DE102022212238A1 - Method for characterizing the dynamic transfer behavior of a system, especially for microlithography - Google Patents

Abstract

The invention relates to a method for characterizing the dynamic transmission behavior of a system, in particular for microlithography, the method having the following steps: determining an amplitude response as a frequency dependency of the amplitude of a transmission function of the system, determining, for this amplitude response, the third derivative of the amplitude according to the frequency, and characterizing the dynamic transmission behavior of the system by determining at least one natural frequency (f0) and at least one damping (D) on the basis of this third derivative.

Description

BACKGROUND OF THE INVENTION

field of invention

The invention relates to a method for characterizing the dynamic transmission behavior of a system, in particular for microlithography.

State of the art

Microlithography is used to produce microstructured components such as integrated circuits or LCDs. The microlithographic process is carried out in a so-called projection exposure system, which has an illumination device and a projection lens. The image of a mask (= reticle) illuminated by means of the illumination device is projected by means of the projection objective onto a substrate (e.g. a silicon wafer) coated with a light-sensitive layer (photoresist) and arranged in the image plane of the projection objective in order to project the mask structure onto the light-sensitive coating of the to transfer substrate.

In a projection exposure system designed for EUV (ie for electromagnetic radiation with a wavelength below 15 nm), mirrors are used as optical components for the imaging process due to the lack of presence of transparent materials. These mirrors can be attached to a support frame and designed to be at least partially manipulable in order to allow the respective mirror to move, for example, in six degrees of freedom (ie with regard to displacements in the three spatial directions x, y and z and with regard to rotations R _x , R _y and R _z around the corresponding axes) to allow.

In the operation of projection exposure systems, especially in EUV systems, dynamic aspects play an increasingly important role for the optical performance of the system. Mechanical disturbances caused by vibrations adversely affect the positional stability of the optical components. Weakly damped mechanical resonances in the system lead to a local increase in the interference spectrum in the range of resonance frequencies and an associated deterioration in the position stability of passively mounted components as well as actively controlled components. Furthermore, in the case of controlled systems, resonances can lead to instability of the control loop.

In practice, the mechatronic design of a complex mechatronic system such as an optical system of a projection exposure system is based on calculated values of natural frequencies and associated damping (as well as assumed tolerances). It is important for the proper operation of the system in question to verify the essential natural frequencies and associated damping in reality and, if necessary (e.g. in the case of significant deviations from the nominal values), to calibrate control parameters, for example. Furthermore, the determination of the natural frequencies and damping in the relevant system can also be used for error analysis and monitoring, with changes in a manufacturing or assembly process being detected if necessary and corresponding errors being able to be predicted at an early stage in subsequent process steps.

Various approaches are known for determining natural frequencies and associated damping from a measured transfer function or a frequency response of a system. In particular, both the amplitude response (i.e. the frequency dependency of the amplitude of the transfer function) and the respective phase position can be taken into account. Although such a procedure has fundamental advantages in terms of using or identifying the complete information from the complex-valued transfer function, in practice problems can arise from the resulting complexity and in particular from unavoidable changes in the phase position caused by delays that occur.

$$\widehat{D}=\frac{1}{2Q}$$

A comparatively simpler method based solely on the use of the amplitude response is the so-called Q-factor method. A Q-factor, also referred to as a "quality factor", and a damping D are determined using the following equations (1) and (2). , where the frequency f _max denotes that frequency value at which the amplitude assumes its maximum and where f _l and f _r denote the respective frequency values at which the amplitude reaches the value A _max /√2 on both sides of the amplitude maximum A _max (see exemplary scenario according to 7a-7b) . $$Q=\frac{{ƒ}_{max}}{{ƒ}_{\mathrm{right}}-{ƒ}_l}$$

$$\widehat{D}=\frac{1}{2Q}$$

Although the above-mentioned Q-factor method is in good agreement in simple systems This method proves to be faulty in more complex systems and especially when two or more resonances are relatively close together in the amplitude response (ie occur at frequencies that differ only slightly from one another). In this case, for example, like from 8a-8b As can be seen from an exemplary scenario, the case can arise that a single resonance is erroneously assumed and, accordingly, a single erroneous damping value is also calculated.

