Patent Publication Number: US-2019186910-A1

Title: Pattern Measurement Device and Computer Program

Description:
TECHNICAL FIELD 
     The present invention relates to a device and a computer program for measuring a feature amount of a pattern formed on a sample using a signal obtained by an imaging device such as an electron microscope or the like, and more particularly, to a pattern measurement device and a computer program for obtaining a feature amount using a probability density function wherein the feature amount of a pattern obtained by an electron microscope or the like is set as a random variable. 
     BACKGROUND ART 
     Along with the high integration of semiconductor devices, the importance of process management based on measurement and inspection using a scanning electron microscope (SEM) capable of visualizing a pattern having a line width of several tens of nm is increasing. PTL 1 describes a Critical Dimension-SEM (CD-SEM) that measures a line edge roughness (LER) appearing at an edge of a pattern. In PTL 1, it is described that the roughness of a specific frequency component is extracted through frequency analysis on a fluctuation component at the edge portion, and the roughness is set as a feature amount. 
     Furthermore, in order to secure the production yield of the semiconductor integrated circuit (LSI), it is necessary to control the characteristic distribution on a wafer. In this regard, process control for measuring in-plane characteristics and feeding the results back to a manufacturing apparatus is performed in a wafer manufacturing process. Regarding statistical calculation of feedback information from the measurement result to the manufacturing apparatus, studies are being conducted in fields of exposure matching, for example, as described in NPL 1. 
     Meanwhile, according to the recent development of the Internet and cloud servers, it is now possible to analyze big data, and statistical analysis on complicated phenomena, such as a Bayesian estimation method, an MCMC method, and the like is being put into practical use. These statistical analysis methods are described in, for example, NPL 2. 
     CITATION LIST 
     Patent Literature 
     
         
         PTL 1: JP-A-2006-215020 (corresponding U.S. Pat. No. 7,230,723) 
       
    
     Non-Patent Literature 
     
         
         NPL 1: Overlay Improvement using Legendre-Zernike Model-based overlay corrections and monitoring with interpolated metric (Proc. of SPIE Vol. 9424 942420-10 (2015)) 
         NPL 2: “Bayesian Statistics as a Tool” by Yoshiyuki Wakui, published by Nippon Jitsugyo Publishing (2009) 
       
    
     SUMMARY OF INVENTION 
     Technical Problem 
     So far, a pattern measurement technology using a CD-SEM or the like as described in PTL 1 has been aimed at obtaining a measurement result close to an actual feature amount (roughness evaluation value or the like) of a pattern. Meanwhile, a measurement value obtained by a CD-SEM or the like varies depending on various factors. For example, even if a pattern sampled for measurement is accurately measured, it is merely accurate measurement of a pattern to be sampled, and even if process control of a manufacturing apparatus is performed based on the measurement result, it may be unable to perform proper control. Also, in case of a scanning electron microscope for detecting electrons obtained based on scanning of electron beams, for example, an emitting direction or the like of electrons emitted from a sample is random and varies for each measurement, and thus even if the same patterns are measured, it is not always possible to form the exact same signal waveforms, and as a result, there is a possibility that a measurement result for performing appropriate process control cannot be obtained. 
     In order to suppress such variations due to various factors and to obtain measurement results for proper process control, it is considered that statistical processing of a plurality of measurement results is performed to obtain measurement results that do not depend on various variation factors, and a large number of measurement results need to be obtained to obtain highly accurate measurement results. However, since CD-SEMs used for measurement in a mass production process of semiconductors are particularly required to have high throughput, it is difficult to achieve compatibility with highly accurate measurements requiring a large number of measurements. NPL 1 and NPL 2 do not refer to a pattern measurement device that achieves both high throughput and high-accuracy measurement based on a small number of measurement results. 
     Hereinafter, a pattern measurement device and a computer program for achieving both high-throughput measurement with a small number of measurements and high-accuracy measurement using statistical processing are proposed. 
     Solution to Problem 
     According to an aspect for achieving the above object, there is provided a pattern measurement device including a calculation processing device that acquires signal intensity distribution for a plurality of positions included in a scanning region from a signal obtained through beam scanning, substitutes, into a probability density function having the signal intensity distribution as a random variable and coordinates within the scanning region as a variable, the signal intensity distribution based on the signal obtained through the beam scanning, and for the plurality of positions within the scanning region, sets the coordinates within the scanning region at which the probability density function is at the maximum or at which prescribed conditions are met as an edge position. 
     According to another aspect for achieving the above object, there is provided a pattern measurement device including a calculation processing device for measuring feature amounts of a plurality of points on a sample from a signal obtained through beam scanning, wherein the calculation processing device sets a statistical model including a probability density function that includes a distribution model of a feature amount using a function of a hyper parameter and coordinates as a parameter and uses the feature amount as a random variable, substitutes the feature amount obtained through the measurement into the statistical model, calculates the hyper parameter at which the probability density function is at the maximum or at which prescribed conditions are met at the plurality of points, and calculates the distribution model at a plurality of positions based on setting of the obtained hyper parameter in the statistical model. 
