METHOD FOR PROJECTING A BEAM OF PARTICLES ONTO A SUBSTRATE WITH CORRECTION OF SCATTERING EFFECTS

Abstract

A method for projecting a particle beam onto a substrate, the method includes a step of calculating a correction of the scattering effects of the beam by means of a point spread function modelling the forward scattering effects of the particles; a step of modifying a dose profile of the beam, implementing the correction thus calculated; and a step of projecting the beam, the dose profile of which has been modified, onto the substrate, and being wherein the point spread function is, or comprises by way of expression of a linear combination, a two-dimensional double sigmoid function. A method to e-beam lithography is also provided.

Claims

1. A method for projecting a particle beam onto a substrate, said method comprising a step of calculating a correction of the scattering effects of said beam by means of a point spread function modelling the forward scattering effects of said particles; a step of modifying a dose profile of said beam, implementing the correction thus calculated; and a step of projecting the beam, the dose profile of which has been modified, onto said substrate, and being wherein said point spread function is, or comprises by way of expression of a linear combination, a two-dimensional double sigmoid function.

2. The method as claimed in claim 1, wherein said two-dimensional double sigmoid function is defined in a plane XY and is expressed by a product of two one-dimensional double sigmoid functions defined on two orthogonal axes X and Y of said plane.

3. The method as claimed in claim 2, wherein each said one-dimensional double sigmoid function is expressed by a difference between two one-dimensional sigmoid functions, having an offset along said axis.

4. The method as claimed in claim 3, wherein each said one-dimensional double sigmoid function is of logistic type.

5. The method as claimed in claim 3, wherein each said one-dimensional double sigmoid function is symmetric and expressed by a difference between two identical one-dimensional sigmoid functions having an offset along said axis.

6. The method as claimed in claim 1, comprising a step of calibrating by determining a set of geometric parameters of said two-dimensional double sigmoid function, said calibrating step comprising the following substeps: a) projecting said particle beam onto a substrate having a threshold Do of exposure to the beam; b) measuring at least one dimension CD.sub.i of a region, of the substrate, in which said exposure threshold was exceeded; these substeps being repeated a plurality of times, identified by an index i, for various values D.sub.i of exposure dose; and c) determining said parameters on the basis of the measured dimensions; substeps a) and b) being implemented experimentally or by means of numerical simulations.

7. The method as claimed in claim 6, wherein CD.sub.i is the width of said region of the substrate in a direction x, acquired in the i-th repetition of substeps a) and b); said substep c) comprises calculating a linear function of expression CD.sub.i=2k.Math..sub.i+2x.sub.0, by regression based on the measurements of width CD.sub.i carried out in the various repetitions of said substep b), where .sub.i=ln[D.sub.0/(D.sub.0D.sub.i)], k and x.sub.0 being said geometric parameters of said two-dimensional double sigmoid function.

8. An e-beam lithography process comprising a step of projecting an electron beam onto a substrate covered with developable resist, said step being implemented as part of a method according to claim 1.

9. A computer-program product comprising program-code instructions stored on a computer-readable medium and suitable for implementing, when said program is executed by a computer, an operation for correcting scattering effects of a particle beam projected onto a substrate, said correction being carried out by means of a point spread function that is, or comprises by way of expression of a linear combination, a two-dimensional double sigmoid function.

Description

[0034] Other features, details and advantages of the invention will become apparent on reading the description given with reference to the appended drawings, which are given by way of example and which show, respectively:

[0035] FIG. 1, a schematic representation of an e-beam lithography apparatus;

[0036] FIG. 2, graphs of three sigmoid functions characterized by different steepness parameters;

[0037] FIG. 3, graphs of three symmetric one-dimensional double sigmoid functions characterized by different steepness parameters;

[0038] FIG. 4, graph of a symmetric two-dimensional sigmoid function;

[0039] FIGS. 5 and 6, the implementation of a calibrating step of a method according to one embodiment of the invention;

[0040] FIG. 7, the implementation of an electron beam;

[0041] FIGS. 8A-8C and 9A-9D, graphs illustrating a technical effect of the invention.

