CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application claims priority to and benefit of the German patent application DE 10 2018 218 129.2, filed with the German Patent and Trademark Office on Oct. 23, 2018, the entire content of which is incorporated herein by reference.

TECHNICAL FIELD

The present disclosure relates to the field of correcting one or more errors of a photolithographic mask. In particular, the present disclosure refers to a method and an apparatus for determining positions of a plurality of pixels to be introduced in a substrate of a photolithographic mask.

BACKGROUND

As a result of the constantly increasing integration density in the semiconductor industry, photolithographic masks must project smaller and smaller structures onto a photosensitive layer, i.e., a photoresist on wafers. To fulfil this demand, the exposure wavelength of photolithographic masks has been shifted from the near ultraviolet across the mean ultraviolet into the far ultraviolet region of the electromagnetic spectrum. Presently, a wavelength of 193 nm is typically used for the exposure of the photoresist on wafers. Consequently, the manufacturing of photolithographic masks that fulfil the increasing resolution requirements is becoming more and more complex, and thus more and more expensive as well. It is not unusual that photolithographic masks, photomasks or simply masks have defects at the end of their manufacturing process. Because of the time-consuming mask fabrication process, defects of photomasks should be repaired whenever possible.

For example, for transmissive photolithographic masks, the homogeneity of the optical transmission across the mask area is an important parameter. A variation of the optical transmission across the area of the photolithographic mask leads to a corresponding variation of the local optical intensity dose applied to the photoresist on the wafer when illuminating a wafer through the mask. The variation of the locally applied optical intensity dose or simply dose results in a fluctuation or a variation of the structure dimension of a pattern element in the developed photoresist. The uniformity of a pattern element across the area of the photolithographic mask is called critical dimension uniformity (CDU). In the article “Improving wafer level uniformity for logic utilizing mask level metrology & process,” Photomask Technology, Proc. SPIE Vol. 8880, 0888025-1-0888025-7, 2013, the authors A. Cohen et al. present investigations that show that a CDCR (Critical Dimension Correction Ratio) generated by various pixel arrangements show a linear behavior for different feature types of photolithographic masks.

Apart from an optical transmission inhomogeneity, photolithographic masks can have further kinds or types of errors. An important type of defects of photolithographic masks is mask image placement errors or registration errors. This type of error or defect occurs if one or more pattern elements of a pattern arranged on a photolithographic mask are not precisely at their positions that are predetermined by the layout data of the mask. Another type of defect is a bending of the substrate of a photomask. A further important defect category of photolithographic masks is overlay errors.

The applicant has disclosed a method for correcting these and other errors of photolithographic masks by introducing or writing pixels into a substrate of a photomask. For example, this method is described in the U.S. Pat. No. 9,658,527 B2 of the applicant, the entire content of which is incorporated by reference. This method is already routinely used for reliably correcting several error types of photomasks. Nevertheless, there is room for a further improvement of this defect correction process.

SUMMARY

In a general aspect, the present invention provides a method and an apparatus for improving the above-mentioned method of correcting defects of a photolithographic mask.

According to an aspect of the invention, a method is provided for determining positions of a plurality of pixels to be introduced into a substrate of a photolithographic mask by use of a laser system, wherein the pixels serve to at least partly correct one or more errors of the photolithographic mask. The method comprises the steps of: (a) obtaining error data associated with the one or more errors; (b) obtaining first parameters of an illumination system, the first parameters determining an illumination of the photolithographic mask by the illumination system when processing a wafer by illuminating with the illumination system using the photolithographic mask; and (c) determining the positions of the plurality of pixels based on the error data and the first parameters.

Pixels that are introduced into a substrate of a photolithographic mask to correct various types of errors or defects of a photolithographic mask locally modify the optical transmission of the mask substrate. In the following, the benefit of the inventive method is mainly illustrated for the example of correcting critical dimension (CD) errors of a mask. However, the inventive method is not restricted to the correction of CD errors of photolithographic masks.

Pixels introduced or written into a mask substrate to correct for example registration errors, generate small scattering centers for the optical radiation transmitting the mask substrate. For example, the pixels that correct one or more registration defects introduce local inhomogeneities in the optical transmission of the mask when a repaired mask having pixels in its substrate is operated in an illumination system. Thus, the introduction of pixels correcting registration errors would result in a critical dimension (CD) variation across the photomask or a critical dimension uniformity (CDU) problem of the photolithographic mask.

To avoid a CDU problem when correcting registration errors, a distribution of a second type of pixel(s) may be determined simultaneously with the determination of the first type of pixels correcting the registration errors. The second type of pixel predominantly locally scatters the optical radiation impacting on the pixel in a defined manner. The second type of pixel essentially does not locally change the density of the substrate. Typically, the second type of pixel is introduced into a mask substrate together with the first type of pixel that corrects for example the error(s) of the photolithographic mask.

