Dose mapping and substrate rotation for substrate curvature control with improved resolution

Abstract

The method may include generating a residual curvature map of a substrate based on measurements of the substrate. The method may include generating a dose map for processing the substrate using a patterning energy source based on the residual curvature map. The method may also include applying the dose map for processing the substrate using a patterning energy source, the dose map being applied by performing multiple exposures of the substrate to the patterning energy source at a plurality of different twist angles.

Description

SUMMARY OF THE INVENTION

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 341,817, filed May 13, 2022, entitled "DOSE MAPPING AND SUBSTRATE ROTATION FOR SUBSTRATE CURVATURE CONTROL WITH IMPROVED RESOLUTION", and U.S. Provisional Patent Application No. 63 / 425,051, filed November 14, 2022, entitled "DOSE MAPPING AND SUBSTRATE ROTATION FOR SUBSTRATE CURVATURE CONTROL WITH IMPROVED RESOLUTION", which are hereby incorporated by reference in their entirety.

[0002] TECHNICAL FIELD

[0002] Embodiments of this disclosure relate to stress control in a substrate, and more particularly, to stress compensation for reducing out-of-plane distortion in a substrate.

[0003] BACKGROUND ART

[0003] Devices (such as integrated circuits, memory devices, and logic devices) can be fabricated on a substrate (such as a semiconductor wafer) by a combination of deposition processes, etching, ion implantation, annealing, and other processes. In many cases, the entire fabrication of a device and its associated circuitry can involve hundreds of steps, including dozens of lithography steps. In particular, in lithography-related processes, it may be necessary to align a given mask for fabricating a structure in a given region or level with existing structures.

[0004]

[0004] One of the common concerns regarding manufacturing such devices and structures on a substrate such as a semiconductor wafer is the occurrence and progression of in-plane distortion (IPD). This distortion affects the overlay of one layer with respect to a reference layer beneath it. IPD is a complex quantity that is affected by both out-of-plane distortion (OPD) of the wafer and the alignment scheme used in photolithography. OPD is a fundamental wafer quantity, and the signature of the residual OPD is important for achievable overlay. For example, a common type of OPD is global wafer curvature, which can occur in many processing instances as a result of stress accumulating in the wafer during processing steps.

[0005]

[0005] Furthermore, device processing may generate a complex pattern of OPD in the wafer after a given stage of processing that tends to affect subsequent processing steps. As an example, a semiconductor wafer substrate may be patterned into an array of regular die regions corresponding to dies that are cut from the semiconductor wafer. Such an array of die regions may be associated with a pattern of OPD that manifests as a residual curvature with higher-order directionality related to the layout of the die regions. In a particular example, a complex pattern of OPD can cause overlay errors in subsequent lithography masking steps.

[0006]

[0006] In recent years, consideration has been given to ion implantation on the backside of the wafer to modify wafer stress and thus OPD. However, in such an approach, an ion beam with a non-uniform shape may be used, and such an ion beam is not ideally suited to processing complex stress OPD patterns.

[0007]

[0007] Embodiments of the present disclosure are provided in relation to the above and other considerations.

Brief Description of the Drawings

[0008]

Figure 1A

Figure 1B

Figure 1C

Figure 1D

Figure 1E

Figure 1F

Figure 1G

Figure 2

Figure 3A

Figure 3B

Figure 4

Figure 5A

Figure 5B

Figure 5C

Figure 5D

Figure 6A

Figure 6B

Figure 6C

Figure 7

DETAILED DESCRIPTION OF THE INVENTION

[0009]

[0019] Hereinafter, embodiments of the present disclosure will be described in more detail with reference to the accompanying drawings showing some embodiments. The subject matter of the present disclosure may be embodied in many different forms and should not be construed as limited to the embodiments described herein. Instead, these embodiments are provided so that this disclosure will be thorough and complete and will fully convey the scope of the subject matter to those skilled in the art. In the drawings, like numbers refer to like elements throughout.

