymeric material  38  may fill the volume after the desired volume has been obtained.  
         [0018]     Referring to  FIGS. 1 and 2 , system  10  further comprises a source  42  of energy  44  coupled to direct energy  44  along a path  46 . Imprint head  34  and stage  16  are configured to arrange mold  24  and substrate  12 , respectively, to be in superimposition and disposed in path  46 . Either imprint head  34 , stage  16 , or both vary a distance between mold  24  and substrate  12  to define a desired volume therebetween such that mold  24  contacts polymeric material  38  and the desired volume is filled by polymeric material  38 . More specifically, polymeric material  38  of droplets  40  may ingress and fill recesses  30  of mold  24 . After the desired volume is filled with polymeric material  38 , source  42  produces energy  44 , e.g., broadband ultraviolet radiation that causes polymeric material  38  to solidify and/or cross-link conforming to the shape of a surface  48  of substrate  12  and patterning surface  26 , defining a patterned layer  50  on substrate  12 . Patterned layer  50  may comprise a residual layer  52  and a plurality of features shown as protrusions  54  and recessions  56 . System  10  may be regulated by a processor  58  that is in data communication with stage  16 , imprint head  34 , fluid dispense system  36 , and source  42 , operating on a computer readable program stored in memory  60 .  
         [0019]     Referring to  FIGS. 1 and 3 , system  10  further comprises an actuator system  62  surrounding patterned device  28  to facilitate alignment and overlay registration. To that end, actuation system  62  includes a plurality of actuators  64  coupled between a frame  66  and patterned device  20 . Each of actuators  64  are arranged to facilitate generation of a force on one of the four sides  70 ,  72 ,  74  and  76  of patterned device  28 .  
         [0020]     As shown, actuator system  62  comprises sixteen actuators  64   a - 62   p  coupled to patterned device  20 . More specifically, coupled to side  70  of template  18  are actuators  64   a - 64   d ; coupled to side  72  of template  18  are actuators  64   e - 64   h ; coupled to side  74  of template  18  are actuators  64   i - 64   l ; and coupled to side  76  of template  18  are actuators  64   m - 64   p . In a further embodiment, template  18  may have any number of actuators  64  coupled thereto and may have differing number of actuators  64  coupled to each side of template  18 . Template  18  may have any configuration and number of actuators  64  positioned on sides  70 ,  72 ,  74 , and  76  thereof. Actuation system  62  may be in data communication with processor  58 , operating on a computer readable program stored in memory  60 , to control an operation thereof, and more specifically, generate control signals that are transmitted to actuators  64  of actuation system  62 .  
         [0021]     Actuation system  62  facilitates alignment and overlay registration by selectively deforming patterned device  20 . This facilitates correcting various parameters of the pattern shape, i.e., magnification characteristics, skew/orthogonality characteristics, and trapezoidal characteristics. Magnification characteristics may be magnification error, such as where the overall pattern changes from a square shape to a rectangular shape. Skew/orthogonality characteristics may be skew/orthogonality error where adjacent edges form an oblique or obtuse angle with respect to one another instead of an orthogonal angle. Trapezoidal characteristics may be trapezoidal error where as in where a square/rectangular assumes the shape of a trapezium, with trapezium including a trapezoid. To control the pattern shape, patterned device  20  may be selectively deformed by actuators  64  to minimize, if not cancel, the distortions present, thereby reducing overlay errors. To that end, patterned device  20  is inspected employing known image placement or image registration systems, e.g., LMS IPRO3 available from Leica Microsystems of Bannockburn, Ill.  
         [0022]     Referring to  FIGS. 1, 3  and  4 , measured information  78  concerning the location of the features on patterned device  20  would be mapped into memory  60 . The features that measured information  78  represents are reference marks present on patterned device  20  to facilitate overlay and alignment techniques. The features may include any known alignment mark, such as box-in-box; cross-in-cross and/or vernier scale marks, referred to as overlay features. The overlay features are usually positioned at differing regions of patterned device  20  as room permits and are arranged in a polygonal, if not rectangular grid. As shown in  FIG. 4 , alignment marks  80  are positioned in the corners of mold  24 .  
         [0023]     Referring to  FIGS. 1 and 3 , loaded into memory  60  would be reference information  82  against which measured information  78  would be compared. Reference information  82  would include information concerning an optimal, or desired, location of overlay features and, therefore, the pattern on patterned devices  20 . This information may be obtained from an existing reference patterned device (not shown) that may be employed as a standard against which patterned device  20  is measured. Alternatively, reference information  82  may be obtained from a GDS file that is employed to form the pattern on patterned device  20 . Considering that errors, or distortion, in the pattern on the patterned device  20  may be attributed to the writing and etch processes used to form patterned device  20 , computer data of the type employed in computer aided design software may provide reference information  82  with the most accurate reflection of the optimal pattern. Exemplary computer data is that employed by CATS™ software sold by Synopsis, Inc., of Mountain View, Calif.  
         [0024]     Referring to  FIGS. 3 and 5 , also stored in memory  60  is a routine  84  that facilitates comparison of measured information  78  with reference information  82 . Routine  84  includes X and Y positional variations between features in measured information  78  with respect to corresponding features in reference information  82  and generates image placement variation data shown in below in Table 1:  
                                           TABLE 1                           Image Placement Variation            Point   X (μm)   Y (μm)                    1   0.01   −0.012       2   0   −0.003       3   −0.003   −0.001       4   0.013   −0.013       5   0.016   −0.016       6   0.018   −0.014       7   0.012   −0.012       8   −0.001   −0.001       9   −0.012   −0.004       10   −0.001   −0.007       11   0.005   −0.014       12   0.009   −0.013       13   −0.004   −0.004       14   −0.017   0.005       15   −0.02   0.01       16   −0.01   −0.002       17   −0.007   −0.008       18   0   −0.007       19   −0.008   0.007       20   −0.022   0.013       21   −0.024   0.017       22   −0.011   0.012       23   −0.005   0       24   0.001   0       25   0.01   −0.001       26   −0.006   0.006       27   −0.006   0.012       28   0.003   0       29   0.012   −0.006       30   0.016   −0.005       31   0.011   −0.01       32   0.002   −0.001       33   −0.005   0.004       34   0.011   −0.003       35   0.016   −0.011       36   0.019   −0.006                  
 
