Patent Publication Number: US-2012038899-A1

Title: Exposure apparatus and alignment error compensation method using the same

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
     This application claims the benefit of Korean Patent Application No. 2010-0078481, filed on Aug. 13, 2010 in the Korean Intellectual Property Office, the disclosure of which is incorporated herein by reference in its entirety. 
     BACKGROUND 
     1. Field 
     Embodiments of the present invention relate to a method of estimating and compensating a slippage amount generated during rotation to align a mask and a substrate in an exposure apparatus. 
     2. Description of the Related Art 
     Generally, a method of forming a pattern on a substrate (or a semiconductor wafer) constituting a liquid crystal display (LCD), a plasma display panel (PDP), or a flat panel display (FPD) is performed as follows. First, a pattern material is applied to a substrate and is selectively exposed using a photomask such that a pattern material part, chemical properties of which have been changed, or the remaining part is selectively removed, thereby forming a pattern. 
     However, as substrate size is gradually increased and pattern precision is also gradually increased, a maskless exposure process for forming a pattern on a substrate (or a semiconductor wafer) without using a photomask has been developed. The maskless exposure eliminates costs for manufacturing/cleaning/maintaining a mask, allows a panel to be freely designed, and eliminates mask manufacturing time, thereby reducing lead time. Since mask defects are not an issue, the effects of mask defects on a fabrication process are eliminated. Because a hybrid layout is used, production flexibility is increased. 
     A plurality of layers is stacked on a substrate. The layers form a pattern on the substrate through an exposure process. The higher the pattern precision, the higher the number of layers each having the pattern. When a plurality of layers is stacked on one substrate, it may be desirable to align a mask and a substrate (or a semiconductor wafer) before exposure. The alignment between the mask and the substrate may be performed through a motion having three degrees of freedom (X, Y, θ). 
     SUMMARY 
     At least one example embodiment provides a method of estimating and compensating a slippage amount due to radial runout generated during planar rotation (θ) to align a mask and a substrate. 
     Additional aspects of the embodiments will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention. 
     In one embodiment, an alignment error compensation method to align a mask and a substrate using a stage to transfer the substrate in at least one direction and a rotational body to rotate the substrate includes measuring a position of a fiducial mark during rotation of the rotational body to align the mask and the substrate, acquiring a position of a center of rotation of the rotational body using the measured position of the fiducial mark, estimating a slippage amount of the rotational body by determining a relative difference in position between the position of the center of rotation of the rotational body and the measured position of the fiducial mark, and compensating a movement amount of the stage based on the estimated slippage amount to align the mask and the substrate. 
     The fiducial mark may include at least one fiducial mark provided at the rotational body or at least one fiducial mark provided at the substrate. 
     The alignment between the mask (including a physical mask and a virtual mask) and the substrate may be performed through planar motions having three degrees of freedom (X, Y, θ). 
     The measuring may measure the position of the alignment mark before and after planar rotation (θ), the acquiring may acquire the position of the center of rotation before and after the planar rotation (θ), and the estimating may estimate the slippage amount by determining the relative difference between the position of the center rotation of the rotational body and the measured position of the alignment mark before and after the planar rotation (θ). 
     In one embodiment the method further includes performing a planar rotation (θ) of the rotational body to estimate the slippage amount of the rotational body. 
     The measuring may measure the positions of the fiducial mark before and after the planar rotation (θ) of the rotational body. 
     The acquiring the center of rotation and estimating the slippage amount of the rotational body may be performed simultaneously. 
     The acquiring may acquire the position of the center of rotation of the rotational body according to a positional change of the fiducial mark before and after rotation of the rotational body. 
     The estimating may estimate the slippage amount of the rotational body according to a positional change of the fiducial mark before and after rotation of the rotational body. 
     In another embodiment, an exposure apparatus to form a mask pattern on a substrate includes a stage configured to move the substrate in at least one direction, a rotational body stacked on the stage and configured to rotate the substrate, an alignment unit configured to measure a position of a fiducial mark during rotation of the rotational body to align the mask and the substrate, and a controller. The controller is configured to estimate a center of rotation and a slippage amount of the rotational body using the measured position of the fiducial mark, and is configured to compensate a movement amount of the stage based on the estimated slippage amount, thereby performing alignment between the mask and the substrate. 
