Method for calibration of a CD-SEM characterisation technique

Abstract

A calibration method for a CD-SEM technique, includes determining a match function converting at least one parameter obtained by modelling a measurement supplied by the CD-SEM technique into a function of at least one parameter representative of a measurement supplied by a characterisation technique different from the CD-SEM technique, the match function being characterised by a plurality of coefficients; performing measurements on a plurality of patterns chosen to cover the desired validity range for the calibration, the measurements being done using both the CD-SEM technique to be calibrated and the reference technique; determining, from the measurements, a set of coefficients of the match function minimising the distance between the functions of the parameters measured using the reference technique and applying the match function to the parameters obtained by modelling measurements supplied by the CD-SEM; using the set of coefficients during the implementation of the calibrated CD-SEM technique.

Claims

1. A method of calibrating a CD-SEM technique of characterisation by scanning electron microscopy in order to determine a critical dimension, said method comprising: determining a mapping function transforming at least one parameter obtained by modelling a measurement provided by the CD-SEM technique into a function of at least one parameter representative of a measurement provided by a reference characterisation technique different from the CD-SEM technique, wherein the mapping function is characterised by a plurality of coefficients; making measurements on a plurality of physical patterns chosen to cover a field of validity desired for the calibration, wherein the measurements on the plurality of physical patterns are made using both the CD-SEM technique of characterisation by scanning electron microscopy requiring calibration and the reference characterisation technique; determining, from the measurements, a set of coefficients of the mapping function minimising a distance between the functions of the parameters measured by the reference characterisation technique and applying the mapping function to the parameters obtained by modelling the measurements provided by the CD-SEM technique; using the set of coefficients when using the calibrated CD-SEM technique.

2. The method according to claim 1, wherein said reference characterisation technique is chosen from among one of the following techniques: atomic force microscopy; scatterometry; transmission electron microscopy.

3. The method according to claim 1, wherein the number of measurements is greater than or equal to the number of coefficients to be determined in the mapping function.

4. The method according to claim 1, wherein the mapping function is a linear function of the different coefficients characterising the function.

5. The method according to claim 4, wherein the mapping function relating parameter q representative of a measurement provided by the reference characterisation technique to parameter(s) p obtained by modelling of measurement provided by the CD-SEM technique is written as follows: q=.sup.K.sub.i=1f.sub.i(p).c.sub.i, where the K coefficients c.sub.i designate the plurality of coefficients characterising the mapping function, and functions f.sub.i(p) designate functions for pre-processing parameter(s) p.

6. The method according to claim 5, wherein the mapping function is a polynomial function of the parameter(s) obtained by modelling a measurement provided by the CD-SEM technique.

7. The method according to claim 4, wherein the mapping function relating parameter q representative of a measurement provided by the reference characterisation technique to parameter(s) p obtained by modelling of measurement provided by the CD-SEM technique is written as follows: g(q)=.sup.K.sub.i=1f.sub.i(p).c.sub.i, where the K coefficients c.sub.i designate the plurality of coefficients characterising the mapping function, function g(q) designates a functioning for pre-processing of parameter q, and functions f.sub.i(p) designate functions for pre-processing parameter(s) p.

8. The method according to claim 1, wherein the mapping function is a non-linear function of the different coefficients characterising the function.

9. The method according to claim 1, wherein the mapping function transforms at least one parameter obtained by modelling of a measurement provided by the CD-SEM technique into at least one parameter representing a measurement provided by a reference characterisation technique.

Description

(1) Other characteristics and advantages of the invention will become clear from the description which is given of it below, by way of example and non-restrictively, with reference to the appended figures, in which:

(2) FIG. 1 illustrates diagrammatically a secondary electron intensity profile as a function of the profile of a pattern obtained by means of instrumentation of the CD-SEM type;

(3) FIGS. 2a and 2b illustrate a first example of parametric mathematical model of a CD-SEM image;

(4) FIG. 3a illustrates diagrammatically the intensity of a CD-SEM profile for an expanding pattern profile;

(5) FIG. 3b illustrates diagrammatically the intensity of a CD-SEM profile for a narrowing pattern profile;

(6) FIG. 4 represents the different steps of the method of the invention;

(7) FIG. 5 illustrates a second example of mathematical modelling

(8) of a CD-SEM image;

(9) FIG. 6 is a CD-SEM image of a line of resin on silicon;

(10) FIG. 7 represents two diffraction peaks of the secondary electrons at the edges of the line imaged by CD-SEM represented in FIG. 6.