         [0025]     From the data in Table 1, distortion vectors  86  are generated. Distortion vectors  86  are vectorized representations of the differences in spatial location of the overlay features associated with measured information  78  with respect to corresponding overlay features associated with reference information  82 . As a result, distortions vectors  86  comprise data  88 , mapped into memory  60 , concerning a set of spatial locations  90  of features of the pattern on patterned device  20 . An exemplary distortion vector  86  generated from image placement variation data would be mapped into memory as a series starting with feature  1  and ending with feature  36  as identifying the x and y variations of each of the features as follows: {0.01, −0.012, 0, −0.003, . . . 0.019, and −0.006}. Distortion vectors  86  may further represent, inter alia, magnification errors, orthogonal errors, and other errors.  
         [0026]     Spatial locations  90  represent the spatial location of the overlay features on patterned device  20 . Data  88  includes directional and magnitude characteristics of the differences between measured information  78  and reference information  82 . Specifically, data  88  includes information concerning the distance, along two orthogonal axes, between spatial locations  90  of each of the overlay features on patterned device  20  with respect to spatial locations of the corresponding overlay feature of the optimal/desired pattern.  
         [0027]     To that end, actuator system  62  facilitates alignment and overlay registration by selectively deforming patterned device  20  by applying forces upon patterned device  20  by actuators  64 . The forces upon patterned device  20  by actuators  64  must satisfy the following equilibrium and moment conditions: 
 
ΣF x =0;  (1) 
 
ΣF y =0; and  (2) 
 
ΣM z =0;  (3) 
 
         [0028]     where F x  are forces in the x direction, F y  are forces in the y direction and M z  are moments about the z axis. To that end, equations (1), (2), and (3) may be modeled as follows: 
 
[ K]×{f}={ 0}  (4) 
 
