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

Abstract

A method may include generating a residual curvature map of a substrate, the residual curvature map based on measurements of the substrate. The method may include generating a dose map based on a residual curvature map for processing a substrate using a patterned energy source. The method may include applying a dose map to process a substrate using a patterned energy source, wherein the dose map is applied by performing multiple exposure of the substrate to the patterned energy source at a plurality of different twist angles.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. Provisional Patent Application No. 63 / 341,817, filed on May 13, 2022, entitled “DOSE MAPPING AND SUBSTRATEROTATION FOR SUBSTRATE CURVATURE CONTROL WITH IMPROVED RESOLUTION,” and U.S. Provisional Patent Application No. 63 / 425,051, filed on November 14, 2022, entitled “DOSE MAPPING AND SUBSTRATE ROTATION FOR SUBSTRATE CURVATURE CONTROL WITH IMPROVED RESOLUTION,” and the entirety of the above patent applications are incorporated herein by reference. Technical Field

[0003] The present embodiments relate to stress control in a substrate, and more particularly to stress compensation to reduce out-of-plane distortion in a substrate. Background Art

[0004] Devices (such as integrated circuits, memory devices, and logic devices) may be fabricated on substrates (such as semiconductor wafers) by a combination of deposition processes, etching, ion implantation, annealing, and other processes. Often, the complete fabrication of a device and associated circuitry may require hundreds of operations, including dozens of photolithography operations. In particular, the photolithography operations may require that a given mask used to fabricate a structure in a given area or level be aligned with a pre-existing structure.

[0005] A common problem with fabricating such devices and structures on substrates such as semiconductor wafers is the generation of in-plane distortion (IPD), which affects the overlay of a layer relative to an underlying reference layer. IPD is a complex quantity that is affected by the out-of-plane distortion (OPD) of the wafer and the alignment scheme employed in lithography. OPD is a fundamental wafer quantity, and the signature of residual OPD is critical to achievable overlay. For example, a commonly encountered type of OPD is global wafer curvature, which can occur in many processing situations due to stress buildup in the wafer caused by processing operations.

[0006] Furthermore, device processing may produce complex OPD patterns on the wafer after any given processing stage, which often affects subsequent processing operations. As an example, a semiconductor wafer substrate may be patterned into a regular array of die areas corresponding to dies to be cut from the semiconductor wafer. The array of die areas may be associated with a pattern of OPD, which itself exhibits a residual curvature with a high degree of directionality related to the layout of the die areas. In a specific example, the complex OPD pattern may generate overlay errors in subsequent photolithography mask operations.

[0007] Recently, ion implantation of the backside of the wafer has been explored to modify wafer stress, and thus modify OPD. However, such methods may employ ion beams with non-uniform shapes that are poorly suited for processing complex stress OPD patterns.

[0008] In view of these and other considerations, the present embodiments are provided. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] 1A-1G depict various stages in the generation of a residual curvature map of a substrate according to an embodiment of the present disclosure;

[0010] FIG. 2 illustrates an exemplary beam profile that may be used to create a blur kernel to be applied to the residual curvature map of FIG. 1G ;

[0011] FIG3A depicts a blurred residual curvature map according to some embodiments;

[0012] FIG3B illustrates an exemplary filtered residual curvature map generated by filtering the blurred residual curvature map of FIG3A ;

[0013] FIG. 4 provides an exemplary ion dose diagram according to an embodiment of the present disclosure;

[0014] 5A depicts a schematic top view of an ion implantation system for controlling substrate OPD according to an embodiment of the present disclosure;

[0015] FIG. 5B illustrates components of the ion implantation system of FIG. 5A , according to an embodiment;

[0016] FIG5C depicts a side view of operations for implanting a dose map in accordance with an embodiment of the present disclosure;

[0017] FIG5D presents an exemplary dose distribution to illustrate the variation of ion dose with radial position of the wafer;

[0018] 6A-6C depict different configurations for reducing residual substrate curvature using a scanned ion beam according to an embodiment of the present disclosure; and

[0019] FIG7 depicts an exemplary process flow. DETAILED DESCRIPTION

[0020] Embodiments of the present invention will now be described more fully below with reference to the accompanying drawings, in which some embodiments are shown. The subject matter of the present disclosure can be embodied in many different forms and should not be construed as being limited to the embodiments set forth herein. On the contrary, since these embodiments are provided, the present 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, the same numbers always refer to the same elements.

