Dose mapping for compensating out-of-plane strain using substrate curvature

Abstract

The method may include generating a residual curvature map of a substrate based on measurements of the surface 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, and applying the dose map to process the substrate using the patterning energy source.

Description

SUMMARY OF THE INVENTION

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 341,797, filed May 13, 2022, entitled “DOSE MAPPING USING SUBSTRATE CURVATURE TO COMPENSATE FOR OUT-OF-PLANE DISTORTION,” and U.S. Provisional Patent Application No. 63 / 425,060, filed Nov. 14, 2022, entitled “DOSE MAPPING USING SUBSTRATE CURVATURE TO COMPENSATE FOR OUT-OF-PLANE DISTORTION,” the entire contents of which are incorporated herein by reference.

[0002] TECHNICAL FIELD

[0002] Embodiments of this disclosure relate to stress control in a substrate, and more specifically, 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 at a given region or level with existing structures.

[0004]

[0004] One of the common concerns regarding the fabrication of such devices and structures on a substrate such as a semiconductor wafer is the occurrence and progression of in-plane distortion (IPD). This IPD 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 residual OPD is important for achievable overlay. For example, a certain type of OPD that is often seen 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 can generate a complex pattern of OPD in the wafer after a given stage of processing that may tend to affect subsequent processing steps. In one particular example, a complex pattern of OPD can cause overlay errors in subsequent lithography masking steps.

[0006]

[0006] Embodiments of the present disclosure are provided in relation to the above considerations and other considerations.

Brief Description of the Drawings

[0007]

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Embodiments for Carrying Out the Invention

[0008]

[0015] Hereinafter, embodiments of the present disclosure will be described more comprehensively 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, by providing these embodiments, this disclosure will be comprehensive 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.

[0009]

[0016] The embodiments described in this book relate to techniques and apparatus for improved out-of-plane strain control in a substrate and for controlling the effect of OPD on substrate processing steps (such as device manufacturing) related thereto. 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 main surface of the substrate.

[0010]

[0017] In various embodiments detailed in this book, a substrate (often referred to as a "wafer") can be measured to determine a substrate OPD map. This OPD map can then be used to extract a spherical OPD, the entity of which is defined as a paraboloid that best fits the measured OPD. Further calculations are performed to extract a spherical substrate curvature component corresponding to this paraboloid, while the residual OPD is used to extract a local substrate curvature or a remaining substrate curvature.

[0011]

[0018] Figures 1A through 1C illustrate the operating principle of an embodiment of the present disclosure. Referring to FIG. 1A, a three-dimensional graph showing the wafer surface with respect to a Cartesian coordinate system is illustrated. Specifically, the graph of FIG. 1A shows a nominally circular and flat wafer shape, where the X-Y plane represents the nominal main plane of the wafer, or equivalently, the ideal plane of the flat platen supporting the wafer. The units shown in the X-Y plane can be, for example, millimeters. The Z-axis represents the perpendicular to the main surface of the substrate, and thus, any location on the substrate that is not in the X-Y plane where z = 0 can be considered to represent the OPD. Note that the Z-axis is dimensionless and is normalized to "1". In semiconductor wafer processing, stresses can be introduced that tend to impart a spherical curvature (e.g., the parabolic shape shown in FIG. 1B) to the substrate due to stress accumulation during the manufacture of layers, devices, etc. This shape can be associated with biaxial tensile or compressive stress. In some non-limiting examples, the level of the OPD can reach a maximum value (such as 100 μm, 200 μm, 300 μm, 400 μm, etc.) over a range of several hundred micrometers.