     The alignment unit may include a scope to measure position coordinates of the fiducial mark before and after rotation of the rotational body. 
     The controller may be configured to estimate a center position of the rotational body according to the position coordinates of the fiducial mark before and after rotation of the rotational body, and may be configured to calculate a relative position between the center position of the rotational body and the measured position of the fiducial mark to estimate the slippage amount of the rotational body. 
     The controller may be configured to reflect the estimated slippage amount of the rotational body in a movement command value of the stage as a compensation value to compensate the slippage amount. 
     In a further embodiment, a method of estimating a center of rotation and a slippage amount of a rotational body that performs planar rotation includes measuring a position of a fiducial mark formed on the rotational body during the planar rotation of the rotational body, estimating a position of the center of rotation of the rotational body using the measured position of the fiducial mark, and determining a relative position between the position of the center of rotation of the rotational body and the measured position of the fiducial mark to estimate the slippage amount of the rotational body. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and/or other aspects of the embodiments will become apparent and more readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which: 
         FIG. 1  is an overall construction view of an exposure apparatus according to an embodiment; 
         FIG. 2  is an operation conceptual view of the exposure apparatus according to the embodiment; 
         FIG. 3  is a control construction view of the exposure apparatus according to the embodiment; 
         FIG. 4  is a view illustrating a process of estimating a center of rotation during planar rotation for alignment in the exposure apparatus according to an embodiment; 
         FIG. 5  is a view illustrating a process of estimating a center of rotation and a slippage amount due to radial runout during planar rotation for alignment in the exposure apparatus according to an embodiment; and 
         FIG. 6  is an operation conceptual view of a measurement system according to another embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Detailed example embodiments are disclosed herein. However, specific structural and functional details disclosed herein are merely representative for purposes of describing example embodiments. Example embodiments may, however, be embodied in many alternate forms and should not be construed as limited to only the embodiments set forth herein. 
     Accordingly, while example embodiments are capable of various modifications and alternative forms, embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit example embodiments to the particular forms disclosed, but to the contrary, example embodiments are to cover all modifications, equivalents, and alternatives falling within the scope of example embodiments. Like numbers refer to like elements throughout the description of the figures. 
     It will be understood that, although the terms first, second, etc. may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element, without departing from the scope of example embodiments. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. 
     It will be understood that when an element is referred to as being “connected” or “coupled” to another element, it may be directly connected or coupled to the other element or intervening elements may be present. In contrast, when an element is referred to as being “directly connected” or “directly coupled” to another element, there are no intervening elements present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between”, “adjacent” versus “directly adjacent”, etc.). 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises”, “comprising,”, “includes” and/or “including”, when used herein, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     It should also be noted that in some alternative implementations, the functions/acts noted may occur out of the order noted in the figures. For example, two figures shown in succession may in fact be executed substantially concurrently or may sometimes be executed in the reverse order, depending upon the functionality/acts involved. 
       FIG. 1  is an overall construction view of an exposure apparatus  10  according to an embodiment, and  FIG. 2  is an operation conceptual view of the exposure apparatus  10  according to the embodiment. Referring to  FIGS. 1 and 2 , the exposure apparatus  10  includes a moving table  100  on which a substrate (a sample, such as a semiconductor wafer or glass, on which a desired or predetermined pattern is to be formed) W is placed and an alignment unit  140  mounted above the moving table  100  to measure a position and posture of the substrate W placed on the moving table  100 . The alignment unit  140  is mounted to a gantry  170  such that the alignment unit  140  moves in X-, Y- and Z-directions. 