[0010]

[0020] The embodiments described in this book relate to techniques and apparatus for the control of stress in a substrate and related out-of-plane strain, and for the control of the impact of OPD on subsequent substrate processing steps (such as device manufacturing). In the embodiments of this book, a new technique can be used to determine a dose map applied to a compensation layer of a substrate by a patterning energy source so as to better correct the OPD and thus reduce or minimize in-plane strain (IPD) that affects device manufacturing and other patterning procedures. Non-limiting examples of patterning energy sources include an ion beam or a laser beam that can be scanned with respect to the major surface of the substrate. In various embodiments, the patterning energy source may transfer a dose pattern to the substrate, which involves a non-uniform direct write process. In this context, a "direct write" process (including a "direct write implantation process") may refer to a process that uses the relative movement of an ion beam or other beam without using a mask to generate a non-uniform dose pattern across the substrate surface. In some embodiments, the direct write process using a patterning energy source may involve exposure to electrons (such as an electron beam) or photons (such as a laser beam) and can be used to locally adjust the curvature of the wafer.

[0011]

[0021] In the various embodiments detailed in this book, apparatus and techniques are provided for reducing substrate OPD by using a patterning energy source in combination with the rotation of the substrate. In certain embodiments, the residual curvature in the substrate is reduced using a dose map that is implemented to selectively process regions of relatively high curvature using a combination of scanning of the ion beam and translation and rotation of the substrate. In certain embodiments, the dose map can be implemented by determining to perform a series of implantation exposures at a series of different twist angles of the substrate so as to improve the resolution of the implantation procedure.

[0012]

[0022] Figures 1A through 1G illustrate a series of steps applied to determine a dose map for processing a substrate to compensate for a substrate OPD in accordance with an embodiment of the present disclosure. Specifically, the process progression shown in Figures 1A through 1G illustrates an approach for eliminating residual curvature on the substrate surface.

[0013]

[0023] Figures 1A through 1C show details for determining the spherical curvature of a substrate to generate a spherical curvature map. Figure 1A shows a three-dimensional representation of a wafer surface shown as a substrate surface map or an initial surface map, which surface may represent, for example, measurements of the substrate using known measurement tools. Figure 1B shows a two-dimensional representation of the surface of Figure 1A, where, substantially, the surface is generally parabolic. The range of OPD in this example is exemplary only and may vary depending on processing conditions, as would be recognized by one of ordinary skill in the art. For example, it is intended to remove spherical curvature by using a blanket processing step to form a blanket film on the back side, but due to process variations, this is not always possible, and a residual component of spherical curvature may remain on the wafer. This spherical curvature still has a parabolic OPD signature and appears as concentrated curvature near the center of the wafer in the curvature map.

[0014]

[0024] Figure 1C shows a spherical curvature map representing the values of curvature according to the x, y coordinates across the entire wafer surface based on the initial surface maps of Figures 1A and 1B. The unit κ is inverse kilometers. According to various non-limiting embodiments of the present disclosure, the spherical curvature may be based on a Gaussian curvature model or an average curvature model. This modeling may be based on the use of two principal planes of curvature that are orthogonal to each other and extend perpendicular to the tangent plane of the surface, as shown in Figure 1D. In the Gaussian model, the product of the maximum curvature and the minimum curvature is obtained, where κ is given by the following equation: κ = κ 1 κ 2 Equation (1) In the average model, the average value of the principal curvatures (the maximum curvature and the minimum curvature) is obtained, where κ is given by the following equation: κ = 1 / 2(κ 1 + κ 2 ) Equation (2) As shown in FIG. 1C, the spherical curvature is substantially uniform across the entire wafer.

[0015]

[0025] FIG. 1E shows a three-dimensional representation of the residual wafer surface after extracting the spherical curvature corresponding to the same wafer as shown in FIG. 1A with a spherical surface. In this example, the curvature model used to extract the spherical curvature component was the mean curvature model. FIG. 1F shows a two-dimensional surface representation of FIG. 1E after extracting the parabolic term of the OPD.

[0016] FIG. 1G shows a residual curvature map representing the curvature values according to the x, y coordinates across the entire wafer surface for the surfaces of FIGS. 1E and 1F. According to an embodiment of the present disclosure, in order to generate the residual curvature map of FIG. 1G based on the OPD map of FIG. 1F, the procedure for modeling the spherical curvature outlined above may be used. As shown in FIG. 1G, the pattern of the residual curvature shows a set of complex features. Generally speaking, the curvature values are relatively low over most of the wafer, while there is a doughnut-shaped region of negative curvature towards the center of the wafer. In a part of the area of the wafer (such as the area along the periphery of the wafer), the curvature has a positive value, and along the doughnut-shaped region, the curvature has a negative curvature value.