[0021] Embodiments described herein relate to techniques and apparatus for improving control of stress and associated out-of-plane distortion in a substrate and control of the effects of OPD on subsequent substrate processing operations, such as device fabrication. Embodiments of the present invention may employ novel techniques to determine a dose map applied by a patterned energy source to a compensation layer of a substrate in order to better correct the OPD and thereby reduce or minimize in-plane distortion (IPD) that affects device fabrication and other patterning processes. Non-limiting examples of patterned energy sources include ion beams or laser beams that can be scanned relative to a principal plane of the substrate. In various embodiments, the patterned energy source may transfer a pattern or dose pattern into a substrate, which involves a non-uniform direct write process. In this context, a "direct write" process includes a direct write implant process, which may refer to a process that employs relative movement of an ion beam or other beam without the use of a mask to produce a non-uniform dose pattern on a substrate surface. In some embodiments, a direct write process using a patterned energy source may involve exposure to electrons (such as an electron beam) or photons (such as a laser beam) that can be used to locally adjust the curvature of a wafer.

[0022] In various embodiments detailed herein, apparatus and techniques are provided for reducing substrate OPD using a patterned energy source in combination with substrate rotation. In a particular embodiment, residual curvature in a substrate is reduced using a dose map implemented to selectively treat selected areas of relatively high curvature using a combination of ion beam scanning, substrate translation, and substrate rotation. In a particular embodiment, a dose map may be implemented by determining a series of implant exposures to be performed at a series of different twist angles of the substrate in order to improve the resolution of the implant process.

[0023] Figures 1A to 1G depict a series of operations that will be applied to determine a dose map for processing a substrate to compensate for substrate OPD according to an embodiment of the present disclosure. In particular, the process illustrated in Figures 3A to 5C illustrates a method for eliminating residual curvature in a substrate surface.

[0024] In Figures 1A to 1C, details for determining the global curvature of a substrate in order to generate a global curvature map are shown. Figure 1A depicts a three-dimensional representation of a wafer surface, which is shown as a substrate surface map or an initial surface map, for example, the surface may represent a measurement of a substrate using a known surface measurement tool. Figure 1B depicts a two-dimensional representation of the surface of Figure 1A, wherein the surface is substantially parabolic in nature. In this example, the OPD range is only exemplary and may vary depending on processing conditions, as will be appreciated by those skilled in the art. Although it is intended to remove the global curvature, such as by forming a blanket film on the back side using a blanket processing operation, this may not always be done due to processing variations, so that a residual component of the global curvature remains on the wafer. This global curvature still has a parabolic OPD feature mark and looks like a concentrated curvature near the middle of the wafer on the curvature map.

[0025] FIG. 1C depicts a global curvature map based on the initial surface maps of FIG. 1A and FIG. 1B , which represents the curvature values ​​as a function of x,y coordinates on the wafer surface. Units is the inverse of km. According to various non-limiting embodiments of the present disclosure, the global curvature may be based on a Gaussian curvature model or a mean curvature model. Modeling may be based on the use of two mutually orthogonal principal planes of curvature extending perpendicular to the tangential plane of the surface, as shown in FIG. 1D. In the Gaussian model, the product of the maximum curvature and the minimum curvature is taken, where Given by the following formula

[0026]

[0027] In the mean model, the mean of the principal (maximum and minimum) curvatures is taken, where Given by the following formula

[0028]

[0029] As shown in FIG. 1C , the global curvature is nearly uniform across the wafer.

[0030] FIG. 1E depicts a three-dimensional representation of the residual wafer surface corresponding to the same wafer showing the global surface in FIG. 1A after extracting the global curvature. In this example, the curvature model used to extract the global curvature component is a mean curvature model. FIG. 1F depicts a two-dimensional representation of the surface of FIG. 1E after extracting the parabolic term of the OPD.

