Method for manufacturing single crystal silicon ingot with reduced radial resistivity variation

Abstract

A method for producing a single crystal silicon ingot with reduced radial resistivity variation is disclosed. During growth of the single crystal silicon ingot, the silicon melt is counterdoped in multiple batches. The amount of the second dopant in each batch is controlled to be less than a maximum amount determined based on a predetermined target minimum resistivity, a predetermined target maximum radial resistivity gradient, and a baseline radial resistivity gradient that would occur without counterdoping.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001]

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 486,126, filed February 21, 2023, the contents of which are incorporated herein by reference. [Technical Field]

[0002]

[0002] The present disclosure relates to a method for producing a single crystal silicon ingot with reduced radial resistivity variation, and a method for determining the batch number of counter dopants to add to the silicon melt from which the single crystal silicon ingot is grown in order to increase the saleable number of wafers cut from the single crystal silicon ingot. [Background technology]

[0003] Counterdoping is used in various solar and semiconductor applications to improve throughput and prime yield of single crystal silicon ingots. For example, in N-type IGBT applications, counterdoping with P-type dopants may be performed to achieve relatively tight resistivity tolerances (e.g., within a range of less than ±13%). While counterdoping protocols may include methods to meet relatively tight resistivity tolerances, such methods do not address the control of radial resistivity variations during the counterdoping transition.

[0004]

[0004] Radial resistivity variations can depend on the interface geometry (i.e., the boundary between the growing crystal and the melt) and the selected growth conditions (e.g., melt flow conditions, hot zone configuration, magnet type, etc.). The addition of counterdopants during crystal growth induces transient radial resistivity changes (depending on melt flow dynamics, interface, magnetic field type and strength, crucible rotation, seed crystal rotation, etc.) on time scales of tens of minutes. During this period, the radial resistivity gradient of the material can exceed customer specifications, resulting in the material being discarded. The discarded length depends on this transient time scale and the pull rate and can exceed tens of millimeters.

[0005] A need exists for a counterdoping method that reduces or eliminates the axial ingot portion that contains out-of-specification radial resistivity gradients.

[0006] This section is intended to introduce the reader to various aspects of technology that may be related to various aspects of the disclosure described below. This discussion is believed to be helpful in providing the reader with background information to better understand the various aspects of the present disclosure. As such, it should be read in this light, and not as admissions of prior art. Summary of the Invention

[0007]

[0007] One aspect of the present disclosure relates to a method for producing a single crystal silicon ingot with reduced radial resistivity variation during counterdoping. The single crystal silicon ingot has a predetermined target minimum resistivity (Rmin) and a predetermined target maximum radial resistivity gradient (RRGmax). An initial charge of solid silicon is added to a crucible. The crucible containing the initial charge of solid silicon is heated to form a silicon melt in the crucible. A first dopant is added to the crucible to form a doped silicon melt. A silicon seed crystal is contacted with the doped silicon melt. The silicon seed crystal is pulled to grow a single crystal silicon ingot. During growth of the single crystal silicon ingot, a second dopant is added to the silicon melt in multiple batches to counterdope the silicon melt. The amount of the second dopant added in each batch does not exceed a maximum amount (Mmax). The maximum amount Mmax is determined by determining the base radial resistivity gradient (RRGbase) that occurs without counterdoping. The maximum resistivity (Rmax) to which the resistivity of the silicon melt can be increased during counterdoping without exceeding RRGmax is given by the following formula: Rmax=((RRGmax-RRGbase)*Rmin)+Rmin is determined by Mmax is the maximum amount of second dopant that can be added to the batch without increasing the resistivity of the melt above Rmax.

[0008]

[0008] Yet another aspect of the present disclosure is a method for determining the number of batches of counterdopant to add to a silicon melt from which a single crystal silicon ingot is grown in order to increase the number of saleable wafers cut from the single crystal silicon ingot. The batches of counterdopant are added during the growth of the single crystal silicon ingot. The silicon melt is doped with a first dopant, different from the counterdopant, before ingot growth. The maximum amount of counterdopant that can be added in each batch (Mmax) is determined by determining the base radial resistivity gradient (RRGbase) that occurs without counterdoping. The maximum resistivity (Rmax) that can be increased during counterdoping without the resistivity of the silicon melt exceeding RRGmax is determined by the following formula: Rmax=((RRGmax-RRGbase)*Rmin)+Rmin is determined by. where Rmin is the predetermined target minimum resistivity of the single crystal silicon ingot; RRGmax is the predetermined target maximum radial resistivity gradient of the single crystal silicon ingot; Mmax is the maximum amount of counterdopant that can be added in a batch without increasing the resistivity of the melt beyond Rmax as determined from the model; and the number of counterdopant batches is selected to maximize the length of the ingot with a resistivity above Rmin without exceeding the predetermined total counterdopant concentration (Cmax).

