Method of generating projection pattern of plurality of projections of substrate table

Abstract

A pattern of protrusions in a surface portion of a substrate table for supporting a wafer is generated by: providing a repelling perimeter potential acting in a support plane and including an outer perimeter potential assigned to an outer perimeter point surrounding the surface portion; providing an initial distribution of protrusions, each protrusion having a variable protrusion position in the surface portion; providing a repelling projection potential which acts in the support plane and is assigned to the projection; calculating a local force acting on each of the protrusions based on a superposition of the perimeter potential at the location of the protrusions and the protrusion potential; and generating the pattern of protrusions to be acquired by repeatedly shifting the positions of the protrusions and calculating a local force acting on each of the protrusions at the current position until a predetermined optimization criterion is satisfied. In addition, a substrate table and manufacturing of a substrate table are described.

Description

[0001] Cross - reference to related applications

[0002] This application claims priority to European Application No. 22216358.6, filed on Dec. 23, 2022, the entire content of which is incorporated herein by reference. TECHNICAL FIELD

[0003] The present invention relates to a method for generating a protrusion pattern for a plurality of protrusions of a substrate table for supporting a substrate (such as a semiconductor wafer), in particular a method for determining the protrusion positions of the plurality of protrusions, and particularly to a method for designing a protrusion arrangement of a substrate table to be manufactured. Furthermore, the present invention relates to a method for manufacturing a substrate table and a substrate table. The application of the present invention can be used, for example, in the design of substrate tables (such as electrostatic chucks or vacuum chucks). BACKGROUND ART

[0004] It is well known that a substrate table (which is also referred to as a clamping device, a chuck, a platen, a substrate support, a substrate holder, a wafer table, or a wafer plate) has a plate-shaped base body for receiving a substrate. The plate-shaped base body can be configured as a single plate or can be constructed from a plurality of plates clamped together, and protruding protrusions (also referred to as pins or bosses) are provided on at least one surface. The planar front surface (or: upper surface, end face) of the protrusions forms a support plane for receiving the substrate. The front surface defines the support plane for the component to be clamped.

[0005] When operating the substrate table, the substrate is pressed against the support plane under the action of a holding force. Depending on the type of clamping force used, the chuck can be configured, for example, as an electrostatic chuck (ESC) or a vacuum chuck. Providing a support plane with high flatness and mechanical stability is crucial for the application of the chuck, for example, when processing semiconductor wafers in a lithography process. The front surfaces of the protrusions must be aligned, and an arrangement of protrusions with high uniformity is required.

[0006] The arrangement of the protrusions (also referred to as a protrusion pattern or a protrusion map) is restricted by the outer perimeter (also referred to as the outer contour) of the surrounding surface portion in which the substrate will be held. Furthermore, the protrusion pattern can include at least one interruption (a portion of the surface portion without protrusions) generated by at least one functional element (also referred to as a feature) of the substrate table within the surface portion. Depending on the type of functional element, the interruption can be a 0-dimensional gap (point-like), a 1-dimensional gap (line-like), or a 2-dimensional gap (patch-like or area-like). Each functional element forms an inner perimeter (also referred to as an inner contour) of the surface portion.

[0007] It is challenging to provide a protrusion pattern with a uniform distribution of protrusions along the outer perimeter and optionally at least one inner perimeter of a surface portion that accommodates a substrate. In particular, generating a protrusion pattern for a fixture (e.g., for an ESC or a wafer plate) with optimized flatness involves two problems: The first challenge is to determine a locally optimized protrusion pattern for flatness along portions of the outer perimeter with complex geometries, such as corners or transitions from chamfered portions of the perimeter to straight portions, and / or to determine a locally optimized protrusion pattern for flatness around functional elements, such as e-pins (or: lift pin holes) or notches. This is typically achieved via finite element analysis and numerical optimization. However, due to the complexity of the task, this is limited to as small an area as possible (limited to, for example, the immediate area around the functional elements). The second challenge is to integrate the (multiple) individual local protrusion patterns along the (multiple) portions of the outer perimeter with complex geometries and / or around the (multiple) functional elements into an overall protrusion pattern for the entire surface portion and to adapt these patterns to each other. Based on conventional methods, this is extremely difficult or even impossible, especially for closely spaced features.

[0008] In conventional practice, the first method for generating a protrusion pattern involves manually arranging the protrusions based on the designer's visual considerations and intuition. This concept has significant drawbacks in terms of the time required and limited reliability. Secondly, a known software-based method for designing a protrusion arrangement includes the following steps: creating an initial protrusion pattern with a predefined protrusion pitch (usually a circular pattern, using a scripting language), which only includes the outer perimeter and can divide the surface portion into two electrode segments, subsequently mainly manually cutting the positions of the functional elements, such as at or within the cores of non-nuclear features (e.g., e-pins), and finally fitting the areas between the local patterns at the functional elements to the overall protrusion pattern (via scripting).

[0009] As a major drawback, fitting the (multiple) local patterns is very complex and time-consuming, for example, up to a month, especially in cases where there are multiple functional elements. In particular, the spatial proximity of the functional elements quickly reaches the limit of this method because each local pattern is generated independently and there must be sufficient space for the transition regions. As another limitation, fitting the local patterns is limited to using a given initial protrusion pattern, which may not be optimized for the given geometries and arrangements of the functional elements and / or the outer perimeter.