(11) FIG. 4 illustrates diagrammatically the different steps of method 100 of the invention.

(12) Calibration method 100 of the invention seeks to calibrate a CD-SEM technique for the determination of a critical dimension, CD, of any pattern (holes, pins, line, trench, . . . ) belonging to a microelectronic circuit. The material of the pattern can also be any material. This pattern can be, for example, an isolated pattern or a pattern belonging to a network of patterns which are repeated periodically. It may be a pattern obtained after any step (lithography, etching, . . . ) of a manufacturing process.

(13) According to a step 101 of method 100 of the invention, reference samples representative of the field of validity of the calibration which it is desired to implement must be selected.

(14) For example, if it is desired to calibrate the CD-SEM on samples of lines of resin which are 20 nm to 100 nm wide, and which have an inclination angle of 70 to 90, a number N of samples will be taken which are sufficiently varied to cover the range of variation of both parameters, for example through use of a focus/dose or focus/expo matrix (i.e. The exposure parameters of the lithographic process are varied spatially on the wafer; in one direction the deposited optical power - i.e. the dose - and in the other direction the focus height - i.e. the focus). It will easily be understood that a delicate determination of parameters with a non-robust CD-SEM model and AFM, TEM or OCD reference technique will imply that the number of samples is concentrated in the critical area where accuracy is sought; it will also be possible to oversample over a range in which it is known that accuracy is low (for example on small CDs). It will also easily be understood that the larger the range of validity the greater will be the number of samples.

(15) This type of technique of sample selection comes within the field of Design of Experiments (DOE). Pre-established sampling can, for example, be determined by a regular meshing or by a meshing of the Latin Square type. There are also iterative techniques, such as response functions of the Kriging type, which can indicate, as the technique progresses, the sampling points as a function of the variability of the observed parameters. As an illustration of the present embodiment we shall remain with the case in which there is a DOE technique enabling us to determine correctly the parameters of the model. We shall thus suppose in this case that we have N calibration patterns which sufficiently vary the parameters of interest. It should be noted that in what follows it will be supposed that the number of measurements is greater than or equal to the number of coefficients which must be determined. It should be noted that the number of measurements is not necessarily equal to the number of samples.

(16) Step 101 of selection of the reference samples representative of the field of validity of the calibration is accompanied by a step 102, which consists in determining the mapping function (designated equally below by the terms law or model) relating the parameter(s) obtained by modelling a measurement provided by the CD-SEM technique to the parameter(s) representative of a measurement provided by a reference characterisation technique different from the CD-SEM technique, preferably one of the AFM, TEM or OCD techniques.

(17) As we shall see below, different types of models exist which can be used to relate the CD-SEM mathematical parameters p to the physical parameters q of the reference technique.

(18) According to the invention, at least one parameter p and at least one parameter q will be required, bearing in mind that it is possible to have several parameters p and several parameters q, and that the numbers of parameters p and of parameters q are not necessarily identical. Similarly, parameters q can be representative of measurements derived from one or more reference techniques; for example, certain parameters may be derived from AFM, and certain other parameters may be derived from OCD or TEM.

(19) The term physical parameter of the reference technique is understood to mean a parameter representative of the measurement provided by the reference characterisation technique. Thus, as an illustration only, the reference physical parameter can be the width, such as the width of a line of resin on silicon obtained by means of an AFM technique. Without being exhaustive, the reference physical parameters can also be the height of the patterns, the angle of inclination, etc.

(20) Similarly, the term CD-SEM mathematical parameter is understood to be unit of data obtained by mathematical modelling of the CD-SEM image.

(21) It will be noted that the examples given below relate to the case of a mathematical model of a CD-SEM image, on the understanding that the invention could also apply to the case of a physical model of the CD-SEM image.