     Advantageous Effects of Invention 
     According to the above configuration, it is possible to achieve both high-throughput measurement using a small number of measurements and high-accuracy measurement using statistical processing. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram illustrating an example of a semiconductor measurement system including an SEM. 
         FIG. 2  is a diagram illustrating an outline of a scanning electron microscope. 
         FIG. 3  is a flowchart illustrating processes of calculating a feature amount of a pattern using a detection signal by the SEM and a statistical model. 
         FIG. 4  is a diagram illustrating a process of estimating a true edge position using a detection signal by the SEM and a statistical model, and a display example of an estimation result thereof. 
         FIG. 5  is a diagram illustrating a process of obtaining in-plane distribution of a measurement value (CD value) and in-plane distribution of LCDU using a detection signal by the SEM and a statistical model, and display examples of the in-plane distributions. 
         FIG. 6  is a flowchart illustrating processes of calculating a feature amount of a pattern using a detection signal by the SEM and a statistical model. 
         FIG. 7  is a diagram illustrating an example of a signal waveform obtained when a structure formed on a sample is irradiated with an electron beam. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     In a semiconductor integrated circuit (LSI), high performance and high integration are advanced according to miniaturization of a circuit pattern. At present, a line width of a minimum circuit pattern of the most advanced LSI is less than or equal to 20 nm (nanometer), and in order to secure performance of the LSI, it is necessary to strictly manage these circuit dimensions (for example, with the accuracy in which an allowable value of a variation change in dimensions is equal to or less than 10% to 20% of a design value). The circuit dimensions are preferably measured using an SEM.  FIG. 2  is a diagram illustrating an outline of an SEM. An electron beam  3  is extracted from an electron source  201  by a lead electrode  202  and accelerated by an acceleration electrode that is not illustrated. The accelerated electron beam  203  is narrowed by a condenser lens  204  as a form of a focusing lens, and then deflected by a scanning deflector  205 . As a result, the electron beam  203  scans a sample  209  one-dimensionally or two-dimensionally. The electron beam  203  incident on the sample  209  is decelerated by a negative voltage applied to an electrode embedded in a sample stage  208  and focused at the same time due to a lens effect of an objective lens  206 , so that a surface of the sample  209  is irradiated with the electron beam. 
     Electrons  210  (secondary electrons, backscattered electrons, and the like) are emitted from an irradiated portion of the sample  209 . The emitted electrons  210  are accelerated in a direction of the electron source  201  by an acceleration effect based on the negative voltage applied to the electrode embedded in the sample stage  208 . The accelerated electrons  210  strike a conversion electrode  212  to generate secondary electrons  211 . The secondary electrons  211  emitted from the conversion electrode  212  are captured by a detector  213 , and an output I of the detector  213  is changed due to the amount of the captured secondary electrons. Luminance of a display device that is not illustrated is changed according to the change of the output I. For example, if a two-dimensional image is to be formed, an image of a scanning region is formed by synchronizing a deflection signal to the scanning deflector  205  and the output I of the deflector  213 . In the scanning electron microscope of  FIG. 2 , an electron detector  215  for detecting secondary electrons  216  is disposed in the objective lens  206 . 
     In the configuration example of  FIG. 2 , an example is illustrated in which the electrons  210  emitted from the sample  209  are detected after being converted into the secondary electrons  211  at the conversion electrode  212 , but obviously, the configuration is not limited thereto, and for example, a configuration in which a detection surface of an electron multiplier tube or a detector is arranged on a path of accelerated electrons may be adopted. A control device  214  supplies a control signal necessary for each optical element of the SEM according to an operation program called an imaging recipe for controlling the SEM. 
     Next, a signal detected by the detector  213  is converted into a digital signal by an A/D converter  217  and transmitted to an image processing unit  218 . 
     The image processing unit  218  includes an image memory storage medium  220  for temporarily storing a digital image and a CPU  219  for calculating feature amounts (a dimension value of a line or hole width, a roughness index value, an index value indicating a pattern shape, an area value of a pattern, a pixel position as an edge position, and the like) from an image in the image memory. For example, edge position information is calculated by thinning processing of a white band included in the image, and calculating a center position of the white band as the edge position. 
     Further, a storage medium  221  is provided for storing a measurement value of each pattern, a luminance value of each pixel, and the like. An overall control is performed by a workstation  222 . A necessary operation of a device, confirmation of a detection result, or the like, may be realized by a graphical user interface (hereinafter, referred to as GUI). Also, the image memory is configured to store an output signal (a signal proportional to an amount of electrons emitted from the sample) of the detector in synchronization with a scanning signal supplied to the scanning deflector  205 , in a corresponding address (x, y) on the memory. Also, the image processing unit  218  operates as a calculation processing device that generates a line profile from the luminance values stored in the memory, determines the edge position using a threshold method or the like, and measures the dimension between edges. 