         [0029]     Matrix [K] may be determined by the spatial relationship between actuators  64  and patterned device  20 . In the present example,  
               [   K   ]     =     
     ⁢     [           x   ⁢           ⁢   1           x   ⁢           ⁢   2           x   ⁢           ⁢   3           x   ⁢           ⁢   4           x   ⁢           ⁢   5           x   ⁢           ⁢   6           x   ⁢           ⁢   7           x   ⁢           ⁢   8           x   ⁢           ⁢   9           x   ⁢           ⁢   10           x   ⁢           ⁢   11           x   ⁢           ⁢   12           x   ⁢           ⁢   13           x   ⁢           ⁢   14           x   ⁢           ⁢   15           x   ⁢           ⁢   16               y   ⁢           ⁢   1           y   ⁢           ⁢   2           y   ⁢           ⁢   3           y   ⁢           ⁢   4           y   ⁢           ⁢   5           y   ⁢           ⁢   6           y   ⁢           ⁢   7           y   ⁢           ⁢   8           y   ⁢           ⁢   9           y   ⁢           ⁢   10           y   ⁢           ⁢   11           y   ⁢           ⁢   12           y   ⁢           ⁢   13           y   ⁢           ⁢   14           y   ⁢           ⁢   15           y   ⁢           ⁢   16               m   ⁢           ⁢   1           m   ⁢           ⁢   2           m   ⁢           ⁢   3           m   ⁢           ⁢   4           m   ⁢           ⁢   5           m   ⁢           ⁢   6           m   ⁢           ⁢   7           m   ⁢           ⁢   8           m   ⁢           ⁢   9           m   ⁢           ⁢   10           m   ⁢           ⁢   11           m   ⁢           ⁢   12           m   ⁢           ⁢   13           m   ⁢           ⁢   14           m   ⁢           ⁢   15           m   ⁢           ⁢   16           ]             (   5   )             
 
         [0030]     where x i , y i , and m i  are the coefficients of f i  in equations (1), (2), and (3), respectively. To that end, in the present example, the matrix [K] may be defined as follows:  
               [   K   ]     =     
     ⁢     [         1                   1                   1                   1                   0                   0                   0                   0                     -   1                       -   1                       -   1                       -   1                     0                   0                   0                   0           0                   0                   0                   0                   1                   1                   1                   1                   0                   0                   0                   0                     -   1                       -   1                       -   1                       -   1               -   3                       -   1                     1                   3                     -   3                       -   1                     1                   3                     -   3                       -   1                     1                   3                     -   3                       -   1                     1                   3         ]             (   6   )             
 
         [0031]     The force vector {f} is the forces associated with actuators  64 . In the present example, the force vector {f} may be defined as follow: 
 
{f}={f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13, f14, f15, f16} T   (7) 
 
         [0032]     where f1 is the force associated with actuator  64   a ; f2 is the force associated with actuator  64   b ; f3 is the force associated with actuator  64   c ; f4 is the force associated with actuator  64   d ; f5 is the force associated with actuator  64   e ; f6 is the force associated with actuator  64   f , f7 is the force associated with actuator  64   g ; f8 is the force associated with actuator  64   h ; f9 is the force associated with actuator  64   i ; f10 is the force associated with actuator  64   j ; f11 is the force associated with actuator  64   k ; f12 is the force associated with actuator  64   l ; f13 is the force associated with actuator  64   m ; f14 is the force associated with actuator  64   n ; f15 is the force associated with actuator  64   o ; and f16 is the force associated with actuator  64   p.    
         [0033]     To that end, from equation (4), the nullspace basis vectors may be determined. In the present example, there are 16 independent forces from actuators  64  and there are 3 equilibrium conditions, resulting in 13 independent force vectors. To that end, employing equations (6) and (7) with equation (4), the orthonormal basis of the matrix [K] may be determined using well-known linear algebraic methods and may be defined as follows:  
               [   nK   ]     =     [         1       2       1       2       3         -   2           -   1         0       1       3         -   2           -   1         0             -   2           -   3           -   1           -   2           -   3         3       2       1       0       3       2       1       0           1       0       0       0       0       0       0       0       0       0       0       0       0           0       1       0       0       0       0       0       0       0       0       0       0       0           0       0         -   1           -   1           -   1         0       0       0       0       1       1       1       1           0       0       1       0       0       0       0       0       0       0       0       0       0           0       0       0       1       0       0       0       0       0       0       0       0       0           0       0       0       0       1       0       0       0       0       0       0       0       0           0       0       0       0       0       1       0       0       0       0       0       0       0           0       0       0       0       0       0       1       0       0       0       0       0       0           0       0       0       0       0       0       0       1       0       0       0       0       0           0       0       0       0       0       0       0       0       1       0       0       0       0           0       0       0       0       0       0       0       0       0       1       0       0       0           0       0       0       0       0       0       0       0       0       0       1       0       0           0       0       0       0       0       0       0       0       0       0       0       1       0           0       0       0       0       0       0       0       0       0       0       0       0       1         ]             (   8   )             
 