[0009] Various refinements exist for the features of the above-described aspects of the present disclosure. Furthermore, additional features may be incorporated into the above-described aspects of the present disclosure. These refinements and additional features may exist alone or in any combination. For example, the various features discussed below in connection with any of the illustrated embodiments of the present disclosure may be incorporated into any of the above-described aspects of the present disclosure, alone or in any combination. [Brief explanation of the drawings]

[0010] [Figure 1] FIG. 1 is a cross-sectional view of an ingot pulling apparatus before growing a silicon ingot. [Figure 2] FIG. 2 is a cross-sectional view of an ingot pulling apparatus during silicon ingot growth. [Figure 3] FIG. 3 is a graph showing the change in resistivity and radial resistivity gradient upon counterdoping in a single batch. [Figure 4] FIG. 4 is a graph showing the relationship between radial resistivity and edge exclusion after counterdoping in a single batch. [Figure 5] FIG. 5 is a graph of resistivity and radial resistivity gradient for wafers sliced ​​from a single batch of counterdoped ingot. [Figure 6] FIG. 6 is a graph of resistivity variation across the wafer radius at different radial locations for wafers sliced ​​from a single batch of counterdoped ingots. [Figure 7] FIG. 7 is a graph of the resistivity of the multiple batch counterdoped ingots. [Figure 8] FIG. 8 is a graph of the modeled boron concentration along the body length of a multiple batch counterdoped ingot.

[0011]

[0018] Corresponding reference characters indicate corresponding parts throughout the drawings. DETAILED DESCRIPTION OF THE INVENTION

[0012]

[0019] The present disclosure provides a method for determining the batch number and / or size of a counterdopant to be added to a silicon melt based on a target maximum radial resistivity gradient of wafers cut from the single crystal silicon ingot. The disclosed method can be generally implemented in any ingot puller configured to pull a single crystal silicon ingot. An exemplary ingot puller (or simply "ingot puller") is generally designated "100" in FIG. 1. The ingot puller 100 includes a crucible 102 that holds a melt 104 of semiconductor or solar-grade material, such as silicon, supported by a susceptor 106. The ingot puller 100 includes a crystal pulling housing 108 that defines a growth chamber 152 for pulling a silicon ingot 113 (FIG. 2) from the melt 104 along a pulling axis A.

[0013]

[0020] Crucible 102 includes a bottom surface 129 and a sidewall 131 extending upwardly from bottom surface 129. Sidewall 131 is generally vertical. Bottom surface 129 includes a curved portion of crucible 102 that extends below sidewall 131. Within crucible 102 is a silicon melt 104 having a melt surface 111 (i.e., a melt-ingot interface).

[0014]

[0021] In some embodiments, the crucible 102 has a layered structure. For example, the crucible 102 can be composed of a quartz base layer and a synthetic quartz liner disposed on the quartz base layer.

[0015]

[0022] The susceptor 106 is supported by a shaft 105. The susceptor 106, crucible 102, shaft 105, and ingot 113 (FIG. 2) have a common longitudinal axis A or "pulling axis" A.

[0016]

[0023] The ingot pulling apparatus 100 includes a pulling mechanism 114 for growing and pulling an ingot 113 from the melt 104. The pulling mechanism 114 includes a pulling cable 118, a seed crystal holder or chuck 120 connected to one end of the pulling cable 118, and a silicon seed crystal 122 connected to the holder or chuck 120 as a seed crystal for initiating crystal growth. One end of the pulling cable 118 is connected to a pulley (not shown) or drum (not shown) or other suitable lifting mechanism, such as a shaft, and the other end is connected to the chuck 120, which holds the seed crystal 122. In operation, the seed crystal 122 is lowered until it contacts the melt 104. The pulling mechanism 114 is activated to raise the seed crystal 122, thereby pulling a single crystal ingot 113 ( FIG. 2 ) from the melt 104.