[0042] Generally, a sigmoid function is a function that has two horizontal asymptotes, which pass gradually from one to the other and that have an inflection point. In a more restricted sense, this term designates the function

[00002]$\begin{array}{cc}S\left(x\right)=\frac{1}{1+e^{-\left(x-x_0\right)/k}}& \left(2\right)\end{array}$

which is also called the logistic function.

[0043] Function (2) depends on two parameters: x.sub.0, which determines the position of the inflection point, and k, which determines the steepness of the transition region between the two asymptotes (more precisely, 1/k is the value of the derivative of S(x) at the inflection point x=x.sub.0). FIG. 2 shows graphs of three logistic curves with x.sub.0=100 and k=1 (curve S.sub.1), k=5 (curve S.sub.5) and k=10 (curve S.sub.10).

[0044] A double sigmoid function is given by the difference between two sigmoid functions of the type of equation (2). In particular, if the difference between two sigmoid functions having the same parameter k and parameters x.sub.0 of same absolute value but opposite sign is calculated, a function is obtained that is what may be called a symmetric sigmoid function DSS:

[00003]$\begin{array}{cc}\mathrm{DSS}\left(x\right)=\frac{1}{1+e^{-\left(x+x_0\right)/k}}-\frac{1}{1+e^{-\left(x-x_0\right)/k}}& \left(3\right)\end{array}$

[0045] FIG. 3 shows graphs of three symmetric double sigmoid curves with x.sub.0=100 and k=1 (curve DSS.sub.1), k=5 (curve DSS.sub.5) and k=10 (curve DSS.sub.10). For small k, this function tends toward a rectangle or top-hat function, and, as k increases, becomes increasingly rounded so as to tend towards a bell shape for high k. The full width at half-maximum of the rectangle does not depend on k, but is equal to 2x.sub.0. In contrast, functions defining bell curves (Gaussian, Voigt function, etc.) have a full width at half-maximum that is related to the steepness of their slopes.

[0046] The invention proposes to use, as PSF, a two-dimensional double sigmoid function, and preferably a symmetric two-dimensional double sigmoid function that may be defined by the following equation:

[00004]$\begin{array}{cc}\mathrm{DSS}.Math..Math.2.Math.D\left(x\right)=\left[\frac{1}{1+e^{-\left(x+x_0\right)/k_x}}-\frac{1}{1+e^{-\left(x-x_0\right)/k_x}}\right].Math..Math.\left[\frac{1}{1+e^{-\left(y+y_0\right)/k_y}}-\frac{1}{1+e^{-\left(y-y_0\right)/k_y}}\right]& \left(4\right)\end{array}$

[0047] FIG. 4 shows the graph DSS2D of a symmetric two-dimensional double sigmoid function with: x.sub.0=10, k.sub.x=0.5, y.sub.0=20, k.sub.y=1.

[0048] In certain cases, it will possibly be useful to consider an asymmetric two-dimensional double sigmoid function that may be defined by:

[00005]$\mathrm{DSS}.Math..Math.2.Math.D\left(x\right)=\left[\frac{1}{1+e^{-\left(x+x_0\right)/k_{x,1}}}-\frac{1}{1+e^{-\left(x-x_0\right)/k_{x,2}}}\right].Math..Math.\left[\frac{1}{1+e^{-\left(y+y_0\right)/k_{y,1}}}-\frac{1}{1+e^{-\left(y-y_0\right)/k_{y,2}}}\right]$

[0049] where k.sub.x,1k.sub.x,2 and/or k.sub.y,1k.sub.y,2.

[0050] This generalization allows an asymmetrywhich is most often undesiredin the source of the electron beam to be taken into account. Implementation thereof represents no particular difficulty and therefore, below, only the symmetric case will be considered.

[0051] Furthermore, below, for the sake of simplicity, the case of a symmetric one-dimensional double sigmoid function (equation 3) will be considered, even though a physical PSF is necessarily two-dimensional.