Both, the first type of pixel and the second type of pixel are not homogeneously distributed in the substrate of a photolithographic mask to be corrected. Hence, the optical radiation that transmits the mask substrate to generate an image of a pattern element arranged at different positions across a mask is subjected to different disturbances by the pixels in the mask substrate at various positions across the photolithographic mask. This means that the various numbers of pixels “seen” by the optical radiation imaging pattern elements at various positions across the mask result in a CD variation of pattern elements across the photolithographic mask.

But, even if the pixels would be homogeneously distributed within the mask substrate, different illumination settings of the illumination system would cause that the respective optical radiation distribution “sees” a different amount of scattering centers or pixels along its path through a photolithographic mask so that different illumination settings of the illumination system would generate different CD variations of the pattern elements arranged on a mask substrate.

The inventive method considers the optical radiation distribution of an illumination system that is used for illuminating a corrected photolithographic mask in a manufacturing process of semiconductor devices when determining the positions of the pixels to be introduced into the mask substrate in order to correct the one or more defects of the photolithographic mask. Hence, the optical radiation distribution the corrected photomask will see afterwards during its operation is taken into account when determining the positions of the pixels. Thus, the inventive method improves or optimizes the defect correction for a specific optical radiation distribution used during the operation of the corrected photolithographic mask in an illumination process of wafers for fabricating devices, as for example integrated circuits.

The at least one first type of pixel can locally modify a density of the substrate, and the at least one second type of pixel can locally modify the optical transmission of the substrate. The locally modified density and/or the local optical transmission of the photolithographic mask can be discontinuously modified within a plurality of small volumes of the photolithographic mask, wherein each small volume is called pixel. It is also possible that the locally modified density and/or in particular the optical transmission distribution of the photolithographic mask is continuously modified across the photolithographic mask by the plurality of pixels.

The one or more errors can comprise at least one of: at least one registration error of a pattern of the photolithographic mask, at least one optical transmission error of the substrate of the photolithographic mask, at least one planarity error of the substrate of the photolithographic mask, and at least one overlay error.

Apart from registration errors, the inventive method allows the correction of other types of defects of photolithographic masks in a similar process as illustrated above for registration errors. Typically, the correction of registration errors, planarity errors of the mask substrate, and/or overlay errors requires at least two different types of pixels. The correction of optical transmission defects of a photomask can be corrected using a single pixel type, although two or more types of pixels can also be used for correcting optical transmission inhomogeneities of mask substrates.

For the fabrication of devices, as for example integrated circuits, normally a series of photolithographic masks is successively applied on the same wafer during the manufacturing process in order to generate complex structures on the wafer. In many applications, the absolute location of the pattern elements with respect to the optical axis is not as important as the orientation of two or more masks relative to each other so that overlay critical structures can reliably be printed onto the wafer at predetermined positions. Thus, it is not the target to bring a first photolithographic mask to the best match with a predetermined design, but to obtain the best match with a second mask. The inventive method can also be used to correct such overlay errors of two or more photolithographic masks. In particular, the positions of the plurality of pixels to be introduced into the substrate of a first photomask and the plurality of pixels to be introduced into the substrate of the second photomask can be determined in a joint optimization process.

For registration defects, the error data may comprise differences between determined positions of pattern elements of the photolithographic mask and predetermined positions of the pattern elements that are associated with the layout data. For overlay errors, the error data may comprise differences between determined positions of pattern elements on the wafer illuminated with the illumination system using the photolithographic mask and predetermined positions of the pattern elements on the wafer. Further, for optical transmission errors or CD errors, the error data may comprise differences between a determined optical transmission distribution of the photolithographic mask and a predetermined optical transmission distribution of the photolithographic mask. It is also possible that the error data may comprise differences between the determined optical transmission distribution of the photolithographic mask, as for example a varying size of a feature element across the mask, and the predetermined optical transmission distribution of the photolithographic mask, as for example a uniform size of the feature element across the mask, and differences between determined positions of pattern elements on the photolithographic mask or on the wafer illuminated using the photolithographic mask and predetermined positions of the pattern elements on the photolithographic mask or on the wafer illuminated with the illumination system using the photolithographic mask.

The plurality of pixels can be arranged in a common layer of the substrate, and step c. of the inventive method can comprise determining a depth of the layer within the substrate.

For correcting registration errors, pixels are typically introduced in a layer that is arranged in the center or the half of the depth of the mask substrate. Pixels arranged in the center of the mask substrate essentially do not lead to a bending of the mask substrate. On the other hand, this means that a photolithographic mask that has a bending substrate may be corrected by introducing pixels in the upper half or the lower half of the substrate depending on the kind of bending. Of course, it is also possible to correct a photolithographic mask that has registration errors and a bended substrate at the same time. This means that two or more error types can simultaneously be corrected by a plurality of pixels.

As the scattering effect of the pixels depends on their arrangement in the mask substrate, it is necessary to know both, the depth (z-position) in which the pixels are to be introduced into the substrate as well as their positions within the layer (x- and y-position) in the mask substrate.