[0017]

[0026] According to embodiments of the present disclosure, the residual curvature map of FIG. 1G may form a basis for a patterning ion implantation process, which reduces or eliminates the OPD that generates the features of the residual curvature map. Specifically, on the back side of the wafer, a selective ion implantation pattern, represented by the curvature map of FIG. 1G, may be performed. This selective ion implantation pattern may be designed to mimic the features of the residual curvature map, whereby the stress is proportionally reduced according to the degree of curvature at any location on the wafer surface. As will be described in more detail below, in various embodiments of the present disclosure, the selective ion implantation pattern may be implemented at various wafer twist angles in a series of exposures, thereby improving the resolution of the implantation pattern when implemented on the substrate, particularly in regions of high substrate curvature, and thus improving the resolution of OPD reduction.

[0018]

[0027] In various embodiments of the present disclosure, the residual curvature map may be converted in a series of steps to generate a dose map. This dose map defines the ion dose implanted into the substrate according to the x, y positions on the wafer surface. The dose map may then be implemented at various wafer twist angles in a series of ion beam exposures, where the ion beam is generally scanned along the x-axis in a given exposure, and optionally, the substrate is translated parallel to the y-axis therewith.

[0019]

[0028] FIG. 2 shows an exemplary ion beam profile that may be used for actual ion implantation. This profile is used by a blur kernel process to create a blur kernel to be applied to the residual curvature map of FIG. 1G, which may be regarded as an untreated residual curvature map. This process may be used to create a blur kernel to be applied to the residual curvature map of FIG. 1G to attenuate the influence of high spatial frequencies on the ion implantation apparatus. This blur kernel may then be used to generate the blurred residual curvature map shown in FIG. 3A.

[0020]

[0029] The blurred residual curvature map of FIG. 3A presents the same qualitative pattern of positive and negative curvature regions as the non-blurred residual curvature map, but the width of the regions is slightly wider and the curvature values in the regions are also slightly different. This blurred curvature map may be more suitable for implementation by a patterning energy source (such as a scanned ion beam) considering the finite size of the ion beam.

[0021]

[0030] Referring now to FIG. 3B, a filtered residual curvature map is illustrated. By filtering the blurred residual curvature map of FIG. 3A to remove all positive terms of curvature (positive curvature cannot be affected by the ion beam), a filtered residual curvature map is generated. As a result, there remain two main linear regions with relatively high negative curvature and some regions with low negative curvature protruding from these linear regions. This map may then be converted, for example, into a dose map for a scanable ion beam, where the total ion dose to be applied over the two-dimensional surface (x-y plane) of the wafer is based on the characteristics of the curvature map of FIG. 3B. Thus, the ion dose pattern of a suitable dose map may exhibit features having the same shape as the characteristics of the curvature map.

[0022]

[0031] FIG. 4 presents an exemplary ion dose map based on the filtered curvature map of FIG. 3B, and this dose map shows a pattern qualitatively similar to the curvature map of FIG. 3B. In this example, two parallel linear regions of the dose map corresponding to the high-curvature linear regions of the filtered curvature map will receive a substantially higher dose than the overall "background" region. For example, while the background region covering most of the wafer surface receives a relative ion dose within a range of 15%, the linear regions will receive a relative ion dose ranging from approximately 50% to 85%.

[0023]

[0033] FIG. 5A shows a schematic top view of an ion implantation system for controlling the OPD of a substrate according to an embodiment of the present disclosure. This ion implantation system (referred to as ion implantation apparatus 300) represents a process chamber that includes an ion source 304 for generating an ion beam 308 and a series of beam line components. The ion source 304 may include a chamber for receiving a gas flow and generating ions. The ion source 304 may also include a power supply and an extraction electrode assembly (not shown) disposed in the vicinity of the chamber. The beam line components may include, for example, an analyzing magnet 320, a mass resolving slit (MRS) 324, an operation / focusing component 326, and an end station 330 including a substrate holder 331.