[0010] Object of the present invention

[0011] The object of the present invention is to provide an improved method for generating a pattern of protrusions for a substrate table for supporting a substrate (such as a semiconductor wafer), wherein the method is capable of avoiding the limitations or drawbacks of conventional techniques. In particular, the method is capable of generating a pattern of protrusions in a substrate table to be manufactured and has improved uniformity, less time consumption, an improved ability to adapt the arrangement of the protrusions to the geometry of the substrate table to be manufactured and its (optional) functional elements, and / or a reduced or even no dependence on the designer's manual experience-based design decisions. Another object of the present invention is to provide an improved method for manufacturing a substrate table for holding a substrate (in particular a semiconductor wafer) and / or an improved substrate table capable of avoiding the limitations or drawbacks of conventional techniques. Summary of the Invention

[0012] These objects are solved by a method for generating a pattern of protrusions of a substrate table, a method for manufacturing a substrate table, and / or a substrate table, which methods and substrate table include the features of the independent claims. Advantageous embodiments and applications of the present invention are defined in the dependent claims.

[0013] According to a first general aspect of the present invention, the above object is solved by a method for generating a pattern of protrusions of a substrate table configured to support a substrate, in particular a semiconductor wafer, in a surface portion of the substrate table, wherein the end face of the protrusion defines a support plane (or: holding plane, plane of the surface portion), and the surface portion is surrounded by a plurality of outer perimeter points in the support plane.

[0014] The protrusions include protrusions on the surface of the substrate table. The protrusions have a planar end face, for example, a cylindrical or cubic shape. Preferably, all the protrusions have the same shape and size. Compared with the planar dimensions of the surface portion, the cross-sectional dimensions (e.g., its diameter) of the protrusions are preferably small enough to be negligible. The protrusions can be regarded as points or tips, and the position of each protrusion is provided by the position of the center of the protrusion cross-section in the surface portion. These protrusions are arranged at intervals from each other. Generating the protrusion pattern (pattern of protrusion positions) particularly includes determining the protrusion positions for the arrangement of the protrusions of the substrate table to be manufactured. The surface portion includes a part of the substrate table surface in which the substrate will be supported by the protrusions.

[0015] The outer perimeter points, in particular their positions, are input parameters of the method for generating the protrusion pattern, and they include predetermined positions on the surface of the substrate table to be designed. The outer perimeter points are located at fixed positions around the surface portion. The term "outer" means that the protrusions of the surface portion are arranged within a closed line along all the continuous outer perimeter points.

[0016] According to the present invention, the method includes providing a plurality of two-dimensional repulsive perimeter electric potentials in a support plane. The perimeter electric potentials include an outer perimeter electric potential, each outer perimeter electric potential being assigned to one of the outer perimeter points and being represented by an outer perimeter electric potential function f o (r), where r is the radial distance from the outer perimeter point. The outer perimeter electric potential is an additional input parameter of the method for generating the protrusion pattern. Thus, each outer perimeter point is considered to have a position and an associated outer perimeter electric potential, which is a function having rotational symmetry within the support plane and whose magnitude decreases with the distance from the considered outer perimeter point.

[0017] According to the present invention, the method further includes providing an initial distribution of a plurality of protrusions, each protrusion having an initially variable protrusion position in a surface portion. In addition, a plurality of two-dimensional repulsive protrusion electric potentials acting in the support plane are provided, where each protrusion electric potential among the protrusion electric potentials is assigned to one of the protrusions, and where for each of the protrusions, the assigned protrusion electric potential is represented by a protrusion electric potential function f p (r), where r is the radial distance from the protrusion. The protrusion electric potential is an additional input parameter of the method for generating the protrusion pattern. Similarly, each protrusion electric potential is preferably a function having rotational symmetry within the plane of the surface portion and whose magnitude decreases with the distance from the protrusion.

[0018] Preferably, the initial protrusion distribution does not cover the entire surface portion in which the protrusions are to be arranged, but only one or more parts thereof. Thus, the protrusions of the initial protrusion distribution have a local density that is greater than the global density of the protrusion arrangement to be obtained. At the starting point of the method for generating the protrusion pattern, the initial protrusion distribution is denser than the final protrusion arrangement, and thus advantageously allows the protrusions to be filled into the regions of the surface portion. After the generation of the protrusion pattern is completed, the distribution of the protrusions has a lower global density within the region of the surface portion.

[0019] According to the present invention, the method further includes calculating a local force acting on each of the protrusions, where each local force is calculated based on the superposition of the perimeter electric potential and the protrusion electric potential at the protrusion position of the considered protrusion, in particular based on the superposition of their spatial gradients. The local force acting on the considered protrusion is the resultant (vector sum) of the partial forces between the considered protrusion and each of the other protrusions and the outer perimeter points. Each partial force is calculated as the force in the pairwise superposed electric potential acting on the considered protrusion and one of the other protrusions or outer perimeter points, i.e., based on the gradient of the pairwise superposed electric potential. As an example, if the perimeter electric potential and the protrusion electric potential are proportional to 1 / r, as the electric potential of a charge, the partial force is calculated as the repulsive force between charges, with the charges represented by the protrusions.

[0020] The effective range of the perimeter potential and the protrusion potential depends on the potential shape, in particular the slope of its distance dependence. Basically, all external perimeter potentials and all protrusion potentials can affect the local force on each considered protrusion. In practice, only the closely adjacent perimeter points or protrusions, in particular the nearest neighbors of the considered protrusion, can determine the local force acting on the considered protrusion.