(22) In the examples mentioned below, by way of example only, we have considered only the position parameters of the diffraction peaks in the CD-SEM image, but the invention applies to all types of parameter which may be obtained by means of modelling of the CD-SEM images (width of the peaks, etc.), or even measuring conditions (enlargement of the images, doses, measuring times, etc.).

(23) According to a first embodiment of step 102, it is supposed that the law relating reference parameters q to CD-SEM mathematical parameters p is a polynomial p law.

(24) The analysis of the measurements of the CD-SEM technique and of the AFM/OCD reference technique of a pattern i produces the row vector of parameter p.sub.i, and q.sub.i. In this case the CD-SEM and AFM/OCD reference techniques give for each sample parameters m and n (respective lengths of row vectors p.sub.i: and q.sub.i:). If it is supposed, for example, that these quantities are related by a polynomial of order 2 with cross-terms, this therefore gives, with m=2 (i.e. two CD-SEM parameters p.sub.i1 and p.sub.i2 for pattern i) and, considering first term q.sub.i1 of row vector q.sub.i:

(25) $\begin{array}{c}{q}_{i,1}{c}_0+p_{i1}{c}_1+p_{i2}{c}_2+p_{i1}^2{c}_3+p_{i2}^2{c}_4+p_{i1}{p}_{i2}{c}_5\\ \left(\begin{array}{cccccc}1& p_{i1}& p_{i2}& p_{i1}^2& p_{i2}^2& p_{i1}{p}_{i2}\end{array}\right)\left(\begin{array}{c}{c}_0\\ c_1\\ c_2\\ c_3\\ c_4\\ c_5\end{array}\right)\end{array}$
with K (in this case K=5) coefficients c.sub.i characterising the mapping function

(26) Thus, by making N measurements (step 103), the following matrix relationship is obtained:

(27) $q_{:,1}\left(\begin{array}{cccccc}1& p_{11}& p_{12}& p_{11}^2& p_{12}^2& p_{11}{p}_{12}\\ 1& p_{21}& p_{22}& p_{21}^2& p_{22}^2& p_{21}{p}_{22}\\ \mathrm{.Math.}& & & & & \\ 1& p_{i1}& p_{i2}& p_{i1}^2& p_{i2}^2& p_{i1}{p}_{i2}\\ \mathrm{.Math.}& & & & & \\ 1& p_{N1}& p_{N2}& p_{N1}^2& p_{N2}^2& p_{N1}{p}_{N2}\end{array}\right)\left(\begin{array}{c}{c}_0\\ c_1\\ c_2\\ c_3\\ c_4\\ c_5\end{array}\right)=P_{1}c$

(28) Consistently with the previously used notation, this therefore gives P.sub.c=P.sub.1c, where function P.sub.c is the mapping function which transforms p into q (q P.sub.c(p)). It is observed in this case that the chosen model is a linear model as a function of coefficients c.sub.i (vector c), such that there is a matrix relationship between the vector of parameters q and vector c of coefficients c.sub.i,. and that the said matrix depends solely on parameters p and not on coefficients c.sub.i.

(29) According to step 104, the coefficients of parameters vector c are determined by minimising the distance:
c=Argmin.sub.cq.sub.:,1P.sub.1c

(30) If it is desired to minimise according to least squares norm L2, the solution is given directly by the following relationship:

(31) c=P.sub.1q.sub.:1

(32) where P.sub.1 is the pseudo-inverse matrix of P.sub.1.

(33) When the distance is not L2 (L1 for example), or alternatively if constraints are added to the system (constraints at intervals to which the parameters must belong, linear or non-linear constraints of the parameters, addition of penalty functions, etc.), other algorithms known to those skilled in the art can be used for the resolution of the linear system.

(34) According to step 105, when vector c has been determined after minimisation, knowledge of function P.sub.c to which this set of coefficients c is applied enables the parametric CD-SEM method to be used alone (i.e. without the assistance of another characterisation technique), both for R&D and for production, across the full range of validity: in other words, mathematical CD-SEM parameters p are measured, the line of P.sub.1 is then formed, and physical quantity of interest q.sub.1 is found using the simple formula q.sub.1=P.sub.1c.