For obtaining a large local pixel density, it is possible to arrange the pixels in two or more layers within the depth of the substrate. The minimum distance between two layers along the depth can be about 10 μm for CDC (Critical Dimension Correction) type pixels and approximately 25 μm for RegC (Registration Correction) type pixels.

Step c. of the method defined above can comprise determining the positions of the plurality of pixels based on the error data, the first parameters, and second parameters associated with a linear imaging transformation performed by the illumination system.

The correction of mask defects can be improved by considering and using the flexibility of an illumination system used for illuminating a wafer through a photolithographic mask.

The positions of the plurality of pixels and the second parameters can be determined in a joint optimization process.

The simultaneous variation of the positions of the plurality of pixels and the second parameters of the illumination system expand the dimension of the variation space of the optimization process compared to a step-wise or successive optimization of these parameters. The expanded solution space of a joint optimization process for the positions of the pixels and the second parameters of the illumination system results in a reduction of the remaining error of the photolithographic mask that is not attainable with two successive separate optimization processes for the determination of the positions of the plurality of pixels and the determination of the second parameters of the illumination system. Thus, the application of the defined defect correction method increases the yield of a photomask fabrication process.

The positions of the plurality of pixels and the first parameters can be determined in a joint optimization process.

Typically, the first parameters of the illumination system are predetermined parameters and are not varied when determining the positions of the pixels in order to correct the error(s) of the photolithographic mask, whereas the second parameters of the illumination system are varied in order to determine the best defect correction of the photolithographic mask. However, if the first parameters of the illumination system are not fixed by constraints of the photomask illumination process, the first parameters can be determined in a joint optimization process together with the positions of the plurality of pixels to optimize the defect correction for the photolithographic mask. Therefore, the errors of a photolithographic mask are optimally corrected for a specific illumination setting of the illumination system used for illuminating a wafer through the corrected photomask.

The positions of the plurality of pixels, the first parameters and the second parameters can be determined in a joint optimization process.

By simultaneously determining the positions of the plurality of pixels as well as the first and the second parameters of the illumination system the best error correction can be achieved for a defective photolithographic mask. This is at the expense of the flexibility of the illumination system; the corrected photolithographic mask optimally works when illuminated with the illumination system having specific sets of the first and the second parameters.

The first parameters can comprise at least one of: an on-axis illumination setting, an annular setting, a dipole setting, a quadrupole setting, a disar setting, a quasar setting, and a source mask optimization (SMO) setting.

An illumination setting of the illumination system determines the illumination or the optical radiation distribution used for imaging the pattern elements of a photomask. Thus, the illumination setting influences the number of pixels the optical radiation passes on its way through the mask substrate. Additionally, the illumination setting or an aperture of the illumination system determining the illumination influences the angle under which the pixels are illuminated, and thus has an effect on the action of the pixels in a mask substrate.

The second parameters associated with the linear imaging transformation can comprise two parameters for a photolithographic mask shift, two parameters for a photolithographic mask scaling, and two parameters for a generalized rotation of the photolithographic mask.

The illumination of the illumination system can comprise at least one aperture determining the illumination of the photolithographic mask. In the following the illumination is also called illumination setting.

Obtaining the first parameters may comprise: deconvoluting an error correction map with a deconvolution kernel comprising an illumination pixel transmission.

The illumination pixel transmission may comprise the illumination and a pixel transmission.

For the following considerations, it is assumed that a photomask has several errors or defects, as for example CD (Critical Dimension) defects. The CD defects can be summarized in a CD error map CDE(x,y). The xy-plane is a plane that is parallel to a surface of the photolithographic mask. The z-direction is a direction that is perpendicular to the surface of the photolithographic mask or that is parallel to an optical axis of the illumination system. A CD error correction map CDC(x,y) correcting the CD defects of the mask is determined from the CD error map CDE(x,y). In the following, a plurality of pixels or pixel arrangements are determined which cause a CD change ΔCD(x,y) which reproduces the error correction map CDC(x,y) at the best. Thus, CDC(x,y) or ΔCD(x,y) describe the correction to be done.

As outlined in the second section, it has been found out that the introduction of one or more pixel arrangements having a pixel density PD(x,y,z) leads to a proportional variation of the optical transmission ΔT(x,y) of the photolithographic mask when the photomask is illuminated by optical radiation at the actinic wavelength. As already indicated above, the pixels can be written everywhere within a depth d or a height of the mask depending on the error type to be corrected by the pixel writing process. For the correction of registration and/or CD errors, the pixels are typically written into half of the depth of the mask substrate to avoid a bending of the mask substrate, i.e. z=d/2. The side of the mask substrate on which the pattern elements are arranged is in following denoted the plane z=0. The pixel density PD(x,y,z) is also called writing map PD(x,y,z) in the following. It is the task to determine a writing map that generates the ΔCD(x,y) or ΔCDC(x,y).