[0021] Furthermore, according to the present invention, the protrusion pattern to be obtained is generated by repeatedly and incrementally offsetting the protrusion positions according to a position offset increment and calculating the local force acting on each protrusion in the protrusions at the current protrusion positions until a predetermined optimization criterion is met. The optimization criterion is typically given by at least one predefined parameter, which is a characteristic of the uniformity of the distribution of the protrusions. The inventors have found that at least one parameter for changing the protrusion position can be used as the optimization criterion.

[0022] In particular, the protrusion positions are changed and the local force acting on each protrusion at the changed protrusion positions is calculated. If the optimization criterion is met, the change stops and the current protrusion pattern is the protrusion pattern to be obtained. If the optimization criterion is not met, the change continues. Thus, the protrusion pattern to be obtained is generated by an iterative process in which the protrusion positions are changed and the local forces are determined, where the iteration process is completed when the optimization criterion is met. The optimization criterion can be tested after each change at a single protrusion position or a subgroup of all protrusion positions. Preferably, the optimization criterion can be tested after the change of all protrusion positions, i.e., all protrusion positions are offset by the position offset increment in each iteration loop and then it is tested whether the optimization criterion is met.

[0023] The position offset increment includes the offset direction in the support plane and the amount of the position offset increment. Preferably, the offset direction of the considered protrusion is opposite to the direction of the net potential obtained from the superposition of the perimeter potential and the protrusion potential at the protrusion position of the considered protrusion. The amount of the position offset increment can be constant during the entire iterative process or can be changed according to the convergence of the iterative process.

[0024] Preferably, the stepwise offset of the protrusion positions can be performed with a position offset increment that depends on the magnitude of the total potential of each protrusion. The initially used increment can be relatively large, for example in the range of 500 μm to 1500 μm, while as the optimized protrusion arrangement is approached, the position offset increment for changing the protrusion positions can be decreased, for example to the range of 5 μm to 20 μm. Advantageously, using a gradually decreasing increment accelerates the process of finding the optimized protrusion arrangement.

[0025] According to a second general aspect of the present invention, the above object is solved by a method of manufacturing a substrate table configured to hold a substrate, in particular a semiconductor wafer, the method comprising the steps of: creating a pattern of the protrusion positions of the protrusions of the substrate table by the method according to the above first general aspect of the present invention or an embodiment thereof, and manufacturing protrusions at the positions provided by the pattern of the protrusion positions on the substrate table. Manufacturing the protrusions on the substrate table can be carried out by additive or subtractive processes known in the conventional art.

[0026] According to a third general aspect of the present invention, the above object is solved by a substrate table configured to hold a substrate, in particular a semiconductor wafer, and comprising a plurality of protrusions, wherein the end faces of the protrusions define a support plane for holding the substrate in a surface portion of the substrate table, and wherein the protrusions are arranged according to a protrusion pattern of the protrusion positions, the protrusion pattern being generated by the method according to the above first general aspect of the present invention or an embodiment thereof. Preferably, the substrate table is manufactured by the method according to the above second general aspect of the present invention or an embodiment thereof. Preferably, the pattern of the protrusion positions of most of the protrusions matches a hexagonal pattern. In particular, the pattern of the protrusion positions includes a plurality of hexagonal sub-patterns (hexagonal domains).

[0027] Other independent aspects of the present invention include a device for data processing (in particular a protrusion pattern generator) comprising a computer device configured to execute the method according to the above first general aspect of the present invention or an embodiment thereof, and other independent aspects of the present invention further include a computer program product comprising instructions that cause a computer to execute the method according to the above first general aspect of the present invention or an embodiment thereof when processing a program.

[0028] The present invention is based on the concept of treating the protrusions as repulsive point charges and letting them distribute themselves according to their interactions within the electric potential of the external perimeter points (and optionally internal perimeter points, as described below). The iterative design process of the present invention uses the self-organization of the protrusions to create the protrusion pattern: instead of the designer distributing the protrusions, the protrusions distribute themselves by gradually changing the protrusion positions and calculating the local forces, while being held within the allowed surface portion by a given framework (the external perimeter and optionally at least one internal functional element, as described below). Advantageously, this distribution automatically provides the most uniform distribution of the protrusions. As another substantial advantage, the method of the present invention only requires a small fraction of the time required to design the protrusion arrangement in the prior art, for example less than 10 hours, especially as low as 3 hours.

[0029] Advantageously, the protrusion pattern to be obtained is generated with a very low workload for the designer, while having maximum flexibility in terms of geometry (the shape of the external perimeter and the number and shape of the optional internal functional elements or perturbations).

[0030] Preferably, all the outer perimeter points of the outer perimeter may have the same outer perimeter potential. Alternatively, the outer perimeter potential may vary along the line of the outer perimeter points. By adopting the latter embodiment, the advantage of adapting the protrusion arrangement to the specific geometric features of the outer perimeter and / or parts thereof can be obtained. The same or different potentials may remain constant throughout the design process or may be changed during the design process.

[0031] According to a preferred embodiment of the present invention, each of the outer perimeter points is represented by an outer perimeter protrusion having a fixed position. The outer perimeter protrusion includes the outermost row of protrusions that surround the substrate portion and provide the outer perimeter. Advantageously, the available knowledge about the optimal position of the outermost row of protrusions can be directly incorporated into the method of the present invention. The fixed position of the outer perimeter protrusion can be obtained by general simulation or experimentation or by the design of the substrate stage.

