CN102798342A - Fitting error interpolation based library matching method for optical scattering measurement - Google Patents
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Abstract
The invention discloses a fitting error interpolation based library matching method for optical scattering measurement. The method comprises the following steps of: determining the variation range of to-be-measured structure parameters of a sample and carrying out discretization treatment on the variation range of the to-be-measured structure parameters of the sample, and storing obtained discretization gridding points and theory spectrum value corresponding to the discretization gridding points into a spectrum library; acquiring a measurement spectrum of the to-be-measured sample, calculating fitting errors between the measurement spectrum value corresponding to the discretization gridding points and the theory spectrum value, and then storing the fitting errors similarly to the spectrum library; setting threshold for the fitting errors and carrying out rough searching, constructing a candidate parameter set by utilizing the discretization gridding points corresponding to the searched fitting errors, and carrying out multidirectional interpolation treatment on the fitting errors, thereby obtaining a corresponding fitting error interpolation function; and carrying out fine searching and finding out the overall optimal point based on the fitting error interpolation function, wherein the parameter value corresponding to the overall optimal point is the finally determined optimal parameter. According to the invention, the measurement resolution and accuracy of the library matching process can be improved, and the fitting error interpolation based library matching method for optical scattering measurement has the advantage of being rapid and convenient in operation.
Description
Technical field
The invention belongs to field of optical measuring technologies, more specifically, relate to a kind of storehouse matching process that optical scattering is measured that is used for based on the error of fitting interpolation.
Background technology
Optical scattering measuring method (optical scatterometry); Be also referred to as the optics critical size and measure (optical critical dimension metrology) method; Its ultimate principle is that the polarized light with a branch of particular polarization state is projected to the testing sample surface; Through measuring the zero order diffracted light of testing sample; Obtain the variation of reflective light intensity or the polarized light ratio polarization state before and after reflection under the different polarization states thus, and then therefrom extract the information such as live width, line height, side wall angle, alignment error of optical grating construction in for example photoetching of structural parameters, etching and the nano impression figure of testing sample.Compare with the mode that adopts scanning electron microscope or atomic force microscope; The optical scattering measuring method has that speed is fast, cost is low, contactless; Non-destruction and be easy to online advantage such as integrated, thereby obtained widespread use at advanced process-monitor and optimization control field.
Except measuring equipment itself, the optical scattering measuring method also depends on the gordian technique of following two aspects: the one, and the modeling of forward optical characteristics; The 2nd, reverse parameter extraction.The modeling of forward optical characteristics can adopt rigorous couple-wave analysis (RCWA) method, finite element method (FEM), boundary element method (BEM), finite time-domain method of difference methods such as (FDTD) to realize; The method that reverse parameter extraction can adopt comprises non-linear regression method and based on the parameter extracting method of storehouse coupling etc.Wherein, Owing in each iterative process, all need call the optical property model of forward based on the parameter extracting method of non-linear regression; And the calculating of forward characteristic model is a relative time-consuming procedure; Especially for the three-dimensional periodic structure of complicacy, therefore be difficult to satisfy the requirement of on-line measurement process; For parameter extracting method based on the storehouse coupling; Though the foundation of library of spectra is a very time-consuming procedure; Fortunately whole process of building the storehouse can be carried out by off-line, in case after library of spectra built up, remaining parameter extraction process only was equivalent to a data library inquiry problem; Can guarantee that parameter extraction accomplishes in the quite short time, so this method has obtained in the optical scattering measuring process to use widely.
In setting up the process of library of spectra, must carry out discretize to the structural parameters of testing sample and handle.The step sizes that is adopted in the discretize process has directly determined final Measurement Resolution of library of spectra (measurement resolution) and accuracy of measurement (measurement accuracy).Generally speaking, step-length is more little, and Measurement Resolution that the storehouse matching process is final and accuracy of measurement are just high more.Yet step-length is more little, and the scale of library of spectra also can be the growth of geometry level several times thereupon, will need more time and memory headroom to create and store library of spectra like this.Therefore, how not increase extra storage space and to exceed under the prerequisite that influences Measuring Time, the Measurement Resolution and the accuracy of measurement that improve the storehouse matching process are very challenging problems.To this problem; The researcher has proposed some solutions both at home and abroad; For example: can adopt linear regression model (LRM) to replace the optical property model of forward, and pass through the method acquisition linear regression model (LRM) of multivariate statistical analyses such as for example principal component analysis (PCA), discriminatory analysis or PLS; Yet this method often can only be used in very limited parameter area, in case exceed this scope, just causes bigger measuring error easily, and the linear regression model (LRM) that is obtained accurate match calibration data often.People such as Zhang Chuanwei have proposed the parameter extraction that a kind of method based on artificial neural network combination Levenberg-Marquardt algorithm is used for the optical scattering measuring process in the paper of " Improved model-based infrared reflectrometry for measuring deep trench structures "; Though this combinational algorithm can be used to improve the accuracy of measurement of library of spectra; But owing in the algorithm iteration process, need repeatedly call the modeling of forward optical characteristics, thereby be difficult to satisfy in the on-line measurement process to measuring the requirement of aspects such as timeliness.