     The SEM that performs dimension measurement based on such line profile acquisition is called CD-SEM, and is used to measure various feature amounts in addition to line width measurement of a semiconductor circuit. For example, unevenness called line edge roughness exists at the edge of the circuit pattern, which causes the circuit performance to change. The CD-SEM can be used to measure the LER. 
     In a manufacturing process of an electronic component, such as a semiconductor integrated circuit, a high-frequency element, or the like, a pattern of an electronic component, such as a transistor, a memory element, or the like or of a circuit or the like connecting the elements is formed on a substrate such as a semiconductor or the like, or a thin film on the substrate using lithography and etching techniques. In a most advanced device where miniaturization is advanced, the dimensions of the above each element and circuit reach 10 nm, and are expected to be further reduced in the future. Since the performance of the electronic component largely depends on a planar shape of the element and circuit, the planar shape is generally required to be formed with the accuracy of about ⅓ of a design dimension of the pattern. 
     However, due to various factors, in reality, a deviation occurs between a design pattern and a pattern of an element and circuit actually manufactured. Such a deviation may be quantitatively handled by measuring EPE (Edge Placement Error), i.e. the deviation between a position of an edge of the pattern of the element or circuit actually manufactured and a position of an edge where the pattern should correctly be. Factors that cause EPE are classified into the following three categories. 
     The three categories are (1) a deviation of a pattern average dimension, a change in a dimension that is thickened or thinned in overall, (2) a deviation of pattern average position, an overall position deviation, and (3) a local change of a pattern edge or pattern dimension. 
     (1) is caused by thinning/thickening of a resist pattern transferred by an exposure device and side etching, and may be corrected by adjusting exposure conditions, such as an exposure amount/focus position, and the like, or process conditions, such as etching and the like. (2) is caused by, for example, misalignment between a base pattern and a mask transfer image by an exposure device, and may be corrected by adjusting the alignment of the exposure device. It is assumed that (3) is caused by a random or local edge position change that is difficult to be controlled by the above, for example, a stochastic variation of a chemical reaction or dissolution phenomenon in materials in an intermediate region of exposure intensity near an edge of a projected image. Since the amplitude of the variation reflects, for example, the width of an intermediate exposure range=the sharpness of an exposure profile (defocusing conditions or the like), the amplitude varies according to effective projection image contrast. 
     The deviations of (1) to (3) vary due to spatial and temporal variations, and the influence on the element performance depends on a relationship between a spatial period of the variation and the element dimension. When the variation period is smaller than the element dimension, the variation deteriorates the element performance itself. On the other hand, when the variation period is larger than the element dimension, the variation becomes the element performance deviation. Generally, the variation spatial period of (3) is smaller than (1) and (2), but with the miniaturization of a pattern, an effect of (3) having the small spatial period on an element performance variation is being surfaced. In a so-called 7 nm node logical integrated circuit, approximate guidelines of (1) to (3) are considered to be about 1 nm, 2 nm, and 2 nm respectively. 
     Also, in recent years, a multi-patterning technique, such as SAQP (Self Aligned Quadruple Patterning), or the like, has been introduced in semiconductor manufacturing processes so as to form a minute pattern exceeding a resolution limitation of the exposure device. In such a patterning method called a side wall patterning, the quality of a side wall (sidewall) greatly affects the performance of a semiconductor device. In addition, since patterning is performed a plurality of times, factors causing dimension and positional deviation, such as the deposited thickness of the side wall or the like, are further complicated, but the factors of occurrence of EPE are also basically classified into (1) to (3) above. In any case, in order to suppress the deviation caused by the manufacturing processes and materials, it is necessary to accurately measure and correct a process condition/a device knob using a measuring device capable of highly accurate measurement, such as CD-SEM or the like. 
     Next, the spatial and temporal deviations of (1) to (3) include both a systematic variation and a random variation of the spatial distribution and a temporal change. (1) and (2) systematically change spatially/temporally by reflecting wafer in-plane distribution and chip in-plane distribution in the process condition/device state and the time change thereof, and at the same time, the process condition/device state varies randomly. 
     (3) is essentially stochastic, but the amplitude thereof also tends to be systematic spatially/temporally by reflecting the process condition/device state. In order to properly perform feedback control on an exposure device or other process devices, it is necessary to grasp both the tendency of spatial distribution or time change and the size of random variation. It is possible to provide proper feedback of manufacturing processes by separating each factor of the deviation or variation and measuring a feature amount. 
     The dimension of a certain pattern on a wafer is a definite value having a true value under the definition of a certain dimension. In the measurement using CD-SEM or the like, it is desirable to obtain a measurement value as close to the true value as possible. However, many patterns that may be measurement targets exist on a semiconductor wafer, and actual measurement is performed by sampling only a small portion of the many patterns. However, as described above, a feature amount of the measurement target varies. In this case, even if a true value is obtained by performing “perfect” measurement with respect to the true value of the sampled certain pattern, the obtained true value is for only one of the various patterns, and thus it may not be appropriate to control a process or device based thereon. 