[0031] FIG. 1G depicts a residual curvature graph representing the curvature values ​​of the surfaces of FIG. 1E and FIG. 1F as a function of the x, y coordinates on the wafer surface. According to an embodiment of the present disclosure, the residual curvature graph of FIG. 1F may be generated based on the OPD graph of FIG. 1G using a process for modeling global curvature as generally summarized above. As shown in FIG. 1G, the pattern of residual curvature shows a set of complex features. Generally, the curvature values ​​on most of the wafer are relatively low, while there is a donut-shaped region of negative curvature toward the center of the wafer. In some regions on the wafer, the curvature has positive values, such as around the periphery of the wafer, while along the donut-shaped region, the curvature has negative values.

[0032] According to embodiments of the present disclosure, the residual curvature map of FIG. 1G may form the basis of a patterned ion implantation process to reduce or eliminate the OPD of features that generate the residual curvature map. In particular, a selective ion implantation pattern may be performed on the back side of the wafer represented by the curvature map of FIG. 1G. The selective ion implantation pattern may be designed to mimic the features of the residual curvature map so as to reduce stress proportionally to the degree of curvature at any location on the wafer surface. As described in further detail below, in various embodiments of the present disclosure, the selective ion implantation pattern may be implemented in a series of exposures at different wafer twist angles so as to improve the resolution of the implantation pattern when implemented in a substrate, and thereby improve the resolution of the OPD reduction, particularly in areas of high substrate curvature.

[0033] In various embodiments of the present disclosure, the residual curvature map may be transformed in a series of operations to generate a dose map that defines the ion dose to be implanted into the substrate as a function of x,y position on the wafer surface. The dose map may then be implemented in a series of ion beam exposures at different wafer twist angles, where in a given exposure the ion beam is generally scanned along the x-axis, with the substrate optionally translated along the y-axis.

[0034] FIG2 shows an exemplary ion beam profile that can be used for actual ion implantation. This profile is used by a blur kernel operation to create a blur kernel to be applied to the residual curvature map of FIG1G , which can be considered as the original residual curvature map. This operation can be used to create a blur kernel to be applied to the residual curvature map of FIG1G to attenuate the effects of high spatial frequencies on the implanter. This blur kernel can then be used to generate a blurred residual curvature map (shown in FIG3A ).

[0035] The blurred residual curvature map of FIG3A presents positive and negative curvature regions of the same qualitative pattern, but the width of the region is wider and the curvature values ​​within the region are slightly different from the corresponding residual curvature map without blurring. Given the limited size of the ion beam, this blurred residual curvature map may be more suitable for implementation by a patterned energy source (such as a scanning ion beam).

[0036] Turning to FIG. 3B , there is shown a filtered residual curvature map generated by filtering the blurred residual curvature map of FIG. 3A so as to remove all positive curvature terms that cannot be affected by the ion beam. Thus, there remain two main, slightly linear regions of relatively higher negative curvature, and some regions of lower negative curvature protruding from the linear regions. This map can then be converted into a dose map (such as for a scannable ion beam) in which the total ion dose applied over the two-dimensional surface (xy plane) of the wafer is based on the curvature map features of FIG. 3B . Thus, the pattern of ion dose of a suitable dose map can exhibit features having the same shape as the features of the curvature map.

[0037] FIG4 provides an exemplary ion dose map based on the filtered curvature map of FIG3B , wherein the dose map exhibits a qualitatively similar pattern to the curvature map of FIG3B . 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 much higher dose than the general "background" region. For example, the background region over most of the surface of the wafer will receive a relative ion dose in the range of 15%, while the linear region will receive a relative ion dose in the range of between about 50% and 85%.

[0038] FIG5A depicts a schematic top view of an ion implantation system for controlling substrate OPD according to an embodiment of the present disclosure. The ion implantation system, referred to as an ion implanter 300, represents a processing chamber containing an ion source 304 for generating an ion beam 308 and a series of beam-line components and other 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 source and an extraction electrode assembly (not shown) disposed near the chamber. The beam-line components may include, for example, an analyzer magnet 320, a mass resolution slit (MRS) 324, a steering / focusing component 326, and an end station 330 (including a substrate holder 331).