[0024]

[0033] Ion implantation apparatus 300 further includes a beam scanner 336 disposed between MRS 324 and end station 330 along beam line 338. Beam scanner 336 is configured to receive ion beam 308 as a spot beam and scan ion beam 308 along a high-speed scan direction, for example, parallel to the X-axis in the Cartesian coordinate system in the figure. It should be noted that while ion beam 308 is scanned to reciprocate along the X-axis, substrate 332 can be scanned along the Y-axis, so that a given ion treatment can be applied to a given region of substrate 332. Ion implantation apparatus 300 may also have additional components (not shown for clarity), such as a collimator known in the art, for guiding the ions of ion beam 308 to substrate 332 along a series of tracks that are parallel to each other after scanning. In various embodiments, the ion beam can be scanned at frequencies of several Hz (10 Hz, 100 Hz, up to several thousand Hz or more). For example, beam scanner 336 can scan ion beam 308 using a magnetic scan element or an electrostatic scan element, as is known in the art.

[0025]

[0034] By rapidly scanning the ion beam 308 back and forth along, for example, the X-axis over the high-speed scan direction, the ion beam 308 configured as a spot beam can supply the target ion dose to any given region of the substrate in the x-y plane. Ions suitable for the ion beam 308 may include, according to some non-limiting embodiments, ions such as phosphorus, boron, argon, indium BF 2 , and any ion species capable of inducing a stress change at a suitable ion energy, including ions such as these, and the ion energy is customized in strict accordance with the ion species used. To implement the dose map, the scan speed of the ion beam along the x-axis can be adjusted to supply different amounts of ion dose to different locations at different locations on the substrate 332 according to the dose map. Generally, the ion beam 308 may be scanned back and forth between both ends of the substrate over any suitable number of scan times until the target dose specified by the dose map is received in the reach region on the substrate 332, and accordingly, the substrate can also be scanned in a direction orthogonal to the beam scan direction.

[0026]

[0035] For example, the ion implantation apparatus 300 may further include a controller 340 that is also connected to the beam scanner 336 to adjust the operation of the beam scanner 336 and is also connected to the substrate platen or substrate stage 331. As further shown in FIG. 5A, the ion implantation apparatus 300 may also include a user interface 342 that is connected to the controller 340. The user interface 342 may be embodied as a display and may further include a user selection device (including touch screens, on-screen display menus, buttons, knobs, and other devices known in the art). According to various embodiments, the user interface 342 may send commands to the controller 340 to generate an appropriate implantation pattern capable of implementing a dose map suitable for the substrate 332.

[0027]

[0036] Further, as shown in FIG. 5B, the controller 340 may include a processor 352 (such as a microprocessor of a known type, an application specific processor chip, a general purpose processor chip, or a similar device). The controller 340 may further include a memory or memory unit 354 coupled to the processor 352, and this memory unit 354 contains a dose map routine 356. The dose map routine 356 may operate on the processor 352 to manage the scan of the ion beam 308 and the substrate 332 in order to apply the dose map calculated on the substrate 332. The memory unit 354 may include an article of manufacture. In one embodiment, the memory unit 354 may include some non-transitory computer-readable medium or machine-readable medium (such as an optical, magnetic, or semiconductor storage device). This storage medium may store various types of computer-executable instructions for implementing one or more of the logic flows described herein. Examples of computer-readable storage media or machine-readable storage media may include any tangible medium capable of storing electronic data (including volatile memory or non-volatile memory, removable or non-removable memory, erasable or non-erasable memory, writable or non-writable memory, etc.). Examples of computer-executable instructions may include any suitable type of code (such as source code, compiled code, interpreted code, executable code, static code, dynamic code, object-oriented code, visual code, etc.). Embodiments are not limited to this context.

[0028]

[0038] In one implementation form, the dose map may be used to perform implantation into the stress control layer on the back side of the substrate. FIG. 5C shows a side view of the process of implanting the dose map, where the ion beam is scanned along a determined direction, for example, to implant implant species into a stress compensation layer of silicon nitride. By selectively controlling parameters (such as the beam scan speed), the ion dose applied to various locations of the stress compensation layer can be varied, and thus, depending on the position on the wafer and the associated substrate curvature, the change in stress varies.