     Guide bar type moving members  171 ,  172  and  173  are mounted to the gantry  170  such that the moving members  171 ,  172  and  173  move in the X-, Y- or Z-direction. The alignment unit  140  is coupled to the moving members  171 ,  172  and  173  such that the alignment unit  140  is moved in the X-, Y- or Z-direction. The alignment unit  140  has three degrees of freedom (X, Y, Z), which is the most common configuration. The degrees of freedom may be restricted. For example, the alignment unit  140  may have a degree of freedom in the X-, Y- or Z-direction. 
     The alignment unit  140  has three degrees of freedom (X, Y, Z) in which the alignment unit  140  moves in the X-, Y- and Z-directions according to the movements of the moving members  171 ,  172  and  173 . The moving table  100 , on which the substrate W is placed, has two degrees of freedom (X, Y) in which the moving table  100  moves in the X- and Y-directions according to the movement of an XY stage (hereinafter, referred to as a stage)  110 . 
     Also, a θ stage (hereinafter, referred to as a rotational body)  120  is stacked on the moving table  100  such that the rotational body  120  performs rotation θ to align a mask or a virtual mask VM and the substrate W before exposure. 
       FIG. 3  is a control construction view of the exposure apparatus  10  according to the embodiment. 
     Referring to  FIG. 3 , the exposure apparatus  10  includes a stage  110 , a rotational body  120 , a light source unit  125 , a projection unit  130 , an alignment unit  140 , a mark capturing unit  150 , and a controller  160 . 
     The stage  110  is a device to move the moving table  100 , on which a substrate W to be exposed is placed, in the X- and Y-directions. The stage  110  translates the moving table  100  in the X- and Y-directions according to an instruction from the controller  160  when the virtual mask VM and the substrate W are aligned before exposure. 
     The rotational body  120  is a device stacked on the moving table  100  of the stage  110 , which translates in the X- and Y-directions, to perform rotation θ according to an instruction from the controller  160  when the virtual mask VM and the substrate W are aligned before exposure. The rotational body  120  has at least one fiducial mark (FM). 
     The light source unit  125  outputs laser light for exposure. The light source unit  125  includes a semiconductor laser or an ultraviolet lamp. The laser light is output to the substrate W placed on the rotational body  120  through the projection unit  130 . 
     The projection unit  130  is fixed to one side of the stage  110  to divide a pattern, which forms light to form a VM pattern, into a plurality of spot beams and project the spot beams onto the substrate W. 
     The projection unit  130  includes a light modulation element  131  to modulate light output from the light source unit  125  into light having a VM pattern, a first projection lens  132  to enlarge the light modulated by the light modulation element  131 , a multi lens array (MLA)  133  including a plurality of lenses configured in the form of an array to split the light having the VM pattern enlarged by the first projection lens  132  into a plurality of lights and to condense the lights, and a second projection lens  134  to adjust a resolution of the light condensed by the MLA  133  and to allow the condensed light to pass therethrough. 
     The light modulation element  131  includes a spatial light modulator (SLM). For example, the light modulation element  131  may be any of a micro electro mechanical systems (MEMS)-type digital micro-mirror device (DMD), a two-dimensional grating light valve (GLV), an electro-optical element formed of lead zirconate titanate (PLZT), which is a translucent ceramic material, a ferroelectric liquid crystal (FLC), etc. Generally, DMDs are used. For convenience of description, the embodiments assume that the optical modulation element  131  is formed of the DMD. 
     The DMD is a mirror device including a memory cell and a plurality of micromirrors arranged on the memory cell in an L×M matrix. Based on a control signal generated in response to image data, the DMD changes angles of individual micromirrors, reflects and transmits a desired light to the first projection lens  132 , and transmits the remaining light at a different angle such that the remaining light is blocked. 
     When a digital signal is recorded in a memory cell of the light modulation element  131  constituted by the DMD, each micromirror is inclined in the range of a desired (or, alternatively, a predetermined) angle (for example, 12°) on the basis of a diagonal line. On/off control operations of individual micromirrors are controlled by the controller  160 , which will be described later. The light reflected from the ON-status micromirrors exposes an exposure target (generally, a photoresist PR) placed on the substrate W, and the light reflected from the OFF-status micromirrors does not expose the exposure target placed on the substrate W. 
     The first projection lens  132  is constituted by a double telecentric optical system. The first projection lens  132  magnifies an image output from the light modulation element  131 , for example, at a magnification of approximately 4× and forms the magnified image on an aperture plane of the MLA  133 . 