[0032] According to a particularly advantageous embodiment of the present invention, the surface portion includes at least one internal functional element surrounded by a plurality of inner perimeter points, and the perimeter potential further includes an inner perimeter potential, each inner perimeter potential being assigned to one of the inner perimeter points and represented by an inner perimeter potential function fi(r), where r is the radial distance from the inner perimeter point. Advantageously, the inner perimeter potential only provides an additional perimeter potential to be considered when calculating the local forces acting on the protrusions. The method of the present invention allows at least one internal functional element to be integrated into the generation of the protrusion pattern. In addition, due to the self-organizing method of the present invention, the local regions of two adjacent features can be very close to each other because the entire pattern is a "transition zone".

[0033] Each potential of the inner perimeter points (i.e., each inner perimeter potential) is preferably a function that has rotational symmetry in the plane of the surface portion and whose amplitude decreases with the distance from the considered inner perimeter point. The term "inner" means that at least one internal functional element defines a closed shape within the surface portion that has a certain distance from the outer perimeter. Preferably, at least one internal functional element has no protrusions.

[0034] In this preferred embodiment, at least one internal functional element, in particular its position and dimensions, and the external perimeter points, in particular their positions, can be given to characterize the substrate table to be designed, i.e., they are located at fixed positions within and / or around the surface portion. The internal perimeter points are predefined around at least one internal functional element. The internal lines connecting the internal perimeter points of the series-connected internal functional elements completely enclose the internal functional elements within the surface portion of the substrate table. The internal lines can be regarded as the internally-protruding arranged internal perimeter. The internal perimeter points, in particular their positions, are additional input parameters of the method for generating the protruding pattern, and they include the predefined positions on the surface of the substrate table to be designed. The internal perimeter points are located at fixed positions around at least one internal functional element.

[0035] Preferably, all the internal perimeter points can have the same internal perimeter electric potential, or the internal perimeter electric potential can change along the line connecting the series-connected internal perimeter points. Thus, the advantage of adapting the protruding arrangement to at least one specific internal functional element and / or a part thereof can be obtained. Like the external perimeter points, the same or different electric potentials can remain constant throughout the design process, or can be changed during the design process.

[0036] Preferably, each of the internal perimeter points among the internal perimeter points is represented by an internal perimeter protrusion having a fixed position. Thus, the internal perimeter protrusions preferably include the last row of internal protrusions that surround at least one internal functional element and provide the internal perimeter. Similar to the external perimeter protrusions, the available knowledge about the optimal position of the outermost row of protrusions can be directly incorporated into the method of the present invention. The fixed positions of the internal perimeter protrusions can be obtained by general simulation or experiment or by the design of at least one internal functional element. Advantageously, the protrusions providing the internal perimeter boundary can be directly used as parameters for the generation of the overall pattern. As another advantage, the method of the present invention can also distribute other patterns in the existing pattern (e.g., the glue pads between the protrusions).

[0037] Another advantage of the present invention is that it can be used with different types of internal functional elements, such as including at least one of the dot-like gaps, line-like gaps, and plane-like gaps where the protrusions are located in the arrangement. The dot-like internal functional element can include, for example, small holes with a protrusion pitch size, such as gas inlets. The line-like internal functional element can include, for example, the dividing lines of the functional regions, such as the electrodes on a bipolar electrostatic chuck. The area of the plane-like functional element is much larger than the protrusion pitch (e.g., lift pin holes, forklift cutouts, etc.).

[0038] According to another preferred embodiment of the present invention, the local force acting on each protrusion can be determined by the linear superposition of the perimeter electric potential and the protrusion electric potential at the protrusion position of the protrusion under consideration. The advantage of the linear superposition is that the calculation is simple, fast, and has a fast convergence to find the optimal protrusion pattern.

[0039] Advantageously, various optimization criteria can be used to test the uniformity of the distribution of the protrusions. According to a first variant, the optimization criterion is determined by the sum of the local forces acting on the protrusions, and the step of offsetting the protrusion positions is performed until a protrusion pattern is created, which is created when the sum of the local forces acting on the protrusions is minimized, particularly zero. Advantageously, since the sum of the local forces acting on all the protrusions is minimized, particularly vanishes, the positions of all the protrusions are uniformly distributed within the surface portion.

[0040] Alternatively or additionally, according to a second variant, during the stepwise offset of the protrusion positions, the position offset increments can be changed, particularly decreased, and the optimization criterion can be determined by the sum of all the current position offset increments, and the step of offsetting the protrusion positions is performed until the sum of all the current position offset increments is below a predetermined threshold, particularly preferably zero. Advantageously, the position offset increments can be easily tested during the iterative generation of the protrusion pattern to be obtained.

[0041] If, according to another preferred embodiment of the present invention, a step of analyzing the convergence of the protrusion positions by means of a statistical analysis of the average distance between the protrusions and the variance of the average distance between the protrusions is provided, the advantage can be obtained that appropriate position offset increments are set when changing the protrusion positions. Advantageously, the step of changing the protrusion positions can be performed according to the position offset increments that depend on the convergence of the protrusion positions. By increasing the convergence, the position offset increments can be reduced. Alternatively or additionally, the perimeter potential (i.e., the external perimeter potential and / or the optional internal perimeter potential) can be changed during the design process. In particular, the perimeter potential can be adjusted according to the convergence of the protrusion positions.

