Method for computer modeling and simulation of negative-tone-developable photoresists

Abstract

In some embodiments, a method may include improving a development process of a photoresist. The method may include simulating a negative-tone development process of a photoresist. The method may include determining a reaction of a developer with a soluble photoresist surface. Determining the reaction of the developer may include applying a reaction rate constant at a power of a reaction order to a blocked polymer concentration to yield a resist dissolution rate of soluble resist comprising the dissolution-limited regime of development. The method may include determining a flux of the developer into exposed and partially soluble resist. Determining the flux of the developer may include applying a vector valued diffusion coefficient of the developer dependent upon the blocked polymer concentration to a gradient of developer concentration to an expansion rate of insoluble resist comprising the expansion-controlled regime of development. The method may include optimizing an illumination source and a mask on a full chip.

Claims

1. A method of optimizing a mask, comprising: simulating a development process of a photoresist, comprising: determining a reaction of a developer with a soluble photoresist surface comprising: applying a reaction rate constant at a power of a reaction order to a blocked polymer concentration to yield a resist dissolution rate of soluble resist comprising a dissolution-limited regime of development; and determining a flux of the developer into exposed and partially soluble resist comprising: applying a vector valued diffusion coefficient of the developer dependent upon the blocked polymer concentration to a gradient of developer concentration to an expansion rate of insoluble resist comprising a expansion-controlled regime of development; providing a result from the simulation for use in optimizing and forming a mask.

2. The method of claim 1, wherein the development process of photoresists comprises a negative-tone development process.

3. The method of claim 1, wherein the development process of photoresists comprises a positive-tone development process.

4. The method of claim 1, wherein the development process of photoresists comprises photoresists used for 248 nm (KrF), 193 nm (ArF) or 13.5 nm (EUV) lithography.

5. The method of claim 1, wherein applying the reaction rate constant (k.sub.R) to the blocked polymer concentration (m)k.sub.R(m).sup.n at the power of the reaction order (n) to the dissolution rate of soluble resist (R.sub.D) is equivalent to k.sub.R(m).sup.n.

6. The method of claim 1, wherein determining a flux (j.sub.S(r,t)) of the developer into exposed and partially soluble resist is determined by applying a vector-valued diffusion coefficient (D.sub.S) of the developer dependent upon the blocked polymer concentration (m) is applied to a gradient of the developer concentration (?S) at a point r and time t such that
j.sub.S(r,t)=?D.sub.S(m)?S(r,t).

7. The method of claim 1, wherein the dissolution-controlled regime applies when the photoresist surface is at least partially soluble to the developer.

8. The method of claim 1, wherein the expansion-controlled regime applies when the photoresist surface is substantially insoluble to the developer.

9. The method of claim 1, wherein the expansion-controlled regime applies when the photoresist surface is substantially insoluble to the developer such that the developer continues to absorb into the photoresist increasing the volume of the photoresist.

10. The method of claim 1, wherein the dissolution-controlled regime is transitioned to the expansion-controlled regime at a gel point dose.

11. The method of claim 1, wherein the method of simulating the development process of photoresists comprises stochastically simulating the development process of photoresists.

12. The method of claim 1, further comprising correcting the optical proximity of the mask on a full chip.

13. The method of claim 1, further comprising placing and verifying of the mask sub-resolution assist feature on a full chip.

14. The method of claim 1, further comprising repairing the mask sub-resolution assist feature.

15. The method of claim 1, further comprising inspecting the mask on a full chip.

16. The method of claim 1, further comprising identifying, diagnosing, and/or repairing the mask hot-spot on a full chip.

17. The method of claim 1, further comprising optimizing an illumination source on a full chip.

18. A method of forming an integrated circuit, comprising: simulating a negative-tone development process of a photoresist, comprising: determining a reaction of a developer with a photoresist surface comprising: applying a reaction rate constant at a power of a reaction order to a blocked polymer concentration to yield a resist dissolution rate of soluble resist comprising the dissolution-limited regime of development; and determining a flux of the developer into exposed and partially soluble resist comprising: applying a vector valued diffusion coefficient of the developer dependent upon the blocked polymer concentration to a gradient of developer concentration to an expansion rate of insoluble resist comprising the expansion-controlled regime of development; and optimizing an illumination source and a mask on a full chip using a result from the simulation.

19. The method of claim 18, further comprising forming an integrated circuit using an optimized illumination source and mask on a full chip.

20. The method of claim 18, wherein the development process of photoresists comprises a negative-tone development process.

21. The method of claim 18, wherein the development process of photoresists comprises a positive-tone development process.

22. The method of claim 18, wherein the development process of photoresists comprises photoresists used for 248 nm (KrF), 193 nm (ArF) or 13.5 nm (EUV) lithography.