     The second projection lens  134  is also constituted by a double telecentric optical system. The second projection lens  134  magnifies a plurality of spot beams formed on a focal plane of the MLA  133 , for example, at a magnification of approximately 1× and forms the magnified spot beams on the substrate W. In this embodiment, the first projection lens  132  has a magnification of 4× and the second projection lens  134  has a magnification of 1×. However, embodiments are not limited thereto. For example, magnifications of the first and second projection lenses  132  and  134  may be optimally adjusted according to the size of a spot beam and the minimum feature size of a pattern to be exposed. 
     In the MLA  133 , a plurality of micro-lenses corresponding to micromirrors of the light modulation element  131  are two-dimensionally arranged. For example, assuming that the light modulation element  131  includes 1920×400 micromirrors, 1920×400 microlenses are also provided. A pitch of the microlens arrangement may be substantially identical to a value calculated when the magnification of the first projection lens  132  is multiplied by the micromirror arrangement pitch of the light modulation element  131 . 
     The projection unit  130  generates a virtual mask VM having a pattern formed by the spot beams projected through the second projection lens  134 . 
     The exposure apparatus  10  with the above-stated construction outputs light through the light source unit  125 , and allows the light modulation element  131  to modulate the light output from the light source unit  125  into light having a VM pattern. The first projection lens  132  magnifies the VM-patterned light modulated by the light modulation unit  131 . The MLA  133  splits the magnified VM-patterned light into a plurality of spot beams and allows the spot beams to be condensed. The second projection lens  134  adjusts a resolution of the light condensed by the MLA  133  and allows the condensed light to penetrate therethrough, thereby performing exposure. 
     The alignment unit  140  may be a scope provided above the stage  110  to measure the position of a fiducial mark FM formed on the rotational body  120  to perform overlay alignment. 
     The mark capturing unit  150  is provided above the alignment unit  140  to capture the fiducial mark FM formed on the rotational body  120  and transmit the captured image to the controller  160 . At this time, the movement of the rotational body  120  is controlled according to an instruction from the controller  160  until the fiducial mark FM is captured by the mark capturing unit  150 . 
     The controller  160  estimates a center of rotation and a slippage amount of the rotational body  120  using a fiducial mark FM measured by the alignment unit  140 , reflects the estimated slippage amount in a movement command value of the stage  110  as a compensation value, and moves the moving table  100  to compensate the slippage amount. 
     In this way, the controller  160  estimates and compensates the center of rotation and the slippage amount of the substrate W using the fiducial mark FM when the virtual mask VM and the substrate W are aligned before exposure. 
     In this embodiment, the exposure apparatus  10  is a maskless exposure apparatus using a virtual mask VM. However, embodiments are not limited thereto. For example, the exposure apparatus  10  may be a mask exposure apparatus. 
     Hereinafter, a method of estimating a center of rotation and a slippage amount using a fiducial mark FM when a virtual mask VM and a substrate W are aligned for overlay during exposure will be described. 
       FIG. 4  is a view illustrating a process of estimating a center of rotation during planar rotation for alignment in the exposure apparatus according to the embodiment. 
     Referring to  4 , at least one fiducial mark FM is formed on the rotational body  120 . Alternatively, as least one fiducial mark FM may be formed on the substrate W. In this embodiment, two or more fiducial marks FM are formed on the rotational body  120 . 
     The fiducial marks FM formed on the rotational body  120  (or alignment marks formed on the substrate; hereinafter, referred to as fiducial marks for convenience of description) in a field of view F.O.V. of the alignment unit  140  are measured. Physical quantities defined to measure the fiducial marks FM are as follows. 
     The following physical quantities may be regarded as a two-dimensional vector amount (position vector). 
     Although the physical quantities are denoted by a three-dimensional vector (X, Y, Z), XY-plane leveling is performed in such a manner that all Z-axis coordinates are identical to one another. Therefore, Z is a constant, and thus Z is denoted by ‘0’ (Z=0) for convenience of description. 
     Σ O  is a fiducial coordinate system related to an overlay to achieve alignment between the virtual mask VM and the substrate W. 
     Σ C  is a body fixed coordinate system of the rotational body  120  (hereinafter, referred to as a rotation coordinate system). 
       O r C  is the position of a center of rotation of the rotational body  120 . 
     When a substrate (semiconductor wafer or glass) W is placed on the moving table  100  and several layers L (L 1 , L 2  . . . ) are stacked on the substrate W, the position of at least one fiducial mark FM ( O r ij ,  O r ij+1 ), formed on the rotational body  120  so as to achieve alignment between the substrate W and the virtual mask VM before exposure, is measured using the alignment unit  140  as represented by Equation 1 (see  FIG. 4 ). 
         O r ij   =X   ij   ,Y   ij    
         O r ij+1   =X   ij+1   ,Y   ij+1   [Equation 1]
 