[0042] Particularly preferably, the perimeter potential and the protrusion potential are represented by a potential function f(r) proportional to 1 / r p where p is an odd number selected in the range from 1 to 11. When the parameter p approaches 1, the range of the potential is relatively large and the local force is relatively weak, and when the parameter p increases towards 12, the potential range is relatively short and the local force is relatively strong. Therefore, for embodiments using internal functional elements with a large distance, a smaller parameter p is preferably employed, while for embodiments using internal functional elements with a small distance, a larger parameter p is preferably employed.

[0043] According to a further preferred variant of the invention, the number of protrusions can be changed during the stepwise offset of the protrusion positions, in particular at a preselected local position. Changing the number of protrusions can include increasing and / or decreasing the total number of protrusions. For example, the first stage of generating the protrusion pattern can be performed with fewer protrusions compared to the second stage of generating the protrusion pattern (and finding an optimized protrusion pattern). As another example, during the process of generating the protrusion pattern, the number of protrusions can be locally reduced, for example, at certain geometric features. Advantageously, through these embodiments, the required computational processing power can be reduced and / or the convergence of the process can be improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The following describes other advantages and details of the invention with reference to the drawings, which schematically show:

[0045] Figure 1 : A flowchart of a method for generating a protrusion pattern according to an embodiment of the invention;

[0046] Figure 2 and Figure 3 : Figure 1 A schematic diagram of the distribution of protrusions during the process of; and

[0047] Figure 4 : A flowchart of a method for manufacturing a substrate table, including Figure 1 The method for determining the protrusion positions in. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0048] DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS OF THE INVENTION

[0049] The features of the embodiments of the invention are described below with particular reference to the generation of the protrusion pattern of the substrate table. The invention is preferably implemented using an ESC, a vacuum chuck or another fixture known per se. Therefore, the details of the substrate table and its operation are not described as they are known from the prior art. Note that these figures only include schematic diagrams. In practice, the substrate table can have, for example, a circular surface part with approximately 100 to 50,000 protrusions, and the diameter of each protrusion is in the range of, for example, 10 μm to 10 mm. Embodiments using a substrate table with a plurality of internal functional elements are referenced. The invention is not limited to these embodiments and can also be implemented using a substrate table without internal functional elements.

[0050] The inventive method for generating a protrusion pattern can be implemented, for example, with a scripting language (such as "GNU Octave" (a Matlab clone)) or assembly software. The input parameters of the method for generating a protrusion pattern (e.g., the fixed positions of the outer perimeter points and optionally inner perimeter points, the perimeter electric potential of each of the outer perimeter points and optionally inner perimeter points, the protrusion electric potential of the protrusions considered movable during pattern generation, the position offset increment, and the optimization criteria for testing) are selected based on numerical simulations, reference applications, and / or the geometric characteristics of the substrate portion under consideration.

[0051] Figure 1 FIG. schematically illustrates an embodiment of a method 100 for generating a protrusion pattern of a plurality of protrusions of a substrate stage, as Figure 2 and Figure 3 schematically shown. Figure 2 and Figure 3 illustrate the ESC during different stages of the protrusion pattern generation method 100. As a first example, Figure 2 shows half of a bipolar ESC, where the substrate stage 10 of the complete bipolar ESC has a circular surface portion 11 and has a D-shaped electrode (a semi-circular electrode, not shown) in each half of the surface portion 11. As another example, Figure 3 the substrate stage 10 of

[0052] includes a unipolar ESC whose surface portion 11 has a shape with a circular cross-section. Figure 2 D and Figure 3 D (partially shown) the substrate stage 10 having a protrusion pattern 1 to be generated has a surface portion 11 that has a plurality of protrusions 12 and a plurality of internal functional elements 13. The end faces of the protrusions 12 within the surface portion 11 define a support plane (parallel to the drawing plane) for supporting a substrate (e.g., a silicon wafer (not shown)). The outer perimeter 14 surrounds the surface portion 11 within the support plane. The outer perimeter 14 is defined by a line connecting outer perimeter points in series, and as described below, an outer perimeter electric potential is assigned to the outer perimeter points. Preferably, the outer perimeter points are preset fixed outer perimeter protrusions, i.e., the outer perimeter 14 is provided by a line connecting the outermost row of protrusions (see the outer perimeter protrusions 12A in Figure 2 B). Optionally, additional preset outer perimeter points (not shown in detail) can be provided between the protrusions 12B along the outer perimeter 14.

[0053] The outer perimeter 14 is set as the limitation of the surface portion 11 and is used to obtain the convergence of the iterative protrusion pattern generation method 100. The outer perimeter 14 forms a repulsive frame within which the protrusions 12 are held in the allowed region defined by the repulsive frame. The repulsive frame consists of the outer contour of the surface portion 11 translated into points (point charges). By choosing the mutual spacing of adjacent point charges, the calculation time of the last row of protrusions (the number of outer perimeter points other than the protrusions) and the optimization of rasterization can be achieved. In addition, the electric potential of the outer protrusions and optionally the electric potential of other perimeter points at the optional outer perimeter 14 (region boundary) can be changed with respect to the electric potential of the variable protrusions 12, so that the distance of the last row of protrusions to the outer perimeter 14 can be adjusted. In this way, the distance of the protrusions 12 closest to the repulsive frame (the boundary of the allowed region) can be freely selected, and thus the flatness of the perimeter can be adjusted.