[0012]

[0019] According to an embodiment of the present disclosure, the OPD represented by the wafer shape of FIG. 1A can be compensated in a series of steps. In the first step represented in FIG. 1B, the bulk of the OPD can be compensated using a uniform stress compensation layer applied to the back surface of the substrate, which is opposite to the front surface of the substrate on which the device is manufactured. The graph of FIG. 1B represents the spatial distribution in the X-Y plane of the amount of strain or deformation (represented along the Z-axis) of the wafer that can be generated by the uniform stress compensation layer. Again, since the relative amount of deformation along the Z-axis is normalized, in the graph of FIG. 1, the maximum deformation is at the center of the wafer. The step of FIG. 1B can be useful for compensating for a relatively large spherical OPD that develops in the wafer, as represented by the parabolic shape of FIG. 1B. Such an OPD shape can be line-symmetric about the Z-axis, and by the step of FIG. 1B, the stress can be reduced overall in a line-symmetric manner. Thus, after the step represented in FIG. 1B is applied, the spherical curvature of the wafer can be significantly reduced.

[0013]

[0020] Referring to FIG. 1C, a graph is shown representing an example of a residual pattern of OPD that can be superposed on the pattern of FIG. 1B. Depending on the layout of the die on the wafer and the nature of other patterning features, even after removing the spherical OPD pattern as shown in FIG. 1B, there may exist a complex “residual” OPD pattern that is not necessarily line symmetric and may be more local. From this, the embodiments of the present document address this phenomenon by applying a so-called dose map to a patterning energy source (such as a scanable ion beam, electron beam, or laser beam). As will be described later, the patterning energy source can be applied to the substrate (e.g., within an existing compensation layer) in a non-uniform manner to remove the residual OPD pattern.

[0014]

[0021] FIGS. 2A through 2F show different representations of the OPD of a wafer at various stages of processing according to embodiments of the present disclosure. In these examples, similar to FIG. 1A, the units of the X-axis and Y-axis in FIGS. 2A, 2C, and 2E, which represent, for example, a 300 mm wafer, are shown in millimeters. The unit along the Z-axis is nm. Thus, these graphs present a three-dimensional representation of the wafer surface, where ideally the Z-axis coordinates of the flat wafer are constant (e.g., 0) across the entire X-Y plane.

[0015]

[0022] In the example of FIG. 2A, the wafer surface is represented by a three-dimensional array of points. This wafer surface may be characterized as having a parabolic shape and is characterized in that the Z-axis coordinates range from -150,000 nm to +150,000 nm (equal to a maximum OPD of 300,000 nm or 300 μm). A side cross-sectional view of a micro portion of the substrate of FIG. 2A (which may be silicon in some embodiments) is shown in FIG. 2B in a very generalized form. In this example, the front surface is represented by the upper surface in the figure, and there may or may not be additional device layers on this upper surface, but they are omitted for simplicity. The surface shown in FIG. 2A may represent the wafer surface before performing the process for reducing the OPD according to the embodiments of the present disclosure.

[0016]

[0023] Referring to FIG. 2C, the wafer surface corresponding to the wafer of FIG. 2B is shown after the process of depositing a stress compensation layer (see FIG. 2D) on the back side of the wafer. The stress compensation layer can be deposited by a known apparatus such as a physical vapor deposition (PVD) apparatus, a chemical vapor deposition (CVD) apparatus, or other film deposition systems according to various non-limiting embodiments. Examples of materials suitable for the stress compensation layer include silicon nitride, silicon oxide, silicon oxynitride, a layer containing any combination of Si-O-N-C, or other known materials. At this point, the wafer surface may be characterized as having an irregular shape, where the absolute value of the OPD is significantly reduced, so that the maximum value of the OPD is on the order of only a few micrometers.