     In Equation 1,  O r ij  is an i-th measured (before rotation) position of an i-th fiducial mark FM on the fiducial coordinate system Σ O , and  O r ij+1  is an (i+1)-th measured (after rotation) position of the i-th fiducial mark FM on the fiducial coordinate system Σ O . 
     The position of a center of rotation  O r C  is calculated through Equation 2 using the coordinate values of  O r ij  and  O r ij+1  measured at the i-th fiducial mark FM as represented by Equation 1. 
     
       
         
           
             
               
                 
                   
                     
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                                 - 
                                 
                                   x 
                                   21 
                                   2 
                                 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   y 
                                   21 
                                   2 
                                 
                                 - 
                                 
                                   y 
                                   22 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   x 
                                   
                                     2 
                                      
                                     n 
                                   
                                   2 
                                 
                                 - 
                                 
                                   x 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   y 
                                   
                                     2 
                                      
                                     n 
                                   
                                   2 
                                 
                                 - 
                                 
                                   y 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                      
                     
                         
                     
                     ] 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     4 
                   
                   ] 
                 
               
             
           
         
       
     
     Equation 4 shows examples of measured mark positions of two fiducial marks r i=1,2  before and after rotation when the rotational body  120  is rotated n−1 times. 
     Equation 4 may be simply expressed as represented by Equation 5. 
     
       
         
           
             
               
                 
                   
                     A 
                     · 
                     
                       r 
                       C 
                         
                         
                         
                       O 
                     
                   
                   = 
                   
                     
                       b 
                       ⇒ 
                       
                         r 
                         C 
                           
                           
                           
                         O 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 x 
                                 C 
                               
                             
                           
                           
                             
                               
                                 y 
                                 C 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         
                           
                             A 
                             + 
                           
                           · 
                           b 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   A 
                                   T 
                                 
                                  
                                 A 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                            
                           
                             
                               A 
                               T 
                             
                             · 
                             b 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     5 
                   
                   ] 
                 
               
             
           
         
       
     