[0052] According to the invention, it is assumed that the dose deposited in the resist 12 by the beam 21 follows a profile that is identical to the intensity profile of the beam, which is given by a symmetric double sigmoid function that is centered on the point x=0. The expression of the dose D deposited at the point x is:

[00006]$\begin{array}{cc}D\left(x\right)=D.Math.\left[\frac{1}{1+e^{-\left(x+x_0\right)/k}}-\frac{1}{1+e^{-\left(x-x_0\right)/k}}\right]& \left(5\right)\end{array}$

[0053] Furthermore, the resist 12 will be considered to be exposed when D(x)D.sub.0, and not exposed elsewhere.

[0054] Under these conditions, it is possible to measure the parameters x.sub.0 and k experimentally (more precisely: to determine experimentally the best values of x.sub.0 and k, such that expression (5) fits, as closely as possible, in the sense of error variance, the actual profile of the electron beam). To do this, the beam is projected onto the resist with various dose values D, and the critical dimension (width) of the pattern thus transferred to the resist is measured. This is illustrated in FIG. 5, in which D.sub.1 and D.sub.2 are two dose values (D.sub.1<D.sub.2) and CD.sub.1, CD.sub.2 the corresponding critical dimensions (CD.sub.1<CD.sub.2).

[0055] It will easily be understood that, if the values of the parameters x.sub.0 and k of the profile (5) were known, the critical width CD of the pattern obtained with a dose D could be calculated by solving, with respect to x, the equation:

[00007]$\begin{array}{cc}\left(x\right)=D.Math.\left[\frac{1}{1+e^{-\left(x+x_0\right)/k}}-\frac{1}{1+e^{-\left(x-x_0\right)/k}}\right]=D_0& \left(6\right)\end{array}$

and by setting: CD=2x.

[0056] It is found that:

[00008]$\begin{array}{cc}\frac{\mathrm{CD}}{2}=x_0+k.Math.\ln .Math..Math.\left(-\frac{D_0-D}{D_0}\right)& \left(7\right)\end{array}$

[0057] In other words, there is a linear relationship between CD and

[00009]$=\ln .Math..Math.\left(-\frac{D_0-D}{D_0}\right).$

[0058] Thus, to estimate the parameters k and x.sub.0 it is enough to measure the critical dimensions CD.sub.i for various values D.sub.i of the dose, and therefore for various values of , then to perform a linear regression, as is illustrated in FIG. 6. It is preferable to use relatively small dose values D.sub.i, such that backscattering is insignificant. Specifically, a symmetric-double-sigmoid-function PSF mainly takes into account forward-scattering effects, which dominate when the intention is to produce patterns of nanoscale size.

[0059] Instead of determining the parameters k and x.sub.0 experimentally, precise numerical simulations, for example of Monte-Carlo type, will possibly be used.

[0060] According to one variant of the invention, the PSF may be expressed by a linear combination of a plurality of functions, at least one of which is a double sigmoid function. The coefficients of the linear combination and the various parameters characterizing these functions may be determined by regression, in general non-linear regression.

[0061] FIG. 7 illustrates the implementation of an electron beam 21 by means of a diaphragm 50, formed by two metal shields 51, 52 of L shape. Moving these two shields with respect to each other makes the shape (square or a rectangle of greater or lesser elongation) and size of the aperture 500 through which the beam passes change. The profile 60 of the beam, measured at the surface of the substrate 10, has a shape that may more easily be approximated by a symmetric double sigmoid function than by a Gaussian, in particular when the cross section of the beam reaches the limiting dimensions of the e-beam lithography apparatus, typically 20 nm to 50 nm. The reference 70 designates the pattern transferred to the resist 12 by the beam 21 shaped by the diaphragm 50.

[0062] There are diaphragms allowing beams of shape other than rectangular, for example circular or triangular, to be produced.

[0063] Generally, the dose profile deposited in the resist may be defined as the product of convolution of a function defining the desired pattern (for example, a succession of crenels) and the PSF. This dose profile is converted into an exposure pattern by a specific transfer function of the resist, which may for example be a simple threshold function.

[0064] FIGS. 8A to 9D allow the results of simulations of transfer of patterns to a resist using a Gaussian PSF (curve G in FIG. 8A) and a symmetric-double-sigmoid-function PSF (curve DSS) to be compared.

