Methods and system for lithography process window simulation

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First Claim
1. A computerimplemented method of simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features, said method comprising:
 determining, using a computer, a function for generating a simulated image, said function having one or more variables accounting for process window variations from a nominal process condition associated with said lithographic process; and
generating, using the computer, said simulated image utilizing said function;
said simulated image representing an imaging result of said target design for said lithographic process.
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Accused Products
Abstract
A method of efficient simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features. The method includes the steps of determining a function for generating a simulated image, where the function accounts for process variations associated with the lithographic process; and generating the simulated image utilizing the function, where the simulated image represents the imaging result of the target design for the lithographic process. In one given embodiment, the function for simulating the aerial images with focus and dose (exposure) variation is defined as:
I(x,f,1+ε)=I_{0}(x)+└ε·I_{0}(x)+(1+ε)·a(x)·(f−f_{0})+(1+ε)·b(x)·(f−f_{0})^{2}┘
where I_{O }represents image intensity at nominal focus and exposure, f_{O }represents nominal focus, f and ε represent an actual focusexposure level at which the simulated image is calculated, and parameters “a” and “b” represent first order and second order derivative images with respect to focus change.
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22 Claims
 1. A computerimplemented method of simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features, said method comprising:
determining, using a computer, a function for generating a simulated image, said function having one or more variables accounting for process window variations from a nominal process condition associated with said lithographic process; and generating, using the computer, said simulated image utilizing said function;
said simulated image representing an imaging result of said target design for said lithographic process. View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9)
 10. A computerimplemented method of estimating feature edge shift or critical dimension (CD) change due to process window parameter variation of a lithographic process, said method comprising the steps of:
determining, using a computer, a function for generating a simulated image, said function having one or more variables accounting for process window variations from a nominal process condition associated with said lithographic process; and generating, using the computer, said simulated image utilizing said function;
said simulated image representing an imaging result of said target design for said lithographic process; andestimating, using the computer, said feature edge shift or CD change by analyzing said simulated image.  View Dependent Claims (11)
 12. A computer program product having a nontransitory computer readable medium bearing a computer program for simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features, the computer program, when executed, causing a computer to perform the steps of:
determining a function for generating a simulated image, said function having one or more variables accounting for process window variations from a nominal process condition associated with said lithographic process; and generating said simulated image utilizing said function;
said simulated image representing the imaging result of said target design for said lithographic process. View Dependent Claims (13, 14, 15, 16, 17, 18, 19, 20)
 21. A computer program product having a nontransitory computer readable medium bearing a computer program for estimating feature edge shift or critical dimension (CD) change due to process window parameter variation of a lithographic process, the computer program, when executed, causing a computer to perform the steps of:
determining a function for generating a simulated image, said function having one or more variables accounting for process window variations from a nominal process condition associated with said lithographic process; generating said simulated image utilizing said function;
said simulated image representing an imaging result of a target design for said lithographic process; andestimating said feature edge shift or CD change by analyzing said simulated image.  View Dependent Claims (22)
1 Specification
The present application claims priority to U.S. Provisional Application No. 60/992,546 filed 5 Dec. 2007, the contents of which is incorporated herein by reference in it'"'"'s entirety.
The technical field of the present invention relates generally to a method and program product for performing simulation of the imaging results associated with a lithography process, and more specifically to a computationally efficient simulation process that accounts for parameter variations over a process window.
Lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In such a case, the mask may contain a circuit pattern corresponding to an individual layer of the IC, and this pattern can be imaged onto a target portion (e.g. comprising one or more dies) on a substrate (silicon wafer) that has been coated with a layer of radiationsensitive material (resist). In general, a single wafer will contain a whole network of adjacent target portions that are successively irradiated via the projection system, one at a time. In one type of lithographic projection apparatus, each target portion is irradiated by exposing the entire mask pattern onto the target portion in one go; such an apparatus is commonly referred to as a wafer stepper. In an alternative apparatus, commonly referred to as a stepandscan apparatus, each target portion is irradiated by progressively scanning the mask pattern under the projection beam in a given reference direction (the “scanning” direction) while synchronously scanning the substrate table parallel or antiparallel to this direction. Since, in general, the projection system will have a magnification factor M (generally <1), the speed V at which the substrate table is scanned will be a factor M times that at which the mask table is scanned. More information with regard to lithographic devices as described herein can be gleaned, for example, from U.S. Pat. No. 6,046,792, incorporated herein by reference.
In a manufacturing process using a lithographic projection apparatus, a mask pattern is imaged onto a substrate that is at least partially covered by a layer of radiationsensitive material (resist). Prior to this imaging step, the substrate may undergo various procedures, such as priming, resist coating and a soft bake. After exposure, the substrate may be subjected to other procedures, such as a postexposure bake (PEB), development, a hard bake and measurement/inspection of the imaged features. This array of procedures is used as a basis to pattern an individual layer of a device, e.g., an IC. Such a patterned layer may then undergo various processes such as etching, ionimplantation (doping), metallization, oxidation, chemomechanical polishing, etc., all intended to finish off an individual layer. If several layers are required, then the whole procedure, or a variant thereof, will have to be repeated for each new layer. Eventually, an array of devices will be present on the substrate (wafer). These devices are then separated from one another by a technique such as dicing or sawing, whence the individual devices can be mounted on a carrier, connected to pins, etc.
For the sake of simplicity, the projection system may hereinafter be referred to as the “lens”; however, this term should be broadly interpreted as encompassing various types of projection systems, including refractive optics, reflective optics, and catadioptric systems, for example. The radiation system may also include components operating according to any of these design types for directing, shaping or controlling the projection beam of radiation, and such components may also be referred to below, collectively or singularly, as a “lens”. Further, the lithographic apparatus may be of a type having two or more substrate tables (and/or two or more mask tables). In such “multiple stage” devices the additional tables may be used in parallel, or preparatory steps may be carried out on one or more tables while one or more other tables are being used for exposures. Twin stage lithographic apparatus are described, for example, in U.S. Pat. No. 5,969,441, incorporated herein by reference.
The photolithographic masks referred to above comprise geometric patterns corresponding to the circuit components to be integrated onto a silicon wafer. The patterns used to create such masks are generated utilizing CAD (computeraided design) programs, this process often being referred to as EDA (electronic design automation). Most CAD programs follow a set of predetermined design rules in order to create functional masks. These rules are set by processing and design limitations. For example, design rules define the space tolerance between circuit devices (such as gates, capacitors, etc.) or interconnect lines, so as to ensure that the circuit devices or lines do not interact with one another in an undesirable way. The design rule limitations are typically referred to as “critical dimensions” (CD). A critical dimension of a circuit can be defined as the smallest width of a line or hole or the smallest space between two lines or two holes. Thus, the CD determines the overall size and density of the designed circuit. Of course, one of the goals in integrated circuit fabrication is to faithfully reproduce the original circuit design on the wafer (via the mask).
As noted, microlithography is a central step in the manufacturing of semiconductor integrated circuits, where patterns formed on semiconductor wafer substrates define the functional elements of semiconductor devices, such as microprocessors, memory chips etc. Similar lithographic techniques are also used in the formation of flat panel displays, microelectro mechanical systems (MEMS) and other devices.
As semiconductor manufacturing processes continue to advance, the dimensions of circuit elements have continually been reduced while the amount of functional elements, such as transistors, per device has been steadily increasing over decades, following a trend commonly referred to as ‘Moore'"'"'s law’. At the current state of technology, critical layers of leadingedge devices are manufactured using optical lithographic projection systems known as scanners that project a mask image onto a substrate using illumination from a deepultraviolet laser light source, creating individual circuit features having dimensions well below 100 nm, i.e. less than half the wavelength of the projection light.