In addition; A kind of method of spectrum being carried out interpolation is disclosed among US6768967B2, the US7043387B2; Wherein set up library of spectra and come the spectrum interpolation model is calibrated, the prior like this spectrum that is not stored in the library of spectra can obtain through the spectrum of having deposited in the library of spectra is carried out interpolation; The forward optical property model is replaced by the spectrum interpolation model, and is used for the spectrum that the non-linear regression process is calculated the structural parameters correspondence, wherein can be final measurement with the pairing structural parameters of interpolation spectrum that measure spectrum is mated most.Yet, in the method, in order to obtain a complete curve of spectrum, need carry out interpolation one by one to each wavelength for the spectral signal of wavelength resolution type, need carry out interpolation one by one to each angle for the spectral signal of angle-resolved type; Especially; If the output form of spectral signal is Stokes vector or Mueller matrix; Then need carry out interpolation one by one to each element in vector or the matrix, therefore in general the Interpolation Process of this method is very complicated, causes the operation inconvenience of actual measurement process.
Summary of the invention
To the defective and the technical need of prior art, the object of the present invention is to provide a kind of storehouse matching process that optical scattering is measured that is used for based on the error of fitting interpolation.This method at first makes up the error of fitting interpolating function based on library of spectra of having set up and the measure spectrum that is obtained; And then convert finding the solution of inverse problem in the Scattering Measurement into find the solution the error of fitting interpolating function optimal value problem, and the corresponding structural parameters in optimal value place are and the pairing structural parameters to be measured of measure spectrum.Through the present invention, can not influenced by the concrete output form of spectrum and library of spectra under the situation of discrete grid block resolution limit etc., extract parameter to be measured more accurately and rapidly, and be specially adapted to the on-line measurement process.
According to the present invention, a kind of storehouse matching process based on the error of fitting interpolation that optical scattering is measured that is used for is provided, this method comprises the following steps:
(a) confirm the excursion of testing sample structural parameters based on process conditions; This number range is carried out discretization according to preset step-length δ handle obtaining a plurality of discrete grid block points, and these discrete grid block points and each self-corresponding theoretical spectral value thereof are stored in the library of spectra;
(b) utilize the optical scattering measurement mechanism that testing sample is measured to obtain measure spectrum; Calculate the error of fitting between each self-corresponding measure spectrum value of said discrete grid block point and the theoretical spectral value, be stored into these errors of fitting in the library of spectra equally then;
(c) the error of fitting setting threshold that is calculated for step (b) is searched for out with the error of fitting that falls into this threshold value in the library of spectra, and the pairing discrete grid block point of the error of fitting of utilizing these to search out makes up the candidate parameter collection; In addition, said error of fitting is carried out multi-dimensional interpolation handle and obtain corresponding error of fitting interpolating function, thus for setting up the function corresponding relation between testing sample structural parameters and the error of fitting;
(d) search for to find out its global optimum's point based on said error of fitting interpolating function; This global optimum puts pairing parameter value and is the final testing sample structural parameters of confirming, the said process of finding out global optimum's point specifically realizes through one of following dual mode:
(d1) the discrete grid block point of concentrating with the constructed candidate parameter of step (c) is that center and radius do not exceed said preset step-length δ as the hunting zone; In this hunting zone, carry out the discretize processing to compare littler step-length δ ' with said step-length δ; Obtain a plurality of new discrete grid block points thus and obtain its corresponding error of fitting, wherein the pairing point of the minimum value in the error of fitting is global optimum's point; Perhaps
(d2) the discrete grid block point concentrated of candidate parameter that step (c) is constructed is as the primary iteration value, and employing constrained optimization algorithm is directly obtained global optimum's point.
As further preferably, in step (b), go out the error of fitting between each self-corresponding measure spectrum value of said discrete grid block point and the theoretical spectral value through the root-mean-square error algorithm computation.