[0017]

[0024] During heating and crystal pulling, a crucible drive unit 107 (e.g., a motor) rotates the crucible 102 and susceptor 106. A lift mechanism 112 raises and lowers the crucible 102 along the pull axis A during the growth process. For example, the crucible 102 is positioned at its lowest point (near the bottom heater 126) and an initial source of solid polycrystalline silicon, previously charged into the crucible 102, is melted. Crystal growth begins by contacting the melt 104 with a seed crystal 122 and lifting the seed crystal 122 with the pull mechanism 114. As the ingot grows, the silicon melt 104 is consumed and the melt height within the crucible 102 decreases. The crucible 102 and susceptor 106 can be raised to maintain the melt surface 111 at or near the same position relative to the ingot puller 100 (FIG. 2).

[0018]

[0025] The crystal drive unit (not shown) can also rotate the pull cable 118 and ingot 113 ( FIG. 2 ) in a direction opposite (e.g., counter-rotation) to the direction in which the crucible drive unit 107 rotates the crucible 102. In embodiments using uniform rotation, the crystal drive unit can rotate the pull cable 118 in the same direction as the crucible drive unit 107 rotates the crucible 102. Additionally, the crystal drive unit raises and lowers the ingot 113 relative to the melt surface 111 as needed during the growth process.

[0019]

[0026] The ingot puller 100 may also include an inert gas system that introduces and exhausts an inert gas, such as argon, into the growth chamber 152. The ingot puller 100 may also include a dopant delivery system (not shown) for introducing dopants into the melt 104.

[0020]

[0027] According to the Czochralski single crystal growth process, solid silicon (e.g., polycrystalline silicon) is loaded into a crucible 102. The initial semiconductor- or solar-grade material introduced into the crucible is melted by heat provided by one or more heating elements to form a silicon melt within the crucible. The ingot puller 100 includes bottom insulation 110 and side insulation 124 for retaining heat within the puller. In the illustrated embodiment, the ingot puller 100 includes a bottom heater 126 positioned below the crucible bottom surface 129. The crucible 102 can be moved into relatively close proximity to the bottom heater 126 to melt the polycrystalline silicon loaded into the crucible 102.

[0021]

[0028] To form an ingot, a seed crystal 122 is brought into contact with the surface 111 of the melt 104. A pulling mechanism 114 is activated to pull the seed crystal 122 from the melt 104. Referring to FIG. 2 , the ingot 113 includes a crown portion 142 that transitions and tapers outward from the seed crystal 122 until it reaches a target diameter. The ingot 113 includes a constant diameter portion 145 or cylindrical "body" of the crystal that is grown by increasing the pulling rate. The body 145 of the ingot 113 has a relatively constant diameter. The ingot 113 includes a tail or end cone portion (not shown) that tapers in diameter behind the body 145. Once the diameter is sufficiently small, the ingot 113 is separated from the melt 104.

[0022]

[0029] The ingot pulling apparatus 100 includes a side heater 135 and a susceptor 106 surrounding the crucible 102 to maintain the temperature of the melt 104 during crystal growth. The side heater 135 is positioned radially outward relative to the crucible sidewall 131 as the crucible 102 moves up and down along the pulling axis A. The side heater 135 and the bottom heater 126 can be any type of heater operable as described herein. In some embodiments, the heaters 135, 126 are resistive heaters. The side heater 135 and the bottom heater 126 are controlled by a control system (not shown) to control the temperature of the melt 104 throughout the pulling process.

[0023]

[0030] The ingot pulling apparatus 100 can include a heat shield 151. The heat shield 151 covers the ingot 113 and can be positioned within the crucible 102 during crystal growth (FIG. 2).

[0024]

[0031] According to an embodiment of the present disclosure, the silicon melt 104 is doped with a first dopant (i.e., by adding the first dopant to the solid silicon before or during melting, or to the silicon melt after melting), and then counterdoped as the single crystal silicon ingot is pulled from the melt. The single crystal silicon ingot has a predetermined target minimum resistivity (Rmin) (labeled "Res LSL" in FIG. 3 ), and the growth process (e.g., by counterdoping) is adapted to increase the amount of the single crystal silicon ingot above Rmin (i.e., the saleable portion of the ingot). As shown in FIG. 3 , the resistivity of the melt can decrease during ingot growth due to differences in the segregation factors of N-type and P-type species. Counterdoping during ingot growth increases the resistivity, increasing the prime portion of the ingot (i.e., the portion from which saleable wafers can be sliced). The ingot has a predetermined maximum resistivity (labeled "Res USL" in FIG. 3 ), and the portion of the ingot above this maximum resistivity can be non-prime.