Further, it has also been detected that a change or a variation of the CD used for correcting CD errors, i.e. ΔCD(x,y) induced by the photomask having the one or more pixel arrangements is proportional in a wafer to the pixel density PD(x,y,z):
PD(x,y,z)∝ΔT(x,y)∝ΔCD(x,y)  (1)

As already indicated above, the induced CD variation, i.e. ΔCD(x,y), is selected so that it essentially reproduces the error correction map CDC(x,y).

Based on equation (1) an equation (2) can be formulated:
ΔCD(x,y)=CDCR·ΔT(x,y)  (2)
wherein the constant of proportionality CDCR (Critical Dimension Correction Ratio) has a unit of nm per % of optical attenuation, and wherein ΔCD(x,y) describes the critical dimension variation at the mask level.

The variation of the optical transmission ΔT(x,y) results in a variation of the size of the images generated by feature elements of the mask on a wafer, precisely in a photoresist layer xW, yWarranged on the wafer, which is multiplied by the magnification or reduction factor of the projection objective used to image the pattern of feature elements of the mask onto the wafer. Typically, a reduction factor of 4 or 5 are presently used. When a reduction factor of four is assumed, this leads to the relation:
ΔCD(x,y)=4·ΔCDW(4·xW,4·yW)  (3)

In another general aspect of the invention, a feature of the method described in this application is to determine a writing map PD(x,y,z) that describes the positions of one or more arrangements of pixels that correct the one or more errors of a photolithographic mask, for example described by the CD error map CDE(x,y), when the one or more pixel arrangements are written or introduced into the substrate of a photolithographic mask.

An optical illumination Iopt′(xp,yp) determined by an aperture of the illumination system generally has two effects on the writing map PD(x,y,z) of pixels when transmitting a photolithographic mask: (I) The pixel density PD(x,y,z=0) describes the optical attenuation of the illumination Iopt′(xp,yp) when the pixels would be arranged in the same plane as the pattern elements of the mask substrate. PD(x,y,z=0) or PD(x,y) has the dimension of an aerial density or pixel per area, and z=0 is the plane of the photolithographic mask at which the pattern elements are arranged. (II) The function pixel transmission PT′(xp,yp) accounts for the angle dependence of the scattering behavior of the pixels in a mask substrate when the pixels are written somewhere in a mask substrate. The pixel transmission varies across the pupil and is independent of the illumination setting. The unit of PT′(xp,yp) is percent optical attenuation per pixel aerial density.

In the previous paragraph, the optical illumination Iopt′(xp,yp) and the pixel transmission PT′(xp,yp) are indicated in dimensionless pupil coordinates: xp=n·sin Θx, yp=n·sin Θy, wherein n is the refraction index of the mask substrate, and Θxand Θyare the angles of the illumination with respect to the optical axis or the z-axis. Θmasand Θminare fixed by the illumination setting. The optical illumination Iopt′(xp,yp) and the pixel transmission PT′(xp,yp) can be converted to mask coordinates x, y (see the description ofFIG. 7below):

Iopt⁡(x,y)=Iopt′⁡(d·tan⁡(arcsin⁢xpn),d·tan⁡(arcsin⁢ypn)),and(4)PT⁡(x,y)=PT′⁡(d·tan⁡(arcsin⁢xpn),d·tan⁡(arcsin⁢ypn)),(5)
wherein d is the height within the mask substrate as indicated above. It can be seen from equation (5) that for d=0, e.g. the pixels are arranged in the plane of the pattern elements and the pixel transmission has a constant numerical value:
PT(x,y)=PT′(0,0)=Const.  (6)

The variation of the optical transmission ΔT(x,y) caused by the writing map PD(x,y,z) can be expressed by an equation combining the quantities illumination setting, pixel transmission, and pixel density:
T(x,y)=[Iopt(x,y)·PT(x,y)]*PD(x,y,z)=IPT(x,y)*PD(x,y,z),  (7)

“*” denotes a convolution and wherein
IPT(x,y)=Iopt(x,y)·PT(x,y).  (8)

IPT is called illumination pixel transmission. This quantity describes a weighted illumination, or it can be considered as an integral of the transmitted optical intensity. It can also be considered as a convolution kernel of the illumination illuminating the pixel density PD(x,y,z). In case the illumination Ioptand the pixel transmission PT are available in form of matrices, the multiplication in equation (8) is an element-by-element multiplication: IPT(x,y)=Iopt(x,y)⊙PT(x,y). Equation (8) expresses that the optical transmission variation ΔT(x,y) is the convolution of the illumination pixel transmission with the pixel density or the writing map PD(x,y,z).

In equation (7) ΔT(x,y) can be replaced using equation (2).

Δ⁢⁢CD⁡(x,y)CDCR=Δ⁢⁢T⁡(x,y)=IPT⁡(x,y)*PD⁡(x,y,z)(9)

Equation (9) shows that the pixel density or the writing map PD(x,y,z) can be obtained by deconvoluting the convolution kernel IPT(x,y) with the optical transmission variation ΔT(x,y) or the quotient of the CD variation ΔCD(x,y) and the constant CDCR.