This process, in which features with dimensions smaller than the classical resolution limit of an optical projection system are printed, is commonly known as lowk_{1 }lithography, according to the resolution formula CD=k_{1}×λ/NA, where λ is the wavelength of radiation employed (currently in most cases 248 nm or 193 nm), NA is the numerical aperture of the projection optics, CD is the ‘critical dimension’—generally the smallest feature size printed—and k_{1 }is an empirical resolution factor. In general, the smaller k_{1}, the more difficult it becomes to reproduce a pattern on the wafer that resembles the shape and dimensions planned by a circuit designer in order to achieve particular electrical functionality and performance. To overcome these difficulties, sophisticated finetuning steps are applied to the projection system as well as to the mask design. These include, for example, but not limited to, optimization of NA and optical coherence settings, customized illumination schemes, use of phase shifting masks, optical proximity correction in the mask layout, or other methods generally defined as ‘resolution enhancement techniques’ (RET).
As one important example, optical proximity correction (OPC, sometimes also referred to as ‘optical and process correction’) addresses the fact that the final size and placement of a printed feature on the wafer will not simply be a function of the size and placement of the corresponding feature on the mask. It is noted that the terms ‘mask’ and ‘reticle’ are utilized interchangeably herein. For the small feature sizes and high feature densities present on typical circuit designs, the position of a particular edge of a given feature will be influenced to a certain extent by the presence or absence of other adjacent features. These proximity effects arise from minute amounts of light coupled from one feature to another. Similarly, proximity effects may arise from diffusion and other chemical effects during postexposure bake (PEB), resist development, and etching that generally follow lithographic exposure.
In order to ensure that the features are generated on a semiconductor substrate in accordance with the requirements of the given target circuit design, proximity effects need to be predicted utilizing sophisticated numerical models, and corrections or predistortions need to be applied to the design of the mask before successful manufacturing of highend devices becomes possible. The article “FullChip Lithography Simulation and Design Analysis—How OPC Is Changing IC Design”, C. Spence, Proc. SPIE, Vol. 5751, pp 114 (2005) provides an overview of current ‘modelbased’ optical proximity correction processes. In a typical highend design almost every feature edge requires some modification in order to achieve printed patterns that come sufficiently close to the target design. These modifications may include shifting or biasing of edge positions or line widths as well as application of ‘assist’ features that are not intended to print themselves, but will affect the properties of an associated primary feature.
The application of modelbased OPC to a target design requires good process models and considerable computational resources, given the many millions of features typically present in a chip design. However, applying OPC is generally not an ‘exact science’, but an empirical, iterative process that does not always resolve all possible weaknesses on a layout. Therefore, postOPC designs, i.e. mask layouts after application of all pattern modifications by OPC and any other RET'"'"'s, need to be verified by design inspection, i.e. intensive fullchip simulation using calibrated numerical process models, in order to minimize the possibility of design flaws being built into the manufacturing of a mask set. This is driven by the enormous cost of making highend mask sets, which run in the multimillion dollar range, as well as by the impact on turnaround time by reworking or repairing actual masks once they have been manufactured.
Both OPC and fullchip RET verification may be based on numerical modeling systems and methods as described, for example in, U.S. Pat. No. 7,003,758 and an article titled “Optimized Hardware and Software For Fast, Full Chip Simulation”, by Y. Cao et al., Proc. SPIE, Vol. 5754, 405 (2005).
While fullchip numerical simulation of the lithographic patterning process has been demonstrated at a single process condition, typically best focus and best exposure dose or best ‘nominal’ condition, it is well known that manufacturability of a design requires sufficient tolerance of pattern fidelity against small variations in process conditions that are unavoidable during actual manufacturing. This tolerance is commonly expressed as a process window, defined as the width and height (or ‘latitude’) in exposuredefocus space over which CD or edge placement variations are within a predefined margin (i.e., error tolerance), for example ±10% of the nominal line width. In practice, the actual margin requirement may differ for different feature types, depending on their function and criticality. Furthermore, the process window concept can be extended to other basis parameters in addition to or besides exposure dose and defocus.
Manufacturability of a given design generally depends on the common process window of all features in a single layer. While stateoftheart OPC application and design inspection methods are capable of optimizing and verifying a design at nominal conditions, it has been recently observed that processwindow aware OPC models will be required in order to ensure manufacturability at future process nodes due to everdecreasing tolerances and CD requirements.
Currently, in order to map out the process window of a given design with sufficient accuracy and coverage, simulations at N parameter settings (e.g., defocus and exposure dose) are required, where N can be on the order of a dozen or more. Consequently, an Nfold multiplication of computation time is necessary if these repeated simulations at various settings are directly incorporated into the framework of an OPC application and verification flow, which typically will involve a number of iterations of fullchip lithography simulations. However, such an increase in the computational time is prohibitive when attempting to validate and/or design a given target circuit.
As such, there is a need for simulation methods and systems which account for variations in the processwindow that can be used for OPC and RET verification, and that are more computationally efficient than such a ‘bruteforce’ approach of repeated simulation at various conditions as is currently performed by known prior art systems.
In addition, calibration procedures for lithography models are required that provide models being valid, robust and accurate across the process window, not only at singular, specific parameter settings.
Accordingly, the present invention relates to a method which allows for a computationally efficient technique for considering variations in the process window for use in a simulation process, and which overcomes the foregoing deficiencies of the prior art techniques.
More specifically, the present invention relates to a method of simulating imaging performance of a lithographic process utilized to image a target design having a plurality of features. The method includes the steps of determining a function for generating a simulated image, where the function accounts for process variations associated with the lithographic process; and generating the simulated image utilizing the function, where the simulated image represents the imaging result of the target design for the lithographic process. In one given embodiment, the function is defined as:
I(x,f)=I_{0}(x)+a(x)(f−f_{0})+b(x)(f−f_{0})^{2 }
where I_{O }represents image intensity at nominal focus, f_{O }represents nominal focus, f represents an actual focus level at which the simulated image is calculated, and parameters “a” and “b” represent first order and second order derivative images.
The present invention provides significant advantages over prior art methods. Most importantly, the present invention provides a computational efficient simulation process with accounts for variations in the process window (e.g., focus variations and exposure dose variations), and eliminates the need to perform the ‘bruteforce’ approach of repeated simulation at various conditions as is currently practiced by known prior art methods. Indeed, as further noted below, when considering N process window conditions for purposes of the simulation, the computation time of the present method is approximately 2T, whereas the prior art method would require approximately NT, where T denotes the computation time required for simulating one process window condition.
The method of the present invention is also readily applied to other applications such as, but not limited to, model calibration; lithography design inspection; yield estimates based on evaluation of common process windows; identification of hot spots (or problem spots) and correction of such hotspots by utilizing process window aware OPC; and modelbased process control corrections (e.g., to center the common process window for a given lithography layer in the lithography process).
Although specific reference may be made in this text to the use of the invention in the manufacture of ICs, it should be explicitly understood that the invention has many other possible applications. For example, it may be employed in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, liquidcrystal display panels, thinfilm magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms “reticle”, “wafer” or “die” in this text should be considered as being replaced by the more general terms “mask”, “substrate” and “target portion”, respectively.
In the present document, the terms “radiation” and “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultraviolet radiation, e.g. having a wavelength in the range 520 nm).