(35) As was mentioned above, the mathematical CD-SEM parameters can be of different types. Let us take, for example, the CD-SEM model of a rectangle (illustrated in FIG. 5), written with parameters x.sub.1, x.sub.2, a, b, w,y.sub.( ), which represent respectively the two abscissae of the secondary electron intensity peaks, the lower and upper signal levels, the expansion of the exponential functions, and the level of the background signal.

(36) The profile can then be modelled, for example, by the function y(x), which is broken down into three sub-functions, depending on whether x is less than x.sub.1, between x.sub.1 and x.sub.2 or greater than x.sub.2:

(37) $\begin{array}{c}y\left(x\right)=y_0+\left(a+b\right)\exp \left(-{\left(x-x_1\right)}^2/2/w^2\right)\mathrm{si}x<x_1\\ =y_0+a+b\exp \left(-{\left(x-x_1\right)}^2/2/w^2\right)+\\ b\exp \left(-{\left(x-x_2\right)}^2/2/w^2\right)\mathrm{si}x\left[x_1,x_2\right]\\ =y_0+\left(a+b\right)\exp \left(-{\left(x-x_2\right)}^2/2/w^2\right)\mathrm{si}x>x_2\end{array}$

(38) The mathematical parameters of the model are therefore, in this case, (making y.sub.0=0):
x.sub.1, x.sub.2, a, b, w.

(39) With reference to the document CD characterization of nanostructures in SEM metrology (C. G. Frase, E. Buhr, and K. Dirscherl, Meas. Sci. Technol., vol. 18, no 2, p. 510, February. 2007) cited previously in reference to function M(x) illustrated in FIG. 2b, the CD-SEM mathematical parameters can be: x.sub.1, x.sub.2, x.sub.3, x.sub.4, x.sub.5, S.sub.0, S.sub.1, X.sub.2, S.sub.3, t.sub.0, t.sub.1, t.sub.2, t.sub.3, t.sub.4, t.sub.5, A, B, C, D.

(40) In this latter case, bearing in mind the number of parameters which must be determined, the number of samples must be higher and/or constraints relating to the parameters must be imposed.

(41) One possibility to reduce the number of parameters can consist in:

(42) $\mathrm{given}\text{:}{u}_1=\frac{x_2+x_1}{2},u_1=\frac{x_5+x_4}{2},\mathrm{imposing}:$$x_3=\left(u_1+u_2\right)/2$$S_1=S_3,t=t_i{V}_i.$

(43) The mathematical parameters to be calibrated then become:
u.sub.1, u.sub.2, S.sub.0, S.sub.1, S.sub.2, t, A, B, C, D.

(44) Without altering the general character of the linear approach (i.e. there is always a linear system which can be written according to the matrix relationship: q=Pc), according to step 102, it can be observed that the formation of matrix P.sub.1 in order to determine physical parameter q.sub.1 from parameters p is arbitrary. Indeed, the CD-SEM parameters p can equally be pre-processed, in order to form matrix P.sub.1; in other words, it is not necessary to use parameters p directly. The system shown above can then be written in the following form:

(45) $\begin{array}{cc}{f}_q\left(q_{:,1}\right)(\begin{array}{cccccc}{f}_0\left(p_{1:}\right)& f_1\left(p_{1:}\right)& f_2\left(p_{1:}\right)& f_3\left(p_{1:}\right)& f_4\left(p_{1:}\right)& f_5\left(p_{1:}\right)\\ f_0\left(p_{2:}\right)& f_1\left(p_{2:}\right)& f_2\left(p_{2:}\right)& f_3\left(p_{2:}\right)& f_4\left(p_{2:}\right)& f_5\left(p_{2:}\right)\\ \mathrm{.Math.}& & & & & \\ f_0\left(p_{i:}\right)& f_1\left(p_{i:}\right)& f_2\left(p_{i:}\right)& f_3\left(p_{i:}\right)& f_4\left(p_{i:}\right)& f_5\left(p_{i:}\right)\\ \mathrm{.Math.}& & & & & \\ f_0\left(p_{N:}\right)& f_1\left(p_{N:}\right)& f_2\left(p_{N:}\right)& f_3\left(p_{N:}\right)& f_4\left(p_{N:}\right)& f_5\left(p_{N:}\right)\end{array})\left(\begin{array}{c}{c}_0\\ c_1\\ c_2\\ c_3\\ c_4\\ c_5\end{array}\right)=P_{1}c& \left(\mathrm{relationship}1\right)\end{array}$
where the K functions f.sub.i(p) (where K is equal to the number of coefficients c.sub.1 characterising the mapping function) are pre-processing functions, which the user can choose according to the specific characteristics of the problem. In the above example a pre-processing function f.sub.q is also applied to vector q (i.e. both parameters p and q can be pre-processed). In this case, in order to obtain q after this, function fq must be reversed; in other words, we shall obtain q=f.sub.q.sup.1 (f.sub.q(q)).