     Furthermore, as described above, the measurement of a planar shape is performed using an electron microscope (SEM). In particular, in processes of manufacturing a semiconductor integrated circuit, a CD-SEM specialized in pattern dimension measurement on a wafer is often used. However, an observation result by the SEM is not deterministic as generally believed, but includes a variation due to a measurement error. Thus, it may not be necessarily appropriate to control the process or device based thereon. 
     Next, the following two representative examples will be described as specific examples of variations in SEM measurement. 
     (1) Variation in Edge Position 
     First, an edge position of a line pattern is to be measured by the SEM. As illustrated in  FIG. 7 , secondary electrons and backscattered electrons, which are obtained by scanning a focused electron beam such that a path of the electron beam is perpendicular to a longitudinal direction of an edge (a side wall of a structure), are detected. A signal intensity profile (line profile) is obtained by storing a detection signal in association with an irradiation position of the electron beam for obtaining the detection signal. Since signal intensity is the maximum when incident near the edge according to an edge effect, the edge position may be defined by, for example, setting an appropriate threshold value for relative signal intensity. 
     However, a measurement result obtained as such includes measurement reproducibility error. A main cause of the error is considered to be particle nature of electrons. In other words, when the secondary electrons or reflected electrons generated as the electron beam is incident on a sample are detected, each process of an incident position of electrons, a scattering/reflection direction inside the sample, generation of the secondary electrons, or the like, is a stochastic process, and as a result, the signal intensity profile is varied. Thus, the edge position obtained from the profile is also varied. Meanwhile, a true value itself at the edge position of the line pattern is also varied according to a scanning position in an edge direction. Accordingly, when the observed line edge position deviates from a correct position, it is unable to distinguish whether the deviation is deviation of the true value or deviation caused by a measurement error. 
     Also, the measurement error caused by the particle nature of the electrons is suppressed by increasing the number of incident electrons (an incident current or a total irradiation time (for example the number of frames)). However, in this case, the measurement target is damaged by the electrons, and a time required for the measurement is increased. 
     (2) Variation in Dimension Values and the Like 
     In general, a semiconductor LSI includes a huge number of identical design patterns on a wafer. For example, a memory LSI or a memory block of a logic LSI includes a large number of memory cells, and each memory cell needs to operate substantially equally within a predetermined variation range. The same is applied to standard cells constituting the logic LSI. In this regard, the identical design patterns need to be uniformly formed regardless of a position of the wafer and a manufacturing time, but due to various factors described in (1) to (3) above, deviation occurs in designs of patterns of elements and circuits that are actually manufactured. Accordingly, the spatial distribution of pattern dimensions/shapes and a time change thereof are important. 
     In the meantime, the spatial distribution of the pattern dimensions/shapes varies depending on a factor of deviation. Dimension variation caused by a wafer process, such as etching, deposition, or the like, has wafer level distribution reflecting in-plane distribution of a device state and process conditions, while variation distribution caused by an exposure device and a mask has chip level distribution. In addition, stochastic edge position variation caused by resist depends on lithography contrast and generates random dimension variation at a local region. Furthermore, there is dimension distribution reflecting a wafer history and a stochastic defect. (For example, a particle defect on a rear surface of a wafer causes variation in a wafer surface height, which leads to a dimension variation due to focus deviation and increase in resist-induced local dimension variation (LCDU) caused by contrast degradation.) As such, the pattern dimension distribution on the wafer shows a complicated distribution shape by overlapping with dimension variation/variation caused by various factors having different spatial periods. For appropriate feedback, it is necessary to separate distribution/variation of measured dimensions and the like according to factors. 
     In addition, these statistical measurements generally require a large number of measurements. However, highly accurate/highly reproducible measurements using an electron beam is low speed, and thus sufficient sampling is not performed within a practical time and statistically sufficient reliability is not obtained. 
     In an embodiment described below, a pattern feature amount measurement device that enables a highly reliable measurement result optimum for process control to be obtained with a small number of measurement samples considering statistical variations due to manufacturing variations and measurement variations, and a computer program for causing a calculation processing device to calculate a feature amount will be described. 
     In the embodiment described hereinafter, a physical amount (feature amount) to be measured, such as a dimension or the like, is treated as a random variable. In other words, when a signal intensity profile or a dimension/shape or an edge position to be measured from the signal intensity distribution is obtained by SEM, the signal intensity profile or the dimension/shape, or the like to be measured is treated as the random variable and a probability density function is obtained. 
     According to an examination result of the inventor, such a phenomenon may be described by a phenomenological statistical model based on a physical model or an experience model. Thus, as the probability density function, a statistical model based on a physical model or an experience model considering a variation due to various factors is assumed. Since the probability density function by such a statistical model generally has a very complicated form, a measurement result may be represented by a parameter or a hyper parameter of the probability density function. 