The convolution can be performed by Fourier transforming each quantity in equation (9) and performing a multiplication of the quantities.

FT⁡(Δ⁢⁢CDCDCR)=FT⁡(IPT)·FT⁡(PD)(10)

Thus, the pixel density PD(x,y,z) can be determined by transforming equation (10):

PD⁡(x,y,z)=FT-1(FT⁡(Δ⁢⁢CDCDCR)FT⁡(IPT))=CDCR-1·FT-1⁡(FT⁡(Δ⁢⁢CD)FT⁡(IPT))(11)
wherein FT denotes the Fourier transform of the quantity in the bracket, and FT−1denotes the inverse Fourier transform. By applying an inverse Fourier transform, the quantities in equation (11) can be retransmitted in the position-space.

To determine the writing map PD(x,y,z) without an approximation, the constant CDCR, the CD variation ΔCD(x,y), and the illumination pixel transmission or the convolution kernel IPT(x,y) have to be determined in a calibration process. Before describing the general case of a calibration process, various approximations are discussed in order to simplify the calibration process.

In order to perform a calibration process in a first approximation, a pixel density PD(x,y,z) is written into a calibration mask substrate having a low variation in the xy-plane of the mask substrate. Low variation of the pixel density PD(x,y,z) means that a variation of the pixel density in a pixel plane or xy-plane can be ignored within the illumination area of the calibration mask defined by the illumination setting.

For a constant pixel density within the area in the illumination of the calibration mask and the additional assumption that the effect of the illumination of the pixels is considered as if the pixels would be arranged in the plane of the pattern elements of a photolithographic mask (i.e. z=0), the pixel transmission has a constant numerical value, PT(x,y)=k, the convolution in equation (7) can be executed:
ΔT(x,y)=IPT(x,y)*PD(x,y)=∫IPT(x′,y′)dx′dy′·PD(x,y,z)=∫(Iopt(x′,y′)·PT(x′,y′))dx′dy′·PD(x,y,z)=k·∫Iopt(x′,y′)dx′dy′·PD(x,y,z)=C1·PD(x,y,z)  (12),
wherein C1is a constant having a dimension: [% att./(pixel/m2)]=[% att.·m2/pixel].

Performing the convolution in equation (7) is equivalent to considering the contribution of each point within the area illuminated by the illumination setting in the pixel plane of the calibration mask.

Equation (12) also follows from the relation (1). This means the relation (1) is based on the assumptions discussed above. Equation (12) allows for a simple transformation between a pixel density PD(x,y,z) and a variation of the optical transmission variation ΔT(x,y) caused by the pixel density arranged within the mask substrate. Or expressed in other words, a pixel density that is essentially uniform across an illumination setting and that is assumed to be arranged in the plane of the pattern elements enables a determination of the constant C1and allows to transform a variation of the optical 1.5 transmission ΔT(x,y) in a related or dedicated pixel density PD(x,y,z) or vice versa. Thus, based on the discussed assumptions, equation (12) allows for a calibration of a pixel writing process (see equation (13)).

Combining equations (2) and (12)
ΔCD(x,y)=CDCR·ΔT(x,y)=C1·CDCR·PD(x,y,z)
leads to the result:

PD⁡(x,y,z)=Δ⁢⁢CD⁡(x,y)C1·CDCR(13)

The product of the constants C1·CDCR has the dimension

[m2·%⁢⁢att.pixel·m%⁢⁢att.]=[m3pixel].

The assumption that the pixels are considered to be arranged in the plane of the pattern elements is now dropped. Rather, it is now assumed that the pixels are arranged somewhere in the substrate of a photolithographic mask and the angular dependence of the illumination setting is taken into account. But, in this second approximation, it is still assumed that the pixels act as perfect scattering centers, i.e. it is assumed that the pixels have a perfect spherical shape. This assumption leads to a pixel transmission PT′(xp,yp) that is constant in the pupil plane of the aperture.

According to equation (5), a constant pixel transmission in the pupil plane results in a constant pixel transmission in the layer where the pixels are arranged:

PT⁡(x,y)=PT′⁡(xp,yp)=PT′⁡(d·tan⁡(arcsin⁢xpn),d·tan⁡(arcsin⁢ypn))=C2

This assumption simplifies equation (7):
ΔT(x,y)=[Iopt(x,y)·PT(x,y)]*PD(x,y)=C2·Iopt(x,y)*PD(x,y,z).  (14)

Obtaining the first parameters may comprise: deconvoluting the error correction map with the deconvolution kernel comprising the illumination of the illumination system.

Then the pixel density or the writing map PD(x,y,z) can be determined using equation (11):

PD⁡(x,y,z)=FT-1(FT⁡(Δ⁢⁢CDCDCR)FT⁡(IPT))=FT-1(FT⁡(Δ⁢⁢CDCDCR)FT⁡(C2·Iopt))=CDCR-1·C2-1·FT-1⁡(FT⁡(Δ⁢⁢CD)FT⁡(Iopt))(15)

In this second approximation, the pixel density or the writing map PD(x,y,z) is determined from the inverse Fourier transform of the quotient of the Fourier transform of the CD variation ΔCD(x,y) and the Fourier transform of the optical intensity Iopt(x,y).