         [0034]     To that end, each column of the matrix [nK] is an independent force vector and may be referred to as λ 1 , λ 2 , . . . ,λ 13 . Force vectors λ 1 , λ 2 , . . . ,λ 13  may be referred to as the nullspace basis vectors of equation (4). More specifically, the matrix [nK] may be defined as follows: 
 
[nK] 16×13 =[λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 , λ 8 , λ 9 , λ 10 , λ 11 , λ 12 , λ 13 , λ 14 , λ 15 , λ 16 ]  (9) 
 
         [0035]     As a result, any force vector {f} may be defined as follow: 
 
{ f}={p   1 λ 1   +p   2 λ 2   +p   3 λ 3   +p   4 λ 4   +p   5 λ 5   +p   6 λ 6   +p   7 λ 7   +p   8 λ 8   +p   9 λ 9   +p   10 λ 10   +p   11 λ 11   +p   12 λ 12   +p   13 λ 13   +p   14 λ 14   +p   15 λ 15   +p   16 λ 16 }  (10) 
 
         [0036]     wherein p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 , p 8 , p 9 , p 10 , p 11 , p 12 , p 13 , p 14 , p 15 , and p 16  are the scalar coefficients of λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 , λ 8 , λ 9 , λ 10 , λ 11 , λ 12 , λ 13 , λ 14 , λ 15 , and λ 16 , respectively.  
         [0037]     Referring to  FIGS. 3 and 5 , to that end, processor  58  operates on routine  84  to process data concerning distortion vectors  86  and generate signals that are sensed by actuators  64  to selectively deform patterned device  20  and attenuate, if not abrogate, differences between measured information  78  and reference information  82 , thereby minimizing overlay variations between the pattern on patterned device  20  with respect to the optimal/desired pattern. The distance between the overlay features associated with measured information  78  from the corresponding overlay features associated with reference information  82  is minimized by creating translational movement of spatial locations  90 . To that end, routine  84  determines the loads to be applied by actuators  64  in order to selectively deform patterned device  20  solving an inverse transform function as follows: 
 
 [A]{p}={u},   (11) 
 
         [0038]     where [A] represents the compliance matrix to be specified for patterned device  20 ; {p} comprises weighting coefficients for the force vectors λ 1 , λ 2 , . . . ,λ 13 ; and {u} represents spatial translation of features associated with measured information  78  must undergo in order to match the spatial location of the corresponding feature in reference information  82 , i.e., {u} represents an additive inverse of the distortion vectors  86 .  
         [0039]     One manner in which to determine the compliance matrix [A] employs finite element analysis (FEA). To that end, an FEA model of patterned device  20 , referred to as modeled device  96  is generated and stored in memory  60 , using any known modeling technique, such as software sold under the trade name Pro/Engineer™ 2001 and finite element solver software sold under the trade name Pro/Mechanica™ 2001.  
         [0040]     Employing FEA, obtained are measurements of the spatial displacement of each of a plurality of data points  98  of the modeled device  96  in response to simulated loading of force vectors λ i  by actuators  64 . Data points  98  represent the spatial location of the overlay features of the pattern on modeled device  96 . To obtain useful information, the overlay features with which data points  98  are associated correspond to same features of patterned device  20  that are associated with spatial locations  90 . In the present example, each of data points  98  is associated with one of spatial locations  90 , such that each of data points  98  corresponds to one of spatial locations  90  that differs from the spatial locations  90  associated with the remaining data points  98 . Once compliance matrix [A] is determined, vector {p} is determined from equation (11), and thus force vector {f} is determined from equation (10). Signals are generated by processor  58  to cause actuators  64  to apply the requisite loads to patterned device  20  that are a function of the force vector {f}. In this fashion, distortions in the patterned device  20  are minimized, if not abrogated.  
         [0041]     For each of data points  98  a displacement along the x and y axes may be defined as follows: 
 