The prior art is only given as an example U.S. 5,960,091 referred.

SUMMARY OF THE INVENTION

The object of the present invention is to provide a method for characterizing the dynamic transmission behavior of a system, in particular for microlithography, which enables reliable characterization while at least partially avoiding the problems described above.

This object is solved according to the features of independent patent claim 1 .

1. a) Bestimmen eines Amplitudengangs als Frequenzabhängigkeit der Amplitude einer Übertragungsfunktion des Systems;
2. b) Bestimmen, für diesen Amplitudengang, der dritten Ableitung der Amplitude nach der Frequenz; und
3. c) Charakterisieren des dynamischen Übertragungsverhaltens des Systems durch Bestimmen wenigstens einer Eigenfrequenz (f₀) sowie wenigstens einer Dämpfung (D) auf Basis dieser dritten Ableitung.

A method according to the invention for characterizing the dynamic transmission behavior of a system, in particular for microlithography, has the following steps:

1. a) determining an amplitude response as a frequency dependency of the amplitude of a transfer function of the system;
2. b) determining, for this amplitude response, the third derivative of amplitude with respect to frequency; and
3. c) Characterizing the dynamic transmission behavior of the system by determining at least one natural frequency (f ₀ ) and at least one damping (D) on the basis of this third derivative.

The invention is based in particular on the concept of characterizing the transmission behavior of a mechatronic system (e.g. an optical system of a microlithographic projection exposure system) in the sense of identifying the existing values of natural frequency(s) and damping(s), in particular also for verification of this previously set values and, if necessary, calibrating the system with regard to any deviations that are present, by determining the frequency dependency of the amplitude (hereinafter also referred to as "amplitude response") in the transfer function of the system, determining the third derivative of the amplitude according to the frequency and then using this suitable criteria for a quantitative determination of said natural frequency(s) or damping(s).

The invention is based, among other things, on the finding that in said third derivation of the amplitude response, a resonance present in the transmission behavior generates a characteristic pattern that is comparatively easy to detect, from which both natural frequency(s) and damping(s) can be derived even with high or can be determined with sufficient accuracy when there are two or more resonances at relatively close frequencies.

The quantitative determination described in more detail below, based solely on the amplitude response or its third derivative, has the advantage over a complete use of both the amplitude and the phase information of the transmission behavior that problems due to an unavoidable (e.g. due to delays that occur) Changing the phase position can be avoided. Furthermore, the quantitative determination of natural frequency(s) and damping(s) according to the invention can be carried out automatically in a clearly reproducible manner and free from subjective error influences. A further advantage of the method according to the invention is that additional knowledge or information relating to the system to be characterized with regard to its transmission behavior is not required.

According to one embodiment, at least one zero point (f _z ) through which this function has a positive gradient is determined in step c) for a function describing the frequency dependency of the third derivative.

According to one embodiment, the at least one natural frequency is determined on the basis of this at least one zero point (f _z ).

According to one embodiment, in step c) for a function describing the frequency dependency of the third derivative, a first frequency value, which is below this natural frequency and at which the function has a local minimum, and a second frequency value, which is above this natural frequency and at which the Function has a local maximum determined.

According to one embodiment, the at least one attenuation is based on the difference (Δf) between the first frequency value and the second frequency value is determined.

According to one embodiment, a plurality of natural frequencies (f _0,i ) and associated damping (D _i ) are determined in step c).

According to one embodiment, the respective function describing the frequency dependency of the third derivative of the amplitude according to the frequency is adapted in this determination in an iterative process after determining a respective natural frequency (f _0,i ) and associated damping (D _i ) and determining a further one Natural frequency (f _0,i+1 ) and damping (D _i+1 ) are taken as a basis.

According to one embodiment, the system is an optical system, in particular of a microlithographic projection exposure system.