     In this way, the positions of the fiducial mark formed on the rotational body  120  before and after rotation may be measured to estimate the position of the center of rotation  O r C  of the rotational body  120 . 
     As shown in  FIG. 4 , the rotational body  120  is ideally rotated about a constant center of rotation  O r C . Generally, however, radial runout is generated in the rotational body  120  mainly due to mechanical causes, with the result that the center of rotation of the rotational body  120  is not uniform. That is, slippage of the rotational body  120  occurs, which will be described with reference to  FIG. 5 . 
       FIG. 5  is a view illustrating a process of estimating a center of rotation and a slippage amount due to radial runout during planar rotation for alignment in the exposure apparatus according to the embodiment.  FIG. 5  shows a case in which slippage occurs when the rotational body  120  is rotated by a desired (or, alternatively, a predetermined) angle to align the substrate W and the virtual mask VM. 
     In  FIG. 5 , the rotational body  120  is shown as a straight line for convenience. 
     A center of rotation  O v C  and a slippage amount Δv C  of the rotational body  120  is calculated using Equation 6. 
       Δ v   C   =v   C   ′−v   C   =[X   S   Y   S ]
 
         O   r′   i =( O   v   C   +Δv   C )+ R (θ)· C   r   i   [Equation 6]
 
     In Equation 6, 
     
       
         
           
             θ 
             = 
             
               
                 arg 
                  
                 
                   ( 
                   
                     
                       r 
                       
                         i 
                         + 
                         1 
                       
                       ′ 
                         
                         
                       O 
                     
                     - 
                     
                       r 
                       i 
                       ′ 
                         
                         
                       O 
                     
                   
                   ) 
                 
               
               - 
               
                 arg 
                  
                 
                   ( 
                   
                     
                       r 
                       
                         i 
                         + 
                         1 
                       
                         
                         
                         
                       O 
                     
                     - 
                     
                       r 
                       i 
                         
                         
                         
                       O 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               R 
                
               
                 ( 
                 θ 
                 ) 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   
                     
                       
                         - 
                         sin 
                       
                        
                       
                           
                       
                        
                       θ 
                     
                   
                 
                 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       θ 
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               
                 r 
                 i 
                   
                   
                   
                 C 
               
               = 
               
                 
                   r 
                   i 
                     
                     
                     
                   O 
                 
                 - 
                 
                   v 
                   C 
                     
                     
                     
                   O 
                 
               
             
             , 
           
         
       
     
       O r i , is a position of the i-th fiducial mark FM measured before rotation on the fiducial coordinate system Σ O ,  O r′ i  is a position of the i-th fiducial mark FM measured after rotation on the fiducial coordinate system Σ O , and  O r j  is a measured position of the i-th fiducial mark FM on the rotation coordinate system Σ C . 
     Meanwhile, Δv C  and  O v C  in Equation 6 are values to be calculated. 
     Coordinate values of  O r i  and  O r′ i  in Equation 6 are measured using the alignment unit  140  as represented by Equation 7 (see  FIG. 5 ). 
     
       
         
           
             
               
                 
                   
                     
                       r 
                       i 
                         
                         
                         
                       O 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               i 
                             
                           
                         
                         
                           
                             
                               Y 
                               i 
                             
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       r 
                       i 
                       ′ 
                         
                         
                       O 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               i 
                               ′ 
                             
                           
                         
                         
                           
                             
                               Y 
                               i 
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       v 
                       C 
                         
                         
                         
                       O 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               C 
                             
                           
                         
                         
                           
                             
                               Y 
                               C 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
     A center of rotation (X C , Y C ) and a slippage amount (X S , Y S ) may be expressed in the form of a matrix vector using Equation 7 as represented by Equation 8. 
         O   r′   i =( O   v   C   +Δv   C )+ R (θ)· C   r   i =( O   v   C   +Δv   C )+ R (θ)·( O   r   i − O   v   C )
 
         X′   i =( X   i  cos θ− Y   i  sin θ)+ X   S   +X   C (1−cos θ)+ Y   C  sin θ
 
         Y′   i =( X   i  sin θ+ Y   i  cos θ)+ Y   S   −X   C  sin θ+ Y   C (1−cos θ)  [Equation 8]
 