[0054] The internally functional element 13 schematically shown (see Figure 2 ) includes, for example, zero-dimensional features such as small holes for gas inlets, and / or two-dimensional dot-like or patch-like features such as lift pin holes. In Figure 2 's case, the electrically insulating line between the semi-circular electrodes can be regarded as a one-dimensional linear internally functional element 13A (see Figure 2 D).

[0055] Each internally functional element 13 is regarded as being surrounded by a line along a plurality of preset internal perimeter points (not shown in detail) that define the internal perimeter 15. The internal perimeter points are provided for defining the internal perimeter electric potential. Preferably, the internal perimeter points of the internally functional element 13 are considered to be provided by a given set of the nearest fixed protrusions 12B surrounding the internally functional element 13, that is, the last row of protrusions 12B surrounding the internally functional element 13 (for example, for illustrative purposes, as shown in Figure 2 B).

[0056] Referring to Figure 1 , the protrusion pattern generation method 100 includes a first step S11 of providing a two-dimensional repulsive perimeter electric potential, where each perimeter electric potential is assigned to one of the protrusions 12A at the outer perimeter 14 of the surface portion 11, optionally assigned to additional outer perimeter points, or assigned to one of the protrusions 12B surrounding the internally functional element 13 at the internal perimeter 15 of the surface portion 11. The protrusions 12A, 12B at the outer perimeter 14 and the internal perimeter 15 and the optional additional outer perimeter points have fixed positions. The fixed positions are obtained, for example, from a memory storing the geometric data of the substrate stage 10 to be designed and / or by collecting and processing an image of the surface portion 11 of the substrate stage 10 to be designed (for example, according to Figure 3 A).

[0057] The outer repulsive perimeter electric potential or the inner repulsive perimeter electric potential fo / fi is represented, for example, by the following equation:

[0058] f(r) = 1 / r (1)

[0059] Thus, all the peripheral electric potentials are equal. Alternatively, in order to further improve flexibility and possible flatness, several factors and / or functions of r can even be introduced for the internal peripheral electric potential and the external peripheral electric potential and / or within each set of internal peripheral electric potential and / or external peripheral electric potential. Advantageously, this allows taking into account the different mechanical behaviors of the silicon wafer held at the external periphery and internal features.

[0060] Subsequently, through step S12, an initial distribution 2 of a plurality of protrusions 12 is created, wherein each protrusion 12 of the initial distribution 2 has a repulsive protrusion electric potential fp and a variable protrusion position in the surface portion 11. An example of the initial distribution 2 is Figure 2 A and Figure 3 shown in B. The number of protrusions of the initial distribution 2 can be equal to the total number of protrusions 12 to be arranged. Alternatively, a smaller number of protrusions (e.g., 3 to 9 times) can be set as the initial distribution 2. As an example, the initial distribution 2 can include 6000 to 9000 protrusions.

[0061] The protrusion electric potential fp assigned to each protrusion 12 is also represented by equation (1). Thus, the protrusions 12 are similarly regarded as equal electric point charges and thus repel each other. However, the repulsive protrusion electric potential can also be different from the point charge field, for example, extending to an electric field having about 1 / r to, for example, about 1 / r5, to achieve special effects, such as in terms of uniformity or protrusion density. The peripheral electric potential and the protrusion electric potential can be equal or different. All the protrusion electric potentials can be equal, or the protrusion electric potentials of different protrusions can be different.

[0062] According to the subsequent step S13, the initial local force F acting on each protrusion 12 is calculated, wherein the local force is determined by the superposition of the internal peripheral electric potential and the external peripheral electric potential and the protrusion electric potential at the protrusion position of each considered protrusion 12. According to equation (1), the internal peripheral electric potential and the external peripheral electric potential and the protrusion electric potential are represented by potential functions. For example, the local force F in the field is calculated according to the following equation:

[0063]

[0064] The internal peripheral electric potential is fi, the external peripheral electric potential is fo, and the protrusion electric potential is fp. The force Fj on protrusion j is based on the superposition of all fixed (peripheral) protrusions k, l and all "movable" protrusions m. Weighting factors a, b, and c (positive numbers) are selected to achieve the easiest convergence. r is the distance from each of the fixed or movable protrusions to protrusion j. As a result of step S13, the local force is assigned to each protrusion 12 of the initial distribution 2.

[0065] Through additional processing, the protrusion positions of all the protrusions 12 are changed, and the local forces on each protrusion 12 at the new positions are calculated using equation (2) (step S14). Changing the protrusion positions means shifting the protrusions 12 by a certain position offset increment, including direction and a certain amount of position offset. The displacement direction within the support plane can be obtained from the superposition of the electric potentials at the current protrusion positions. The superposed electric potential represents a specific effective field (net field), whose direction depends on the sum of the electric potentials. The displacement is performed in a direction opposite to the direction of this effective field (or: opposite to the direction of the effective force). The amount of the position offset increment of the displacement is selected according to the phase of the iterative process in method 100. At the beginning, the amount of the position offset increment is, for example, 100 μm, and in the later stage of the process, the amount of the position offset increment can be adjusted to be lower, for example, 10 μm, as shown in step S18 below.