The term mask as employed in this text may be broadly interpreted as referring to generic patterning means that can be used to endow an incoming radiation beam with a patterned crosssection, corresponding to a pattern that is to be created in a target portion of the substrate; the term “light valve” can also be used in this context. Besides the classic mask (transmissive or reflective; binary, phaseshifting, hybrid, etc.), examples of other such patterning means include:
a programmable mirror array. An example of such a device is a matrixaddressable surface having a viscoelastic control layer and a reflective surface. The basic principle behind such an apparatus is that (for example) addressed areas of the reflective surface reflect incident light as diffracted light, whereas unaddressed areas reflect incident light as undiffracted light. Using an appropriate filter, the said undiffracted light can be filtered out of the reflected beam, leaving only the diffracted light behind; in this manner, the beam becomes patterned according to the addressing pattern of the matrixaddressable surface. The required matrix addressing can be performed using suitable electronic means. More information on such mirror arrays can be gleaned, for example, from U.S. Pat. No. 5,296,891 and U.S. Pat. No. 5,523,193, which are incorporated herein by reference.
a programmable LCD array. An example of such a construction is given in U.S. Pat. No. 5,229,872, which is incorporated herein by reference.
The invention itself, together with further objects and advantages, can be better understood by reference to the following detailed description and the accompanying schematic drawings.
Prior to discussing the present invention, a brief discussion regarding the overall simulation and imaging process is provided.
In a lithography simulation system, these major system components can be described by separate functional modules, for example, as illustrated in
More specifically, it is noted that the properties of the illumination and projection optics are captured in the optical model 32 that includes, but not limited to, NAsigma (σ) settings as well as any particular illumination source shape. The optical properties of the photoresist layer coated on a substrate—i.e. refractive index, film thickness, propagation and polarization effects—may also be captured as part of the optical model 32. The mask model 30 captures the design features of the reticle and may also include a representation of detailed physical properties of the mask, as described, for example, in U.S. Pat. No. 7,587,704. Finally, the resist model 34 describes the effects of chemical processes which occur during resist exposure, PEB and development, in order to predict, for example, contours of resist features formed on the substrate wafer. The objective of the simulation is to accurately predict, for example, edge placements and CDs, which can then be compared against the target design. The target design, is generally defined as the preOPC mask layout, and will be provided in a standardized digital file format such as GDSII or OASIS.
In general, the connection between the optical and the resist model is a simulated aerial image within the resist layer, which arises from the projection of light onto the substrate, refraction at the resist interface and multiple reflections in the resist film stack. The light intensity distribution (aerial image) is turned into a latent ‘resist image’ by absorption of photons, which is further modified by diffusion processes and various loading effects. Efficient simulation methods that are fast enough for fullchip applications approximate the realistic 3dimensional intensity distribution in the resist stack by a 2dimensional aerial (and resist) image. An efficient implementation of a lithography model is possible using the following formalism, where the image (here in scalar form, which may be extended to include polarization vector effects) is expressed as a Fourier sum over signal amplitudes in the pupil plane. According to the standard Hopkins theory, the aerial image may be defined by:
where, I(x) is the aerial image intensity at point x within the image plane (for notational simplicity, a twodimensional coordinate represented by a single variable is utilized), k represents a point on the source plane, A(k) is the source amplitude from point k, k′ and k″ are points on the pupil plane, M is the Fourier transform of the mask image, P is the pupil function, and TCC_{k′,k″}=Σ_{k}A(k)^{2}P(k+k′)P*(k+k″). An important aspect of the foregoing derivation is the change of summation order (moving the sum over k inside) and indices (replacing k′ with k+k′ and replacing k″ with k+k″), which results in the separation of the Transmission Cross Coefficients (TCCs), defined by the term inside the square brackets in the third line in the equation. These coefficients are independent of the mask pattern and therefore can be precomputed using knowledge of the optical elements or configuration only (e.g., NA and σ or the detailed illuminator profile). It is further noted that although in the given example (Eq. 1) is derived from a scalar imaging model, this formalism can also be extended to a vector imaging model, where TE and TM polarized light components are summed separately.
Furthermore, the approximate aerial image can be calculated by using only a limited number of dominant TCC terms, which can be determined by diagonalizing the TCC matrix and retaining the terms corresponding to its largest eigenvalues, i.e.,
where λ_{i }(i=1, . . . , N) denotes the N largest eigenvalues and φ_{i}(•) denotes the corresponding eigenvector of the TCC matrix. It is noted that (Eq. 2) is exact when all terms are retained in the eigenseries expansion, i.e., when N is equal to the rank of the TCC matrix. However, in actual applications, it is typical to truncate the series by selecting a smaller N to increase the speed of the computation process.
Thus, (Eq. 1) can be rewritten as:
Using a sufficiently large number of TCC terms and a suitable model calibration methodology allows for an accurate description of the optical projection process and provides for ‘separability’ of the lithographic simulation model into the optics and resist models or parts. In an ideal, separable model, all optical effects such as NA, sigma, defocus, aberrations etc. are accurately captured in the optical model module, while only resist effects are simulated by the resist model. In practice, however, all ‘efficient’ lithographic simulation models (as opposed to firstprinciple models, which are generally too slow and require too many adjustable parameters to be practical for fullchip simulations) are empirical to some extent and will use a limited set of parameters. There may in some cases be ‘lumped’ parameters that account for certain combined net effects of both optical and resist properties. For example, diffusion processes during PEB of resist can be modeled by a Gaussian filter that blurs the image formed in resist, while a similar filter might also describe the effect of stray light, stage vibration, or the combined effect of highorder aberrations of the projection system. Lumped parameters can reproduce process behavior close to fitted calibration points, but will have inferior predictive power compared with separable models. Separability typically requires a sufficiently detailed model form—in the example above, e.g., using 2 independent filters for optical blurring and resist diffusion—as well as a suitable calibration methodology that assures isolation of optical effects from resist effects.
While a separable model may generally be preferred for most applications, it is noted that the description of throughprocess window “PW” aerial image variations associated with the method of the present invention set forth below does not require strict model separability. Methods for adapting a general resist model in order to accurately capture throughPW variations are also detailed below in conjunction with the method of the present invention.
The present invention provides the efficient simulation of lithographic patterning performance covering parameter variations throughout a process window, i.e., a variation of exposure dose and defocus or additional process parameters. To summarize, using an imagebased approach, the method provides polynomial series expansions for aerial images or resist images as a function of focus and exposure dose variations, or other additional coordinates of a generalized PW. These expressions involve images and derivative images which relate to TCCs and derivative TCC matrices. Linear combinations of these expressions allow for a highly efficient evaluation of the image generated at any arbitrary PW point. In addition, edge placement shifts or CD variations throughout the PW are also expressed in analytical form as simple linear combinations of a limited set of simulated images. This set of images may be generated within a computation time on the order of approximately 2 times the computation time for computing a single image at NC (Nominal Condition), rather than N× by computing images at N separate PW conditions. Once this set of images is known, the complete throughPW behavior of every single edge or CD on the design can be immediately determined.
It is noted that the methods of the present invention may also be utilized in conjunction with model calibration, lithography design inspection, yield estimates based on evaluating the common PW, identification of hot spots, modification and repair of hot spots by PWaware OPC, and modelbased process control corrections, e.g., to center the common PW of a litho layer.