As further preferably; In step (c); The detailed process of said structure candidate parameter collection comprises: for the error of fitting that is stored in the library of spectra is set the error of fitting threshold value, then with in the library of spectra corresponding error of fitting be used to make up the candidate parameter collection less than all discrete grid block points of this setting threshold; Perhaps calculate the uncertainty of all local optimum points in the library of spectra and set corresponding uncertainty threshold value, then pairing uncertainty in the library of spectra is not less than all discrete grid block points that set threshold value and is used to make up the candidate parameter collection.
As further preferably, in step (c), adopt the mode of multidimensional linearity or multidimensional spline to carry out said multi-dimensional interpolation processing.
As further preferably, in step (d1), with said discrete grid block point be center and radius do not exceed said preset step-length δ 1/2 as the hunting zone.
As further preferably, in step (d2), said constrained optimization algorithm comprises active set method, interior point method or Sequential Quadratic Programming method.
As further preferably, the structural parameters to be measured of said sample comprise live width, line height, side wall angle, alignment error etc.
As further preferably, the spectral signal that said library of spectra is stored is reflectivity, ellipsometric parameter, Stokes vector or Mueller matrix.
In general, according to the storehouse matching process based on the error of fitting interpolation of the present invention compared with prior art, mainly possess following advantage:
1, because through the error of fitting interpolation is obtained the error of fitting interpolating function; Compare match calibration data more accurately with linear regression model (LRM); And in Interpolation Process, needn't consider the concrete form of output spectrum; Also needn't therefore each wavelength or incident angle interpolation one by one be had more versatility and be convenient to operation;
2, can improve the Measurement Resolution and the accuracy of conventional libraries matching process; Simultaneously can not increase extra storage space; Except building the storehouse link; All the other flow processs all need not called the forward optical property model, so measuring process and data by MoM and MEI more save time, and are applicable to the on-line measurement process by it;
3, the technical matters of measuring to the optics critical size among the present invention; Through adopting technological means such as making up library of spectra, discretize processing, multi-dimensional interpolation processing and search optimum point; Should be able to obtain to extract accurately and quickly the effect of parameter to be measured mutually, therefore be expected in scatterometry, to obtain widespread use.
Description of drawings
Fig. 1 is the process flow diagram according to the storehouse matching process based on the error of fitting interpolation of the present invention;
Fig. 2 is the data structure synoptic diagram of the library of spectra set up according to the present invention;
Fig. 3 is used to show the synoptic diagram that makes up the candidate parameter collection according to the present invention.
Embodiment
In order to make the object of the invention, technical scheme and advantage clearer,, the present invention is further elaborated below in conjunction with accompanying drawing and embodiment.Should be appreciated that specific embodiment described herein only in order to explanation the present invention, and be not used in qualification the present invention.
The storehouse matching process based on the error of fitting interpolation that is used for the optical scattering measurement that the present invention proposes mainly is the consideration that can determine final measurement from the error of fitting between theoretical spectral and the measure spectrum.Generally speaking, measurement result is corresponding to the theoretical spectral that has minimum error of fitting in the library of spectra with measure spectrum.From this angle, we think to the direct interpolation of error of fitting want spectrum carry out interpolation processing more directly, more meaningful.Correspondingly; Through error of fitting is carried out interpolation processing and is obtained the error of fitting interpolating function; Can be for setting up the function corresponding relation between measurement result and the error of fitting; And find out global optimum's point of error of fitting interpolating function through search, thereby so that the mode of operation draws corresponding with this optimum point also is to have the theoretical spectral of minimum error of fitting in the library of spectra as measurement result.
Fig. 1 is according to the storehouse matching process process flow diagram based on the error of fitting interpolation of the present invention.As shown in fig. 1; At first; In first step; Confirm the testing sample structural parameters according to process conditions, for example comprise the variation range of the live width, line height, side wall angle, alignment error etc. of optical grating construction, and this numerical range is carried out discretize according to preset step-length δ handle and obtain a plurality of discrete grid block points thus.Then, calculate each discrete grid block and put pairing theoretical spectral value, and these discrete grid block points and each self-corresponding theoretical spectral value thereof are stored in the library of spectra.
For example, can structural parameters to be measured be designated as the capable vector x=(x of a n dimension
1, x
2..., x
n), and confirm the span Ω of parameter x to be measured according to process conditions.Generally speaking, the variation meeting of parameter to be measured fluctuates about 10% up and down in its design load.For this span, can carry out discretize according to preset step-length δ and handle, obtain a plurality of discrete grid block points thus; Default step-length δ can adjust as required.