23. The method of claim 18, wherein applying the reaction rate constant (k.sub.R) to the blocked polymer concentration (m)k.sub.R(m).sup.n at the power of the reaction order (n) to the dissolution rate of soluble resist (R.sub.D) is equivalent to k.sub.R(m).sup.n.

24. The method of claim 18, wherein determining a flux (j.sub.S(r,t)) of the developer into exposed and partially soluble resist is determined by applying a vector-valued diffusion coefficient (D.sub.S) of the developer dependent upon the blocked polymer concentration (m) is applied to a gradient of the developer concentration (?S) at a point r and time t such that
j.sub.S(r,t)=?D.sub.S(m)?S(r,t).

25. The method of claim 18, wherein the dissolution-controlled regime applies when the photoresist surface is at least partially soluble to the developer.

26. The method of claim 18, wherein the expansion-controlled regime applies when the photoresist surface is substantially insoluble to the developer.

27. The method of claim 18, wherein the expansion-controlled regime applies when the photoresist surface is substantially insoluble to the developer such that the developer continues to absorb into the photoresist increasing the volume of the photoresist.

28. The method of claim 18, wherein the dissolution-controlled regime is transitioned to the expansion-controlled regime at a gel point dose.

29. The method of claim 18, wherein the method of simulating the development process of photoresists comprises stochastically simulating the development process of photoresists.

30. The method of claim 18, further comprising correcting the optical proximity of the mask on a full chip.

31. The method of claim 18, further comprising placing and verifying the mask sub-resolution assist feature on a full chip.

32. The method of claim 18, further comprising repairing the mask sub-resolution assist feature.

33. The method of claim 18, further comprising inspecting the mask on a full chip.

34. The method of claim 18, further comprising identifying, diagnosing, and/or repairing a mask hot-spot on a full chip.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Advantages of the present invention may become apparent to those skilled in the art with the benefit of the following detailed description of the preferred embodiments and upon reference to the accompanying drawings.

(2) FIG. 1 depicts a diagram of a flowchart of a method of improving a development process of a photoresist.

(3) FIG. 2 depicts predicted lithographic results, ADI, using the improved approach vs. the classical approach for simulating negative-tone development. The improved method is shown to predict much larger process latitude, especially with respect to depth-of-focus, than the classical method.

(4) While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and may herein be described in detail. The drawings may not be to scale. It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims.

(5) The headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description. As used throughout this application, the word may is used in a permissive sense (i.e., meaning having the potential to), rather than the mandatory sense (i.e., meaning must). The words include, including, and includes indicate open-ended relationships and therefore mean including, but not limited to. Similarly, the words have, having, and has also indicated open-ended relationships, and thus mean having, but not limited to. The terms first, second, third, and so forth as used herein are used as labels for nouns that they precede, and do not imply any type of ordering (e.g., spatial, temporal, logical, etc.) unless such an ordering is otherwise explicitly indicated. For example, a third die electrically connected to the module substrate does not preclude scenarios in which a fourth die electrically connected to the module substrate is connected prior to the third die, unless otherwise specified. Similarly, a second feature does not require that a first feature be implemented prior to the second feature, unless otherwise specified.

(6) Various components may be described as configured to perform a task or tasks. In such contexts, configured to is a broad recitation generally meaning having structure that performs the task or tasks during operation. As such, the component can be configured to perform the task even when the component is not currently performing that task (e.g., a set of electrical conductors may be configured to electrically connect a module to another module, even when the two modules are not connected). In some contexts, configured to may be a broad recitation of structure generally meaning having circuitry that performs the task or tasks during operation. As such, the component can be configured to perform the task even when the component is not currently on. In general, the circuitry that forms the structure corresponding to configured to may include hardware circuits.

(7) Various components may be described as performing a task or tasks, for convenience in the description. Such descriptions should be interpreted as including the phrase configured to. Reciting a component that is configured to perform one or more tasks is expressly intended not to invoke 35 U.S.C. ? 112 paragraph (f), interpretation for that component.

(8) The scope of the present disclosure includes any feature or combination of features disclosed herein (either explicitly or implicitly), or any generalization thereof, whether or not it mitigates any or all of the problems addressed herein. Accordingly, new claims may be formulated during prosecution of this application (or an application claiming priority thereto) to any such combination of features. In particular, with reference to the appended claims, features from dependent claims may be combined with those of the independent claims and features from respective independent claims may be combined in any appropriate manner and not merely in the specific combinations enumerated in the appended claims.

(9) It is to be understood the present invention is not limited to particular devices or biological systems, which may, of course, vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting. As used in this specification and the appended claims, the singular forms a, an, and the include singular and plural referents unless the content clearly dictates otherwise. Thus, for example, reference to a linker includes one or more linkers.