[0029]

[0038] FIG. 5D presents an exemplary dose profile showing ion doses corresponding to the radial position along the wafer. This dose profile is implemented by selectively changing parameters (such as scan speed) to selectively increase the ion dose in the peak region of the ion profile while scanning the ion beam from left to right between both ends of the wafer. To implement a two-dimensional dose map (such as the dose map of FIG. 4), the ion beam is selectively scanned along the beam scan direction, and at the same time, the wafer can be scanned in a wafer scan direction orthogonal to the beam scan direction. In addition, according to embodiments of the present disclosure, as detailed below, the substrate can be rotated about the z-axis between exposures of a series of exposures to the scanning ion beam in order to transfer the pattern of the dose map to an implantation pattern of a substrate with improved resolution.

[0030]

[0039] Figures 6A - 6C show various configurations for reducing residual substrate curvature using a scanned ion beam, according to embodiments of the present disclosure. These figures present, along with the pattern of the scan of the ion beam illustrated in a circuitous path, a composite image showing a two - dimensional representation of the above - mentioned filtered residual curvature map. Generally, the ion beam is scanned at high speed along a given axis during a given exposure, and simultaneously, the substrate may or may not be scanned. In the example of FIG. 6A, the ion beam is scanned along the x - axis of the Cartesian coordinate system shown. The substrate may be moved intermittently or continuously along the direction of the thick arrow (parallel to the y - axis in this example) during the scan of the ion beam, such that the entire substrate can be exposed to the ion beam. The scan speed of the ion beam along the x - axis, when the scan distance is ~450 mm, may, in some examples, be in the range of several hundred to several thousand Hz, which should be noted can be converted to scan times in the range of several hundred microseconds. At the same time, the scan speed of the substrate along the y - axis can be on the order of several micrometers per second. Thus, during the time elapsed for a single individual scan, the substrate can be considered quasi - stationary.

[0031]

[0040] The dose map calculated to theoretically eliminate the illustrated residual curvature may show the same geometric pattern as the residual curvature map (as can be seen by comparing FIGS. 3B and 4), and it should also be noted that the ion dose to be implanted at a particular point on the substrate represented by x, y coordinates is proportional to the intensity of the residual curvature at that point. Thus, for clarity of explanation, the residual curvature map of FIG. 6A can be considered a proxy for the dose map (FIG. 4) used to eliminate the residual curvature of FIG. 6A.

[0032]

[0041] During the scanning of the ion beam, the total ion dose injected at a given x-y point will be determined by the ion beam current density at that point and the ion beam scan speed at that point in each of up to thousands of scans that can cover that point. Further, the ion beam current density at a given point will typically be affected by the non-uniform beam shape or beam profile. Since the residual curvature pattern of FIG. 6A shows several areas with non-uniform degrees of curvature, in order to maximize the elimination of these areas, the scan speed of the ion beam exposure can be changed according to the position while the substrate is being scanned along the y-axis. As shown in FIG. 6A, two main areas with high residual curvature extend along a direction showing a rotation or twist of approximately 30 degrees with respect to the x-axis. Thus, in the configuration of FIG. 6A where the substrate is scanned parallel to the y-axis and the wafer is oriented such that the beam is scanned parallel to the x-axis, the scan direction of the ion beam does not coincide with the longitudinal direction of the features with high curvature. To implement a dose map that matches the residual curvature map, that is, to precisely control the variation of the scan speed of the ion beam so that the ion beam is scanned at a twist angle of 30 degrees with respect to the longitudinal direction of these features while applying an appropriate ion dose to the above-mentioned high-curvature regions, it can be quite difficult. Further, since the beam profile is non-uniform, the resolution of the ion beam for transferring a desired dose map to the substrate using the scan configuration of FIG. 6A can be sub-optimal.