According to one embodiment, the optical system is designed for operation in EUV.

Further configurations of the invention can be found in the description and in the dependent claims.

The invention is explained in more detail below with reference to exemplary embodiments illustrated in the attached figures.

character list

• 1a-2 Diagramme zur Erläuterung des der vorliegenden Erfindung zugrundeliegenden Konzepts;
• 3 ein Flussdiagramm zur Erläuterung einer möglichen Ausführungsform des erfindungsgemäßen Verfahrens;
• 4a-4c Diagramme zur Verdeutlichung der Leistungsfähigkeit des erfindungsgemäßen Verfahrens;
• 5-6 Diagramme zur Erläuterung einer weiteren möglichen Ausgestaltung des erfindungsgemäßen Verfahrens;
• 7a-8b Diagramme zur Erläuterung eines herkömmlichen Verfahrens zur Charakterisierung des dynamischen Übertragungsverhaltens eines Systems nach der Q-Faktor-Methode; und
• 9 eine schematische Darstellung einer Projektionsbelichtungsanlage für die EUV-Projektionslithographie im Meridionalschnitt.

Show it:

• 1a-2 Diagrams for explaining the concept underlying the present invention;
• 3 a flowchart to explain a possible embodiment of the method according to the invention;
• 4a-4c Diagrams to illustrate the performance of the method according to the invention;
• 5-6 Diagrams to explain a further possible embodiment of the method according to the invention;
• 7a-8b Diagrams explaining a conventional method for characterizing the dynamic transmission behavior of a system using the Q-factor method; and
• 9 a schematic representation of a projection exposure system for EUV projection lithography in the meridional section.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

A possible course of a method according to the invention for characterizing the dynamic transmission behavior of a system, such as an optical system for microlithography, is described below using exemplary embodiments and with reference to in 3 shown flowchart and the other diagrams according to 1a-1c , 2 , 4a-4c and 5-6 explained.

1a shows the respective amplitude response as a frequency dependency of the amplitude for an exemplary second-order system with a natural frequency f ₀ = 200 Hz with varying damping (with values between D = 0.02 and D = 2). This amplitude response is according to the flow chart of 3 after the start of the method according to the invention (step S10), it is first determined or measured in a step S20, with optional filtering being able to be carried out (in particular in the case of a significant noise component).

Then, in step S30, the third derivation of the amplitude according to the frequency is determined for this amplitude response. 1b shows the course of this third derivation for the above mentioned damping values. Here are from the diagram of 1b characteristic properties of this course can be seen:

On the one hand, the third derivative at the amplitude maximum has a single zero point with a positive slope (hereinafter referred to as f _z ), which migrates to lower frequencies with increasing damping. The relationship between natural frequency f ₀ , zero point f _z and damping D is given by: $${ƒ}_{\mathrm{e.g}}={\mathrm{<figure-callout\; id="0"\; label="ƒ"\; filenames="DE102022212238A1\_0007.png,DE102022212238A1\_0008.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_0\sqrt{1-2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="DE102022212238A1\_0010.png,DE102022212238A1\_0012.png"\; state="\{\{state\}\}">D</figure-callout>}}^2}$$

Since the remaining zeros of the third derivative are traversed with a negative gradient, the natural frequency f ₀ can be determined unambiguously on the basis of this zero point f _z , which is traversed with a positive gradient.

If one also considers the difference Δf between the respective frequency values below and above said natural frequency f ₀ , for which the third derivation according to 1b has an absolute minimum or an absolute maximum, this difference Δf according to 1c linearly dependent on the damping D (so that, for example, if the damping D is tripled, the differential frequency Δf also has a triple value). Furthermore, it is also evident that the slope of the describes the above linear dependency ends straight lines in the diagram of 1c in turn is linearly dependent on the natural frequency f ₀ . In other words, said differential frequency Δf increases with a change in damping D as from 2 As can be seen, the greater the natural frequency f ₀ is, the stronger it is, with this relationship also being linear.