The basic approach of the method can be understood by considering throughfocus changes in resist line width (or edge placement) of a generic resist line. It is well known that the CD of the resist line typically has a maximum or minimum value at best focus, but the CD varies smoothly with defocus in either direction. Therefore, the throughfocus CD variations of a particular feature may be approximated by a polynomial fit of CD vs. defocus, e.g. a secondorder fit for a sufficiently small defocus range. However, the direction and magnitude of change in CD will depend strongly on the resist threshold (dose to clear), the specific exposure dose, feature type, and proximity effects. Thus, exposure dose and throughfocus CD changes are strongly coupled in a nonlinear manner that prevents a direct, general parameterization of CD or edge placement changes throughout the PW space.
However, the aerial image is also expected to show a continuous variation through focus. Every mask point may be imaged to a finitesized spot in the image plane that is characterized by the point spread function of the projection system. This spot will assume a minimum size at best focus but will continuously blur into a wider distribution with both positive and negative defocus. Therefore, it is possible to approximate the variation of image intensities through focus as a secondorder polynomial for each individual image point within the exposure field:
I(x,f)=I_{0}(x)+a(x)·(f−f_{0})+b(x)·(f−f_{0})^{2} (Eq. 4)
where f_{0 }indicates the nominal or best focus position, and f is the actual focus level at which the image I is calculated. The secondorder approximation is expected to hold well for a sufficiently small defocus range, but the accuracy of the approximation may easily be improved by including higherorder terms if required (for example, 3^{rd }order and/or 4^{th }order terms). In fact, (Eq. 4) can also be identified as the beginning terms of a Taylor series expansion of the aerial image around the nominal best focus plane:
which can in principle be extended to an arbitrarily sufficient representation of the actual throughfocus behavior of the aerial image by extension to include additional higherorder terms. It is noted that the choice of polynomial base functions is only one possibility to express a series expansion of the aerial image through focus, and the methods of the current invention are by no means restricted to this embodiment, e.g., the base functions can be special functions such as Bessel Functions, Legendre Functions, Chebyshev Functions, Trigonometric functions, and so on. In addition, while the process window term is most commonly understood as spanning variations over defocus and exposure dose, the process window concept can be generalized and extended to cover additional or alternative parameter variations, such as variation of NA and sigma, etc.
Comparison of (Eq. 4) and (Eq. 5) reveals the physical meaning of the parameters “a” and “b” as first and secondorder derivative images. These may in principle be determined directly as derivatives by a finite difference method for every image point and entered into (Eq. 4) and (Eq. 5) to interpolate the image variations. Alternatively, in order to improve the overall agreement between the interpolation and the actual through focus variation over a wider range, the parameters a and b can be obtained from a least square fit of (Eq. 4) over a number of focus positions {f_{1}, f_{2}, . . . , f_{L}} for which aerial images are explicitly calculated as {I_{1}, I_{2}, . . . , I_{L}}. The parameters “a” and “b” are then found as solutions to the following system of equations in a least square sense (assuming here that L>3, in which case the system of equations is overdetermined).
Without loss of generality, it is assumed that f_{0}=0 so as to simplify the notation. Then for a fixed image point,
I_{1}=I_{0}+a·f_{1}+b·f_{1}^{2 }
I_{2}=I_{0}+a·f_{2}+b·f_{2}^{2 }
. . .
I_{L}=I_{0}+a·f_{L}+b·f_{L}^{2} (Eq. 6)
where I_{0 }is the aerial image at nominal conditions (NC), i.e. f=f_{0}. The solution to the above set of equations minimizes the following sum of squared differences, with the index l referring to the L different focus conditions:
where W_{l }is a userassigned weight to defocus f_{l }(l=1, 2, . . . , L). Through {W_{1}, W_{2}, . . . , W_{L}}, it is possible to assign different weights to different focuses. For example, in order to make the 2^{nd }order polynomial approximation have a better match at PW points closer to NC, it is possible to assign a larger weight close to NC and a smaller weight away from NC; or if it is desired for all focus points to have equal importance, one can simply assign equal weights, i.e., W_{1}=W_{2}= . . . =W_{L}=1. For large deviations in focus and dose relative to the nominal condition, many patterns become unstable in printing and the measurements of CDs become unreliable, in such cases it may be desirable to assign small weights to such process window conditions.
To solve (Eq. 7), it is noted that the best fit will fulfill the conditions:
(Eq. 8) can be solved analytically, resulting in immediate expressions for “a” and “b” as the linear combination or weighted sum of the {I_{l}}, as shown below. The coefficients of this linear combination do not depend on the pixel coordinate or pattern, but only on the values of the {f_{l}} and {W_{l}}. As such, these coefficients can be understood as forming a linear filter for the purpose of interpolation in the space of f, and the particular choice of polynomials as base functions gives rise to the specific values of the coefficients, independent of the mask pattern. More specifically, the calculation of these coefficients is performed once the values of {f_{l}} and {W_{l}} are determined, without knowing the specific optical exposure settings or actually carrying out aerial image simulations.
With regard to solving (Eq. 8), (Eq. 7) can be rewritten as:
As a result, (Eq. 8) can be expanded as:
Note that:
As is made clear below, this property will be useful in the resist model section. The above set of equations can be readily generalized to accommodate a higherorder polynomial fitting.
The benefit of introducing the derivative images “a” and “b” is that using (Eq. 4), the aerial image can be predicted at any point of the process window by straightforward scaling of the a and b images by the defocus offset and a simple addition, rather than performing a full image simulation (i.e., convolution of the mask pattern with the TCCs) at each particular defocus setting required for a PW analysis. In addition, changes in exposure dose can be expressed by a simple upscaling or downscaling of the image intensity by a factor (1+ε):
I(x,f,1+ε)=(1+ε)·I(x,f) (Eq. 11)
where I(x,f) is the aerial image at the nominal exposure dose, while ε is the relative change in dose.
Combining this with (Eq. 4) yields the general result:
where ΔI will typically be small perturbations within a reasonable range of PW parameter variations.
The foregoing method is exemplified by a flow diagram in
It is noted that if a sufficient coverage of the process window requires evaluation at N process window points, and L<N images are used for fitting the derivative images a and b, the reduction in computation time will be close to L/N, since scaling the predetermined images I_{0}, a and b requires significantly less computation time than an independent recalculation of the projected image at each new parameter setting. The foregoing method is generally applicable, independent of the specific details of the aerial image simulation. Furthermore, it is also applicable to both the aerial image as well as to the resist image from which simulated resist contours are extracted.
The foregoing method also does not depend on any specific model or implementation used for simulating the set of aerial images {I_{1}, I_{2}, . . . , I_{L}} at varying defocus. However, the foregoing method requires a number L>2 of individual images to be simulated for each mask layout under consideration. In a second embodiment of the method of the present invention, an even more efficient solution is made possible by the TCC formalism introduced in (Eq. 1).
From (Eq. 1), each aerial image at focus f_{l }(l=0, 1, . . . , L) can be defined as:
I_{l}(x)=Σ_{k′}Σ_{k″}TCC_{l,k′,k″}M(k′)M*(k″)exp(−j(k′−k″)x)
where TCC_{l }is the TCC at focus f_{l }and TCC_{l,k′,k″} is the matrix element of TCC_{l}, and M(•) represents the mask image, which is independent of the focus.
Combining this with (Eq. 9) and exchanging the order of summation,
Thus if two new TCCs are defined as linear combinations of TCC_{l }(l=0, 1, . . . , L) in the following way:
then “a” and “b” are “aerial images” which can be computed directly from A and B, i.e.,
are the matrix elements of A and B, respectively. This also implies that a linear combination of aerial images of different planes can be computed using a single linear combination of TCCs corresponding to those planes.