[0039] The ion implanter 300 further includes a beam scanner 336 positioned along a beam line 338 between the MRS 324 and the end station 330. The beam scanner 336 can be arranged to receive the ion beam 308 as a point beam and scan the ion beam 308 along a fast scan direction (such as parallel to the X-axis in the Cartesian coordinate system shown). It is worth noting that the substrate 332 can be scanned along the Y-axis, so when the ion beam 308 is scanned back and forth along the X-axis at the same time, a given ion treatment can be applied to a given area of ​​the substrate 332. The ion implanter 300 can have additional components, such as collimators known in the art (not shown for clarity), to guide the ions of the ion beam 308 along a series of mutually parallel trajectories to the substrate 332 after scanning, as shown in Figure 5A. In various embodiments, the ion beam can be scanned at a frequency of several Hz, 10 Hz, 100 Hz, up to several thousand Hz or more. For example, the beam scanner 336 can use magnetic or electrostatic scanning elements to scan the ion beam 308, as known in the art.

[0040] By rapidly scanning the ion beam 308 in a fast scan direction (such as scanning back and forth along the x-axis), the ion beam 308 configured as a spot beam can deliver a target ion dose to any given area of ​​the substrate in the xy plane. According to some non-limiting embodiments, suitable ions for the ion beam 308 may include any ion species capable of inducing stress changes at a suitable ion energy, including ions such as phosphorus, boron, argon, indium BF2, etc., where the ion energy is customized according to the exact ion species used. In order to implement the dose map, the scanning speed of the ion beam along the x-axis can be adjusted at different locations of the substrate 332 according to the dose map so as to deliver different ion doses at different locations. In general, the ion beam 308 can scan back and forth on the substrate for any suitable number of scans, and is accompanied by scanning the substrate in a direction orthogonal to the beam scan direction until the target dose specified by the dose map is received at the arrival area on the substrate 332.

[0041] For example, the ion implanter 300 may further include a controller 340 coupled to the beam scanner 336 to coordinate the operation of the beam scanner 336, and a substrate platen or substrate stage 331. As further shown in FIG. 5A, the ion implanter 300 may include a user interface 342 coupled to the controller 340. The user interface 342 may be embodied as a display and may include user selection devices including a touch screen, a displayed menu, buttons, knobs, and other devices known in the art. According to various embodiments, the user interface 342 may send instructions to the controller 340 to generate an appropriate implant pattern, wherein the pattern may implement an appropriate dose map for the substrate 332.

[0042] As further shown in FIG. 5B , the controller 340 may include a processor 352, such as a known type of microprocessor, a dedicated processor chip, a general purpose processor chip, or the like. The controller 340 may further include a memory or memory unit 354 coupled to the processor 352, wherein the memory unit 354 contains a dose map routine 356. The dose map routine 356 may be operable on the processor 352 to manage the scanning of the ion beam 308 and the substrate 332 so as to transfer the calculated dose map to the substrate 332. The memory unit 354 may include an article of manufacture. In one embodiment, the memory unit 354 may include any non-transitory computer-readable medium or machine-readable medium, such as an optical, magnetic, or semiconductor storage device. The storage medium may store various types of computer-executable instructions to implement one or more logic flows described herein. Examples of computer-readable or machine-readable storage media may include any tangible media capable of storing electronic data, including volatile memory or non-volatile storage, removable or non-removable memory, erasable or non-erasable memory, writable or rewritable memory, and the like. 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, and the like. The embodiments are not limited in this context.

[0043] In one implementation, a dose map can be used to perform an implant into a stress control layer on the back side of a substrate. FIG5C depicts a side view of an operation for implanting a dose map, where an ion beam is scanned in a fixed direction to implant an implant species into a stress compensating layer such as silicon nitride. By selectively controlling parameters such as beam scan rate, the ion dose applied to different locations of the stress compensating layer can be varied, thereby varying the stress variation (and associated substrate curvature) as a function of location on the wafer.