Overall, this results in the following bilinear relationship: $$\Delta ƒ\left({\mathrm{<figure-callout\; id="0"\; label="ƒ"\; filenames="DE102022212238A1\_0007.png,DE102022212238A1\_0008.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_0,D\right)=\frac{1}{k}{ƒ}_{0}D$$

$$D=\frac{k}{{ƒ}_0}\Delta ƒ\frac{1}{\sqrt{1+\frac{2k^2}{{ƒ}_z^2}\Delta {ƒ}^2}}\approx \frac{k}{{ƒ}_z}\Delta ƒ$$

The constant k can be calculated from any non-zero reference value. The desired value of the damping D can thus be calculated for a given profile of the third derivative of the amplitude response according to equations (5) and (6) below. $$D=\frac{\mathrm{<figure-callout\; id="0"\; label="k\; ƒ"\; filenames="DE102022212238A1\_0007.png,DE102022212238A1\_0008.png"\; state="\{\{state\}\}">k</figure-callout>}}{{ƒ}_{}}\Delta ƒ$$

$$D=\frac{\mathrm{<figure-callout\; id="0"\; label="k\; ƒ"\; filenames="DE102022212238A1\_0007.png,DE102022212238A1\_0008.png"\; state="\{\{state\}\}">k</figure-callout>}}{{ƒ}_{}}\mathrm{<figure-callout\; id="1"\; label="\Delta \; ƒ"\; filenames="DE102022212238A1\_0007.png,DE102022212238A1\_0010.png"\; state="\{\{state\}\}">\Delta </figure-callout>}ƒ\frac{1}{\sqrt{1+\frac{2{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102022212238A1\_0010.png,DE102022212238A1\_0012.png"\; state="\{\{state\}\}">k</figure-callout>}}^2}{{ƒ}_{\mathrm{e.g}}^2}\mathrm{<figure-callout\; id="2"\; label="\Delta \; ƒ"\; filenames="DE102022212238A1\_0010.png,DE102022212238A1\_0012.png"\; state="\{\{state\}\}">\Delta </figure-callout>}{ƒ}^{}{}^2}}\approx \frac{k}{{ƒ}_{\mathrm{e.g}}}\Delta ƒ$$

The above-described determination of the natural frequency and damping for a resonance occurring in the transfer function or in the associated amplitude response using the described analysis of the frequency dependency of the third derivative of this amplitude response can, according to the invention, generally be carried out with sufficient accuracy even if there are several resonances (possibly also with comparatively slightly different frequency values). Referring again to 3 Therefore, in step S40, all zeros traversed with a positive slope are generally determined for the respective third derivation, the number of zeros being denoted here by N and correspondingly an index denoted by i being passed from i=1 to i=N below. In each case, the above-mentioned differential frequency is determined in step S60 and the damping and natural frequency based thereon are calculated in step S70.

4a-c show diagrams to illustrate the performance of the method according to the invention. It shows first 4a a measured amplitude response, which has a first resonance with a natural frequency of 200 Hz and damping of 3% and a second resonance with a natural frequency of 225 Hz and damping of 5%. According to 4b As already explained in the introduction, the conventional Q-factor method incorrectly delivers a single attenuation value of 7.67%. On the other hand, based on the in 4c Determination of the third derivative shown near the frequency for the amplitude response of 4a the two existing resonances are identified with high accuracy, whereby in the present example a natural frequency of 200.8 Hz with a damping of 3% is determined for the first resonance and a natural frequency of 224.3 Hz with a damping of 4.97% is determined for the second resonance.