     In Equation 8, the center of rotation (X C , Y C ) and the slippage amount (X S , Y S ) are unknown, and two equations are indeterminate, with the result that a unique solution is not obtained. 
     Therefore, the equation is solved using a least squares method (LSM) as represented by Equation 9. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                           
                               
                           
                         
                         O 
                       
                        
                       
                         r 
                         i 
                         ′ 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             v 
                             C 
                               
                               
                               
                             O 
                           
                           + 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               v 
                               C 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         
                           R 
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                         · 
                         
                           r 
                           i 
                             
                             
                             
                           C 
                         
                       
                       + 
                       
                         ( 
                         residual 
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           
                             
                               
                                 ⇒ 
                                 
                                   ( 
                                   residual 
                                   ) 
                                 
                               
                               = 
                                 
                                
                               
                                 e 
                                 i 
                               
                             
                           
                         
                         
                           
                             
                               = 
                                 
                                
                               
                                 
                                   r 
                                   i 
                                   ′ 
                                     
                                     
                                   O 
                                 
                                 - 
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           v 
                                           C 
                                             
                                             
                                             
                                           O 
                                         
                                         + 
                                         
                                           Δ 
                                            
                                           
                                               
                                           
                                            
                                           
                                             v 
                                             C 
                                           
                                         
                                       
                                       ) 
                                     
                                     + 
                                     
                                       
                                         R 
                                          
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                       · 
                                       
                                         r 
                                         i 
                                           
                                           
                                           
                                         C 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         
                           
                             
                               = 
                                 
                                
                               
                                 
                                   ( 
                                   
                                     
                                       r 
                                       i 
                                       ′ 
                                         
                                         
                                       O 
                                     
                                     - 
                                     
                                       v 
                                       C 
                                         
                                         
                                         
                                       O 
                                     
                                   
                                   ) 
                                 
                                 - 
                                 
                                   [ 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         v 
                                         C 
                                       
                                     
                                     + 
                                     
                                       
                                         R 
                                          
                                         
                                           ( 
                                           θ 
                                           ) 
                                         
                                       
                                       · 
                                       
                                         ( 
                                         
                                           
                                             r 
                                             i 
                                               
                                               
                                               
                                             O 
                                           
                                           - 
                                           
                                             v 
                                             C 
                                               
                                               
                                               
                                             O 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                        
                       
                         
 
                       
                        
                       
                           
                       
                       ∴ 
                       
                           
                       
                        
                       
                         minimizing 
                          
                         
                             
                         
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                             
                                
                               
                                 e 
                                 i 
                               
                                
                             
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         
                           e 
                           i 
                           T 
                         
                         · 
                         
                           e 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                      
                     
                         
                     
                      
                     9 
                   
                   ] 
                 
               
             
           
         
       
     
     In Equation 9, each component is 
         e   i | x   =r   x =( X′   i   −X   C )−[ X   S +( X   i   −X   C )cos θ−( Y   i   −Y   C )sin θ]
 
         e   i | y   =r   y =( Y′   i   −Y   C )−[ Y   S +( X   i   −X   C )sin θ+( Y   i   −Y   C )cos θ]
 
     In Equation 9, an optimization problem to minimize the sum of the square of a residual is solved to calculate the center of rotation  O v C =[X C , Y C ] and the slippage amount Δv C =[X S , Y S ] of the rotational body  120 . 
     In addition, the position of the center of rotation  O r C = O v C =[X C , Y C ] of the rotational body  120  calculated through Equation 1 to Equation 5 may be introduced to calculate the slippage amount Δv C =[X S , Y S ] of the rotational body  120  as represented by Equation 10. 
         X   S   =X′   i   −X   i  cos θ+ Y   i  sin θ− X   C (1−cos θ)− Y   C  sin θ
 
         X   S   =Y′   i   −X   i  sin θ− Y   i  cos θ+ X   C  sin θ− Y   C (1−cos θ)  [Equation 10]
 