[0025]

[0032] The single crystal silicon may have a predetermined target maximum radial resistivity gradient (RRGmax). In some embodiments, RRGmax is determined from the difference between the resistivity at the central axis of the crystalline silicon ingot and the resistivity at an edge (e.g., 6 mm from the edge) divided by the resistivity at the center ((CE) / C). In some embodiments, the predetermined (i.e., preselected prior to growth of the single crystal silicon ingot) minimum resistivity, maximum resistivity, and maximum radial resistivity gradient are preselected (i.e., selected prior to ingot growth, and ingot growth parameters are adjusted to avoid exceeding the specification limits) as customer specifications for single crystal silicon wafers sliced ​​from the single crystal silicon ingot. For example, Rmin may be 10 ohm-cm or greater, 20 ohm-cm or greater, 30 ohm-cm or greater, 50 ohm-cm or greater, 100 ohm-cm or greater, or 200 ohm-cm or greater (10 ohm-cm to 1000 ohm-cm, 10 ohm-cm to 500 ohm-cm, or 10 ohm-cm to 250 ohm-cm). Alternatively, or in addition, RRGmax may be 10% or less, 6% or less, or 5% or less.

[0026]

[0033] After the doped melt is prepared and stabilized, ingot growth is initiated by contacting a silicon seed crystal 122 with the doped silicon melt 104 and pulling the silicon seed crystal 122 to grow a single crystalline silicon ingot 113 (FIG. 2). As the ingot 113 is pulled from the melt 104, the melt 104 is counterdoped with a second dopant (also referred to herein as a "counterdopant") in multiple batches (e.g., not continuously, but in discrete amounts with time intervals between batches during which the second dopant is not added).

[0027]

[0034] The first dopant used for initial doping of the ingot is of a first type (i.e., P-type or N-type), and the second dopant added to the melt during ingot growth is of a second type opposite the first type (i.e., if the first dopant is P-type, the second dopant is N-type, and vice versa). For example, the first dopant may be N-type, such as phosphorus, and the second dopant may be P-type, such as boron. The first dopant is introduced into the ingot puller 100 as a solid (directly added to the melt or solid added to the growth chamber 152 where it sublimes and outgasses due to the heat of the ingot puller) or as a gas. In some embodiments, the second dopant is added in a batch process as a solid (e.g., boron-doped silicon chips). The second dopant is added between the crucible sidewall 131 and the growing crystal. The second dopant migrates toward the center of the melt, creating a second dopant concentration gradient between the crystal's central axis and its periphery. The dopant pathway in the melt depends on the melt flow and diffusion, which in turn depend on the hot zone and magnet. The dopant pathway determines whether the dopant is taken up from the crystal's edge or center.

[0028]

[0035] According to embodiments of the present disclosure, the amount of dopant in each batch of added second dopant does not exceed a maximum amount (Mmax), which, as described below, prevents the radial resistivity gradient from exceeding a predetermined maximum value (RRGmax).

[0029]

[0036] In one embodiment of the present disclosure, Mmax is determined by first determining the maximum resistivity (Rmax) that a batch addition can increase the resistivity of the melt (i.e., the solidified portion of the ingot) without exceeding RRGmax. For example, Rmax can be calculated using the following formula: Rmax=((RRGmax-RRGbase)*Rmin)+Rmin (1) It may be determined by: where R is a predetermined target minimum resistivity (R), and R is the reference radial resistivity gradient (RR) that occurs without counterdoping (i.e., the radial resistivity gradient that occurs during ingot growth with doping with a first dopant but without adding a batch containing a second dopant).

[0030]

[0037] Once Rmax is known, Mmax can be determined. Mmax is the maximum amount of second dopant that can be added to a batch without increasing the resistivity of the melt above Rmax. The maximum amount of second dopant that can be added to a batch without increasing the resistivity of the melt above Rmax (Mmax) can be determined, for example, from a model that correlates dopant amount with ingot resistivity.