A significant advantage of using TCC_{0}, A, and B in place of the L throughfocus images is that the TCC_{0}, A, and B can be precomputed, independently of the actual mask pattern, for known illumination and projection parameters, giving rise to the possibility of further reduction of computing time (down from L throughfocus simulations for each mask pattern), which will be further explained below. It is noted that the generation of A and B neither requires computation of a set of aerial images at different defocus conditions nor requires calibration from this set of aerial images. Once TCC_{0}, A, and B have been calculated, these terms can be generally applied to predict the throughfocus imaging performance for any specific mask design using (Eq. 15) and (Eq. 4). Besides the throughfocus variation, a variation of exposure dose around nominal condition can be applied to the TCC terms by the same linear scaling as described by (Eq. 11) and (Eq. 12) above.
Calculating the derivative images a and b from TCCs A and B allows a further reduction of computation time by using only the dominant terms of A and B, as in the discussions related to (Eq. 2). More specifically, suppose the diagonalization of TCC_{0}, A and B is:
where λ_{0,i }(i=1, . . . , N_{0}) denotes the N_{0 }largest eigenvalues and φ_{0,i}(•) denotes the corresponding eigenvector of the TCC matrix TCC_{0}; λ_{A,i }(i=1, . . . , N_{A}) denotes the N_{A }largest eigenvalues and φ_{A,i}(•) denotes the corresponding eigenvector of the TCC matrix A; and λ_{B,i }(i=1, . . . , N_{B}) denotes the N_{B }largest eigenvalues and φ_{B,i}(•) denotes the corresponding eigenvector of the TCC matrix B.
Then, from (Eq. 3), for mask image M(•),
where I_{0 }is the nominal aerial image,
Utilizing a larger number of TCC terms generally improves the accuracy of the optical model and the separability of optical and resist model components. However, since the image or TCC derivatives relate to relatively minor image variations within the PW, typically on the order of 10% in CD variation, a smaller number of terms may suffice for the A and B terms than for the Nominal Condition TCC_{0}. For example, if 64 terms are considered for TCC_{0}, (i.e., N_{0}=64), only 32 terms are typically required for each of the A and B terms in order to achieve sufficient CD prediction accuracy, i.e., N_{A}=N_{B}=32. In this case, approximately the same amount of computation time will be required to generate the derivative images a and b as compared to the nominal condition I_{0}. It is noted that, unlike the original TCC matrices, a coefficient TCC matrix such as A or B is in general not nonnegativedefinite, which implies both positive and negative eigenvalues exist for a derivative TCC matrix. Therefore, the leading terms from the eigenseries expansion and truncation should include all eigenvalues with the largest absolute values, both positive and negative.
Similar to (Eq. 5), (Eq. 14) can be derived alternatively from series expansion. More specifically, the variation of TCC matrix elements around nominal or best focus f_{0 }may also be expressed as a series expansion:
Thus, the coefficients of the series expansion may be evaluated directly by a numerical finite difference method, or again from a leastsquare fitting to a number of individually calculated TCC terms corresponding to a set of focus positions, in a manner similar to the throughfocus fitting of aerial images discussed in the previous section. The fitting approach provides a wider range of validity, and introduces weight factors to place more or less emphasis on certain parts of the PW. This approach will follow (Eq. 6)(Eq. 9) after replacing the set of test images I_{l }by their corresponding TCCs in the equations. Consequently, the best fit derivative matrices A and B are obtained from the same linear combination set forth above, also after formally replacing the I_{l }by TCC_{l}, i.e.,
where h_{al }and h_{bl }are again computed using (Eq. 9). It is noted that h_{al }and h_{bl }are constants that do not depend on the patterns or TCC_{l}. Thus, A and B are simply a linear combination of the Nominal Condition TCC_{0 }and a set of TCC'"'"'s at various defocus conditions (TCC_{1 }through TCC_{L}).
Note that (Eq. 19) is the same as (Eq. 14), thus these two alternative approaches lead to the same final formulation. Similarly, (Eq. 4) can also be derived from (Eq. 15), (Eq. 18), and (Eq. 19).
The method of the second embodiment is exemplified by the flow diagram in
In the flowchart of
An additional reduction in computation time may be achieved by further suitable assumptions or a priori knowledge about the physics of the optical system. For example, in the absence of strong aberrations, it can be expected that the throughfocus variation of aerial image intensities will be an even (i.e. symmetrical) function of defocus. Therefore, it can be expected that the firstorder derivatives “A” and “a” will be negligible under these conditions.
This simplification can be further justified by noting that the effect of defocus corresponds to a multiplication of the pupil function by a phase factor p=p_{0 }exp[ja(f−f_{0})^{2}], where the nominal focus is at f_{0}=0. For small defocus the phase shift can be approximated by a Taylor expansion: p=p_{0}. [1+ja(f−f_{0})^{2}], which does not contain a linear term.
All the above methods may also be extended to a generalized process window definition that can be established by different or additional base parameters in addition to exposure dose and defocus. These may include, but are not limited to, optical settings such as NA, sigma, aberrations, polarization, or optical constants of the resist layer (whose effects on the imaging process are included in the optical model, i.e. the TCCs). As one example, including a variation of NA around nominal conditions, the aerial image can be expressed as:
I(f,NA)=I_{0}+a·(f−f_{0})+b·(f−f_{0})^{2}+c·(NA−NA_{0})+d·(NA−NA_{0})^{2}+e·(f−f_{0})·(NA−NA_{0}) (Eq. 20)
where I, I_{0}, a, . . . , e are 2dimensional images and image derivatives, respectively. The additional parameters “c”, “d”, and “e” can be determined by a least square fit to a set of simulated images or a set of simulated TCCs at varying parameter values for f and NA, while the scaling with exposure dose as in (Eq. 11) and (Eq. 12) still applies. It is noted that, similar to (Eq. 9), these parameters (a, b, c, d, and the crossterm coefficient e) are again a linear combination of aerial images {I_{l}}. The coefficients of this linear combination do not depend on the pixel coordinate or pattern, but only on the values of the {f_{l}}, {NA_{l}}, and/or the userassigned weights {W_{l}}.
For this generalized PW model, simplifications based on physical insight are also possible. In case of NA variations, for example, it can be expected that these will have a rather monotonous, linear effect on the image variations, in which case (Eq. 20) can be simplified by dropping the higher order “d” and “e” terms in NA, possibly in addition to the linear term in defocus. Also, for any generalized PW definition, the number of TCC terms used for calculating I_{0 }at Nominal Condition need not be the same as the number of terms used for calculating image variations from the TCC derivatives A, B, . . . . A sufficiently accurate description of minor image variations due to small parameter variations around Nominal Condition may be achieved with a large number of terms for I_{0 }and a significantly smaller number for the derivatives, in order to reduce the overall computation time.
For simplicity purposes, the following discussion will be based on defocus and exposure dose. However, it should be noted that all the disclosures herein can be extended to generalized PW with other parameters such as NA, sigma, aberrations, polarization, or optical constants of the resist layer, as illustrated in (Eq. 20).
In the embodiments set forth above, analytic expressions for the aerial image in the vicinity of best focus for a range of PW parameters were developed. The following descriptions derive similar expressions and methods to calculate the resist image, which forms the basis for extraction of simulated resist contours across the PW.