     The slippage amount may be compensated using the center of rotation  O v C =[X C , Y C ] and the slippage amount Δv C =[X S , Y S ] of the rotational body  120  estimated through Equation 9 and Equation 10. 
     For example, when the substrate W is placed on the moving table  100 , which performs planar motions in three degrees of freedom (X, Y, θ) to process/manufacture/inspect the substrate W, command values, such as X=X cmd , Y=Y cmd , and θ=θ cmd , are transmitted from the controller  160  to the stage  110 . The fiducial mark FM or the alignment mark formed on the substrate W before and after rotation θ=θ cmd  is measured to estimate the slippage amount using the above-mentioned equation. The estimated slippage amount is reflected in the XY movement command value as a compensation value as represented by Equation 11. 
       Command values before compensation:  X=X   cmd   ,Y=Y   cmd   ,θ=θ   cmd    
       Commands values after compensation:  X=X   cmd   −X   S   ,Y=Y   cmd   −X   S ,θ=θ cmd   [Equation 11]
 
     In conclusion, XY translation caused due to radial runout generated during planar rotation θ to align the virtual mask VM and the substrate W may be offset through compensation of the slippage amount as represented by Equation 11. 
     In  FIGS. 1 to 3 , the exposure apparatus  10  uses a physical mask or a virtual mask. However, embodiments are not limited to the exposure apparatus  10 . For example, embodiments may be applied to a rotational body that performs planar rotation, which will be described with reference to  FIG. 6 . 
       FIG. 6  is an operation conceptual view of a measurement system  200  according to another embodiment of the present invention. 
     Referring to  FIG. 6 , the measurement system  200  includes a rotational body  210  on which a substrate (a sample, such as a semiconductor wafer or glass, on which a predetermined pattern is to be formed) W is placed and an alignment unit  220  mounted above the rotational body  210  to measure a position and posture of the substrate W placed on the rotational body  210 . The alignment unit  220  is mounted to a gantry  230  such that the alignment unit  220  moves in X-, Y- and Z-directions. 
     Guide bar type moving members  231 ,  232  and  233  are mounted to the gantry  230  such that the moving members  231 ,  232  and  233  move in the X-, Y- or Z-direction. The alignment unit  220  is coupled to the moving members  231 ,  232  and  233  such that the alignment unit  220  is moved in the X-, Y- or Z-direction. The alignment unit  220  has three degrees of freedom (X, Y, Z), which is the most common configuration. The degrees of freedom may be restricted. For example, the alignment unit  220  may have a degree of freedom in the X-, Y- or Z-direction. 
     The alignment unit  220  has three degrees of freedom (X, Y, Z) in which the alignment unit  220  moves in the X-, Y- and Z-directions according to movement of the moving members  231 ,  232  and  233 . 
     The rotational body  210 , on which the substrate W is placed, includes an upper plate  211  and a lower plate  212 . The upper plate  211  is a rotor that performs planar rotation θ, and the lower plate  212  is a stator. 
     In the measurement system  200  with the above-stated construction, the alignment unit  220  is moved during planar rotation θ of the device that performs planar rotation θ (specifically, the rotational body) to measure the position of a fiducial mark FM formed on the rotational body  210  or the position of an alignment mark (referred to as a fiducial mark FM for convenience) formed on the substrate W, thereby estimating a center of rotation and a slippage amount of the rotational body  210 . A process of estimating the center of rotation and a slippage amount of the rotational body  210  is identical to that shown in  FIGS. 4 and 5 . 
     As is apparent from the above description, the exposure apparatus and the alignment error compensation method using the same, estimate and compensate a center of rotation and a slippage amount, when slippage occurs due to radial runout during planar rotation θ to align a mask (including a virtual mask) and a substrate, thereby improving overlay performance. 
     Although a few embodiments have been shown and described, it would be appreciated by those skilled in the art that changes may be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the claims and their equivalents.