DETAILED DESCRIPTION

Definitions

(10) Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art.

(11) The term connected as used herein generally refers to pieces which may be joined or linked together.

(12) The term coupled as used herein generally refers to pieces which may be used operatively with each other, or joined or linked together, with or without one or more intervening members.

(13) The term directly as used herein generally refers to one structure in physical contact with another structure, or, when used in reference to a procedure, means that one process effects another process or structure without the involvement of an intermediate step or component.

(14) Improving Negative and Positive Tone Development in a Chemically-Amplified Photoresist:

(15) The classical method for simulation of resist dissolution, derived predominantly by Mack has been applied to PTD resists, typically with very good results, and more lately to NTD resists, with much poorer results. The recent use of the classical method for the simulation of the NTD resist dissolution process is generally referred to as the inverse Mack method.

(16) In the classical method as applied to PTD resist processes, three steps are considered

(17) 1. The diffusion of the developer to the photoresist surface

(18) 2. The reaction of the developer with the photoresist at the resist surface

(19) 3. The diffusion of the dissolved photoresist back into the developer.

(20) The first step, the rate of the diffusion of developer to the resist surface, is described
r.sub.D=k.sub.dS(5)
where r.sub.D is the rate of diffusion of the developer to the resist surface, S is the concentration of the active basic molecule in the aqueous developer solution at the resist surface (typically tetramethylammonium hydroxide, TMAH), and k.sub.d is the rate constant. As described above, in a PTD resist process, higher concentrations of blocked sites on the resin polymer [M] act as dissolution inhibitors and n inhibitor sites in the exposed region must be removed for a molecule of the resin to fully dissolve, in the sense that to be fully dissolved, a molecule of the resist resin must be surrounded by molecules of the developer.

(21) The second step, reaction of the developer with the resist at the resist surface is described
r.sub.R=k.sub.rS(M.sub.0?M).sup.n(6)
where r.sub.R is the rate of reaction of the developer with the resist at the resist surface, S is the concentration of developer at the resist surface, k.sub.r is the reaction rate constant, M.sub.0 is the initial concentration of the inhibitor and n is the reaction order.

(22) The third step, diffusion of the dissolved resist back into developer, occurs quickly enough that this step may be ignored. The rate equations occur in series, step two follows step one, and the two steps will come to a steady state such that the overall dissolution rate R.sub.PTD for a PTD resist is
R.sub.PTD=r.sub.D=r.sub.r(7)

(23) Equating the two rate equations allows cancellation of S and the PTD dissolution rate R.sub.PTD can be written

(24) $\begin{array}{cc}{R}_{\mathrm{PTD}}=\frac{r_r}{r_D}=\frac{k_r{S\left(M_0-M\right)}^n}{k_{d}S}=\frac{k_r}{k_d}{\left(M_0-M\right)}^n={k_R\left(M_0-M\right)}^n& \left(8\right)\end{array}$
where

(25) $k_R=\frac{k_r}{k_d}.$
Letting m=M/M.sub.0, where m is the relative dissolution inhibitor concentration, the equation for R.sub.PTD becomes
R.sub.PTD=k.sub.R(1?m).sup.n(9)
where the rate is proportional to (1?m) and where (1?m) represents the extent of polymer de-blocking.

(26) The classical theory has more recently been applied to modern NTD resist systems with very only minor changes to the PTD dissolution rate equation. Called the inverse Mack method, m simply acts as a dissolution promoter, giving
R.sub.NTD=k.sub.R(m).sup.n(10)
where the negative tone dissolution rate R.sub.NTD is proportional to m and where m represents the extent of polymer blocking.
Improved Method of Simulating Negative and Positive Tone Development in a Chemically-Amplified Photoresist:

(27) In NTD systems, higher values of m produce faster dissolution rates, while lower values of m, where light intensity is greater, produce slower dissolution rates. As the light intensity or more appropriately, the exposure dose increases, the NTD dissolution rate slows until the gel-point dose is encountered. At the gel point dose, the resist film becomes partially insoluble. At doses above the gel-point dose, an increasing fraction of the resist is insoluble, but the insoluble resist may continue to take up developer, purely driven by the effects of diffusion of developer into partially insoluble resist. The rate that developer diffuses into insoluble resist is proportional to the local value of m and therefore to the exposure dose and the local acid concentration.

(28) When applied to NTD processes, the classical method, as described above, does not account for the lithographic effects produced by the continued diffusion of the developer into insoluble resist. The failure to account for at least these effects in computational lithography models of NTD processes severely degrades modeling accuracy, rendering the classical method inadequate for simulation of NTD resist processes.