Separable, Linear Resist Model
Although the response of photo resist to illumination by the projected aerial image may be strongly nonlinear, having a thresholding behavior, many processes occurring in the resist layer, such as diffusion during PEB, can be modeled by convoluting the aerial image with one or more linear filters before applying the threshold. Such models will be generally referred to as ‘linear’ resist models, and the latent resist image for such models may be expressed schematically as:
R(x)=P{I(x)}+R_{b}(x) (Eq. 21)
here, P{ } denotes the functional action of applying a linear filter (i.e. generally a convolution), while R_{b }is a mask loading bias that is independent of the aerial image. The resist threshold is understood to be included in R_{b }such that resist contours correspond to locations where R(x)=0.
Applying this model to the general, scaled, interpolated aerial image derived above, i.e., (Eq. 12, assuming f_{0}=0 without loss of generality), results in
where R_{0 }is the resist image at Nominal Condition (NC). All corrections due to changes in exposure dose and focus (or, other PW parameters) are derived by applying the same filter to the derivative images a, b as to the image I_{0 }at NC, and simple scaling and summation of the correction terms.
Moreover, the effect of a linear filter may be included in the imaging TCC formalism, since the convolution with a filter in the space domain is equivalent to a multiplication with the filter'"'"'s Fourier series components in the frequency domain. Starting from an aerial image expression (Eq. 1):
I(x)=Σ_{k′}Σ_{k″}TCC_{k′,k″}M(k′)M*(k″)exp(−j(k′−k″)x)
It is shown that the TCC matrix element at k′, k″ contributes to the (k′−k″) frequency component of I(x) by the amount TCC_{k′,k″}M(k′)M*(k″). Therefore, a resist image defined by:
I(x)g(x)
where g(x) is a spatial filter with the Fourier transform being G(k), can be expressed as:
with a new TCC matrix defined as
TCC^{new}_{k′,k″}=TCC_{k′,k″}G(k′−k″)
With this procedure, the linear filter is incorporated into the bilinear TCC matrix, so all the computational procedures applicable to a purely optical aerial image may be applied to the linearly filtered aerial image. This property allows a significant reduction in overall computation time, since the complete resist image can be generated by a single evaluation of (Eq. 1), with the only modification of adding weight factors corresponding to the Fourier coefficients of the filter P. For any given mask design input, this formulation would allow to generate directly, in one pass each, the images P{I_{0}}, P{a}, P{b} from the precomputed, filteradjusted TCC_{0}, A, and B matrices. (Eq. 22) then defines the actual resist image for any arbitrary PW point as a linear combination of these three images.
NonSeparable, Linear Resist Model
In the preceding discussion, it was implicitly assumed that all parameters of the linear filters establishing the resist model are constant across the variations of the process window parameters. This equates to one condition for an overall separable lithography model: resist model parameters are independent of optical model parameters. A pragmatic test for separability is the ability to accurately calibrate the model and fit test data across the complete extent of the PW. In practice, the semiempirical nature of models suitable for fullchip lithography simulation may preclude perfect separability and may require resist model parameters that are allowed to vary with PW parameters such as defocus, NA or sigma settings. For a physically motivated model, it should be expected (or required as a constraint), though that the model parameters vary smoothly under variation of the PW variables. In this case, the series expansion of the resist image may include derivative terms of the resist model parameters.
For illustration purposes, consider defocus as the only PW parameter. If the linear resist model is equivalent to a convolution with a linear filter, (or a multitude of linear filters), a separable model may be described by:
R(x,f)=P(x)I(x,f)+R_{b}(x) (Eq. 23)
while a nonseparable model may require an explicit fdependence of the filter
R(x,f)=P(x,f)I(x,f)+R_{b}(x) (Eq. 24)
Now, considering throughfocus changes, a proforma series expansion may be applied to (Eq. 24), for illustration herein only up to first order:
If the resist model parameters are found to vary continuously across the PW space, similar series expansion and fitting as introduced above for the AI and TCCs can be applied to the resist model parameters during model calibration. In this case a linear, derivative filter a_{P }can be calculated and be used in (Eq. 25), which may also be extended in a straightforward way to include higherorder terms. In this situation, resist model parameters as well as aerial image variations are smoothly interpolated across the complete PW area. Both P and a_{P }can be determined in a throughPW model calibration step based on experimental wafer data from test or gauge patterns.
However, even if resist model parameters appear to vary nonmonotonously across the PW, any piecewise interpolation in between calibration points could provide ‘bestguess’ resist model parameters for arbitrary PW points.
General Resist Model
For a general resist model that may include nonlinear operations such as truncations of the aerial or resist image, the straightforward separation into nominal condition and derivative terms, as shown in (Eq. 22) will be no longer valid. However, there are three alternative methods to deal with the nonlinear operations.
i) Associated Linear Filter
First, it is assumed that the general variation of the resist image through PW can be approximated formally by the second line in (Eq. 22), with the reinterpretation that the linear filter P{ } will no longer correctly describe the resist model at NC (Normal Condition). Instead, linear filter P{ } will be chosen to reproduce the best representation of differential resist image changes relative to the NC. While a nonlinear model may ensure the most accurate model fitting at the NC, it may require significantly more computation time than a linear model. By relying on such an associated linear filter to emulate the differential throughPW behavior, only a single evaluation of the nonlinear model will be required to generate R_{0}(x), while PW analysis at a multitude of PW conditions can be based on more efficient evaluation of P{I_{0}}, P{a}, P{b}.
The coefficients of the nominal condition resist model as well as of the associated filter may be determined from a unified model calibration procedure based on calibration test patterns and wafer gauge data covering pattern variations and process window variations, as an extension of the method described in U.S. P App. No. 60/719,837.
Further, once a valid unified PW model (FEM) has been generated and calibrated in the manner set forth in U.S. P App. No. 60/719,837, it will provide the best prediction of throughPW changes of the resist image. The parameters of the optimum associated filter may then be determined by minimizing the overall (RMS (root mean square)) difference between the simplified model using the associated filter and the complete, calibrated model, without any need for additional experimental calibration data.
Using the full model, for any suitable number and range of test structures, including e.g. 1D (line/space) and 2D (line ends etc) patterns, ‘correct’ resist images and contours can be simulated for any number of PW points. In addition, the values of the derivative images a and b can be calculated in the vicinity of the resist contours. For each pattern, the change of R(x) throughPW will be calculated at patternspecific gauge points, e.g. the tip of a line for a lineend test pattern, or along any point of the NC resist contour. At each of these evaluation points x_{i }through
ΔR(x_{i},ε,f)=R(x_{i},ε,f)−R(x_{i},ε=0,f=f_{0})=R(x_{i},ε,f) (Eq. 27)
since x_{i }is assumed to be on a resist contour, where R(x_{i},ε=0,f=f_{0})=0.
ΔR(x_{i},ε,f) should be well approximated by
ΔR_{a}(x_{i})=ε·P{I_{0}(x_{i})}+(1+ε)·f·P{a(x_{i})}+(1+ε)·f^{2}·P{b(x_{i})} (Eq. 28)
Therefore, the optimal associated filter will minimize the sum of squared differences between (Eq. 27) and (Eq. 28), and can be determined by a variety of known optimization algorithms. It is noted that evaluation of (Eq. 27) and (Eq. 28) during the associated filter fitting should be performed at resist contours, so that the resulting filter most closely reproduces changes close to edge positions. Performance of the associated filter—in terms of accurately predicting changes in the resist image level—far away from edge positions is generally not required. After this fitting routine, the fullPW behavior of the resist images is again described as
R(x,ε,f)=R_{0}(x)+ΔR_{a}(x,ε,f) (Eq. 29)
where the filtered differential images can be efficiently calculated within the TCC formalism, the ΔR constitutes relatively small perturbations, and the resist images at any arbitrary PW point can be predicted from a simple linear combination of the four images R_{0}, P{I_{0}}, P{a}, and P{b}.
ii) Embedded Linearization
The above approach presents a linearized filter (i.e., the associated filter) which is optimal in that it is the single linear filter which minimizes the (RMS) difference for all patternspecific gauge points or along any point of the NC (Nominal Condition) resist contour. Next, an alternative approach is discussed which incorporates resist model linearization in the computation of derivative resist images.