Then, can for example calculate each discrete grid block point x through rigorous couple-wave analysis method (RCWA), Finite Element Method (FEM), Element BEM (BEM) or finite time-domain method of difference modes such as (FDTD)
i(i=1,2 ..., n) pairing separately theoretical spectral value f (x
i| Π), wherein Π representes corresponding measurement configuration, for example measures the combination at wavelength, incident angle and position angle etc. etc.; Ellipsometric parameter or Stokes vector that spectral signal can be the reflectivity that records through reflectometer, record through traditional ellipsometer, the Mueller matrix that perhaps records etc. through Muller matrix ellipsometer; The output form of spectral signal can be wavelength resolution type, angle-resolved type or both array configurations etc.Calculating each discrete grid block puts after the pairing theoretical spectral value; These discrete grid block points and corresponding theoretical spectral value thereof are stored in the library of spectra, make up the for example library of spectra of structure shown in Fig. 2 thus: wherein mainly comprised ID numbering, structural parameters to be measured, theoretical spectral and error of fitting etc.Because therefore the calculating of error of fitting need can be made as 0 with error of fitting in advance in conjunction with measure spectrum before using library of spectra.
In second step, utilize the optical scattering measurement mechanism that testing sample is measured to obtain measure spectrum y (x
0| Π), x wherein
0The actual value of representing parameter to be measured can calculate the error of fitting between each self-corresponding measure spectrum value of said discrete grid block point and the theoretical spectral value thus, then these errors of fitting is stored in the library of spectra as shown in Figure 2 equally.The mode that is used to calculate error of fitting has multiple, in a preferred embodiment of the invention, has adopted the root-mean-square error algorithm to calculate.It characterizes function and specifically is defined as:
ω wherein
kBe weights, m representes the sum of measurement configuration, and k representes k measurement configuration.Certainly, also can adopt other suitable modes to calculate error of fitting, only need root-mean-square error be got final product with corresponding sign amount replacement.
In third step, at first library of spectra is carried out coarse search, to find out all possible and error of fitting function χ (x, x in the library of spectra
0| Π) the immediate error of fitting of global optimum's point of (x ∈ Ω) (in simple terms, this global optimum's point is promptly corresponding to the minimum point of multidimensional error of fitting curved surface), and utilize the pairing discrete grid block point of these errors of fitting to make up candidate parameter collection Y.Put immediate discrete grid block point, the actual value x of so obvious parameter to be measured if comprised in the library of spectra all possible and the global optimum error of fitting function among the candidate parameter collection Y
0Satisfy following relational expression:
Γ
γ(x
j)＝{x∈Ω:‖x-x
j‖
∞＜γδ},x
j∈Y
‖ ‖ wherein
∞Represent infinitely great norm, γ is scale factor and 0<γ≤1.The value size of γ has directly determined the size of hunting zone, but too small γ value will be actual value x
0Eliminating influences final accuracy of measurement within the scope of being given.In addition, because the grid resolution in the library of spectra is limited, the corresponding discrete grid block point of only selecting in the library of spectra to be stored of minimum error of fitting is inadequate.With one dimension error of fitting function is example; As shown in Figure 3; The pairing discrete point of being stored in the library of spectra of minimum error of fitting is 303, if but only select discrete point 303, the so corresponding error of fitting Function Optimization point of obtaining will be a local optimum point 304; And in fact, global optimum's point of error of fitting function is positioned at discrete point 307 places.
Concentrate the discrete grid block point be unlikely to omit in the library of spectra near error of fitting function global optimum's point in order to guarantee candidate parameter; Adopted following dual mode to solve this problem in the present invention: the one dimension error of fitting function χ (x) with shown in Figure 3 is an example; A kind of method is to set error of fitting threshold value 310; Then with in the library of spectra corresponding error of fitting make up the candidate parameter collection thus less than all discrete point selected as candidates parameters of this setting threshold, that is: Y={302,303; 305,306}.Another kind method is at first to calculate all local optimum points of library of spectra for example discrete point 301,303 and 306 uncertainty (numerical value of this uncertainty is promptly corresponding to the maximal value of the difference of the error of fitting value of current net point and neighbor mesh points); Set a uncertainty threshold value 312 then; Only select to make up the candidate parameter collection greater than the local optimum point of the uncertain threshold value that sets; That is: Y={303,306}.
Then, the error of fitting that is calculated is carried out multi-dimensional interpolation handle and obtain corresponding error of fitting interpolating function g (x), thus for setting up funtcional relationship one to one between structural parameters to be measured and the error of fitting.When the error of fitting in the library of spectra is carried out interpolation processing, can adopt for example modes such as multidimensional linearity, multidimensional spline of multiple middle interpolating function.With regard to the extraction accuracy of final argument, the parameter extraction accuracy of multidimensional spline interpolation will be higher than the parameter extraction accuracy of multidimensional linear interpolation.