[0066] The amount of the position offset increment can be obtained by interpolating the required step size for the selected protrusion 12 (regarded as a point charge) with the cumulative total electric potential exerted on it by all the other protrusions 12, as well as a reasonable limitation of this step size. Preferably, the amount of the position offset increment of the protrusions 12 is proportional to the distance, which would result if the movement were linearly extrapolated in such a way that the force becomes zero based on the current gradient of the net electric potential. Reducing the amount of the position offset increment through step S18 results in "cooling" of the movement of the protrusions 12 to achieve convergence through progressive iteration. This means that, for example, the possible amount of the position offset increment for each iteration is gradually restricted.

[0067] Subsequently, after calculating the local forces acting on all the protrusions 12, in step S15, it is tested whether the new protrusion positions and the local forces on each protrusion 12 satisfy a predetermined optimization criterion. As an example, the sum of the local forces of all the protrusions 12 is calculated. The sum of the local forces is compared with a predetermined threshold that provides the optimization criterion. The threshold is, for example, the sum of the residual local forces, or even 0.

[0068] If step S15 results in the optimization criterion not being satisfied, the process returns to step S14. Again, the protrusion positions are changed, and the local forces on each protrusion 12 are calculated, and then the optimization criterion is tested.

[0069] Before returning to step S14, various steps can be included, as shown in the optional steps S16, S17, and S18. These optional steps S16 to S18 can be provided individually or in combination. For example, the electric potential and / or the amount of the position offset increment can be adjusted based on predetermined reference information. In particular, when the difference between the current sum of the local forces and the optimization criterion is below a predetermined limit, the amount of the position offset increment can be reduced in step S18.

[0070] Alternatively or additionally, convergence analysis may be included in step S16. The convergence analysis includes a statistical analysis of changes in local forces, for example during the current iteration step or considering multiple previous steps.

[0071] For estimating results and for extremely efficient parameter selection (e.g., for the repulsive electric potential of the perimeter points), real-time statistics may be implemented in step S16 in order to know, during the calculation process, for example, the average distance of the protrusions 12 (a measure of the desired target distance of the pattern) and, for example, the variance of this distance (the uniformity of the pattern), and thus the convergence can be evaluated by the user.

[0072] As the iteration process proceeds, the number of protrusions may be increased in step S17, in particular based on the results of the convergence analysis of step S16. Additionally, the perimeter and / or the protrusion electric potential may be adjusted by step S17. Figure 2 An increase in the number of protrusions 12 is schematically illustrated in C (see the enlarged view). As Figure 2 shown in C, a single protrusion 12 is replaced by an arrangement of, for example, nine protrusions 12. As the number of protrusions 12 increases, the further process of changing the protrusion positions and calculating the local forces on each protrusion 12 continues until the optimization criterion is met.

[0073] Starting from reducing the number of protrusions, the number of protrusions is increased in step S17 to the number required for the substrate stage 10, which has a substantial advantage for reducing the processing power. Due to the technology of the present invention, a large number of calculations are performed together, for example, 30,000 real protrusions 12, and sometimes a similar number of perimeter points. To reduce the processing time, for example, to a few minutes, the number of protrusions is increased in step S17 only after a stable distribution state has been reached. As a result of the convergence analysis S16 or by analyzing the current maximum increment, it can be found that the stable state has been reached. Thus, for the entire set of protrusions 12, only a minimum total movement is required, thereby minimizing the calculation time.

[0074] The iterative algorithm is run until the position does not change within a certain range. The method of the present invention allows the protrusions 12 to be automatically and evenly distributed in the surface portion 11. Thus, if step S15 yields that the optimization criterion is met, the arrangement of the protrusions 12 at the current protrusion positions is considered to be the protrusion pattern to be generated, and the protrusion pattern is output as a protrusion map in step S19. Generally, the application of the method 100 of the present invention produces a protrusion pattern that matches the hexagonal protrusion pattern, including the domain boundaries as Figure 2 shown in D. Thus, the protrusion pattern generation method 100 is completed.

[0075] Figure 3 A to Figure 3D further illustrates the execution of the protrusion pattern generation method 100, which is possible for a typical monopolar ESC within a few minutes. Figure 3 A shows the setting of the outer perimeter 14 and the inner perimeter 15 of the surface portion 11 (step S11). The injection of the protrusions 12 (point charges in the support plane) as the initial distribution 2 is as Figure 3 shown in B. As a result of the change in the protrusion position and the local force calculation on each protrusion 12, the protrusions 12 are distributed stepwise between the outer perimeter and the inner perimeter, as Figure 3 schematically shown in C. Before or after achieving a uniform distribution of the protrusions 12, the electric potential can be adjusted to obtain the optimal distance of the protrusions 12 to the outer perimeter 14, as Figure 3 schematically shown in D.

[0076] Figure 4 An embodiment of a method 200 for manufacturing a substrate table, in particular for manufacturing a protrusion pattern of a substrate table, is schematically illustrated. The manufacturing method 200 includes a preprocessing step, including the method 100 for generating a protrusion pattern, as Figure 1 shown. Thus, in step S11, the input includes the geometries of the inner perimeter and the outer perimeter, as well as the assigned inner perimeter electric potential and outer perimeter electric potential. After providing an initial distribution of a plurality of protrusions including the input of the average distance from protrusion to protrusion and the average distance from protrusions to the inner perimeter and the outer perimeter (according to Figure 1 step S12), the pattern generation of steps S13 to S19 of Figure 1 is performed.