More specifically, after obtaining a and b in (Eq. 2), the goal becomes identifying R_{0}, Ra and Rb such that their linear combination (assuming that f_{0}=0 without loss of generality)
R_{EL}(x,f)=R_{0}(x)+Ra(x)·f+Rb(x)·f^{2} (Eq. 30)
is the best fit for
over a number of focus positions f_{l}={f_{1}, f_{2}, . . . , f_{L}} with possibly a set of weights {W_{1}, W_{2}, . . . , W_{L}}, where R_{0 }is the resist image at NC. (Eq. 31) is essentially applying the resist model R{•} to the aerial image expressed in (Eq. 2). It is noted that the resist model R{•} may be nonlinear, thus Ra and Rb are not necessarily P{a} and P{b} or R{a} and R{b}.
As such:
where h_{al }and h_{bl }are coefficients defined in (Eq. 9). The coefficients only depend on {f_{1}, f_{2}, . . . , f_{L}} and possibly weights {W_{1}, W_{2}, . . . , W_{L}}, and they are independent of R(x, f_{l}) or I(x, f_{l}).
In general, the resist model R{•} can be separated as:
R{I(x)}=P{I(x)}+P_{NL}{I(x)}+R_{b} (Eq. 33)
where R_{b }is a mask loading bias that is independent of the aerial image I(x) or focus, P{ } is the linear filter operation and P_{NL}{ } is some nonlinear operation.
Combining (Eq. 32) and (Eq. 33),
As discussed previously, since P{ } is a linear operation, then
As expected, it is possible to derive the following result with the aid of (Eq. 9) and (Eq. 10) set forth above,
Thus, Ra and Rb can computed from
The benefits of this approach are that it does not attempt to capture the differential throughPW behavior for all gauge points using a single linear filter. Rather, this approach minimizes the (RMS) difference for each pixel, thereby improving the overall accuracy. In addition, this approach does not require identification of patternspecific gauge points or all NC resist contour neighboring points. One drawback is that this approach slightly increases the computation complexity for Ra and Rb. However, since the synthesis of throughPW resist images only require scaling and additions of R_{0}, Ra and Rb, the increase in the computation complexity of the derivative images is generally insignificant compared to the reduction in computation complexity of throughPW resist images, especially for dense PW sampling.
iii) Polynomial Approximation of NonLinear Operations
In a third approach, nonlinear resist model operations are approximated using polynomials. More specifically, for truncation operations on image I(x), for the purpose of emulating acid and base reaction effects, quadratic polynomials of the image provide a sufficient approximation. Another typical nonlinear operation, the linear filtering of the image slope, can be expressed precisely as the linear filtering of a quadratic function of the image gradient G{I(x)}=I(x)−I(x−1), thus the quadratic polynomial of the aerial image I(x) itself. More specifically, letting G{ } be the gradient operation and the linear filter be P_{slope}{•} then this nonlinear operation can be expressed as:
P_{Slope}{G{I(x)}}=P_{slope}{(I(x)−I(x−1))^{2}} (Eq. 38)
To summarize, the resist image from aerial image I(x) can be approximated as:
Once again, P_{1}{•} represents the linear filter for the aerial image term, P_{2}{•} represents the linear filter for the aerial image square term, and P_{slope}{•} represents the linear filter for the aerial image gradient term, while R_{b }is a mask loading bias that is independent of the image pattern. Thus the resist image is expressed as a 4^{th}order polynomial of the defocus value. However, in a typical application, R_{3}(x) and R_{4}(x) are very small and may be ignored to improve the computational efficiency.
As noted above, the goal of lithography design verification is to ensure that printed resist edges and line widths are within a prespecified distance from the design target. Similarly, the size of the process window—exposure latitude and depth of focus—are defined by CDs or edge placements falling within the specified margin. The various methods outlined above provide very efficient ways to determine the change of resist image signal level with variation of focus and exposure dose or other, generalized PW parameters. Each method resulted in an approximate expression of throughPW resist image variations ΔR as perturbation of the NC (Nominal Condition) image R_{0}.
In order to relate these changes in R(x) to changes in edge placement, in most cases a firstorder approximation will suffice, due to the small CD or edge placement tolerances. Therefore, the lateral shift of any resist contour (R=0) (i.e., the edge placement change) is simply approximated by the image gradient G at the original (i.e. NC) contour location and the change in resist image level ΔR due to variation of focus, dose, etc. as:
where both the initial contour location and the gradient are determined from the resist image at NC, i.e. R_{0}(x,y). The 2dimensional edge shift can be calculated separately in x and y direction by the partial image derivative in each direction, or as an absolute shift using an absolute gradient value, i.e. the geometrical sum of S_{x}=R_{0}(x,y)−R_{0}(x−1,y) and S_{y}=R_{0}(x,y)−R_{0}(x,y−1), i.e., the absolute gradient value S=√{square root over (S_{x}^{2}+S_{y}^{2})}.
From the foregoing explanation, the edge shift can be directly expressed as a function of the differential images defined above:
while changes in CD or line widths can be determined from adding the individual edge placement shifts on either side of a line, resulting generally in ΔCD=2·ΔEP. Clearly, (Eq. 41) will be able to reproduce the typical 2^{nd }orderlike throughfocus behavior of CD or EPE curves. More importantly, after the set of images such as [R_{0}, P{I_{0}}, P{a}, P{b}] has been calculated, which may be accomplished with only ˜1× more computation than simulating the single image at NC (assuming that fewer TCC terms are required for sufficient accuracy on the differentials), (Eq. 41) may be applied to map out analytically the complete PW for every single edge position on a design, without the need for any further timeconsuming image simulation. A generic flow diagram to illustrate this method is provided in
Referring to
The methods detailed above, and in particular (Eq. 41) can be applied very flexibly for a wide range of tasks in lithography design inspection. Some of these applications are briefly outlined below. However, it is noted that the present invention is not limited to the applications disclosed herein.
For any particular edge or CD, (Eq. 41) allows straightforward determination of the focus latitude (=DOF (Depth of Focus)) at nominal dose, for a given tolerance of CD, EP or line end variation.
For any particular edge or CD, (Eq. 41) allows straightforward determination of the exposure dose at nominal focus, for a given tolerance of CD, EP or line end variation.
For any particular edge or CD, (Eq. 41) allows straightforward mapping of the shape, center and area of the PW in {F,E} space or a generalized PW space, for a given tolerance of CD, EP or line end variation.
For a set of edges or CDs covering the full chip design and all relevant pattern/feature types, the common process window of the design can be efficiently calculated, and process corrections may be derived in order to center the common PW.
Critical, limiting patterns may be identified that define the inner boundaries of the common PW, by either having offcentered PWs or small PWs.
The common PW area may be mapped out as a function of tolerance specs on EP or CD variations. This sensitivity analysis may provide a yield estimate depending on design sensitivity.