(46) In what follows we shall illustrate the case of the extension of the linear model using simple pre-processing functions.

(47) Parameter q is, in this case, width w of resin lines on silicon obtained by an AFM measurement which is the reference technique in this case. An analysis of the 14 quantities (N=14) gives a vector q.sub.:1 containing 14 line widths w.sub.1 . . . 14=q.sub.:,1. The quantities of the vector are given in nm in the second column of table 1 below.

(48) TABLE-US-00001 TABLE 1 w obtained by AFM Image Delta = x2 xl CDSEM n.sup.o (nm) (nm) (80%) 1 32.84 23.09 32.92 2 43.15 29.45 38.88 3 52.33 37.45 47.40 4 89.02 73.53 81.86 5 100.65 86.11 94.07 6 111.66 95.34 105.90 7 121.54 105.74 114.63 8 129.85 115.43 123.76 9 139.33 123.99 133.81 10 147.30 132.52 142.23 11 157.10 143.87 152.14 12 166.85 153.36 162.53 13 176.56 164.45 173.65 14 279.84 267.50 275.82

(49) Simultaneously with the measurements of the reference technique, CD-SEM images are produced and a parametric analysis of the said image is made: the model used is, for example, the model illustrated in FIG. 5 (approximation of the intensity profile by the function y(x) described in detail above).

(50) By constraining parameters a, b and w and to simplify the explanation of the method of the invention, we consider here only two mathematical parameters, x.sub.1 and x.sub.2, for the images obtained by the CD-SEM technique corresponding to the positions of the two diffraction peaks of the secondary electrons at the edges of the line. These peaks, which are illustrated in FIG. 5 (and which correspond to the CD-SEM image of FIG. 4) are noted for the image: i: x.sub.i.1=p.sub.i.1 and x.sub.i.2=p.sub.i.2 ((it is supposed that x.sub.i.2>x.sub.i.1). At this point it is not known how the positions of the peaks obtained with CD-SEM relate to the widths of the lines obtained by the AFM reference technique. The third column of table 1 shows the value of difference p.sub.i,2p.sub.i,1 (i.e. x.sub.i,2x.sub.i,1) for each measurement i. If the distance between two peaks was equal to the value given by AFM, we would have q.sub.i.1=p.sub.i,1+p.sub.i,2. By comparing the values of the second column and the third column it is observed that this is absolutely not the case. The average quadratic error compared to AFM is in this case equal to 14.1 nm.

(51) As a comparison, the last column of table 1 also contains the measurement of the critical dimension obtained by the non-calibrated CD-SEM analysis technique, i.e. an analysis by studying the contours of the peaks taken with thresholding of 80% of the height of the peak. The average quadratic error compared to the AFM measurement is 5.2 nm: this quadratic error is therefore less than the quadratic error obtained in the case of the model of the peak-to-peak distance, but is still too high; one of the goals of the method of calibration of the invention is to reduce this quadratic error.

(52) For physical reasons it may be supposed initially that the following applies: q.sub.i,1=c.sub.0+c.sub.1f(p.sub.i,1, p.sub.1,2), where f(x.sub.1,x.sub.2)=x.sub.2x.sub.1; by using the previous notation of relationship 1 it is possible to write:
f.sub.0(p.sub.i1)=f.sub.0(X.sub.1, X.sub.2)=1.sup.andf.sub.1(p.sub.i1)=f.sub.1(X.sub.1, X.sub.2)=X.sub.2X.sub.1

(53) The following will subsequently be noted: pi,.sub.2p.sub.i,1=x.sub.i,1x.sub.i,1=.sub.i.