[0017]

[0024] At this stage of the process, it can be said that the spherical signature of the stress state across the wafer (which results in a generally regular paraboloid shape being generated on the wafer at a vertical scale of several hundred micrometers) is removed by the deposition of the stress compensation layer. For example, by depositing a uniform stress compensation layer to cover the surface of the wafer, it can be expected that the average shape of the wafer will be modified according to the well-known Stoney's equation that relates the change in substrate curvature to the stress characteristics of the layer in contact with the substrate. According to embodiments of the present disclosure, the thickness and stress state of the stress compensation layer can be selected to reduce the spherical curvature of the wafer according to the initial level of curvature (see FIG. 2A). As will be described further below, the spherical curvature can be modeled to provide a basis for determining the suitable stress compensation layer characteristics required to remove the spherical curvature using various possible models. Thus, in the examples of FIGS. 2C and 2D, a stress compensation layer having a suitable thickness, a suitable elastic modulus, and a suitable stress state can be selected to remove substantially all of the spherical curvature components that generate the maximum OPD of the 300 mm substrate in FIG. 2A. Thus, in FIG. 2C, what remains is a pattern that can be said to be the irregularity of the OPD, which represents the residual curvature that can be caused by artifacts such as die placement or certain device structures or circuit structures present on the front side of the wafer. This irregular pattern of OPD results in unwanted IPD in various regions of the substrate and can cause problems such as an increase in the misalignment of overlaps in subsequent substrate patterning.

[0018]

[0025] According to the embodiments of FIGS. 2E and 2F, the residual curvature shown by the substrate of FIG. 2C can be removed, reduced, or modified by performing exposure to the aforementioned patterning energy source. In the specific example shown in FIG. 2F, an implantation procedure for generating an implantation layer in the stress compensation layer has already been performed, and this implantation procedure may involve a non-uniform direct write implantation process. In this context, the "direct write" process (including the "direct write implantation process") may refer to a process that uses the relative movement of an ion beam without using a mask to generate a non-uniform pattern of ion doses on the substrate surface. Due to this non-uniform implantation process, the curvature of the wafer can be locally adjusted to reduce the OPD as shown in FIG. 2F. In other embodiments, a direct write process involving exposure to electrons or photons (such as a laser beam) may be used to locally adjust the curvature of the wafer. As detailed below, the patterning process for locally adjusting the substrate curvature can be performed using a dose map that is calculated based on the measured value of the initial surface of the wafer before the stress compensation layer is deposited. It should be noted that the OPD correction applied to modify the residual curvature of FIG. 2C using a suitable dose map can be replaced by the correction of the IPD across the entire wafer as shown in FIG. 2G. This figure shows a two-dimensional map in the x-y plane, and this two-dimensional map shows the intensity and direction of the IPD correction corresponding to the x, y coordinates on the 300 mm wafer.

[0019]

[0026] FIGS. 3A to 5C show a series of steps applied to determine a dose map for processing a substrate to compensate for the substrate OPD according to embodiments of the present disclosure. Specifically, the process progression shown in FIGS. 3A-5C shows an approach for eliminating the residual curvature on the substrate surface.

[0020]

[0027] Figures 3A through 3C show details for determining the spherical curvature of a substrate to generate a spherical curvature map. Figure 3A shows a three-dimensional representation of the wafer surface prior to extraction of the spherical curvature components generally described above in connection with Figures 2A-2E. Figure 3B shows a two-dimensional representation of the surface of Figure 3A, where, substantially, the surface is generally paraboloidal. The OPD ranges shown in this figure of this example are exemplary only and may vary widely as would be recognized by one of ordinary skill in the art. A blanket processing step for depositing a blanket film on the backside of the substrate is intended to remove the spherical curvature, but due to process variations, it is not always possible to do so. Thus, a residual component of the spherical curvature may still remain on the wafer. This spherical curvature still has the OPD signature of a paraboloid, but in the curvature map, the curvature appears to be generally uniform across the wafer.

[0021]

[0028] Figure 3C shows a spherical curvature map representing the values of curvature according to the x, y coordinates across the wafer surface. The unit k is inverse kilometers. According to various non-limiting embodiments of the present disclosure, the spherical curvature may be modeled 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 3D. 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 (maximum curvature and minimum curvature) is obtained, where κ is given by the following equation: κ = 1 / 2(κ 1 + κ 2 ) Equation (2) As shown in Figure 3C, the spherical curvature is generally uniform across the wafer.