[0044] FIG5D presents an exemplary dose profile to plot ion dose as a function of radial position along the wafer, wherein the dose profile may be implemented by scanning the ion beam from left to right across the wafer while selectively varying parameters such as the scan rate to selectively increase the ion dose at the peak region of the profile. To achieve a two-dimensional dose map such as that of FIG4 , the wafer may be scanned in a wafer scan direction orthogonal to the beam scan direction while the ion beam is selectively scanned along the beam scan direction. Additionally, as described in detail below, according to embodiments of the present disclosure, the substrate may be rotated about the z-axis between a series of exposures to the scanned ion beam so as to transfer the pattern of the dose map to an implant pattern in the substrate with improved resolution.

[0045] 6A to 6C depict different configurations for reducing residual substrate curvature using a scanned ion beam according to an embodiment of the present disclosure. These figures present a composite image showing a two-dimensional representation of the filtered residual curvature map discussed previously and a pattern of scanning the ion beam shown in a circuitous path. Generally, in a given exposure, the ion beam is scanned rapidly along a fixed axis while or without scanning the substrate. In the example of FIG. 6A , the ion beam is scanned along the x-axis of the Cartesian coordinate system shown. During the scanning of the ion beam, the substrate may be moved intermittently or continuously in the direction of the thick arrow (in this case, parallel to the y-axis) so that the entire substrate may be exposed to the ion beam. It should be noted that for a scanning distance of ~450 mm, the scanning rate of the ion beam along the x-axis may be in the range of hundreds to thousands of Hz, which in some examples is converted into a scanning period in the range of hundreds of microseconds. At the same time, the scanning rate of the substrate along the y-axis may be on the order of several millimeters per second. Therefore, the substrate can be considered to be quasi-stationary during the time period passed by a single scan.

[0046] It is also noted that the dose map calculated to theoretically eliminate the residual curvature shown may exhibit the same geometric pattern as the residual curvature map (compare FIG. 3B and FIG. 4 ), where the ion dose to be implanted into a particular point on the substrate (represented by x,y coordinates) is proportional to the magnitude of the residual curvature at that point. Therefore, for purposes of clarity of explanation, the residual curvature map of FIG. 6A may be considered representative of the dose map ( FIG. 4 ) used to eliminate the residual curvature of FIG. 6A .

[0047] During the ion beam scan, for each of the up to thousands of scans that may cover a given xy point, the total ion dose injected at that point will be determined by the ion beam current density at that point and the ion beam scanning speed at that point, as well as other factors. In addition, the ion beam current density at a given point will be affected by the beam shape or beam profile, which may generally be non-uniform. Since the residual curvature pattern of FIG. 6A exhibits several regions of non-uniform curvature, the ion beam exposure that best eliminates these regions can vary the scanning speed as a function of position while scanning the substrate along the y-axis. As shown in FIG. 6A, the two main regions of high residual curvature extend in a direction rotated or twisted about 30 degrees relative to the x-axis. Therefore, for the configuration of FIG. 6A in which the wafer is oriented so that the substrate is scanned parallel to the y-axis and the beam is scanned parallel to the x-axis, the scanning direction of the ion beam is not aligned with the longitudinal direction of the high curvature feature. In order to implement a dose map that matches the residual curvature map, that is, in order to apply an appropriate ion dose to these high curvature regions while scanning the ion beam at a 30-degree twist angle relative to the longitudinal direction of the feature, it may be quite challenging to precisely control the variation of the scanning speed of the ion beam. Furthermore, due to the non-uniform beam profile, the resolution of the ion beam for transferring the desired dose map into the substrate may not be optimal when using the scanning configuration of FIG. 6A .