Finally, the assumption is dropped that the pixels act as perfect scattering centers. The non-perfect spherical shape lead to a scattering behavior of the pixels that depends on the angle with which optical radiation incidents on the pixels with respect to the z direction or the optical axis of the illumination system. Thus, the general case considered now takes into account an angle dependence of the scattering behavior of the pixels. It does not assume that the optical radiation transmitting one or more openings of an aperture has the same effect on the pixels of the mask irrespective of the position of the optical radiation within the opening of the aperture. Thus, this aspect considers the position of the optical radiation within one or more openings of an aperture on the effect on the pixels on its path through a photolithographic mask. The relation defined above can also be regarded as a spatial frequency dependence of a pixel response on the illumination or the optical radiation, as angles within an opening of an aperture correspond to spatial frequencies on a photolithographic mask. This effect increases when the angle of the illumination increases with respect to the z-direction, which corresponds to the optical axis of the illumination system. Therefore, there are two effects of using larger illumination settings. A first one is the extension of the illumination in the pixel level or pixel plane that increases with larger angles with respect to the optical axis. A second one is the augmentation of the variation of the pixel transmission PT(x,y) across the illumination pupil for larger angles with respect to the optical axis.

In the general case presently discussed, the equations (7) and (11) are solved without any assumptions. To obtain the writing map PD(x,y,z) for this general case, the illumination pixel transformation or the convolution kernel IPT(x,y) has to be determined without any assumption.

Combining equations (2) and (7) leads to:

Δ⁢⁢CD⁡(x,y)CDCR=Δ⁢⁢T⁡(x,y)=[Iopt⁡(x,y)·PT⁡(x,y)]*PD⁡(x,y,z)=IPT⁡(x,y)*PD⁡(x,y,z)

Determining the illumination pixel transmission may comprise: deconvoluting the pixel density with the optical transmission variation. Determining the illumination pixel transmission may also comprise: deconvoluting the pixel density with the error correction map.

This equation can be transformed:

FT⁡(Δ⁢⁢T)=FT⁡(IPT)·FT⁡(PD)⇒IPT⁡(x,y,z)=FT-1⁡(FT⁡(ΔT)FT⁡(PD))⁢⁢or⁢⁢IPT⁡(x,y,z)=[Iopt⁡(x,y)·PT⁡(x,y)]=FT-1⁡(FT⁡(Δ⁢⁢T)FT(⁢PD))=FT-1(FT⁡(Δ⁢⁢CDCDCR)FT⁡(PD))(16)

Thus, the illumination pixel transmission or the convolution kernel IPT(x,y) can be determined by writing a defined pixel density or writing map PD(x,y,z) in the mask substrate of a calibration mask and measuring the resulting CD variation. For the determination of the convolution kernel it is beneficial write a pixel density into a mask substrate that varies with a steep gradient in one or two directions.

After having fixed the convolution kernel IPT(x,y), the pixel transmission can be determined by dividing the convolution kernel IPT(x,y) by the illumination of the calibration mask Iopt(x,y) determined by the illumination setting. This leads to:

PT⁡(x,y)=IPT⁡(x,y)Iopt⁡(x,y)=FT-1⁡(FT⁡(Δ⁢⁢T)FT⁡(PD))Iopt⁡(x,y)(17)

If the quantities IPT and Ioptexist in the form of matrices, equation (17) transforms to an element-by-element division:

PT⁡(x,y)=IPT⁡(x,y)⁢∅⁢⁢Iopt⁡(x,y)=FT-1⁡(FT⁡(Δ⁢⁢T)FT⁡(PD))⁢∅⁢⁢Iopt⁡(x,y).

Determining the deconvolution kernel of the illumination may comprise: (a) generating a reference mask having a plurality of pixels arranged with a varying pixel density; (b) illuminating several wafers with the reference mask using a monopole setting, wherein each wafer is illuminated with a different angle of the monopole setting relative to an optical axis of the reference mask; and (c) determining the deconvolution kernel of the illumination from a variation of the critical dimension of the several wafers.

It is the beneficial effect of this kind of determining the deconvolution kernel of the illumination or the illumination setting that it allows to calculate the deconvolution kernel for various illumination settings. This means that one calibration process enables the determination of a deconvolution kernel for all typically used illumination settings. But, this approach requires a large number of wafer prints, and thus is an expensive approach.

A dimension of the monopole setting in a pupil plane may comprise a range of 0.1 mm to 100 mm, preferably 0.5 mm to 50 mm, more preferred 1 mm to 20 mm, and most preferred 2 mm to 10 mm. An illumination angle may vary in a range from ±10°, preferably ±15°, and most preferred ±20° with respect to an optical axis of the illumination system. A number of wafer prints with varying illumination angle may comprise a range of 2 to 100, preferable 5 to 50, and most preferred 10 to 30.