[0022]

[0029] FIG. 4A shows a three-dimensional representation of the residual wafer surface after extracting the parabolic term of the OPD corresponding to the same wafer as shown in FIG. 3A with a spherical surface. FIG. 4B shows a two-dimensional representation of the surface of FIG. 4A, where the pattern of the OPD is rather complex.

[0023]

[0030] FIG. 4C shows a residual curvature map representing the values of the curvature according to the x, y coordinates over the entire wafer surface for the surfaces of FIGS. 4A and 4B. According to an embodiment of the present disclosure, in order to generate the residual curvature map of FIG. 4C based on the OPD map of FIG. 4B, the procedure for modeling the spherical curvature outlined above can be used.

[0024]

[0031] As shown in FIG. 4, the pattern of the residual curvature shows a set of complex features. Generally speaking, while the curvature values are relatively low over most of the wafer, there is a donut-shaped region of negative curvature towards the center of the wafer. In some parts of the wafer area (such as the area along the periphery of the wafer), the curvature has a positive value, and along the donut-shaped region, the curvature has a negative curvature value.

[0025]

[0032] The residual curvature map of FIG. 4C may be regarded as an unprocessed residual curvature map, and this unprocessed residual curvature map is further processed to generate a final curvature map that is used to generate a dose map. Subsequently, the wafer of FIG. 3A may be processed to remove the features of the residual curvature, and thus, the dose map may be used to eliminate or reduce the IPD caused by such features. FIG. 5A shows an ion beam profile that may be used for actual wafer implantation. This profile is used to create a blur kernel that should be applied to the residual curvature map of FIG. 4C to attenuate the influence of high spatial frequencies on the ion implantation device in the blur kernel process. This profile may then be used to generate the blurred residual curvature map shown in FIG. 5B. The blurred residual curvature map of FIG. 5B 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 that the ion beam has a finite size.

[0026]

[0033] Referring now to FIG. 5C, a filtered residual curvature map is illustrated. Since the positive terms are not subject to the influence of the ion beam, the filtered residual curvature map is generated by filtering the blurred residual curvature map of FIG. 5B and removing all positive terms of the curvature. 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 entire two-dimensional surface (x-y plane) of the wafer is based on the features of the curvature map of FIG. 5C. Thus, the ion dose pattern of a suitable dose map may exhibit features having the same shape as the features of the curvature map.

[0027]

[0034] FIG. 5D presents an exemplary ion dose map based on the curvature map of FIG. 5C, and this dose map shows a pattern qualitatively similar to the curvature map of FIG. 5C. 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 substantially higher doses 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%.

[0028]

[0035] FIG. 6A 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 can include a chamber for receiving a gas flow and generating ions. The ion source 304 can also include a power supply and an extraction electrode assembly (not shown) disposed in the vicinity of the chamber. The beam line components can 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.

[0029]

[0036] Ion implantation apparatus 300 further includes a beam scanner 336 disposed between MRS 324 and end station 330 along beam line 338. The beam scanner 336 can be configured to receive the ion beam 308 as a spot beam and scan the ion beam 308 along the 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 the ion beam 308 is scanned to reciprocate along the X-axis, the substrate 332 can be scanned along the Y-axis, so that a given ion treatment can be applied to a given area of the substrate 332. The ion implantation apparatus 300 may have additional components (not shown for clarity), such as a collimator known in the art, for guiding the ions of the ion beam 308 to the substrate 332 along a series of tracks parallel to each other after scanning, as proposed in FIG. 6A. 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, the beam scanner 336 can scan the ion beam 308 using a magnetic scan element or an electrostatic scan element, as known in the art.