 X   n   =p   1   x   1n   +p   2   x   2n   + . . . +p   m   x   mn ; and  (12) 
 
 Y   n   =p   1   y   1n   +p   2   y   2n   + . . . +p   m   y   mn ;  (13) 
 
         [0042]     where p i  is the scalar co-efficient from force vector λ i , n denotes the data point and x in , y in  represents the movement of a data point n along x, y directions in terms of millimeters/Newtons in response to loading with force vector λ i . In the present example, n is an integer from 1 to 4 and i is an integer from 1 to 8. An exemplary compliance matrix [A] based upon the conditions set forth in equations 1-3 and 12-13 for 4 overlay features is as follows:  
             A   =     le   -     5   ×     [           -   0.0350           -   0.3316           -   0.6845           -   0.4965         0.4924       0.2550       0.2025         -   0.5387             0.4923       0.2551       0.2028         -   0.5388           -   0.0349           -   0.3316           -   0.6845           -   0.4957             0.0311       0.3313       0.6848       0.4965       0.5387         -   0.2034           -   0.2557           -   0.4926             0.4930       0.2550       0.2026         -   0.5389           -   0.4989           -   0.6846           -   0.3310           -   0.0323               -   0.4992           -   0.6846           -   0.3310           -   0.0329         0.4931       0.2549       0.2025         -   0.5388             0.5385         -   0.2033           -   0.2556           -   0.4925         0.0313       0.3313       0.6848       0.4973           0.4938       0.6847       0.3318       0.0333       0.5393         -   0.2036           -   0.2560           -   0.4925             0.5393         -   0.2034           -   0.2559           -   0.4927         0.4941       0.6846       0.3319       0.0338         ]                 (   14   )             
 
         [0043]     Knowing compliance matrix [A], routine  84  may determine the magnitude of the forces to be generated {f} by actuators by solving for {p}. Specifically, routine  84  solves the force vector {p} from equation (11) as follows: 
 
 {p}=[A]   −1   {u},   (15) 
 
         [0044]     were [A] a square matrix. Were [A] not a square matrix, equation (15) is expressed as follows: 
 
 {p}={A   T   A}   −1   A   T   {u},   (16) 
 
         [0045]     where A T  is the transpose matrix of compliance matrix [A].  
         [0046]     To solve for {p} over the infinity norm, equation (11) may be reforumulated as follows: 
 
 [A]{p}−{u}={e}.   (17) 
 
         [0047]     Hence the problem becomes finding {p} such that the error vector {e} is minimized. [A] is the compliance matrix described above. Routine  84  may minimize the error vector {e} over the infinity norm given by the following: 
 
max(|[ A]{p}−{u} |)  (18) 
 
         [0048]     The reason for selecting to minimize the infinity norm is that it is believed that the magnitude of the absolute value of overlay error that determines a pattern layer&#39;s usefulness. As mentioned above, the maximum overlay error is believed to be less than ⅓ rd  the minimum feature size of the pattern, for the pattern layer to be functional. Hence, it is desired to have routine  84  minimize this maximum absolute error, i.e., the infinity norm as follows: 
 
Min(max|[ A]{p}−{u }|).  (19) 
 
         [0049]     Objective function (19) is convex piecewise linear in terms of the decision variables, i.e. p i . A convex piecewise linear function is, by definition, non-linear. The domain of differences among the set may, therefore, include several local minima. To that end, routine  84  may be required to undertake several iterations with a range of trial/guess starting vectors and to implement a directional search routine. A typical iterative procedure in accordance with the present invention commences from an initial point where a function value is calculated. The procedure proceeds to solutions in which the function has lower values. This results in routine  48  computing information concerning the function until convergence is identified. Routine  48  ends the procedure at a minimum value where no further reduction in the functional value is identified within the tolerance.  
         [0050]     Any known iterative directional search techniques like Newton-Raphson Methods, Conjugate Gradient methods, Quasi-Newton Methods may be employed to get the optimum {p}. One manner in which to implement these techniques is with Microsoft EXCEL, stored in memory  60  and operated on by processor  40  using standard operating systems such as WINDOWS®, available from Microsoft Corporation. The data obtained from the finite element analysis, discussed above, is collated in a matrix form and entered, and the appropriate relationships between the matrices are established, e.g., in accordance with equation (11).  
         [0051]     One manner in which to improve the calculation of {p} is by converting the non-linear formulation (19) into a linear problem. To that end, equation (17) is substituted into equation (19). This allows routine  84  to express equation (19) for the series of data  88 , as follows: 
 