(29) In some embodiments, a method for the simulation of NTD, may include two processes considered

(30) 1. The reaction of the developer with the photoresist at the resist surface

(31) 2. The flux of the developer into exposed and partially insoluble resist

(32) In some embodiments, a method may include improving through, for example, simulating a development process of a photoresist. The method may include simulating a negative-tone development process of a photoresist. The method may include determining a reaction of a developer with a soluble photoresist surface. The method may include determining a flux of the developer into exposed and partially soluble resist. In some embodiments, the method may include development process of photoresists comprises a positive-tone development process.

(33) In some embodiments, the development process of photoresists comprises photoresists used for different wavelength lithography. In some embodiments, the development process of photoresists comprises photoresists used for 248 nm (KrF), 193 nm (ArF) and/or 13.5 nm (EUV) lithography.

(34) In some embodiments, a reaction of the developer with the resist at the resist surface is as described in the classical method above
R.sub.NTD=k.sub.R(m).sup.n(11)
where R.sub.NTD is the rate of reaction of the developer with the resist at the resist surface and is the dissolution rate of soluble resist, k.sub.R is the reaction rate constant, m is the normalized concentration of the NTD dissolution promoter, a function of the exposure dose and the local acid concentration and n is the reaction order. In NTD processes, the developer is, for example, a pure organic non-polar solvent, for example n-butylacetate and not an aqueous base solution as in PTD processes.

(35) In some embodiments, determining the reaction of the developer may include applying a reaction rate constant at a power of a reaction order to a blocked polymer concentration to yield a resist dissolution rate of soluble resist comprising the dissolution-limited regime of development.

(36) In some embodiments, as the exposure dose increases, m decreases, inhibiting or slowing the dissolution rate. Therefore, R.sub.NTD decreases as exposure dose increases until the gel point dose is encountered. At the gel point dose, the photoresist becomes partially insoluble; at and above the gel point dose, larger and larger fractions of the resist become insoluble, yet the developer may continue to enter the resist, driven purely by diffusion. In some embodiments, the rate that developer diffuses into the partially insoluble resist is described as a flux with units of m/s
j.sub.S(r,t)=?D.sub.S?S(r,t)(12)
where j.sub.S is the flux of developer passing through any point r inside the undissolved resist at time t, D.sub.S is the vector-valued diffusion coefficient of the developer in units of m.sup.2/s and ?S is the gradient of the developer concentration at a point r and time t inside the resist. The sign of the diffusion coefficient is chosen to indicate that the direction of diffusion is into the photoresist mass. The diffusion coefficient's value in relation to the concentration of blocked or unreacted sites on the photoresist resin is highly complex yet can be estimated by consideration of both polymer solution thermodynamics and percolation theory.

(37) In some embodiments, determining the flux of the developer may include applying a vector valued diffusion coefficient of the developer dependent upon the blocked polymer concentration to a gradient of developer concentration to an expansion rate of insoluble resist comprising the expansion-controlled regime of development.

(38) For example, Flory-Huggins theory characterizes the thermodynamic compatibility between a polymer and a solvent as
D=const(1??.sub.1).sup.2(1?2??.sub.1)(13)
where D is the mutual binary diffusion coefficient in a polymer/solvent system, ?.sub.1 is the developer volume fraction in the solution and ? is the thermodynamic polymer-solvent interaction parameter. With respect to NTD photoresists, ? represents the thermodynamic compatibility between the polymer resin and the developer, a function of the polymer blocking extent and the molecular volume of the developer species.

(39) Percolation theory is a branch of mathematics which deals with such phenomena as the formation of connected holes and channels called percolation clusters. The concentration of hydrophilic percolation clusters in NTD photoresists is highly dose-dependent. For example, at higher doses, more reactive sites are de-blocked on the photoresist resin polymer by the diffusing photo-acid. The products of the de-blocking reaction are partially volatile, leaving behind free volume, which effectively are holes and channels in the photoresist through which the developer may diffuse.

(40) The boundary between undissolved or insoluble resist above the gel point dose and the liquid developer will be represented by ?; for example, in 2 dimensions, ? is a line; in 3 dimensions, ? is a surface; ? represents the photoresist edge. One is interested in the rate, the direction and the amount ? is displaced over a time interval as a response to the diffusion of the developer into insoluble photoresist.