Determining the deconvolution kernel of the illumination may comprise: (a) illuminating a wafer by a photolithographic mask with an illumination of the illumination system, wherein the photolithographic mask does not have the plurality of pixels, and determining a critical dimension of the wafer; (b) writing a pixel density into the substrate of the photolithographic mask; (c) illuminating the wafer by the photolithographic mask with the illumination of the illumination system, and determining the critical dimension; (d) determining a variation of the critical dimension by subtracting the critical dimension of step (a) from the critical dimension of step (c); and (e) determining the deconvolution kernel by deconvoluting the variation of the critical dimension with the written pixel density.

This aspect has the advantageous effect that only a single wafer print is necessary for the experimental determination of the convolution kernel for a specific illumination or a specific illumination setting. The just described aspect requires one calibration measurement for each illumination setting of the illumination system. Therefore, the calibration effort is proportional to the number of used illumination settings.

In case the illumination pixel transmission IPT(x,y) and the illumination pupil Iopt(x,y) are available in the form of matrices, the optical transmission PT(x,y) can be determined by an element-by-element division of deconvolution kernel and the illumination pupil Iopt(x,y).

It is possible to replace the deconvolution operation by varying the entries of the kernel function using an algorithm in order to obtain a best match between the results of the convolution and the observed CD variation ΔCD(x,y).

Different illuminations can comprise using different illumination settings of the illumination system. Using different illumination settings may comprise using different apertures of the illumination system.

Illuminating the reference mask may be replaced by or may additionally comprise: illuminating an aerial image measurement system (AIMS™) with the reference mask using different illuminations. Illuminating the photolithographic mask may be replaced by or may additionally comprise: determining the effect of the illumination on the pixels by comparing images of the photolithographic mask taken by the aerial image measurement system prior to and after introducing the pixels of varying pixel density into the mask substrate.

The defined method can further comprise the step of determining a type of each of the plurality of pixels.

A type of a pixel is characterized by a specific set of pixel parameters. The specific set of pixel parameters determines the major action of a pixel. For example, a first type of pixel can be designed that essentially locally changes a density of the mask substrate, and thus shifts one or more pattern elements of a photolithographic mask. Further, a second type of pixel having a second set of pixel parameters may essentially locally modify the optical transmission of the mask substrate without essentially locally changing the density of the substrate material.

The term “essentially” means in the context of this application the designation of a measured variable within its error margin when using state of the art metrology tools to measure the measured variable.

The positions of the plurality of pixels, their type, and the first and/or the second parameters can be determined in a joint optimization process.

The simultaneous determination of the type and the position of the plurality of pixels in combination with some or all the parameters of the illumination system provides the prerequisite for an optimal correction of a photolithographic mask.

A joint optimization process can comprise: (a) setting up a target functional comprising error data of the photlithographic mask, position data of the plurality of pixels, and the first parameters of the illumination system; and (b) minimizing the target functional by simultaneously varying the position data of the plurality of pixels.

The target functional can additionally comprise the type of the pixel and/or the second parameters of the illumination system. Further, the target functional can comprise error data of the photolithographic mask, position data of the plurality of pixels, the type of the pixel, the first parameters and/or the second parameters. Minimizing the target functional can comprise simultaneously varying the position data of the plurality of pixels and the type of the pixel. Moreover, minimizing the target functional can comprise minimizing the position data of the plurality of pixels, the type of the pixel, and the second parameters. Finally, minimizing the target functional can comprise minimizing the position data of the plurality of pixels, the type of the pixel, the first parameters, and/or the second parameters.

A Lagrange variational principle can be used to set up the target functional. Details of setting up a target functional and minimizing the target functional are described in the U.S. Pat. No. 9,658,527 B2 of the applicant.

The photolithographic mask may comprise a template for the nanoimprint lithography, or the photolithographic mask may comprise a transmissive optical element.

The defined method is not restricted to photolithographic mask. Rather, it can be applied to all transmissive optical elements that are to be corrected by introducing a plurality of pixels. In particular, the inventive method is important if the corrected transmissive optical element is operated with illumination settings having an off-axis optical intensity distribution.

The defined method can further comprise the step of determining laser beam parameters of the laser system based on the determined positions and the type of each of the plurality of pixels.

As discussed above, each type of pixel is associated with a specific set of pixel parameters. In order to write or to introduce a specific type of pixel into a substrate of a photolithographic mask a laser beam is necessary having a specific set of laser beam parameters associated with the respective type of pixel, i.e. the respective set of pixel parameters.

The laser beam parameters can comprise at least one of: an energy of the laser beam, a numerical aperture, a focus size, a beam polarization, a wave front shape, for example an astigmatism, a pulse length, a repetition rate, a number of pulses directed into one location of the substrate of the photolithographic mask, and a distance between two locations the laser beam is directed into the substrate of the photolithographic mask.