Minimize(Maximum (|e 1 |, |e 2 | . . . |e n |)),  (20) 
 
         [0052]     where, e i  are the elements of error vector {e}. By routine  84  expanding equation (20), obtained is the following: 
 
Minimize(Maximum (e 1 , −e 1 , e 2 , −e 2 , . . . e n , −e n )).  (21) 
 
         [0053]     By routine  84  substituting a variable w for (Maximum e 1 , −e 1 , e 2 , −e 2 , . . . , e n , −e n ), equation (21) may be defined as follows: 
 
Minimize (w).  (22) 
 
         [0054]     Providing the following constraints: 
 
w≧e i   (23) 
 
w≧−e i .  (24) 
 
         [0055]     That is, routine  84  may solve non-linear equation (19) formulated as equation (22) with the following constraints: 
 
 w≧[A]{p}−{u }; and  (25) 
 
 w≧{u}−[A]{p}.   (26) 
 
         [0056]     An advantage with reformulating equation (19) as a linear problem is that the linear problem is likely to converge to the global minimum in a finite number of steps, under pseudo-polynomial algorithms like the Simplex method. This minimizes the computational power required to have routine  84  determine the global minimum. Iterative search techniques can however still be used. Also, most often non-linear programming techniques converge to the local optima, unless careful checks are implemented. This was noticed to happen when EXCEL tried to solve the non-linear problem. As a result, reformulated equation (19) as a linear problem facilitates obtaining the minimum among the set of data  88  while minimizing the computational power required.  
         [0057]     Referring to  FIGS. 6-8 , patterned device  20  is shown. To that end, it may be desired for patterned device  20  to have dimensions to facilitate magnification and distortion thereof, with the dimensions for geometric parameters of patterned device  20  shown below in Table 2.  
                                                                                                           TABLE 2                           Geometric Specifications for Patterned Device                Geometric Parameter       Target   Tolerance                            L 1     64.95   mm   ±0.05           L 2     64.95   mm   ±0.05                Q 1     90°   ±1E−3   Radians           Q 2      0°   ±1E−3   Radians                W 1     0.4   mm   ±0.2   mm           W 2     0.4   mm   ±0.2   mm           d 1 -d 2     0   mm   ±0.05   mm           d 3 -d 3     0   mm   ±0.05   mm           T   6.35   mm   ±0.1   mm           R   1.5   mm   ±1   mm                      
 
         [0058]     The geometric parameters may be defined as follows: L 1  may be defined between side  70  and  74 ; L 2  may be defined between side  72  and  76 ; Q 1  may be defined as the angle between any two sides of sides  70 ,  72 ,  74 , and  76 ; Q 2  may be defined as the angle between any side of sides  70 ,  72 ,  74 , and  76  and a plane  100  perpendicular to a plane  102  in which patterned device  20  lays; w 1  may be defined as the width of a first edge surface  106  defined between first surface  20  and a side of sides  70 ,  72 ,  74 , and  76 ; w 2  may be defined as the width of a second edge surface  108  defined between second surface  22  and a side of sides  70 ,  72 ,  74 , and  76 ; d 1 -d 4  may be defined between mold  24  and a side of sides  70 ,  72 ,  74 , and  76 ; T may be defined as the thickness of template  18  between first and second opposed sides  20  and  22 ; and R may be defined as the radius of curvature of template  18  between any two sides of sides  70 ,  72 ,  74 , and  76 .  
         [0059]     The embodiments of the present invention described above are exemplary. Many changes and modifications may be made to the disclosure recited above, while remaining within the scope of the invention. For example, the method described above is discussed with respect to attenuating, if not eliminating overlay error resulting from both image placement and other characteristics, such as magnification, orthogonality and trapezoidal errors in the case of imprint lithography. Were magnification, orthogonality and/or trapezoidal not present or corrected by other methods, for example in the case of optical lithography, the invention described above can be used to minimize the uncorrected overlay errors. The scope of the invention should, therefore, not be limited by the above description, but instead should be determined with reference to the appended claims along with their full scope of equivalents.