(41) The displacement of ? occurs at a velocity equivalent in magnitude to j.sub.S, yet in a direction opposite j.sub.S
R.sub.IEV(r,t)|.sub.?=?j.sub.S(r,t)(14)
where R.sub.IEV(r,t)|.sub.? represents the insoluble edge velocity calculated at any point and at any time on ?. The displacement of the edge ?.sub.d over a time interval may be expressed as an integral

(42) $\begin{array}{cc}{?}_d={?}_t^{t+?t}R\left(r,t\right){.Math.}_{?}dt& \left(15\right)\end{array}$
Or it may be expressed as an ordinary differential equation subject to the boundary condition ?.sub.d(t)=0

(43) $\begin{array}{cc}\frac{d{?}_d\left(r,t\right)}{dt}=R\left(r,t\right){.Math.}_{?}& \left(16\right)\end{array}$

(44) The overall edge velocity {right arrow over (V)}(r,t) at any point r on ? and at any time t during the negative tone development process can therefore be expressed as the sum of the classical dissolution velocity R.sub.NTD and the insoluble edge velocity R.sub.IEV
{right arrow over (V)}(r,t)|.sub.?=({right arrow over (R)}.sub.NTD(r,t)+{right arrow over (R)}.sub.IEV(r,t))|.sub.?(17)

(45) In some embodiments, significantly below the gel point dose, the photoresist is underexposed, m?1 and the resist is highly soluble. In this case, classical inverse Mack dissolution theory largely determines the location, speed and direction ? propagates during development, since
?R.sub.NTD(r,t)?>>?R.sub.IEV(r,t)?, dose<dose.sub.gel(18)
and the negative tone development process is said to be dissolution-controlled
dissolution?controlled ?{right arrow over (V)}(r,t)???R.sub.NTD(r,t)?, dose<dose.sub.gel(19)

(46) In some embodiments, at and above the gel point dose, the photoresist is much more exposed, has absorbed more incident radiation, has produced more photo-generated acid, m<<1 and is therefore much less soluble. This may cause the classical dissolution rate R.sub.NTD(r,t) to approach zero. However, due to the diffusive processes described above, developer continues to enter the photoresist, increasing the volume of the resist mass and perturbing the location of the edge ?.
?R.sub.NTD(r,t)?<?R.sub.IEV(r,t)?, dose?dose.sub.gel(20)

(47) In some embodiments, the dissolution-controlled regime applies when the photoresist surface is at least partially soluble to the developer.

(48) The negative tone development process switches from the dissolution-controlled regime to the expansion-controlled regime
expansion?controlled ?{right arrow over (V)}(r,t)???R.sub.IEV(r,t)?, dose?dose.sub.gel(21)

(49) In some embodiments, the expansion-controlled regime of NTD development strongly influences the lithographic behavior of NTD processes in a way not observed in PTD processes. Ignoring simulation of the effect severely degrades the accuracy of computational lithography modeling of NTD processes, as is shown below.

(50) In some embodiments, the expansion-controlled regime applies when the photoresist surface is substantially insoluble to the developer. The expansion-controlled regime applies when the photoresist surface is substantially insoluble to the developer such that the developer continues to absorb into the photoresist increasing the volume of the photoresist. In some embodiments, the dissolution-controlled regime is transitioned to the expansion-controlled regime at a gel point dose.

(51) FIG. 1 depicts a diagram of a flowchart of a method of improving a development process of a photoresist. Due to the continuous shrinking in half pitch and critical dimension in wafer processing, maintaining a reasonable process window such as depth of focus and exposure latitude becomes very challenging. With the source mask optimization methodology, the lithography process window can be improved and a smaller mask error enhancement factor can be achieved.

(52) In order to improve resolution performance of a lithographic system, various tools may be used. Recent developments in illumination systems include freely tunable illumination sources. Freely tunable illumination sources may provide illumination shapes that are nearly arbitrarily defined, thereby allowing even finer controls over illumination patterns. Illumination shapes of any complexity may be produced by simple apertures, gray-tone plates, or diffractive optical elements. Projection optics may include optical components for shaping, adjusting and/or projecting radiation from the source before the radiation passes the patterning device, and/or optical components for shaping, adjusting and/or projecting the radiation after the radiation passes the patterning device. In light of recent developments that provide an increasing number of tuning and adjustment options for both illumination sources and masks, approaches for determining the optimal combination of source and mask configuration are desired. In some embodiments, using the simulation methods described herein may allow for improved source-mask optimization SMO of illumination source (a scanner) and mask as discussed for example in U.S. Pat. No. 9,213,783 to Hansen, U.S. patent application publication no. 20160110488 to Hansen, and U.S. Pat. No. 8,786,824 to Hansen, all of which are incorporated by reference herein.

(53) In some embodiments, the method may include optimizing a full chip source-mask. In some embodiments, the method may include forming an integrated circuit using the full chip source-mask.

(54) In some embodiments, the method may include correcting the full-chip optical proximity of a mask. The method may include the placement and verification of a mask sub-resolution assist feature on a full chip. The method may include repairing a mask sub-resolution assist feature. The method may include inspecting a full chip mask inspection. The method may include identifying, diagnosing, and/or repairing a mask hot-spot. The method may include optimizing a full chip source-mask.