Design hot spots may be identified from a fullchip analysis using (Eq. 41), as patterns with PW area, DOF or exposure latitude falling below a certain threshold. The behavior of these critical patterns may then be investigated in detail by fullPW simulations, i.e. using the full simulation model for repeated image and resist contour simulation at many points across the PW.
According to one embodiment of the invention, portions of the simulation process may be performed by computer system 100 in response to processor 104 executing one or more sequences of one or more instructions contained in main memory 106. Such instructions may be read into main memory 106 from another computerreadable medium, such as storage device 110. Execution of the sequences of instructions contained in main memory 106 causes processor 104 to perform the process steps described herein. One or more processors in a multiprocessing arrangement may also be employed to execute the sequences of instructions contained in main memory 106. In alternative embodiments, hardwired circuitry may be used in place of or in combination with software instructions to implement the invention. Thus, embodiments of the invention are not limited to any specific combination of hardware circuitry and software. The term “computerreadable medium” as used herein refers to any medium that participates in providing instructions to processor 104 for execution. Such a medium may take many forms, including but not limited to, nonvolatile media, and volatile media. Nonvolatile media include, for example, optical or magnetic disks, such as storage device 110. Volatile media include dynamic memory, such as main memory 106. Common forms of computerreadable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CDROM, DVD, any other optical medium, a RAM, a PROM, and EPROM, a FLASHEPROM, any other memory chip or cartridge, or any other medium from which a computer can read.
Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor 104 for execution. For example, the instructions may initially be borne on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system 100 can receive the data on the telephone line and use an infrared transmitter to convert the data to an infrared signal. An infrared detector coupled to bus 102 can receive the data carried in the infrared signal and place the data on bus 102. Bus 102 carries the data to main memory 106, from which processor 104 retrieves and executes the instructions. The instructions received by main memory 106 may optionally be stored on storage device 110 either before or after execution by processor 104.
Computer system 100 also preferably includes a communication interface 118 coupled to bus 102. Communication interface 118 provides a twoway data communication coupling to a network link 120 that is connected to a local network 122. For example, communication interface 118 may be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface 118 may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface 118 sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information. Network link 120 typically provides data communication through one or more networks to other data devices. For example, network link 120 may provide a connection through local network 122 to a host computer 124 or to data equipment operated by an Internet Service Provider (ISP) 126. ISP 126 in turn provides data communication services through the worldwide packet data communication network, now commonly referred to as the “Internet” 128. Local network 122 and Internet 128 both use electrical, electromagnetic or optical signals that carry digital data streams. The signals through the various networks and the signals on network link 120 and through communication interface 118, which carry the digital data to and from computer system 100, are exemplary forms of carrier waves transporting the information.
Computer system 100 can send messages and receive data, including program code, through the network(s), network link 120, and communication interface 118. In the Internet example, a server 130 might transmit a requested code for an application program through Internet 128, ISP 126, local network 122 and communication interface 118. In accordance with the invention, one such downloaded application provides for the illumination optimization of the embodiment, for example. The received code may be executed by processor 104 as it is received, and/or stored in storage device 110, or other nonvolatile storage for later execution. In this manner, computer system 100 may obtain application code in the form of a carrier wave.
a radiation system Ex, IL, for supplying a projection beam PB of radiation. In this particular case, the radiation system also comprises a radiation source LA;
a first object table (mask table) MT provided with a mask holder for holding a mask MA (e.g., a reticle), and connected to first positioning means for accurately positioning the mask with respect to item PL;
a second object table (substrate table) WT provided with a substrate holder for holding a substrate W (e.g., a resistcoated silicon wafer), and connected to second positioning means for accurately positioning the substrate with respect to item PL;
a projection system (“lens”) PL (e.g., a refractive, catoptric or catadioptric optical system) for imaging an irradiated portion of the mask MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.
As depicted herein, the apparatus is of a transmissive type (i.e., has a transmissive mask). However, in general, it may also be of a reflective type, for example (with a reflective mask). Alternatively, the apparatus may employ another kind of patterning means as an alternative to the use of a mask; examples include a programmable mirror array or LCD matrix.
The source LA (e.g., a mercury lamp or excimer laser) produces a beam of radiation. This beam is fed into an illumination system (illuminator) IL, either directly or after having traversed conditioning means, such as a beam expander Ex, for example. The illuminator IL may comprise adjusting means AM for setting the outer and/or inner radial extent (commonly referred to as σouter and σinner, respectively) of the intensity distribution in the beam. In addition, it will generally comprise various other components, such as an integrator IN and a condenser CO. In this way, the beam PB impinging on the mask MA has a desired uniformity and intensity distribution in its crosssection.
It should be noted with regard to
The beam PB subsequently intercepts the mask MA, which is held on a mask table MT. Having traversed the mask MA, the beam PB passes through the lens PL, which focuses the beam PB onto a target portion C of the substrate W. With the aid of the second positioning means (and interferometric measuring means IF), the substrate table WT can be moved accurately, e.g. so as to position different target portions C in the path of the beam PB. Similarly, the first positioning means can be used to accurately position the mask MA with respect to the path of the beam PB, e.g., after mechanical retrieval of the mask MA from a mask library, or during a scan. In general, movement of the object tables MT, WT will be realized with the aid of a longstroke module (coarse positioning) and a shortstroke module (fine positioning), which are not explicitly depicted in
The depicted tool can be used in two different modes:
In step mode, the mask table MT is kept essentially stationary, and an entire mask image is projected in one go (i.e., a single “flash”) onto a target portion C. The substrate table WT is then shifted in the x and/or y directions so that a different target portion C can be irradiated by the beam PB;
In scan mode, essentially the same scenario applies, except that a given target portion C is not exposed in a single “flash”. Instead, the mask table MT is movable in a given direction (the socalled “scan direction”, e.g., the y direction) with a speed v, so that the projection beam PB is caused to scan over a mask image; concurrently, the substrate table WT is simultaneously moved in the same or opposite direction at a speed V=Mv, in which M is the magnification of the lens PL (typically, M=¼ or ⅕). In this manner, a relatively large target portion C can be exposed, without having to compromise on resolution.
The concepts disclosed herein may simulate or mathematically model any generic imaging system for imaging sub wavelength features, and may be especially useful with emerging imaging technologies capable of producing wavelengths of an increasingly smaller size. Emerging technologies already in use include EUV (extreme ultra violet) lithography that is capable of producing a 193 nm wavelength with the use of a ArF laser, and even a 157 nm wavelength with the use of a Fluorine laser. Moreover, EUV lithography is capable of producing wavelengths within a range of 205 nm by using a synchrotron or by hitting a material (either solid or a plasma) with high energy electrons in order to produce photons within this range. Because most materials are absorptive within this range, illumination may be produced by reflective mirrors with a multistack of Molybdenum and Silicon. The multistack mirror has a 40 layer pairs of Molybdenum and Silicon where the thickness of each layer is a quarter wavelength. Even smaller wavelengths may be produced with Xray lithography. Typically, a synchrotron is used to produce an Xray wavelength. Since most material is absorptive at xray wavelengths, a thin piece of absorbing material defines where features would print (positive resist) or not print (negative resist).
While the concepts disclosed herein may be used for imaging on a substrate such as a silicon wafer, it shall be understood that the disclosed concepts may be used with any type of lithographic imaging systems, e.g., those used for imaging on substrates other than silicon wafers.
Although the present invention has been described and illustrated in detail, it is to be clearly understood that the same is by way of illustration and example only and is not to be taken by way of limitation, the scope of the present invention being limited only by the terms of the appended claims.