[0031]

[0038] Knowing the maximum amount (Mmax) of second dopant that can be added in a batch without increasing the resistivity of the melt above Rmax allows the number of additions of second dopant (or the number of "batches" to which second dopant is added) to be selected. The number of batches can be selected to maximize the length of the ingot with a resistivity above Rmin. In some embodiments, the number of batches is kept below the number that exceeds a predetermined total second dopant concentration (Cmax) (e.g., the amount by which wafers sliced ​​from the ingot exceed an upper limit for the total second dopant concentration, such as a customer-defined upper limit). The number of batches of second dopant selected to maximize the length of the ingot with a resistivity above Rmin without exceeding a predetermined total second dopant concentration (Cmax) can be determined from a model. In some embodiments, Cmax is 1×10 15 atoms / cm 3 Less than 1×10 14 atoms / cm 3 Less than or equal to 1 x 10 13 atoms / cm 3 is less than.

[0032]

[0039] The disclosed method offers several advantages over conventional techniques. In conventional techniques, the portion of the ingot grown immediately after counterdoping is discarded because the radial resistivity gradient exceeds a predetermined upper limit. This transition time is even worse under low-oxygen controlled conditions. This is because the melt flow rate is slower than typical Czochralski growth conditions, extending the transition time for the radial resistivity change. By using a formula for determining Rmax (e.g., (RRGmax-RRGbase)*Rmin)+Rmin), a counterdoping dose that remains within the radial resistivity gradient limit (and above a predetermined minimum resistivity) can be determined independently of the target resistivity. Counterdoping strategies, including intermittent counterdoping, improve prime yield while reducing the radial resistivity gradient. [Example]

[0033]

[0040] The processes herein are further illustrated by the following examples, which should not be construed as limiting.

[0034] Example 1: Effect of Boron Counterdoping in a Single Batch

[0041] Figure 3 shows the case where an N-type monocrystalline silicon ingot was counterdoped with boron during ingot growth in a single batch. Y1 is resistivity (ohm·cm), Y2 is the radial resistivity gradient (RRG%) between the ingot center axis and 6 mm from the edge, and X is the body position. The prescribed maximum resistivity ("Res USL") and prescribed minimum resistivity ("Res LSL") are also shown. The Y1 axis shows the resistivity measurement of the center slug (white square), and the Y2 axis shows the RRG% (white circle). The RRG% was measured 6 mm from the edge.

[0035]

[0042] A single batch doping with boron resulted in a resistivity increase from the lower spec limit to the upper spec limit, and the RRG% shifted from the typical baseline characteristic to significantly greater than 10%, causing the material to be discarded (e.g., body length 20-40 mm). Once the resistivity had fallen to the lower spec limit, counterdoping to the upper spec limit could be repeated until the end of body growth, with a portion of the ingot discarded as prime loss each time.

[0036] Example 2: Transient Effects of Solid-Phase Boron Counterdoping in a Single Batch

[0043] Ingots were prepared according to the method of Example 1 (single-batch direct solid-phase boron addition). Figure 4 shows the radial resistivity profiles from the center (0 mm) to the edge (140 mm). Wafer 1 is immediately after dopant addition, while wafer 18 is after several tens of minutes of axial crystal growth. In this case, boron counterdopant was added when the resistivity reached near the lower limit, resulting in doping with enough boron to satisfy the upper limit. As shown in Figure 4, the resistivity immediately increased sharply at the radial edge and then gradually shifted radially toward the center. The radial resistivity gradient (RRG%) exceeded 10% for all wafers except wafer 18, whose resistivity stabilized radially to the standard characteristic value.

[0037]

[0044] A second ingot was prepared in the same manner, and the resistivity of the center and edge (140 mm) of consecutive wafers was plotted on the Y1 axis. The upper resistivity limit (USL) and lower resistivity limit (LSL) are also shown. The radial resistivity gradient ((center-edge) / center) is plotted on the Y2 axis. Wafer 5 was doped with boron directly. The edge of the wafer rose to the counterdoping level, and it took approximately 20 wafers for the resistivity transition at the center to be complete. The RRG% immediately reversed from a positive value, where the center resistivity was higher than the edge, to a negative value, where the edge resistivity was higher than the center, and maintained a negative RRG until the transition was complete. Although the resistivity shift was approximately 50% of the specification limit defined by the USL / LSL line, the absolute value of the RRG% shifted outside the standard performance range of the process.