     Further, in an embodiment described below, the probability density function is obtained as a function of position or time. Here, the parameter or the hyper parameter may be the function of position or time. With respect to a change of a measurement target over time, time development by sequentially updating the probability density function according to the measurement result may be obtained. 
     One of general forms of the statistical model in an embodiment described below is that, when measurements are performed on n points at positions ri (i=1 to n) at a time t, the probability density function, in which an overall observation result {xi} is obtained with an observation result xi (i=1 to n) at each position as a random variable, is set to Π_i [P(xi, b(ri, t))] (here, Π_i represents a product regarding i). Here, b (ri, t) is a parameter or hyper parameter depending on a position and time. In particular, for example, the position may be a pixel position in an SEM image and the observation result may be detection intensity of each pixel, or the position may be a position on a wafer and the observation result may be a pattern dimension measured at each position. The probability density function, or the parameter or hyper parameter b(ri, t) that is most likely to provide the overall observation result is obtained. 
     The above statistical model for describing such a phenomenon with sufficient accuracy is generally complicated, and a plurality of parameters or hyper parameters included therein are often difficult to be identified from actual measurement values. Thus, in an embodiment described below, the probability density function is obtained by applying a hierarchical Bayes method, and further applying an MCMC (Markov Chain Monte Carlo) method that is a general solution. Specific procedures are as follows. 
     (1) Observing a measurement target (for example, a pattern dimension and an edge position) using a measurement device (for example, CD-SEM or the like) and storing a result, (2) defining a statistical model of the measurement target, (3) assuming a (hyper) parameter in the statistical model and obtaining a first probability density function, (4) calculating a first likelihood (likelihood=probability of obtaining an observation result in probability density function) for obtaining the observation result in the first probability density function, (5) calculating a second likelihood for obtaining the observation result similarly with respect to a second probability density function in which the (hyper) parameter is randomly perturbed, (6) updating the (hyper) parameter based on a magnitude relationship between the first and second likelihoods, (7) repeating the above steps (3) to (6), and (8) eliminating an initial process in which a change in the parameter is intense and distributing the (hyper) parameters. 
       FIG. 1  illustrates an example of a measurement system including a calculation processing device that calculates a feature amount by substituting, into a probability density function having the feature amount obtainable based on the detection signal described above as a random value, signal intensity at a plurality of positions included in a scanning region or the feature amount obtained at a plurality of positions on a sample, and obtaining a variable at which the probability density function is at the maximum or at which prescribed conditions are met. 
     In the example of  FIG. 1 , a scanning electron microscope  101  that is an imaging system and a calculation processing device  105  (pattern measurement device  102 ) that performs a measurement process based on a detection signal are connected to each other via a network, but the example is not limited thereto, and for example, the image processing unit  218  included in the scanning electron microscope illustrated in  FIG. 2  may perform a calculation process described later. The system illustrated in  FIG. 1  includes an SEM  101 , the pattern measurement device  102  that performs measurement of a pattern or the like based on an obtained signal, and a design data storage medium  103  that stores design data of a semiconductor device or layout data generated according to the design data. 
     The pattern measurement device  102  transmits SEM image data obtained by the SEM  101  to a memory  106  that is an image storage device. The calculation processing device  105  performs following operations using the stored image data as input data. 
     An image processing unit  107  performs various image processes and measurement processes, such as obtaining a signal intensity profile in a certain direction, for example, a direction perpendicular to a pattern edge, or the like, from an image, extracting pattern edge coordinates therefrom according to a predetermined algorithm, calculating a pattern dimension or extracting a contour line indicating a pattern outline from a plurality of pattern edge coordinates extracted as such, calculating the deviation or a pattern feature by further comparing the pattern dimension or outline with the design data stored in the design data storage medium  103 , and the like. A necessary processing result is stored in the memory  106 . 
     A statistical model input unit  108  defines or selects a statistical model, represents the statistical model in an equation, and defines the probability density function as in an embodiment below. A statistical processing unit  109  calculates the probability density function that is most likely to provide an entire processing result output from the image processing unit  107 , or a parameter or hyper parameter thereof. 
     A result processing unit  110  displays the obtained calculation result on, for example, a display device of an input device  104 . 
     According to the measurement system described above, highly accurate estimation is possible from a small number of samples by performing physical/empirical assumption regarding spatial distribution or temporal change. Accordingly, even when highly-accurate/highly reproducible measurement using an electron beam is used, it is possible to control/manage various manufacturing devices and manufacturing processes with high production accuracy within a practical time range, improve production yields of various semiconductor devices and electronic components, and improve performance thereof. 
     Hereinafter, more specifically, a pattern feature amount measurement device that performs a measurement process based on a detection signal or the like obtained by a charged particle beam device such as SEM, a computer program that causes a calculation processing device (computer) to perform the measurement process, and a storage medium that stores the computer program are described. 