[0030]

[0037] By rapidly scanning the ion beam 308 to reciprocate along the high-speed scan direction, for example, along the X-axis, the ion beam 308 configured as a spot beam can supply the target ion dose to any given area of the substrate in the x-y plane. Ions suitable for the ion beam 308, according to some non-limiting embodiments, are phosphorus, boron, argon, indium BF 2, and may include any ion species capable of inducing stress changes at a suitable ion energy, including ions such as these. The ion energy is customized in strict accordance with the ion species used. To implement a dose map, the scan speed of the ion beam along the x-axis can be adjusted according to the dose map to supply different amounts of ion dose to different locations on the substrate 332. Generally, until the target dose specified by the dose map is received in the area of the substrate 332 where it arrives, the ion beam 308 may be scanned back and forth between the two ends of the substrate for any suitable number of scan times. Along with this, the substrate may also be scanned in a direction orthogonal to the beam scan direction.

[0031]

[0038] For example, the ion implantation device 300 may further include a controller 340 that is connected to the beam scanner 336 to adjust the operation of the beam scanner 336 and is also connected to the substrate holder 331. As further shown in FIG. 6A, the ion implantation device 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 user selection devices (including touchscreens, 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.

[0032]

[0039] As further shown in FIG. 6B, 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 encapsulates 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.

[0033]

[0040] Referring now to FIG. 7, a process flow 700 according to an embodiment of the present disclosure is illustrated. At block 702, an initial substrate surface map is received. The substrate surface map can represent the three-dimensional coordinates of a set of points on the substrate surface and a map of the OPD according to the x, y coordinates, where the OPD is represented by the z coordinate of a given surface point with respect to a reference x, y plane.

[0034]

[0041] In block 704, a spherical curvature map is generated from the initial substrate surface map using a model. In some examples, the spherical curvature map may correspond to a surface modeled as a paraboloid using an average model or a Gaussian model, as detailed above.

[0035]

[0042] In block 706, a residual surface is extracted based on the initial substrate surface map and the spherical curvature map. From this, the residual surface may include residual regions or local regions of the OPD at various x, y portions of the substrate.

[0036]

[0043] In block 708, a residual curvature map is generated based on the residual surface. In the residual curvature map, the curvature can be plotted as the inverse of the length according to the x, y locations on the target substrate.

[0037]

[0044] In block 710, a blurred residual curvature map is generated from the residual curvature map using a blurring kernel. The blurred residual curvature map may present a qualitative pattern of the same curvature regions as the residual curvature map, but the width of the regions will be wider and the curvature values may be different from those of the non-blurred residual curvature map. This blurring can be used to reveal size effects, such as the beam size of the scan energy source used to implement the dose map based on the residual curvature map.

[0038]

[0045] In block 712, if there is a positive curvature component in the blurred residual curvature map, it is removed, and a filtered residual curvature map is generated.

[0039]

[0046] In block 714, a dose map for processing the substrate is generated based on the filtered residual curvature map. This dose map can present a pattern qualitatively similar to the filtered residual curvature map, where the relative dose is increased in the x, y regions with relatively higher curvature.

[0040]

[0047] Referring now to FIG. 8, a process flow 800 according to an embodiment of the present disclosure is illustrated. At block 802, the flow continues from block 704. Specifically, based on the spherical curvature map of a given substrate, a stress compensation layer deposition recipe is generated. With this recipe, several parameters, such as the type of layer, deposition conditions, and layer thickness, can be specified.

[0041]

[0048] At block 804, a backside layer is deposited on a given substrate based on the stress compensation layer deposition recipe.

[0042]

[0049] Referring now to FIG. 9, a process flow 900 according to an embodiment of the present disclosure is illustrated. At block 902, the flow proceeds from block 714, and the dose map detailed in process flow 700 is received in a patterning energy tool (such as an ion implantation device).

[0043]

[0050] At block 904, the flow proceeds from block 804, and a substrate having a backside layer based on the stress compensation layer deposition recipe is received into the patterning energy tool.

[0044]

[0051] At block 906, the dose map is applied to the backside layer using the patterning energy source (such as a scanned ion beam) of the patterning energy tool.