(55) Experimental Data

(56) I. Measuring the Agreement Between Simulated Predictions and Experimental Data

(57) Suppose that one is fitting N data points (x.sub.i,y.sub.i),i=1 . . . , N to a model that has M adjustable parameters a.sub.j,j=1 . . . , M. The model predicts a functional relationship between the measured independent and dependent variables
y(x)=y(x|a.sub.1. . . a.sub.M)(22)
where the vertical bar indicates dependence on the parameters on the right side. What does one want to minimize to get fitted values for the a.sub.js? The first thing that comes to mind is the familiar least-squares fit

(58) $\begin{array}{cc}\mathrm{minimize}\mathrm{over}{a}_1\mathrm{.Math.}{a}_M:\underset{i=1}{\overset{N}{.Math.}}{\left[y_i-y\left(x_i.Math.a_1\mathrm{.Math.}{a}_M\right)\right]}^2& \left(23\right)\end{array}$

(59) Data consists of a sample of observations drawn from a parent distribution that determines the probability of making any particular observation. Given a particular set of parameters, what is the probability that the observed data should have occurred, plus or minus some small fixed ?y on each data point? If the probability of obtaining the data set is too small, then one concludes that the parameters under consideration are unlikely to be correct. The data set should not be too improbable for the correct choice of parameters. Suppose each data point y.sub.i has a measurement error that is independently random and distributed normally around the true y(x). For simplicity, suppose that the standard deviation ? of these normal distributions is the same for all data points. Then the probability of the data set is the product of the probabilities of each point:

(60) $\begin{array}{cc}P\left(\mathrm{data}.Math.\mathrm{model}\right)?\underset{i=1}{\overset{N}{.Math.}}\left\{{\exp \left[-\frac{1}{2}\left(\frac{y_i-y\left(x_i\right)}{?}\right)\right]}^2?y\right\}& \left(24\right)\end{array}$

(61) The most probable model is the one which maximizes this equation which is equivalent to minimizing the argument in the exponential

(62) 0$\begin{array}{cc}\left[\underset{i=1}{\overset{N}{.Math.}}\frac{{\left[y_i-y\left(x_i\right)\right]}^2}{2{?}^2}\right]-N\log ?y& \left(25\right)\end{array}$
Since N, ?, and ?y are all constants, minimizing this equation is equivalent to minimizing

(63) $\begin{array}{cc}\underset{i=1}{\overset{N}{.Math.}}{\left[y_i-y\left(x_i.Math.a_1\mathrm{.Math.}{a}_M\right)\right]}^2& \left(26\right)\end{array}$

(64) One therefore identifies the probability of the data given the parameters as the likelihood of the parameters given the data. Parameters derived this way are called maximum likelihood estimators. If each data point (x.sub.i, y.sub.i) has its own, known standard deviation ?.sub.i then the maximum likelihood estimate of the model parameters may be obtained by minimizing the sum in the above formula, commonly referred to as chi-square or, upon dividing by the degrees of freedom, the reduced chi-square

(65) $\begin{array}{cc}{?}^2=\underset{i=1}{\overset{N}{.Math.}}{\left[\frac{y_i-y\left(x_i.Math.a_1\mathrm{.Math.}{a}_M\right)}{{?}_i}\right]}^2{?}_{\mathrm{reduced}}^2=\frac{{?}^2}{v}& \left(27\right)\end{array}$
with v=N?M degrees of freedom. Chi-square has no units, it is a pure number. A rule of thumb is that a typical value for a good fit is ?.sup.2?v or ?.sub.reduced.sup.2?1. More precise is the statement that the ?.sup.2 statistic has a mean v and a standard deviation ?{square root over (2v)} and for large v becomes normally distributed. Taking the derivative of chi-square with respect to the parameters a.sub.j one obtains the equations that must hold at the chi-square minimum, which are a set of M nonlinear equations for the M unknown a.sub.j:

(66) $\begin{array}{cc}0=\underset{i=1}{\overset{N}{.Math.}}\left(\frac{y_i-y\left(x_i\right)}{{?}_i^2}\right)\left(\frac{?y(x_i.Math.\mathrm{.Math.}{a}_j\mathrm{.Math.})}{?a_j}\right)j=1,\mathrm{.Math.},M& \left(28\right)\end{array}$

(67) The root mean square of the error (RMSE) between simulated predictions and experimental data, also a maximum likelihood estimator, is derived similarly, with units of the RMSE identical to the units of the experimental data; smaller values of RMSE indicate better agreement between simulated predictions and experimental data.