Optionally, in further embodiments, an additional increase in accuracy can be achieved by adapting the third derivative of the amplitude response in an iterative process after each determination of a respective natural frequency and associated damping. This procedure, which can be carried out iteratively to increase accuracy, is in 3 represented by the step S120 comprising the sub-steps S121-S123. In each case, a correspondingly adapted curve is used as a basis for determining the next natural frequency and damping (corresponding to the next resonance that occurs). For this purpose, after determining a natural frequency and damping according to 5 used to calculate a corresponding frequency-dependent course of the third derivative of the amplitude response for a simplified system (corresponding to a "one-mass vibration system") (step S121), with the correspondingly simplified calculated curve before determining the next natural frequency and damping (corresponding to the next occurring resonance) is first subtracted from the original third derivative (step S122). In this case, the derivation calculated for the one-mass vibration system is suitably scaled before said subtraction. The scaling (with factor V) is based on the nearest local maximum of the (already adapted) third derivative of the amplitude response, denoted by A _max,i , above the associated zero point f _z,i . Furthermore, the zeros previously determined in step S40 are adjusted accordingly based on the result of step S120 (step S123).

As a result, with the method according to the invention, a highly accurate identification of the natural frequencies and damping present in a given system can be achieved even without more detailed knowledge of the system (in particular with regard to the number of resonances present) and in particular without taking into account the phase position, which avoids the problems described at the beginning become.

9 shows schematically in a meridional section a projection exposure system 1 for the EUV Projection lithography as an example of an optical system in which the invention can be advantageously implemented.

According to 9 the projection exposure system 1 has an illumination device 2 and a projection lens 10 . The illumination device 2 serves to illuminate an object field 5 in an object plane 6 with radiation from a radiation source 3 via an illumination optics 4 . In this case, a reticle 7 arranged in the object field 5 is exposed. The reticle 7 is held by a reticle holder 8 . The reticle holder 8 can be displaced in particular in a scanning direction via a reticle displacement drive 9 . In 9 a Cartesian xyz coordinate system is drawn in for explanation. The x-direction runs perpendicular to the plane of the drawing. The y-direction is horizontal and the z-direction is vertical. The scanning direction is in the 1 along the y-direction. The z-direction runs perpendicular to the object plane 6.

The projection lens 10 is used to image the object field 5 in an image field 11 in an image plane 12. A structure on the reticle 7 is imaged on a light-sensitive layer of a wafer 13 arranged in the region of the image field 11 in the image plane 12. The wafer 13 is Wafer holder 14 held. The wafer holder 14 can be displaced in particular along the y-direction via a wafer displacement drive 15 . The displacement of the reticle 7 via the reticle displacement drive 9 on the one hand and the wafer 13 on the other hand via the wafer displacement drive 15 can be synchronized with one another.

The radiation source 3 is an EUV radiation source. The radiation source 3 emits in particular EUV radiation, which is also referred to below as useful radiation or illumination radiation. The useful radiation has in particular a wavelength in the range between 5 nm and 30 nm. The radiation source 3 can be a plasma source, for example an LPP (laser produced plasma) source or a DPP (Gas Discharged Produced Plasma) source. It can also be a synchrotron-based radiation source. The radiation source 3 can also be a free-electron laser (“free-electron laser”, FEL). The illumination radiation 16 emanating from the radiation source 3 is bundled by a collector 17 . After the collector 17, the illumination radiation 16 propagates through an intermediate focus in an intermediate focal plane 18. The intermediate focal plane 18 can represent a separation between a radiation source module, comprising the radiation source 3 and the collector 17, and the illumination optics 4.

The illumination optics 4 comprises a deflection mirror 19 and a first facet mirror 20 downstream of this in the beam path. If the first facet mirror 20 is arranged in a plane of the illumination optics 4 which is optically conjugate to the object plane 6 as the field plane, this is also referred to as a field facet mirror. The first facet mirror 20 comprises a large number of individual first facets 21, of which 1 only a few are shown as examples. A second facet mirror 22 is arranged downstream of the first facet mirror 20 in the beam path of the illumination optics 4. If the second facet mirror 22 is arranged in a pupil plane of the illumination optics 4, it is also referred to as a pupil facet mirror. The second facet mirror 22 comprises a plurality of second facets 23, which are also different from those in 1 only a few are shown as examples.