[0045]

[0052] The advantages provided by the embodiments of this document are diverse. As a first advantage, the approach of this document enables subsequent devices to progress more accurately (e.g., subsequent lithography steps that require low planar strain). As a second advantage, the approach of this document more accurately reduces regions of high planar strain by targeting residual areas of high substrate curvature for processing at higher energies.

[0046]

[0053] The present 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 the present disclosure will be apparent to those of ordinary skill in the art in addition to the embodiments described herein. Accordingly, such other embodiments and modifications are intended to be included within the scope of the present disclosure. Further, although the present disclosure is described herein in the context of specific implementations for specific purposes in a specific environment, those of ordinary skill in the art will recognize that the utility is not limited to such context, and that the present disclosure may 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 spirit of the present disclosure as described herein.

Claims

1. A method comprising: generating a residual curvature map of the substrate based on measurements of the surface 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.

2. The method of claim 1, wherein generating the residual curvature map comprises: modeling a spherical curvature map based on an initial substrate surface map of out-of-plane strain of the surface of the substrate; extracting the spherical curvature map from the initial substrate surface map to generate an unprocessed residual curvature map; applying a blurring kernel process to the unprocessed residual curvature map.

3. The method of claim 2, wherein generating the residual curvature map further comprises applying a filter to remove positive curvature from the residual curvature map.

4. The method of claim 2, wherein the substrate curvature represented by the spherical curvature map is removable by performing a blanket process step.

5. The method of claim 4, wherein the blanket process step comprises depositing a stress compensation layer on the back side of the substrate.

6. The method of claim 5, wherein the stress compensation layer comprises a layer containing silicon nitride, silicon oxide, silicon oxynitride, or any combination of Si—O—N—C.

7. The method of claim 1, wherein applying the dose map comprises: exposing the 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 pattern without using a mask to transfer the dose map to the substrate.

8. The method of claim 1, wherein the patterning energy source comprises an ion beam, an electron beam, or a laser beam.

9. A method comprising: receiving a substrate surface map of the substrate, the substrate surface map including a map of out-of-plane strain 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 A method comprising: **Claim 10** By extracting the spherical curvature map from the substrate surface map, an untreated residual curvature map is generated, and the method includes Creating a blur kernel using the beam profile of the patterning energy source; Applying the blur kernel to the untreated residual curvature map to generate a blurred residual curvature map The method according to claim 9, further comprising: **Claim 11** The method according to claim 10, further comprising applying a filter to remove positive curvature from the blurred residual curvature map. **Claim 12** The method according to claim 9, wherein the substrate curvature represented by the spherical curvature map can be removed by performing a blanket treatment step. **Claim 13** The method according to claim 12, wherein the blanket treatment step includes depositing a stress compensation layer on the back side of the substrate. **Claim 14** The method according to claim 13, wherein the stress compensation layer includes a layer containing silicon nitride, silicon oxide, silicon oxynitride, or any combination of Si—O—N—C. **Claim 15** Applying the dose map includes Exposing the 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 pattern without using a mask to transfer the dose map to the substrate The method according to claim 9, comprising: **Claim 16** The method according to claim 9, wherein the patterning energy source includes an ion beam, an electron beam, or a laser beam. **Claim 17** A method, comprising: Receiving a substrate surface map of a substrate, the substrate surface map including a map of out-of-plane distortion (OPD) of the substrate based on a set of measured out-of-plane distortions (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 a patterning energy source A method comprising

18. generating the dose map based on the unprocessed residual curvature map, using the beam profile of the patterning energy source to create a blocker kernel, applying the blocker kernel to the unprocessed residual curvature map to generate a blurred residual curvature map, applying a filter to remove positive curvature from the blurred residual curvature map The method according to claim 17, comprising

19. The method according to claim 17, wherein the model comprises a Gaussian curvature model or an average curvature model.

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