     Embodiment 1 
     In the current embodiment, details about specific processes for determining a position of an edge that is a type of a feature amount are described. In the current embodiment, “a signal intensity profile congruent in a direction perpendicular to an average edge for all edge points” is assumed as a physical model with the signal intensity distribution as a random variable, and the signal intensity distribution that is most likely is determined.  FIG. 3  illustrates a rough process flow. 
     First, using a scanning electron microscope as illustrated in  FIG. 2 , secondary electrons and backscattered electrons from a sample are detected by scanning a surface of the sample with a focused electron beam at constant current and scanning speed. The detection signal intensity distribution I(x, y) is obtained from a relationship between detection intensity and an electron beam incident position. Since the current and scanning speed are constant, an average incident electron number per pixel is constant. Here, for convenience, a sample having a step extending in a y-direction was observed as the sample. A step edge position varies in an x-direction by so-called LER. The detection signal intensity distribution I(x, y) is transmitted to the image memory  220  or the memory  106 . 
     The image processing unit  107  receives image data stored in an image storage device, calculates a signal intensity profile in a direction perpendicular to an edge at a plurality of y coordinate positions along the edge of the step pattern, and stores the signal intensity profile in the image memory  220  or the memory  106 . 
     Next, in the statistical model input unit  108 , a random variable, a probability density function, and a parameter/hyper parameter are defined. Herein, the followings are set. First, the random variable is set to the detection signal intensity {Iij} (i=1 to m, j=1 to N) at each pixel. Here, Iij=I(xi, yj), wherein xi, yj are x, y coordinates of a pixel (i, j). 
     Next, a probability density function P having the detection signal intensity distribution {Iij} as a random variable is defined as P=Π_ij pij (Iij) (here, Π_ij is a product regarding all pairs of i and j). pij denotes a probability for obtaining the detection signal intensity Iij at the pixel (i, j) (i=1 tom, j=1 to n) (element probability density function for each pixel). 
     The element probability density function is defined as a normal distribution, such as pij=N(Iij|Ix(xi−xe(yj)), σx(xi−xe(yj))), with respect to the detection signal intensity Iij for each pixel obtained as described above. N is the normal distribution. 
     Here, Ix(x) denotes the ideal signal intensity distribution (depending only on an x coordinate) when an edge extending in a y-direction is at an origin of an x-axis. Here, “ideal” means a limit value approaching when an electron beam irradiation amount is increased if a sample does not change by electron beam irradiation. Herein, a profile when a frame number is increased is used, but a simulation result or the like may also be used. Also, xe(y) is defined as a hyper parameter at an x coordinate of an x-direction edge at a y-direction coordinate y. An x-direction edge position at an edge position y may be estimated by obtaining an average value of the xe(y) distribution for each y. 
     σx(x) denotes a standard deviation of the signal intensity distribution Ix(x) caused by a particle nature of electrons, and is defined as a hyper parameter. In the above, P is represented as the probability density function having the detection signal intensity distribution {Iij} as the random variable, and xe(y) and σx(x) as hyper parameters. Also, herein, the probability density function is the normal distribution N, but another function type may be used. For example, the signal intensity distribution may be obtained as a digital signal by discretely counting a signal electron number. In this case, the element probability density function for the detection signal intensity Iij for each pixel may be Poisson distribution. The statistical model input unit  108  sets other various parameters necessary for an MCMC method. 
     Next, the statistical processing unit  109  specifies the density probability function and the hyper parameters. In particular, from the detection signal intensity distribution I(x, y) stored in the image memory  220  or the memory  106 , and the probability density function P defined by the statistical model input unit  108 , distributions of each hyper parameter of the probability density function are obtained using the above method and MCMC method, and average values and standard deviations thereof are calculated. As a result, xe(yj) and σx(xi) are determined such that P(I(x, y) x, y) is the maximum for all combinations of x and y. In the current embodiment, although Ix(x) is provided as a known function such as an analytical solution or the like with respect to an average value of a plurality of measurement results or an ideal edge, Ix (x) itself may also be an unknown function and obtained as a hyper parameter. 
     Also, in the current embodiment, an edge position is obtained by calculating an average value of distribution of edge positions (xe(y)) at which the probability density function P(I(x, y) x, y) having signal intensity at each position in a scanning region as a random variable is the maximum, but the edge position (xe(y) at which P satisfies predetermined condition) at which the probability density function P(I(x, y) x, y) is, for example, equal to or greater than a predetermined threshold value, or is equal to or greater than the predetermined threshold value and satisfies another condition may be obtained. 
     The result output unit  110  displays the average value as the estimated position of the edge on the display device of the input device  104 , or the like. A typical output result is illustrated in  FIG. 4 . An upper left column of  FIG. 4  is a diagram illustrating a relationship between a coordinate position of an edge portion obtained according to an SEM image and luminance (signal intensity). Also, an upper right column shows an outline of a statistical model. In the current embodiment, two pieces of distribution information of variation distribution of an edge position in the x-direction and the variation distribution of signal intensity are stored as the statistical model, so as to estimate a true edge position using the two pieces of distribution information. Accordingly, the two pieces of distribution information are displayed as the upper right column to be visually determined. Measurement results (the luminance distribution information obtained from the actual SEM image) and the estimated true edge positions obtained based on the statistical model as such are shown, for example, in a lower column of  FIG. 4 . In addition, when it is difficult to be progressed or to perform a calculation in the MCMC method, the intent thereof (error message) may be displayed. Further, not only the estimated edge position information, but also measurement results may be displayed by performing dimension measurement or EPE measurement between the plurality of estimated edge positions. 