The types of each of the plurality of pixels can comprise at least a first type of pixel that is adapted to at least partly correct at least one of: a registration error, a planarity error and an overlay error, and at least one second type of pixel adapted to at least partly correct an optical transmission error of the substrate.

Laser beam parameters for introducing a first type of pixel may comprise: a pulse energy of 0.05 μJ to 5 μJ, a pulse length of 0.05 ps to 100 ps, a repetition rate of 1 kHz to 10 MHz, a pulse density of 1 000 pulses per mm2to 10 000 000 pulses per mm2, a NA of an objective of 0.1 to 0.9, and a magnification of the objective of 5× to 40×. Laser beam parameter for introducing a second type of pixels may comprise: a pulse energy of 0.45 μJ-0.55 μJ, a pulse duration of 5-10 ps, a repetition rate of 10 kHz-100 kHz, a NA of the objective of 0.3-0.4, a magnification of the objective of 10×-20×, and a pulse density of 1 000-100 000 pulses per mm2.

The defined method can further comprise the step of generating a writing map based on the positions and the type of the determined pixels, wherein the writing map may describe a distribution of the plurality of pixels to be introduced in the substrate of the photolithographic mask.

The establishment of a writing map PD(x,y,z) may be the result or the output of the method for determining the positions of the plurality of pixels. The writing map comprises the positions, the type of pixels to be introduced into the substrate of a defective mask and the depth into which the pixels are written into the mask substrate.

Thus, the writing map defines the parameters of the laser beam used for introducing the plurality of pixels into the mask substrate.

The method defined above may further comprise the step of introducing the plurality of pixels into the substrate of the photolithographic mask using ultra-short laser pulses of the laser system.

Introducing the plurality of pixel into the substrate may be based on the writing map.

The writing map may comprise a first plurality of pixel arrangements of the first type or kind of pixel and a second plurality of pixel arrangements of the second type of pixel.

A pixel arrangement comprises a plurality of pixels arranged in a constant density, i.e. the distance of identical pixels in a pixel arrangement is equidistant in one, two or three 1.5 dimensions. A pixel arrangement typically contains pixels of a single pixel type. For example, a pixel arrangement either comprises a first plurality of pixels of the first pixel type or a second plurality of pixels of the second pixel type.

A writing map PD(x,y,z) may comprise pixels of two or more pixel types. Thus, a single writing map may contain the overall information to correct the one or more errors of a photolithographic mask. It is also possible to establish a separate writing map for each type of pixels. For example, the density of the substrate of the photolithographic mask can specifically be modified by locally introducing pixels into the mask substrate according to first writing map having pixels or pixel arrangements of a first pixel type, and the optical transmission distribution of the mask can separately be modified continuously or discontinuously by using a second writing map describing a plurality of pixels of the second pixel type or a plurality of second pixel arrangements having pixels of the second pixel type. Thus, the correction of registration errors and the correction of errors of the optical transmission distribution of a photolithographic mask can also be separated.

The writing map may compensate an averaging effect of an illumination of the photolithographic mask by the illumination system having the first and/or the second parameters on the plurality of pixels when processing the wafer.

Compared to the prior art, the correction method described in a present application determines a modified writing map that takes into account the optical radiation distribution the corrected photomask is afterwards subjected to when determining the distribution of the plurality of pixels that serve for correcting the error(s) of the photolithographic mask.

The defined method may further comprise the step of increasing a density of pixel arrangements in the writing map and/or reducing the pixels density within the pixel arrangements in the writing map for compensating the averaging effect of the illumination of the illumination system. By increasing the density of the pixel arrangements in the writing map higher spatial frequency components are considered in the writing map. These higher frequency components lead to a locally increased or reduced pixel density.

A computer program may comprise instructions for causing a computer system to perform the steps of any of the aspects described above.

In a further embodiment, the above-mentioned problem is solved by an apparatus for determining positions of a plurality of pixels to be introduced into a substrate of a photolithographic mask by use of a laser system, wherein the pixels serve to at least partly correct one or more errors of the photolithographic mask. The apparatus comprises: (a) means for obtaining error data associated with the one or more errors; (b) means for obtaining first parameters of an illumination system, the first parameters determining an illumination of the photolithographic mask by the illumination system when processing a wafer by illuminating with the illumination system using the photolithographic mask; and (c) means for determining the positions of the plurality of pixels based on the error data and the first parameters.

The apparatus may further comprise the laser system adapted to generate ultra-short laser pulses.

The means for obtaining error data may comprise means for measuring the error data of the photolithographic mask.

The defined apparatus may further comprise means for determining an optical transmission variation across the photolithographic mask.

The means for determining the one or more registration errors of a pattern may comprise a PROVE® tool.

The illumination system may comprise a scanner or a stepper of a photolithographic exposure system.

Finally, the above defined apparatus may be adapted to execute the steps of any of the above described aspects.