(54) The method of the invention seeks to determine coefficients c.sub.1 (in this case two coefficients c.sub.o and c.sub.1 characterising the mapping function), and then to use these coefficients when implementing the CD-SEM technique in production. The formation of matrix Pi derived from the analysis of the 14 images gives a matrix of 14 lines (i.e. the number of measurements) and 2 columns (i.e. the number of coefficients). The pseudo-inverse matrix is determined, and the following is obtained experimentally: c.sub.0=14.33 nm and c.sub.1=0.997. The average quadratic error between the model obtained by the method of the invention and the reference measurement (in this case the AFM technique) is 1.68 nm. An average quadratic error is therefore obtained which is far lower than the average quadratic error between the CD-SEM model with thresholding of 80% of the height of the peak and the AFM technique.

(55) According to a first variant, additional polynomial terms can be included by adding two additional coefficients c.sub.2 and c.sub.3:
q.sub.i,1=c.sub.0+c.sub.1.sub.i.sup.2+c.sub.3.sub.i.sup.3

(56) Using the previous notations of relationship 1, it is possible to write:
f.sub.0(p.sub.i1)=f.sub.0(X.sub.i, X.sub.2)=1
f.sub.1(p.sub.i1)=f.sub.1(X.sub.1, X.sub.2)=X.sub.2X.sub.1
f.sub.2)(p.sub.i1)=f.sub.1(X.sub.1, X.sub.2)=X.sub.2X.sub.1).sup.2
f.sub.3(p.sub.i1i)=f.sub.1(X.sub.1, X.sub.2)(X.sub.2X.sub.1).sup.3

(57) Matrix P therefore has 14 lines and 4 columns. The average quadratic error between the model obtained by the method of the invention and the reference measurement (in this case AFM) is in this case 0.75 nm, and is again therefore substantially smaller

(58) According to a second variant, the following model may be supposed:

(59) $q_{i,1}=c_0+c_1{}_i+\frac{c_2}{{}_i}+\frac{c_3}{{}_i^2}$

(60) Using the previous notations of relationship 1, it is possible to write:
f.sub.0(p.sub.i1)=f.sub.0(X.sub.1, X.sub.2)=1
f.sub.1(p.sub.i1)=f.sub.1(X.sub.1, X.sub.2)=X.sub.2X.sub.2)=1
f.sub.2(p.sub.i1)=f.sub.1(X.sub.2X.sub.1)
f.sub.3(p.sub.i1)=f.sub.1(x.sub.1, X.sub.2)=1/(x.sub.2X.sub.1).sup.2

(61) As with the previous case, matrix P has 14 lines and 4 columns. The average quadratic error between the model obtained by the method of the invention and the reference measurement (in this case AFM) is in this case 0.73 nm, and is once again smaller than the first variant.

(62) Through these models the importance and influence of the choice of the model to which the measurements will subsequently be applied will be understood.

(63) According to another embodiment of method 100 of the invention, step 102 may consist of determining a mapping function not based on a linear model; in other words, there is a matrix relationship between the vector of parameters q and vector c of coefficients with a matrix which would depend only on parameters p, and not on coefficients c.sub.i.

(64) If we return to the initial formation, the calibration problem amounts to resolving the following equation
c=Argmin.sub.cq.sub.expP.sub.c(p.sub.exp)
i.e. that c is the estimated value for which distance D between q.sub.exp and P.sub.c(p.sub.exp) is minimal.

(65) The problem considered in terms of its general character still has no single solution, due to the non-linearity of P.sub.c. Several strategies known to those skilled in the art can be envisaged, such as a global optimisation, or selection of convex domains (in which the solution is unique), and a local optimisation in the convex domain.

(66) As an example, the following non-linear model may be used:

(67) $q_{i,1}=c_0+c_1{}_i+c_2\arctan \left(\frac{{}_i}{c_3}\right)$

(68) Non-restrictively, after being calibrated by the method of the invention CD-SEM technology can, for example, be used for techniques to improve OPC (Optical Proximity Correction) models. For this application, reliable information concerning the dimensions of the objects is required in order to develop the behaviour models of the lithography tools.