     Embodiment 2 
     Next, an example in which a feature amount estimation method using a statistical model is used to measure a dimension of a pattern or the like or to calculate wafer in-plane distribution is described. In Embodiment 1, the detection signal intensity distribution in the scanning region is set as the random variable, whereas in the current embodiment, the dimension distribution in a wafer is set as a random variable and the most likely dimension distribution in a wafer is determined by assuming that, as a physical model, “the dimension (CD) distribution in the wafer or random variation (LCDU) distribution thereof each follow an independent function”. 
     In the current embodiment, first, by using the scanning electron microscope illustrated in  FIG. 2 , an SEM image is captured according to a predetermined method with respect to a plurality of regions including n representative points (2-dimensional coordinate positions ri, i=1 to n) pre-set on a wafer substrate including elements and circuit patterns on a surface. The captured SEM image is transmitted to the image memory  220  or the memory  106 . Here, the plurality of regions include the same design patterns (in other words, patterns theoretically having the same dimensions and same shapes). 
     The image processing unit  107  measures a dimension CDi(i=1 to n) of the same design patterns included in a captured region with respect to data of each SEM image according to a pre-set predetermined recipe based on input of data of a plurality of SEM images stored in the image memory  220  or the memory  106 , and stores a measurement result in the image memory  220  or the memory  106 . In the current embodiment, an example in which the calculation is performed using the dimension value CDi as the feature amount of the pattern is described, but an index value indicating another pattern dimension or shape may be used as the feature amount. For example, types of the feature amount include a shape, area, index value of roughness, and the like of the pattern. 
     Next, the statistical model input unit  108  defines a random variable, a probability density function, and a parameter/hyper parameter. Followings are set in the current embodiment. The random variable is set to a dimension {CDi} at each measurement position on a wafer substrate. A probability density function P having the dimension {CDi} as the random variable is defined as P=Π_i pi(CDi) (here, Π_i is a product regarding i). pi denotes a probability for obtaining the dimension CDi at a position ri (an element probability density function). The element probability density function for the dimension CDi at the position ri is defined as a normal distribution, such as pi=N (CDi CDmean (ri), LCDU(ri)) (N denotes normal distribution). CDmean(ri) and LCDU(ri) denote a distribution model using wafer coordinates as a function. Here, ri=(Xi, Yi)=(Ri, ei) denotes 2-dimensional wafer coordinates. 
     Here, CDmean(ri) and LCDU(ri) developed respectively to CDmean(ri)=Σ_j Aj ϕj(ri) (here, Σ_j denotes a sum regarding j) and LCDU(ri)=Σ_j Bjϕj(ri), as functions of wafer coordinates. Since ϕj(ri) is a two-dimensional orthogonal polynomial system and a wafer and a chamber shape of a process device are generally circular, herein, a Zernike polynomial system is used with ri as a polar coordinate system (Ri, θi). 
     Also, the distribution of CDmean or LCDU reflects the distribution of conditions or states of the process device, such as etching, CVD, CMP, and the like, for example, the distribution of shapes, temperatures, gas pressures, electric potentials, or the like of the wafer or a chamber, stage, or the like of the process device. The distributions are well expressed in the Zernike polynomial system, but are not limited thereto, and another distribution model may be used. 
     From the above, P may be expressed as the probability density function having the dimension {CDi} as a random variable, and Aj and Bj as hyper parameters. Also, herein, the normal distribution N is used as the element probability density function, but another function form may be used. 
     The statistical model input unit  108  sets other various parameters necessary for the MCMC method. 
     Next, the statistical processing unit  109  calculates, from the dimension data {CDi} stored in the image memory  202  or the memory  106  and the probability density function defined in the statistical model input unit  108 , each of the hyper parameters {Aj} and {Bj} of the probability density function, and also the distributions of CDmean and LCDU obtained from each of {Aj} and {Bj}, using the method described in the section on the means and the MCMC method. In particular, for example, Aj and Bj are determined such that P=Π_i pi (CDi) is the maximum for all 2-dimensional coordinate positions ri. In addition, Aj and Bj may be determined such that P is, for example, equal to or greater than a predetermined threshold value, or is equal to or greater than the predetermined threshold value and satisfies another condition. Furthermore, average values and standard deviations of CDmean and LCDU at each position are calculated. The widths of CDmean distribution and LCDU distribution obtained via repetition of the MCMC method are a CD value at each point on the wafer and an index value indicating measurement reproducibility of LCDU. 