[0038]

[0045] For the same ingot of wafers 1-25, the radial variation in the counterdoping transition is shown in Figure 6. The radial resistivity gradient was calculated every 10 mm ((center - radius position) / center). The maximum bar indicates ±10% RRG.

[0039] Example 3: Effect of solid-phase boron counterdoping in multiple batches

[0046] Ingots were grown using the process described in Example 2, but with a different counterdoping protocol. It was found that intermittent direct solid-state doping can reduce or completely eliminate prime loss by adjusting the doping concentration to keep the resistivity change (measured at the transition point between center and edge resistivity) below 10% RRG, or within the normal RRG% variation range for the process. Figure 7 shows the addition of small amounts of solid-state boron dopant multiple times (16 times in the figure) during crystal growth.

[0040]

[0047] The resistivity shift due to counterdoping was kept relatively small to maintain the typical RRG% expected for non-counterdoped products. The boron counterdoping addition was set to provide an overall RRG% of less than 6%, and the counterdoping amount was determined according to the method described herein.

[0041]

[0048] Figure 7 also demonstrates the benefit of multiple counterdoping (solid line) over no counterdoping (thin dashed line): counterdoping can extend prime length by nearly two-fold (arrow) while maintaining RRG% within the normal performance range.

[0042]

[0049] To ensure an adequate depletion layer thickness that affects threshold voltage, it is desirable to minimize counterdoping elements, so it is possible to set an overall upper limit on the boron concentration allowed in any wafer sliced ​​from the ingot (e.g., 1 × 10 15 atoms / cm 3 Less than 1×10 14 atoms / cm 3Less than 1×10 or as per customer request 13 atoms / cm 3 To increase the sealable area of ​​the ingot, it is advantageous to delay the start of the boron counterdoping until the crystal resistivity approaches the lower resistivity limit. The boron concentration in the modeled crystal is shown in Figure 8. The overall boron concentration (atoms / cm) versus body position 3 ) is shown. As shown in Figure 8, a counterdoping protocol was designed to keep the boron concentration low.

[0043]

[0050] As used herein, the terms "about," "substantially," "essentially," and "approximately," when used in conjunction with ranges of dimensions, concentrations, temperatures, or other physical or chemical properties, are intended to encompass variations that may exist at the upper and / or lower limits of those property ranges (including, for example, variations due to rounding error, measurement method, and other statistical variations).

[0044]

[0051] When introducing elements of this disclosure or embodiments thereof, the articles "a," "an," "the," "said," etc. mean that there are one or more elements. The terms "comprising," "including," "containing," and "having" are inclusive and mean that there may be additional elements other than the listed elements. The use of specific orientational terms (e.g., "top," "bottom," "side," etc.) is for convenience of description and does not require a specific orientation of the described articles.

[0045]

[0052] Because various changes may be made in the above structures and methods without departing from the scope of the disclosure, all matter contained in the above description and shown in the accompanying drawings is intended to be interpreted as illustrative and not in a limiting sense.

Claims

1. 1. A method for producing a single crystal silicon ingot with reduced radial resistivity variation during counterdoping, the single crystal silicon ingot having a predetermined target minimum resistivity (Rmin) and a predetermined target maximum radial resistivity gradient (RRGmax), the method comprising: adding an initial charge of solid silicon to a crucible; heating a crucible containing an initial charge of solid silicon to form a silicon melt within the crucible; adding a first dopant to the crucible to form a doped silicon melt; contacting the doped silicon melt with a silicon seed crystal; Pulling a silicon seed crystal and growing a single crystal silicon ingot; and Counterdoping a silicon melt during growth of a single crystal silicon ingot by adding a second dopant to the silicon melt in multiple batches, wherein the amount of the second dopant in each batch does not exceed a maximum amount (Mmax), where Mmax is as follows: determining a reference radial resistivity gradient (RRG) that occurs without counterdoping; and The following formula: Rmax=((RRGmax−RRGbase)*Rmin)+Rmin; where Mmax is the maximum amount of second dopant that can be added to the batch without increasing the resistivity of the melt above Rmax; determining the maximum resistivity (Rmax) to which the resistivity of the silicon melt can be increased during counterdoping without exceeding RRGmax; A step determined by A manufacturing method comprising:

2. 10. The method of claim 1, wherein the maximum amount of second dopant that can be added to the batch without increasing the resistivity of the melt above Rmax is determined from a model.