In the 4th step; Search for to find out its global optimum's point based on said error of fitting interpolating function; This global optimum puts pairing parameter value and is the final structural parameters to be measured of confirming sample; This process specifically realizes through one of following dual mode: first kind of mode is to be that center and radius do not exceed preset step-length δ as the hunting zone to make up the discrete grid block point that candidate parameter concentrates; In this hunting zone, carry out again the discretization processing to compare littler step-length δ ' with step-length δ; Obtain a plurality of new discrete grid block points thus and obtain each self-corresponding error of fitting of these new discrete grid block points based on the error of fitting interpolating function, wherein the pairing point of the minimum of a value in the error of fitting is global optimum's point; The second way is to make up the concentrated discrete grid block point of candidate parameter as the primary iteration value, and adopts the constrained optimization algorithm directly to obtain global optimum's point.
At last, can finally definite structural parameters to be measured be shown and export, accomplish whole measuring process thus.
Those skilled in the art will readily understand; The above is merely preferred embodiment of the present invention; Not in order to restriction the present invention, all any modifications of within spirit of the present invention and principle, being done, be equal to and replace and improvement etc., all should be included within protection scope of the present invention.
Claims (8)
1. one kind is used for the storehouse matching process based on the error of fitting interpolation that optical scattering is measured, and this method comprises the following steps:
(a) confirm the excursion of testing sample structural parameters based on process conditions; This scope is carried out discretization according to preset step-length δ handle obtaining a plurality of discrete grid block points, and these discrete grid block points and each self-corresponding theoretical spectral value thereof are stored in the library of spectra;
(b) utilize the optical scattering measurement mechanism that testing sample is measured to obtain measure spectrum; Calculate the error of fitting between each self-corresponding measure spectrum value of said discrete grid block point and the theoretical spectral value, be stored into these errors of fitting in the library of spectra equally then;
(c) the error of fitting setting threshold that is calculated for step (b) is searched for out with the error of fitting that falls into this threshold value in the library of spectra, and the pairing discrete grid block point of the error of fitting of utilizing these to search out makes up the candidate parameter collection; In addition, said error of fitting is carried out multi-dimensional interpolation handle and obtain corresponding error of fitting interpolating function, thus for setting up the function corresponding relation between testing sample structural parameters and the error of fitting;
(d) search for to find out its global optimum's point based on said error of fitting interpolating function; This global optimum puts pairing parameter value and is the final testing sample structural parameters of confirming, the said process of finding out global optimum's point specifically realizes through one of following dual mode:
(d1) the discrete grid block point of concentrating with the constructed candidate parameter of step (c) is that center and radius do not exceed said preset step-length δ as the hunting zone; In this hunting zone, carry out the discretize processing to compare littler step-length δ ' with said step-length δ; Obtain a plurality of new discrete grid block points thus and obtain its corresponding error of fitting, wherein the pairing point of the minimum value in the error of fitting is global optimum's point; Perhaps
(d2) the discrete grid block point concentrated of candidate parameter that step (c) is constructed is as the primary iteration value, and employing constrained optimization algorithm is directly obtained global optimum's point.
2. the method for claim 1 is characterized in that, in step (b), goes out the error of fitting between each self-corresponding measure spectrum value of said discrete grid block point and the theoretical spectral value through the root-mean-square error algorithm computation.
3. according to claim 1 or claim 2 method; It is characterized in that; In step (c); The detailed process of said structure candidate parameter collection comprises: for the error of fitting that is stored in the library of spectra is set the error of fitting threshold value, then with in the library of spectra corresponding error of fitting be used to make up the candidate parameter collection less than all discrete grid block points of this setting threshold; Perhaps calculate the uncertainty of all local optimum points in the library of spectra and set corresponding uncertainty threshold value, then pairing uncertainty in the library of spectra is not less than all discrete grid block points that set threshold value and is used to make up the candidate parameter collection.
4. like any described method of claim 1-3, it is characterized in that, in step (c), adopt the mode of multidimensional linearity or multidimensional spline to carry out said multi-dimensional interpolation processing.
5. like any described method of claim 1-4, it is characterized in that, in step (d1), with said discrete grid block point be center and radius do not exceed said preset step-length δ 1/2 as the hunting zone.
6. like any described method of claim 1-4, it is characterized in that in step (d2), said constrained optimization algorithm comprises active set method, interior point method or Sequential Quadratic Programming method.
7. like any described method of claim 1-6, it is characterized in that the parameter to be measured of said sample comprises live width, line height, side wall angle, alignment error etc.
8. method as claimed in claim 7 is characterized in that, the spectral signal that said library of spectra is stored is reflectivity, ellipsometric parameter, Stokes vector or Mueller matrix.
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