[0048] According to various embodiments of the present disclosure, the resolution of transferring a dose map to an implant pattern in a substrate can be improved by performing a routine comprising a series of exposures to a scanned ion beam, wherein the twist angle of the substrate varies between consecutive exposures. In a given exposure, the ion beam is scanned along a fixed direction, wherein the direction is characterized as a given twist angle relative to a fixed axis (such as the x-axis) of the substrate. In the example of FIG. 6A , the twist angle of the exposure is zero degrees, which means that the ion beam is scanned parallel to the x-axis. Turning to FIG. 6B , a configuration is shown in which the twist angle is about 15 degrees, while in FIG. 6C , the configuration exhibits a twist angle of about 30 degrees, thereby matching the tilt angle of the high curvature feature relative to the x-axis. Thus, a series of exposures at different twist angles can be used to transfer a dose map to an implant pattern in a substrate, as illustrated in FIG. 6A to FIG. 6C , wherein the resolution is improved compared to scanning the ion beam at a single twist angle. Note that to implement the dose map, since convolution is easily simplified to multiplication, calculations can be performed in the frequency domain for each scan profile of the exposed ion beam at different twist angles.

[0049] Turning now to FIG. 7 , a process flow 700 is shown according to some embodiments of the present disclosure. At block 702, a residual curvature map based on measured OPD data of a substrate is received. The residual curvature map may plot the substrate curvature in inverse length form as a function of x, y position on the substrate in question. The residual curvature map may be determined from the residual surface of the substrate extracted after removing the global curvature map.

[0050] Thus, a global curvature map can be generated using the model used to model the initial substrate surface, which plots the OPD magnitude as a function of x,y position on an assumed flat substrate surface. In some examples, the surface can be modeled as a parabola using a mean model or a Gaussian model, as described above.

[0051] At block 704, a dose map for processing the substrate is generated based on the residual curvature map. In some implementations, the dose map may be determined by first applying a blur kernel to the residual curvature map to generate a blurred residual curvature map, e.g., to account for size effects of an ion beam to which the dose map is to be applied. In some cases, the blurred residual curvature map may be further filtered to produce the dose map. For example, a positive curvature filter may be applied to remove positive curvature elements from the blurred residual curvature map because the positive curvature components may not be suitable for processing by an implanted ion beam. A dose map may then be generated based on the filtered residual curvature map, wherein the dose map may exhibit a qualitatively similar pattern to the filtered residual curvature map, wherein the relative dose increases in x,y regions of relatively higher curvature.

[0052] At block 706, a dose map is applied to the substrate by scanning the ion beam along a fixed direction in multiple exposures at multiple different twist angles. The scan speed profile for each of the exposures may be varied in a manner such that the total ion dose applied to the substrate as a function of position matches the dose map.

[0053] The advantages provided by embodiments of the present invention are multifaceted. As a first advantage, the present method allows subsequent devices to proceed with greater accuracy, such as subsequent lithography steps that require low in-plane distortion. As a second advantage, the present method more accurately reduces areas with greater in-plane distortion by targeting residual areas with greater substrate curvature for greater energy treatment. As a third advantage, the present embodiment improves the resolution of the scanning ion beam used to transfer the desired implant pattern into the substrate by rotating the substrate over multiple exposures, thereby providing a more accurate method to reduce residual or localized substrate curvature.

[0054] The present disclosure is not limited in scope by the specific embodiments described herein. In fact, various other embodiments and modifications of the present disclosure will be apparent to those of ordinary skill in the art, in addition to those embodiments and modifications described herein, according to the foregoing description and the accompanying drawings. Therefore, such other embodiments and modifications are intended to fall within the scope of the present disclosure. In addition, the present disclosure has been described herein in the context of a specific implementation in a specific environment for a specific purpose, but it will be appreciated by those of ordinary skill in the art that usefulness is not limited thereto, and the present disclosure can be beneficially implemented in any number of environments for any number of purposes. Therefore, the claims set forth herewith should be interpreted according to the full scope and spirit of the present disclosure described herein.

Claims

1. A method, the method include: generating a residual curvature map of a substrate, the residual curvature map being based on measurements of the substrate; generating a dose map based on the residual curvature map, the dose map being used to process the substrate using a patterned energy source; as well as The dose map is applied to process the substrate using the patterned energy source, wherein the dose map is applied by performing multiple exposures of the substrate to the patterned energy source at multiple different twist angles.

2. The method of claim 1, wherein the step of generating the residual curvature map include: subtracting a global curvature map from an initial surface map of the substrate to generate an original residual curvature map; as well as A blur kernel operation is applied to the original residual curvature map. 3 . The method of claim 2 , the step of generating the residual curvature map further comprising applying a filter to remove positive curvature from the residual curvature map.