     The result output unit  110  displays the average value on the display device of the input device  104 , or the like. A typical output is illustrated in  FIG. 5 .  FIG. 5( a )  illustrates the distribution of measurement points on the wafer. Further,  FIG. 5( b )  illustrates display examples of in-plane distribution of CD values and in-plane distribution of LCDU, which are obtained based on sampling of CD values at each measurement point and the statistical processing. In the in-plane distribution of CD values, identification display (for example, color display) is performed according to the magnitude of measurement value. Also,  FIG. 5( c )  illustrates an example of displaying a histogram showing the variation of measurement results in each chip on the wafer or each shot unit. In addition, when it is difficult to be progressed or to perform a calculation in the MCMC method, the intent thereof (error message) may be displayed. 
     In the above embodiment, the distribution in the wafer is modeled by the Zernike polynomial, but an exposure device and mask-induced distribution may be added to the model. For example, CDmean and LCDU are developed as CDmean(ri)=ΣAj ϕj(ri)+Σak Ψk(rc_i), LCDU(ri)=ΣBjϕj(ri)+Σbk Ψk(rc_i) respectively as functions of wafer coordinates and chip coordinates. rc_i is the coordinates in a chip of a measurement point at the position ri (coordinates in an exposure area). Since the exposure area is rectangular, Ψk(rc_i) is a two-dimensional orthogonal polynomial system, and since the exposure area is rectangular, 2-dimensional Legendre polynomial system is used herein with rc_i as an orthogonal coordinate system (xi, yi). Furthermore, an offset term per chip, the distribution per chip, or the like may be added. As such, by separating the distribution in the wafer and the distribution in the chip, the wafer process, such as etching, depositing, CMP, developing/washing, or the like may be monitored from the former, and the exposure process and the state of the mask may be monitored from the latter, thereby further controlling the device state or process conditions. 
     Also, the probability density function other than the normal distribution, for example, double/triple Gaussian or the like may be assumed to apply a function, such as pi=N1 (CDi|CDmean_1(ri), LCDU_1(ri))+N2 (CDi|CDmean_2(ri), LCDU_2(ri))+ . . . . 
     Also, in the above embodiment, the dimension CD of the same design pattern is measured from the captured SEM image data, but in recent CD-SEM, it is possible to obtain an image including a pattern of two different layers constituting LSI and measure misalignment OVL between the two layers. Accordingly, by substituting the CD in the above embodiment with OVL, it is possible to obtain wafer in-plane distribution OVL(ri) of the misalignment and the variation thereof in the exact same manner. 
     Furthermore, it is possible to estimate the distribution of the EPE in the wafer plane in combination with CDmean(ri), LCDU(ri), or OVL(ri) estimated in the current embodiment, or the measured result of misalignment in the wafer plane measured by another measurer as necessary. Accordingly, it is possible to improve the yield by predicting chip yield distribution in the wafer plane and feeding back to a setting condition of the process device or the like as necessary. 
     Since CDmean(ri), LCDU(ri), and OVL(ri) are random variables, EPE that is the function thereof may also be defined as a random variable. Thus, according to the current embodiment, it is possible to more accurately predict the yield by obtaining the probability density distribution function of EPE. 
     Embodiment 3 
     When the variation progresses temporally, so-called Bayes estimation may be applied. That is, for example, it is assumed that a result {CDi (t+Δt)} is obtained by obtaining a probability density function p(t) at a time t via the method described in Embodiment 2, and then performing observation at a time t+Δt. In this case, a probability density function P(t+Δt) at the time t+Δt may be obtained from P(t+Δt,{CDi})=Π_i N(CDi(t+Δt)|CDmean(t, ri), LCDU(t, ri))P(t, {CDi}). As such, by updating the probability density function with the lapse of time and applying the probability density function as the prior probability distribution as illustrated in  FIG. 6 , the process of change may be accurately grasped without performing measurement on a large number of points. 
     Embodiments 1 and 2 above correspond to a case where uniform probability distribution is assumed for a prior distribution. Generally, a result having higher reliability is obtained when a previous measurement result is applied than when the uniform probability distribution is assumed for the prior distribution, but it may be preferable to assume the uniform probability distribution after feedback is provided to the process device or the like. 
     REFERENCE SIGNS LIST 
     
         
         
           
               101 : scanning electron microscope (SEM) 
               102 : pattern measurement device 
               103 : design data storage medium 
               104 : input device 
               105 : calculation processing device 
               106 : memory 
               107 : image processing unit 
               108 : statistical model input unit 
               109 : statistical processing unit 
               110 : result output unit 
               201 : electron source 
               202 : lead electrode 
               203 : electron beam 
               204 : condenser lens 
               205 : scanning deflector 
               206 : objective lens 
               207 : sample chamber 
               208 : sample stage 
               209 : sample 
               210 : accelerated electron 
               211 : secondary electron 
               212 : conversion electrode 
               213 : detector 
               214 : control device