[0065] The case of the transfer of a periodic pattern made up of crenels of width equal to 100 nm and of 200 nm period is considered. The curves P.sub.G and P.sub.DSS in FIG. 8B show the deposited-dose profiles obtained by convoluting the G and DSS PSFs with this periodic pattern, respectively. The curves are different near their maxima, but have a comparable full width at half-maximum of 50 nm. The Gaussian function is characterized by a dispersion =21.23 nm, whereas the symmetric double sigmoid function is defined by x.sub.0=50 nm and k=2.5.

[0066] FIG. 8C shows the patterns M.sub.G and M.sub.DSS actually transferred to the resist for a threshold equal to 0.5 (arbitrary units). In this case, these patterns are practically identical, and are actually coincident in the figure. This ceases to be true for patterns of smaller critical dimensions. FIGS. 9A/9B relate to the case of a pattern made up of crenels of width equal to 50 nm and of 100 nm period, and FIGS. 9C/9D to the case of a pattern of the same type but of width equal to 32 nm and of 64 nm period. In both cases, the pattern computed using a symmetric-double-sigmoid-function PSF is narrower than the pattern obtained with a Gaussian PSF. It is moreover known that the use of a Gaussian PSF leads to an overestimation of the width of the patterns, and makes corrections necessary: decrease of the dose or of the critical dimensions of the patterns to be transferred. This is no longer necessary if a symmetric-double-sigmoid-function PSF is used. Correction of proximity effects is facilitated thereby.

[0067] Moreover, it is possible to show that a symmetric double sigmoid function allows, with a very small error, a Gaussian function to be approached, whereas the inverse is not true. Thus, the use of a double sigmoid function allows the true dose profile to be approached at least as well as, and generally better than, with a Gaussian PSF.

[0068] Thus, a symmetric-double-sigmoid-function PSF (or a PSF expressed by a linear combination of functions including at least one symmetric double sigmoid function and, for example, a Gaussian describing backscattering) improves the description of the intensity profile of a shaped beam, decreasing the error in correction of proximity effects with respect to use of a Gaussian function. This is at least partially explained by the fact that the width of the curve described by such a function (defined by the parameter x.sub.0) may be adjusted independently of the steepness of its edges (defined by the parameter k).

[0069] Moreover, sigmoid functionsin particular sigmoid functions of the logistic typehave analytical expressions that are easy to integrate into numerical simulation tools; the same goes for symmetric double sigmoid functions. Since their cumulative distribution functions are also analytical, the convolution computations used for the correction of proximity effects have a complexity and a consumption in terms of computational power that are comparable to those of the prior art.

[0070] Moreover, a plurality of PSFs comprising symmetric double sigmoid functions, corresponding to the various beam geometries, pattern sizes and other working conditions achievable with a given piece of shaped-beam lithography equipment, will possibly be determined, in the way described above with reference to FIG. 5.

[0071] Another advantage of the invention resides in the simplification of the proximity-effect-correcting process. Known prior-art software packages (for example PROXECCO (registered trademark), or INSCALE (registered trademark)) are able to combine, to optimize geometry, a dose modulation and a modulation of the geometry of the patterns to be exposed, as is described in patent application EP 2 650 902. When the forward scattering of electrons is described by symmetric double sigmoid functions, only a dose modulation is necessary, this simplifying and decreasing the time taken to perform the associated computations.

[0072] To implement embodiments of the invention in shaped-beam e-beam lithography, it is possible to use an e-beam lithography machine of known type, for example the Vistec SB 3054. Dose-modulation corrections according to the invention will possibly be integrated into commercially available software packages such as PROXECCO (registered trademark), which is distributed by Synopsis, or INSCALE (registered trademark), which is distributed by Aselta Nanographics, or even BEAMER (registered trademark), which is distributed by GeniSys, in order to replace the forward-scattering PSFs of the prior art (Gaussian functions or combinations thereof) with the double-sigmoid PSF described above. For the backscattering PSF, the same functions as in the prior art will possibly be used, i.e. essentially Gaussians or combinations of Gaussians.

[0073] The invention has mainly been described with respect to its application to e-beam lithography. However, it may also be applied to lithography processes using beams of particles other than electrons, and even to processes, in which a beam of particles interacts with a target, other than lithography processes. It may in particular be applied to electron microscopy.