(68) $\begin{array}{cc}\mathrm{RMS}\mathrm{err}={\left(\frac{1}{N}\underset{i=1}{\overset{N}{.Math.}}{\left[y_i-y\left(x_i.Math.a_1\mathrm{.Math.}{a}_M\right)\right]}^2\right)}^{\frac{1}{2}}& \left(29\right)\end{array}$

(69) Comparison of Modeling Error: the Classical Method Vs. the Improved Method for Simulating Negative Tone Development of a Chemically-Amplified Photoresist

(70) Two calibrated computational lithography models differing in the method used for simulating the negative tone development process were used to predict the critical dimensions of the photoresist relief image after completion of the development process (the after-develop image or ADI). The computational lithography models differ in their treatment of the negative tone development processone model includes support for the simulation of both dissolution-controlled and expansion-controlled development (the improved NTD modeling approach), while the other includes support for the simulation of dissolution-controlled development only (the classical inverse Mack approach). The set of experimental data used to evaluate the performance of each approach is exactly identical. The data have been collected by measuring the critical dimensions of lithographic features produced using a state-of-the-art NTD photoresist. The critical dimensions have been collected as a function of scanner exposure dose, scanner focus, mask feature width, mask feature pitch and feature tone. In NTD processes, opaque mask features print as spaces or holes in resist, while transparent mask features print as lines or posts. The set of data describing features collected using a single mask feature width and tone, a single mask feature pitch and multiple dose and focus values is referred to as a focus-exposure matrix (FEM). Each FEM is simulated using both models. The model predictions are then compared to the experimental data using the method of maximum likelihood. The goodness of model fit is quantified and shown in the tables below for the classical inverse Mack method and the improved method for simulating negative tone development, which accounts for both dissolution-controlled and expansion-controlled processes during development. The top row of the table lists the overall model error vs. experimental data, including the RMS error and the reduced chi-square, described above. Table 1 shows the best results obtained for the computational lithography model using the classical inverse Mack method to simulate negative tone development; the RMS error of this approach is 6.78 nm, and the reduced chi-square is 36.62.

(71) TABLE-US-00001 TABLE 1 results using the classical inverse Mack approach Calibration Statistics: Average Chi-Square Output/Group Name Weight Points RMS Error Absolute Error Average Error Max Error Reduced F Reduced Group 402 6.775882 5.803183 ?0.3317238 21.85167 39.61736 39.61736 65S120P_L:Y:ADI CD (nm) 1 45 5.519484 4.692963 ?3.472284 21.26267 32.80157 65S120P_S:Y:ADI CD (nm) 1 45 5.273606 4.31964 3.048798 21.85167 33.4935 66L150P_L:Y:ADI CD (nm) 1 48 6.51887 5.869005 ?4.78983 16.77435 30.38408 66L150P_S:Y:ADI CD (nm) 1 48 6.063706 5.336473 4.125926 17.28848 42.14913 80S200P_L:Y:ADI CD (nm) 1 66 8.004296 6.990122 ?4.121093 19.75455 42.18098 80S200P_S:Y:ADI CD (nm) 1 60 7.720978 6.56885 3.767708 20.77959 39.78538 85L200P_L:Y:ADI CD (nm) 1 45 7.103975 6.136977 ?4.963214 12.59775 38.98001 85L200P_S:Y:ADI CD (nm) 1 45 6.588597 5.729033 4.152123 11.54474 56.35861
Table 1 shows the best results obtained for the computational lithography model using the classical inverse Mack method to simulate negative tone development; the RMS error of this approach is 6.78 nm, and the reduced chi-square is 36.62.

(72) TABLE-US-00002 TABLE 2 results using the improved approach Calibration Statistics: Average Chi-Square Output/Group Name Weight Points RMS Error Absolute Error Average Error Max Error Reduced F Reduced Group 402 2.015629 1.540161 ?0.378552 6.508347 2.886436 2.886436 65S120P_L:Y:ADI CD (nm) 1 45 1.830041 1.421631 0.1654438 6.508347 3.626313 65S120P_S:Y:ADI CD (nm) 1 45 1.832103 1.516736 ?0.5903961 5.810532 3.391773 66L150P_L:Y:ADI CD (nm) 1 48 1.231472 0.7955651 ?0.4842105 5.271576 0.9526621 66L150P_S:Y:ADI CD (nm) 1 48 1.092839 0.7300282 ?0.179841 4.314445 1.667326 80S200P_L:Y:ADI CD (nm) 1 66 2.723076 2.322778 ?0.8973885 5.897751 4.639248 80S200P_S:Y:ADI CD (nm) 1 60 2.714341 2.356389 ?0.09091084 6.342537 4.023414 85L200P_L:Y:ADI CD (nm) 1 45 1.686716 1.28823 ?0.3245471 5.285645 1.799993 85L200P_S:Y:ADI CD (nm) 1 45 1.775586 1.356279 ?0.4865245 6.355682 2.003978
Table 2 shows the best results obtained for the computational lithography model using the improved method to simulate negative tone development; the RMS error of this approach is 2.01 nm, and the reduced chi-square is 2.89. The improved method is shown to reduce the RMS error by a factor of 3.4, meaning that modeling error using the improved method is in this example 3.4 times better than that obtained with the classical method under identical conditions.
II. Applications of the Improved Method for Simulating Negative Tone Development in a Chemically-Amplified Photoresist
Corrected Predictions of Enhanced Lithography Process Window