[0033]

[0042] According to various embodiments of the present disclosure, the resolution for transferring a dose map to an implantation pattern of a substrate may be improved by performing a routine that includes a series of exposures to a scanned ion beam, where the twist angle of the substrate is changed between successive exposures. In a given exposure, the ion beam is scanned along a determined direction, which is characterized by a given twist angle with respect to a determined axis (e.g., the x-axis) of the substrate. In the example of FIG. 6A, the twist angle of the exposure is 0 degrees, i.e., the ion beam is scanned parallel to the x-axis. Referring to FIG. 6B, a configuration with a twist angle of approximately 15 degrees is illustrated, while the configuration of FIG. 6C shows a twist angle of approximately 30 degrees, corresponding to an inclination angle characterized by a high curvature with respect to the x-axis. Thus, as illustrated in FIGS. 6A - 6C, the dose map can be transferred to the implantation pattern of the substrate with an improvement in resolution as compared to a scan of the ion beam at a single twist angle, using a series of exposures at various twist angles. It should be noted that for implementing the dose map, the calculation of each scan profile of the ion beam for exposures at various twist angles can be performed in the frequency domain. This is because convolution is easily reduced to multiplications.

[0034]

[0043] Now referring to FIG. 7, a process flow 700 according to an embodiment of the present disclosure is illustrated. In block 702, a residual curvature map is received based on the measured OPD data of the substrate. In the residual curvature map, the substrate curvature can be plotted as the inverse length according to the x, y locations on the target substrate. The residual curvature map can be determined from the residual surface of the substrate extracted after removal of the spherical curvature map.

[0035]

[0044] For this reason, the spherical curvature map can be generated using a model for modeling the initial substrate surface that plots the intensity of the OPD according to the x, y positions on the presumptively flat substrate surface. In some examples, the surface can be modeled as a paraboloid using an average model or a Gaussian model as detailed above.

[0036]

[0045] In block 704, based on the residual curvature map, a dose map for processing the substrate is generated. The dose map, in some embodiments, is first generated by applying a blurring kernel to the residual curvature map to generate a blurred residual curvature map and can be determined, for example, by revealing the size effect of the ion beam to which the dose map is applied. Optionally, the blurred residual curvature map can be further filtered to generate the dose map. For example, a positive curvature filter can be applied to remove the positive curvature component from the blurred residual curvature map because the positive curvature component may not be suitable for the processing by ion beam implantation. Then, based on the filtered residual curvature map, a dose map can be generated. Here, the dose map can present a pattern qualitatively similar to the filtered residual curvature map, and the relative dose is increased in the x, y regions with relatively high curvature.

[0037]

[0046] In block 706, the dose map is applied to the substrate by scanning the ion beam along a determined direction in multiple exposures at multiple different twist angles. The scan speed profile of each exposure can be changed according to the position so that the total ion dose applied to the substrate matches the dose map.

[0038]

[0047] The advantages provided by the embodiments of this book are diverse. As a first advantage, the approach of this book enables subsequent devices to progress more accurately (e.g., subsequent lithography steps that require low planar distortion). As a second advantage, the approach of this book more accurately reduces regions with large planar distortion by targeting residual areas with large substrate curvature for processing at higher energy. As a third advantage, the embodiments of this book provide a more accurate approach for reducing residual substrate curvature or local substrate curvature by rotating the substrate through multiple exposures to increase the resolution of the scanned ion beam for transferring the desired implantation pattern to the substrate.

[0039]

[0048] This disclosure is not limited in scope by the specific embodiments described herein. Indeed, from the foregoing description and the accompanying drawings, various other embodiments and modifications to this disclosure will be apparent to those skilled in the art in addition to the embodiments described herein. Accordingly, such other embodiments and modifications are also intended to be included within the scope of this disclosure. Further, although this disclosure has been described in the context of specific implementations for specific purposes in a specific environment, those skilled in the art will recognize that the utility is not limited to such context and that this disclosure can be beneficially implemented in any number of environments for any number of purposes. Accordingly, the claims set forth below should be construed in view of the full scope and essence of the disclosure described herein.

Claims

1. A method comprising: generating a residual curvature map of the substrate based on a measurement of the substrate; generating a dose map for processing the substrate using a patterning energy source based on the residual curvature map; applying the dose map to process the substrate using the patterning energy source, wherein the dose map is applied by performing multiple exposures of the substrate to the patterning energy source at a plurality of different twist angles; A method.

2. The method according to claim 1, wherein generating the residual curvature map comprises: generating an unprocessed residual curvature map by removing a spherical curvature map from an initial surface map of the substrate; applying a blurring kernel process to the unprocessed residual curvature map. The method according to claim 1, wherein generating the residual curvature map further comprises applying a filter to remove positive curvature from the residual curvature map.