The projection objective 10 includes a plurality of mirrors Mi, which are numbered consecutively according to their arrangement in the beam path of the projection exposure system 1 . At the in the 9 example shown, the projection lens 10 comprises six mirrors M1 to M6. Alternatives with four, eight, ten, twelve or another number of mirrors Mi are also possible. The penultimate mirror M5 and the last mirror M6 each have a passage opening for the illumination radiation 16. The projection objective 10 is a doubly obscured optical system. The projection objective 10 has a numerical aperture on the image side which is greater than 0.5 and which can also be greater than 0.6 and which can be 0.7 or 0.75, for example.

Reflection surfaces of the mirrors Mi can be designed as free-form surfaces without an axis of rotational symmetry. Alternatively, the reflection surfaces of the mirrors Mi can be designed as aspherical surfaces with exactly one axis of rotational symmetry of the reflection surface shape. Just like the mirrors of the illumination optics 4, the mirrors Mi can have highly reflective coatings for the illumination radiation 16. These coatings can be in the form of multilayer coatings, in particular with alternating layers of molybdenum and silicon.

The projection lens 10 can in particular be anamorphic. In particular, it has different imaging scales β _x , β _y in the x and y directions. The two image scales β _x , β _y of the projection objective 10 are preferably at (β _x , β _y )=(±0.25, ±/0.125). A positive image scale β means an image tion without image reversal. A negative sign for the imaging scale β means imaging with image inversion. The projection lens 10 thus leads to a reduction in the ratio 4:1 in the x-direction, that is to say in the direction perpendicular to the scanning direction. The projection objective 10 leads to a reduction of 8:1 in the y-direction, ie in the scanning direction. Other imaging scales are also possible. Image scales with the same sign and absolutely the same in the x and y directions, for example with absolute values of 0.125 or 0.25, are also possible.

Although the invention has also been described on the basis of specific embodiments, numerous variations and alternative embodiments will become apparent to the person skilled in the art, e.g. by combining and/or exchanging features of individual embodiments. Accordingly, it will be understood by those skilled in the art that such variations and alternative embodiments are intended to be encompassed by the present invention, and the scope of the invention is limited only in terms of the appended claims and their equivalents.

QUOTES INCLUDED IN DESCRIPTION

This list of documents cited by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Patent Literature Cited

• US5960091 [0009]

Claims (9)

Method for characterizing the dynamic transfer behavior of a system, in particular for microlithography, the method having the following steps: a) determining an amplitude response as a frequency dependency of the amplitude of a transfer function of the system; b) determining, for this amplitude response, the third derivative of amplitude with respect to frequency; and c) characterizing the dynamic transmission behavior of the system by determining at least one natural frequency (f ₀ ) and at least one damping (D) on the basis of this third derivative. procedure after claim 1 , characterized in that in step c) at least one zero point (f _z ) through which this function passes with a positive slope is determined for a function describing the frequency dependency of the third derivative. procedure after claim 1 or 2 , characterized in that the at least one natural frequency is determined on the basis of this at least one zero point (f _z ). procedure after claim 3 , characterized in that in step c) for a function describing the frequency dependency of the third derivative, a first frequency value which is below this natural frequency and at which the function has a minimum, and a second frequency value which is above this natural frequency and at which the Function has a maximum can be determined. procedure after claim 4 , characterized in that the at least one attenuation is determined on the basis of the difference (Δf) between the first frequency value and the second frequency value. Method according to one of the preceding claims, characterized in that in step c) a plurality of natural frequencies (f ₀ ) and associated damping (D) are determined. procedure after claim 6 , characterized in that in this determination in an iterative process, the respective function describing the frequency dependence of the third derivative of the amplitude according to the frequency is adapted after determining a natural frequency (f _0,i ) and associated damping (D _i ) and determining a further natural frequency (f _0,i+1 ) and damping (D _i+1 ) is taken as a basis. Method according to one of the preceding claims, characterized in that the system is an optical system, in particular a microlithographic projection exposure system. procedure after claim 8 , characterized in that the optical system is designed for operation in the EUV.

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