[0077] Subsequently, post-processing is performed, including step S21 of deriving the geometry of the generated protrusion pattern. The protrusion pattern created with method 100 is output to the control unit of the manufacturing machine. In addition, the flatness of the protrusion pattern can be evaluated. Finally, a manufacturing machine using a bottom-up process or a top-down process manufactures the protrusion pattern in step S22, as is known per se in the prior art.

[0078] The features of the invention disclosed in the above description, the drawings, and the claims are important for implementing the invention in various embodiments of the invention, whether alone or in combination or sub-combination. The invention is not limited to the above preferred embodiments. On the contrary, various variations and derivatives are possible, which also use the concept of the invention and thus fall within the scope of protection. In addition, the invention also claims the subject matter and features of the dependent claims, regardless of the features and claims to which they refer.

Claims

1. A method for generating a protrusion pattern (1) of a plurality of protrusions (12) of a substrate table (10), the substrate table being configured to support a substrate, in particular a semiconductor wafer, in a surface portion (11) of the substrate table (10), wherein end faces of the protrusions (12) define a support plane, and the surface portion (11) is surrounded by a plurality of outer perimeter points within the support plane, the method comprising the steps of: Providing a plurality of two-dimensional repulsive perimeter electric potentials in the support plane, wherein the perimeter electric potentials include an outer perimeter electric potential, each outer perimeter electric potential being assigned to one of the outer perimeter points and represented by an outer perimeter electric potential function f o (r), where r is the radial distance from the outer perimeter point; providing an initial distribution (2) of the plurality of protrusions (12), each of the plurality of protrusions having a changeable protrusion position in the surface portion (11); Providing a plurality of two-dimensional repulsive protrusion electric potentials in the support plane, wherein each of the protrusion electric potentials is assigned to one of the protrusions (12), and wherein for each of the protrusions (12), the assigned protrusion electric potential is represented by a protrusion electric potential function f p (r), where r is the radial distance from the protrusion (12); calculating a local force acting on each of the protrusions (12), wherein each local force is calculated based on a superposition of the perimeter potential and the protrusion potential at the protrusion position of the protrusion (12); and generating the to - be - obtained protrusion pattern (1) by repeatedly step - by - step offsetting the protrusion positions by a position offset increment and by calculating the local force acting on each of the protrusions (12) at the current protrusion positions until a predetermined optimization criterion is met.

2. The method according to claim 1, wherein each of the outer perimeter points is represented by an outer perimeter protrusion (12A) having a fixed position.

3. The method according to claim 1 or 2, wherein the surface portion (11) comprises at least one internal functional element (13) surrounded by a plurality of inner perimeter points, and The perimeter electric potential further includes an internal perimeter electric potential, each internal perimeter electric potential being assigned to one of the internal perimeter points and represented by an internal perimeter electric potential function f i (r), where r is the radial distance from the internal perimeter point.

4. The method according to claim 3, wherein each of the inner perimeter points is represented by an inner perimeter protrusion (12B) having a fixed position.

5. The method according to claim 3 or 4, wherein the at least one internal functional element (13) comprises at least one of a dot - shaped gap, a line - shaped gap, and a planar - shaped gap in the arrangement of the protrusions (12).

6. The method according to any one of the preceding claims, wherein the local force acting on each protrusion (12) is determined by a linear superposition of the perimeter potential and the protrusion potential at the protrusion position of the protrusion (12), and / or wherein the optimization criterion is determined by the sum of the local forces acting on the protrusions (12), and the step of offsetting the protrusion positions is performed until the protrusion pattern (1) is created, under which the sum of the local forces acting on the protrusions (12) is minimized, in particular zero, and / or wherein the optimization criterion is determined by the sum of all current position offset increments, and the step of offsetting the protrusion positions is performed until the sum of all current position offset increments is below a predetermined threshold.

7. The method according to any one of the preceding claims, comprising the steps of: analyzing the convergence of the protrusion positions by a statistical analysis of the average distance of the protrusions (12) and the variance of the average distance of the protrusions (12).

8. The method according to claim 7, wherein the position offset increment is adjusted according to the convergence of the protrusion positions.

9. The method according to any one of the preceding claims, wherein The perimeter potential and the protrusion potential are represented by a potential function proportional to 1 / r p where p is an odd number in the range from 1 to 11.

10. The method according to any one of the preceding claims, comprising the steps of: changing the number of the protrusions (12) during a stepwise offset of the positions of the protrusions, in particular at a preselected local position.

11. A method of manufacturing a substrate table (10) configured to hold a substrate, in particular a semiconductor wafer, the method comprising the steps of: using the method according to any one of the preceding claims to create a protrusion pattern (1) of the positions of the protrusions (12) of the substrate table (10); and fabricating the protrusions (12) on the substrate table (10) at the positions provided by the protrusion pattern (1).

12. A substrate table (10) configured to hold a substrate, in particular a semiconductor wafer, the substrate table comprising: a plurality of protrusions (12), wherein end faces of the protrusions (12) define a support plane for holding the substrate in a surface portion (11) of the substrate table (10), wherein the protrusions (12) are arranged according to a protrusion pattern (1) of the positions of the protrusions, the protrusion pattern being generated by the method according to any one of claims 1 to 10.

13. The substrate table (10) according to claim 12, wherein the protrusion pattern (1) of most of the protrusions (12) matches a hexagonal pattern.

14. A data processing apparatus comprising a computer device configured to execute the method according to any one of claims 1 to 9.

15. A computer program product comprising instructions that cause a computer to execute the method according to any one of claims 1 to 9 when processing the program.

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