3. 3. The method of claim 1 or 2, wherein the number of batches of the second dopant is selected to maximize the length of the ingot whose resistivity exceeds Rmin without exceeding a predetermined total second dopant concentration (Cmax).

4. Cmax is 1 x 10 １５ atoms / cm ３ Less than 1 x 10 １４ atoms / cm ３ Less than or 1 x 10 １３ atoms / cm ３ The method of claim 3 wherein the

5. 5. The method of claim 3 or 4, wherein the number of batches of the plurality of second dopant batches selected to maximize the length of the ingot whose resistivity exceeds Rmin without exceeding a predetermined total second dopant concentration (Cmax) is determined from the model.

6. The method according to any one of claims 1 to 5, wherein the first dopant is N-type and the second dopant is P-type.

7. 7. The method of claim 6, wherein the first dopant is phosphorus and the second dopant is boron.

8. 8. The method according to any one of claims 1 to 7, wherein RRGmax and RRGbase are based on a value obtained by dividing the difference between the resistivity at the central axis of the crystalline silicon ingot and the resistivity at an end of the crystalline silicon ingot by the resistivity at the central axis.

9. The method according to any one of claims 1 to 8, wherein Rmin and RRGmax are customer specifications.

10. The method according to any one of claims 1 to 9, wherein RRGmax is 10% or less, 6% or less, or 5% or less.

11. 11. The method of any of claims 1 to 10, wherein Rmin is 10 ohm-cm or greater, 20 ohm-cm or greater, 30 ohm-cm or greater, 50 ohm-cm or greater, 100 ohm-cm or greater, or 200 ohm-cm or greater.

12. A method for determining the number of batches of counter dopant to be added to a silicon melt for growing a single crystal silicon ingot in order to increase the number of saleable wafers cut from the single crystal silicon ingot, wherein the batches of counter dopant are added during the growth of the single crystal silicon ingot, and the silicon melt is doped with a first dopant different from the counter dopant before the ingot is grown, comprises the following steps: The maximum amount of counter dopant that can be added in each batch (Mmax) is determining a base radial resistivity gradient (RRG) that occurs without counterdoping; and The maximum resistivity (Rmax) that can be increased in resistivity of the silicon melt during counterdoping without exceeding RRGmax is determined by the following formula: Rmax=((RRGmax−RRGbase)*Rmin)+Rmin; determining by, Rmin is the predetermined target minimum resistivity of the single crystal silicon ingot, RRGmax is the predetermined target maximum radial resistivity gradient of the single crystal silicon ingot, and Mmax is the maximum amount of counter dopant that can be added in the batch without increasing the resistivity of the melt beyond Rmax as determined from the model; and selecting the number of batches of counterdopants that maximizes the length of the ingot having a resistivity above Rmin without exceeding a predetermined total counterdopant concentration (Cmax).

13. 13. The method of claim 12, wherein the number of batches of counterdopants that maximizes the length of the ingot whose resistivity exceeds Rmin without exceeding a predetermined total counterdopant concentration (Cmax) is determined from the model.

14. 14. The method of claim 12 or 13, wherein the first dopant is N-type and the counter dopant is P-type.

15. 15. The method of claim 14, wherein the first dopant is phosphorus and the counterdopant is boron.

16. The method according to any one of claims 12 to 15, wherein RRGmax and RRGbase are each based on a value obtained by dividing the difference between the resistivity at the central axis of the crystalline silicon ingot and the resistivity at the end of the crystalline silicon ingot by the resistivity at the central axis.

17. The method according to any one of claims 12 to 16, wherein Rmin and RRGmax are customer specifications.

18. The method according to any one of claims 12 to 17, wherein RRGmax is 10% or less, 6% or less, or 5% or less.

19. 19. The method of any of claims 12 to 18, wherein Rmin is 10 ohm-cm or greater, 20 ohm-cm or greater, 30 ohm-cm or greater, 50 ohm-cm or greater, 100 ohm-cm or greater, or 200 ohm-cm or greater.

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