3. The method according to claim 2, wherein the substrate curvature represented by the spherical curvature map is removable by a blanket treatment process.

4. The method according to claim 2, wherein the patterning energy source includes an ion beam, an electron beam, or a laser beam.

5. The method according to claim 5, wherein the patterning energy source is scanned along a first direction on a main surface of a substrate platen supporting the substrate during the multiple exposures, and the substrate rotates at a certain twist angle about an axis extending perpendicular to the main surface of the substrate platen between consecutive exposures among the multiple exposures.

6. The method according to claim 1, wherein applying the dose map comprises: exposing a stress compensation layer on the back side of the substrate to the patterning energy source; scanning the patterning energy source over the entire stress compensation layer in a certain pattern without using a mask to transfer the dose map to the substrate.

7. A method comprising: receiving a substrate surface map of the substrate, the substrate surface map including a map of out-of-plane distortion of the substrate; modeling a spherical curvature map from the substrate surface map; generating a residual curvature map after extracting the spherical curvature map from the substrate surface map. ​ ​ ​ ​ Generating a dose map for processing the substrate using a patterning energy source based on the residual curvature map; Applying the dose map to process the substrate using the patterning energy source, wherein the dose map is applied by performing multiple exposures of the substrate to the patterning energy source at a plurality of different twist angles; A method. **Claim 9** Generating the residual curvature map includes: Removing the spherical curvature map from the initial surface map of the substrate to generate an untreated residual curvature map; Applying a blurring kernel process to the untreated residual curvature map; The method according to claim 8. **Claim 10** The method according to claim 9, wherein generating the residual curvature map further includes applying a filter to remove positive curvature from the residual curvature map. **Claim 11** The method according to claim 9, wherein the substrate curvature represented by the spherical curvature map is removable by a blanket treatment process. **Claim 12** The method according to claim 8, wherein the patterning energy source includes an ion beam, an electron beam, or a laser beam. **Claim 13** The patterning energy source is scanned along a first direction on a main surface of a substrate platen supporting the substrate during the multiple exposures, and the substrate rotates at a certain twist angle about an axis extending perpendicular to the main surface of the substrate platen between consecutive exposures among the multiple exposures. The method according to claim 12. **Claim 14** Applying the dose map includes: Exposing a stress compensation layer on the back side of the substrate to the patterning energy source; Scanning the patterning energy source over the entire stress compensation layer in a certain pattern without using a mask to transfer the dose map to the substrate; The method according to claim 8. **Claim 15** A method comprising: Receiving a substrate surface map of a substrate, the substrate surface map including a map of out-of-plane displacement (OPD) of the substrate based on a set of measured out-of-plane displacements (OPDs); Generating a spherical curvature map from the substrate surface map using a model; Extracting a residual surface based on the substrate surface map and the spherical curvature map; generating an untreated residual curvature map based on the residual surface; generating a dose map based on the untreated residual curvature map; applying the dose map to process the substrate using the patterning energy source, wherein the dose map is applied by performing multiple exposures of the substrate to the patterning energy source at a plurality of different twist angles; a method. **Claim 16** generating the dose map based on the untreated residual curvature map includes applying a blurring kernel process to the untreated residual curvature map to generate a blurred residual curvature map The method according to claim 15. **Claim 17** The method according to claim 16, wherein generating the dose map based on the untreated residual curvature map further includes applying a filter to remove positive curvature from the blurred residual curvature map. **Claim 18** The method according to claim 15, wherein the patterning energy source includes an ion beam, an electron beam, or a laser beam. **Claim 19** the patterning energy source is scanned along a first direction on a main surface of a substrate platen supporting the substrate during the multiple exposures, The method according to claim 15, wherein the substrate rotates at a certain twist angle about an axis extending perpendicular to the main surface of the substrate platen between consecutive exposures among the multiple exposures. **Claim 20** applying the dose map includes exposing a stress compensation layer on the back side of the substrate to the patterning energy source; scanning the patterning energy source across the entire stress compensation layer in a certain pattern without using a mask to transfer the dose map to the substrate. The method according to claim 15.

Images