4. The method of claim 2, wherein substrate curvature represented by the global curvature map can be removed by a blanket processing operation. The method of claim 1 , wherein the patterning energy source comprises an ion beam, an electron beam or a laser beam.

6. The method of claim 5, wherein during the plurality of exposures, the patterned energy source is scanned along a first direction within a principal plane of a substrate stage supporting the substrate, and Wherein between successive exposures in the plurality of exposures, the substrate is rotated by a certain twist angle about an axis extending perpendicularly to the main plane of the substrate platen.

7. The method of claim 1, wherein the step of applying the dose map include: exposing the stress compensating layer on the back side of the substrate to the patterned energy source, and The patterned energy source is scanned in a pattern over the stress compensating layer without using a mask to transfer the dose map into the substrate.

8. A method, the method comprising: include: receiving a substrate surface map of a substrate, the substrate surface map comprising a map of out-of-plane distortion of the substrate; modeling a global curvature map based on the substrate surface map; generating a residual curvature map after extracting the global curvature map from the substrate surface map; generating a dose map based on the residual curvature map, the dose map being used to process the substrate using a patterned energy source; as well as The dose map is applied to process the substrate using the patterned energy source, wherein the dose map is applied by performing multiple exposures of the substrate to the patterned energy source at multiple different twist angles.

9. The method of claim 8, wherein the step of generating the residual curvature map include: subtracting the global curvature map from an initial surface map of the substrate to generate an original residual curvature map; as well as A blur kernel operation is applied to the original residual curvature map.

10. The method of claim 9, the step of generating the residual curvature map further comprising applying a filter to remove positive curvature from the residual curvature map.

11. The method of claim 9, wherein substrate curvature represented by the global curvature map is removable by a blanket processing operation.

12. The method of claim 8, wherein the patterning energy source comprises an ion beam, an electron beam, or a laser beam.

13. The method of claim 12, wherein during the plurality of exposures, the patterned energy source is scanned along a first direction within a principal plane of a substrate stage supporting the substrate, and Wherein between successive exposures in the plurality of exposures, the substrate is rotated by a certain twist angle about an axis extending perpendicularly to the main plane of the substrate platen.

14. The method of claim 8, wherein the step of applying the dose map include: exposing the stress compensating layer on the back side of the substrate to the patterned energy source, and The patterned energy source is scanned in a pattern over the stress compensating layer without using a mask to transfer the dose map into the substrate.

15. A method, the method comprising: include: receiving a substrate surface map of a substrate, the substrate surface map comprising a map of out-of-plane distortions (OPDs) of the substrate based on a set of measured OPDs; using a model to generate a global curvature map from the substrate surface map; extracting a residual surface based on the substrate surface map and the global curvature map; generating an original residual curvature map based on the residual surface; generating a dose map based on the original residual curvature map; as well as The dose map is applied to process the substrate using a patterned energy source, wherein the dose map is applied by performing multiple exposures of the substrate to the patterned energy source at multiple different twist angles.

16. The method of claim 15, wherein the step of generating the dose map is based on the original residual curvature map. include: A blur kernel operation is applied to the original residual curvature map to generate a blurred residual curvature map.

17. The method of claim 16, the step of generating the dose map based on the original residual curvature map further comprising applying a filter to filter out positive curvature from the blurred residual curvature map.

18. The method of claim 15, wherein the patterning energy source comprises an ion beam, an electron beam, or a laser beam.

19. The method of claim 15, wherein during the plurality of exposures, the patterned energy source is scanned along a first direction within a principal plane of a substrate stage supporting the substrate, and Wherein between successive exposures in the plurality of exposures, the substrate is rotated by a certain twist angle about an axis extending perpendicularly to the main plane of the substrate platen.

20. The method of claim 15, wherein the step of applying the dose map include: exposing the stress compensating layer on the back side of the substrate to the patterned energy source, and The patterned energy source is scanned in a pattern over the stress compensating layer without using a mask to transfer the dose map into the substrate.

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