(73) Expansion phenomena during the negative-tone development process results in profound improvement to ADI lithography, particularly with respect to spaces or holes. As an example, consider simulation of a lithographic process printing 40 nm spaces in photoresist on 120 nm mask pitch using 193 nm radiation and a state-of-the-art 1.35 NA scanner with off-axis illumination. Shown below is a plot of comparative exposure latitude vs. depth-of-focus, a typical metric used to evaluate the magnitude of the process window, for two computational lithography models: the improved method for simulation of negative tone development, supporting simulation of both the dissolution-controlled and expansion-controlled regimes of development\ vs. the classical inverse Mack method for simulation of negative tone development, supporting simulation of the dissolution-controlled regime only. The models used are calibrated with precision exhibited in Tables 1 and 2. The improved modeling method, able to predict experimental results within a RMS error of about 2 nm, predicts a process with maximum exposure latitude of 13.4%, maximum depth-of-focus (DOF) of ca. 150 nm, at a sizing dose of 35.8 mJ/cm.sup.2; the classical method, able to predict experimental results within a RMS of about 7 nm, predicts maximum exposure latitude of 12.8%, maximum depth-of-focus of ca. 57 nm, at a sizing dose of 51.8 mJ/cm.sup.2. The improved modeling method predicts a process with 2.6? greater focus latitude (DOF) compared to the classical method.

(74) Corrected Predictions of Process Throughput

(75) The throughput or speed of wafer processing depends to a large extent upon the sizing dose of the photoresist, or the amount of energy required to produce the proper dimension of the photoresist feature; the sizing dose of the photoresist is measured in units equivalent to the radiation intensity measured at the wafer per unit area per unit time multiplied by the time required for exposure
sizing dose=I.Math.t(30)
where I is the intensity of the radiation in units of mJ/(cm.sup.2.Math.s) and t is the exposure time in seconds. IN NTD processes, larger values of dose correspond to smaller space CDs and larger line CDsline CD increases and space CD decreases as dose increases. Smaller values of the sizing dose indicate a resist process with faster throughput, since wafers spend less time being exposed in the scanner. A smaller sizing dose therefore increases the rate of high volume manufacturing or the number of wafers processed per unit time. Including the improved method for simulating NTD development predicts a sizing dose of 35.8 mJ/cm.sup.2; the classical method predicts a sizing dose of 51.8 mJ/cm.sup.2 for the same feature.

(76) It is clear that including simulation of the expansion-controlled regime of negative-tone development produces a more accurate computational lithography model, as evidenced by Tables 1 and 2 above. Using the improved method, modeling error is reduced by a factor of 3.5 compared to the classical method. Lower modeling error means more accurate predictions. The improved model can then be exploited to give a truer picture of the process window, in this case a process window much larger than that predicted by the classical method, evidenced by FIG. 2, and a more accurate prediction of faster manufacturing throughout, as evidenced by the prediction of the sizing dose.

(77) In this patent, certain U.S. patents, U.S. patent applications, and other materials (e.g., articles) have been incorporated by reference. The text of such U.S. patents, U.S. patent applications, and other materials is, however, only incorporated by reference to the extent that no conflict exists between such text and the other statements and drawings set forth herein. In the event of such conflict, then any such conflicting text in such incorporated by reference U.S. patents, U.S. patent applications, and other materials is specifically not incorporated by reference in this patent.

(78) Further modifications and alternative embodiments of various aspects of the invention will be apparent to those skilled in the art in view of this description. Accordingly, this description is to be construed as illustrative only and is for the purpose of teaching those skilled in the art the general manner of carrying out the invention. It is to be understood that the forms of the invention shown and described herein are to be taken as the presently preferred embodiments. Elements and materials may be substituted for those illustrated and described herein, parts and processes may be reversed, and certain features of the invention may be utilized independently, all as would be apparent to one skilled in the art after having the benefit of this description of the invention. Changes may be made in the elements described herein without departing from the spirit and scope of the invention as described in the following claims.