Temperature control device design method and temperature control device

Abstract

A temperature control device (1) comprises: a heat diffusion plate (10) that is a member provided with a first surface (S1) facing a controlled object (CO) and a second surface (S2) parallel to the first surface (S1) and facing the opposite direction therefrom, and in which heat is diffused in the surface direction of the first surface (S1); and a heat source (11) that is thermally bonded with the heat diffusion plate (10) at the second surface (S2) and heats or absorbs heat from the heat diffusion plate (10). In this design method, the sizes of the heat diffusion plate (10) and the heat source (11) and the thermal conductivity and overall heat transfer coefficient of the heat diffusion plate (10) are determined on the basis of a computation formula indicating the relationship between the ambient temperature around the controlled object (CO), a target temperature for the controlled object (CO), the amount of heat input to the heat diffusion plate (10) from the heat source (11), the sizes of the heat diffusion plate (10) and the heat source (11), the thermal conductivity and the overall heat transfer coefficient of the heat diffusion plate (10), and variation in temperature on the first surface (S1) so that the difference between the maximum temperature and the minimum temperature of a part of the first surface (S1) in contact with the controlled object (CO) when the ambient temperature, the target temperature, and the input amount of heat serve as given design conditions falls within a permissible value range for in-plane temperature variation in the controlled object (CO).

Description

Temperature control device design method and temperature control device

[0001] The present invention relates to a method for designing a temperature control device and a temperature control device.

[0002] The physical properties of an object and reactions between objects are temperature-dependent. Therefore, in a processing process in which some kind of processing is performed on the entire surface of an object that has an area extending in the planar direction, in order to reduce variations in the physical properties of the object to be processed or the reaction rate of the reaction field that occurs during the processing, it is necessary to reduce temperature variations within the surface of the object. Therefore, for example, in a semiconductor manufacturing process, an assembly is disclosed that incorporates a thermal phase diffuser into a pedestal that supports a wafer to reduce temperature variations within the wafer surface (see Patent Document 1).

[0003] Special Publication No. 2020-526012

[0004] In the semiconductor manufacturing process described above, there is a tolerance for the variation in reaction rate of the reaction field within the wafer surface, and there is also a corresponding tolerance for the variation in temperature within the wafer surface. The above-described assembly can reduce the temperature variation of the wafer. However, Patent Document 1 does not disclose how to suppress the temperature variation within the wafer surface within the tolerance. To suppress the temperature variation within the tolerance, the control capability of the temperature control device that controls the temperature within the wafer surface must be designed to satisfy the tolerance.

[0005] The present invention has been made in light of the above-mentioned circumstances, and aims to provide a design method for a temperature control device and a temperature control device that can suppress temperature variation within the surface of a controlled object that has an extension in the surface direction within an allowable value.

[0006] In order to achieve the above object, a design method for a temperature control device according to a first aspect of the present invention is a design method for a temperature control device, wherein the temperature control device comprises: a heat diffusion plate which is a member provided with a first surface facing a controlled object and a second surface parallel to and facing opposite to the first surface, and which diffuses heat in a planar direction of the first surface; and a heat source which is thermally joined to the heat diffusion plate at the second surface and which heats or absorbs heat from the heat diffusion plate, Based on a calculation formula showing the relationship between the ambient temperature around the controlled object, the target temperature of the controlled object, the amount of heat input from the heat source to the heat diffusion plate, the sizes of the heat diffusion plate and the heat source, the thermal conductivity and overall heat transfer coefficient of the heat diffusion plate, and the temperature variation on the first surface, the sizes of the heat diffusion plate and the heat source, the thermal conductivity and overall heat transfer coefficient of the heat diffusion plate are determined so that the difference between the maximum and minimum temperatures of the part of the first surface that comes into contact with the controlled object when the ambient temperature, the target temperature, and the heat input amount are given design conditions falls within the allowable value for the in-plane variation of the temperature of the controlled object.

[0007] In this case, the heat diffusion plate may comprise: a housing having a sealed internal space and made of any of ceramics, ceramic composites, and inorganic materials excluding metals; and a working fluid located in the internal space, circulating in the internal space along the surface direction of the first surface while repeatedly evaporating due to heat reception and condensing due to heat release, and diffusing heat in the surface direction of the first surface.

[0008] The thermal expansion coefficient of the material of the housing is 8.0 × 10 －６ [1 / K] or less.

[0009] A tolerance for variation in the temperature of the controlled object within a plane may be set based on the temperature characteristics of the physical property value of the controlled object or, when the controlled object is a reaction field, the characteristics of the reaction rate within the plane.

[0010] The heat source may be any one of a heater, a Peltier element, and a cold plate.

[0011] The heat source and the heat diffusion plate may be integrated on the second surface.

[0012] The heat source may be incorporated into the heat diffusion plate.

[0013] The heat source and the heat diffusion plate may each have a disk-like outer shape and be arranged concentrically with each other, and the radius of the heat source may be smaller than the radius of the heat diffusion plate.

[0014] A temperature control device according to a second aspect of the present invention is designed using the design method for a temperature control device according to the first aspect.

[0015] According to the present invention, the sizes of the heat source and the heat diffusion plate, and the thermal conductivity and overall heat transfer coefficient of the heat diffusion plate can be determined so that the difference between the maximum and minimum temperatures on the first surface in contact with the controlled object is less than the allowable value for the temperature variation within the surface of the controlled object, thereby making it possible to keep the temperature variation within the surface of the controlled object within the allowable value.

[0016] FIG. 1 is a side view and a bottom view of a temperature control device according to an embodiment of the present invention. FIG. 2 is a schematic diagram showing the internal structure of a heat diffusion plate. FIG. 3 is a graph showing the temperature characteristics of a reaction rate. FIG. 4 is a graph showing the temperature characteristics of a physical property value. FIG. 5 is a schematic diagram showing how thermal energy is transmitted in the surface direction of a heat diffusion plate. FIG. 6 is a graph showing a dimensionless temperature distribution in the radial direction of a heat diffusion plate. FIG. 7 is a block diagram showing the hardware configuration of an information processing device. FIG. 8 is a flowchart of design processing by an information processing device. FIG. 9 is a graph showing the temperature characteristics of thermal conductivity of a heat diffusion plate. FIG. 10 is a schematic diagram showing a first modified example of the configuration of a temperature control device. FIG. 11 is a schematic diagram showing a second modified example of the configuration of a temperature control device. FIG. 12 is a side view showing the configuration of a substrate holding device incorporating a temperature control device. FIG. 13 is a perspective view showing a cylindrical temperature control device.

[0017] DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings. In the drawings, the same or equivalent parts are designated by the same reference numerals.

[0018] [Basic Configuration of Temperature Control Device] First, the basic configuration of a temperature control device that is the design target of the design method according to this embodiment will be described. As shown in Fig. 1, a temperature control device 1 controls the temperature of a control target CO. The control target CO represents, for example, a disk-shaped object or reaction field. The temperature control device 1 includes a heat diffusion plate 10 having a disk-shaped outer shape and a heat source 11 also having a disk-shaped outer shape.

[0019] Of the circular surfaces of the heat diffusion plate 10 whose normal direction is the z-axis direction, the surface facing the +z direction, i.e., the surface facing the control target CO, is referred to as the first surface S1. The surface whose normal direction is the z-axis direction and facing the -z direction is referred to as the second surface S2. The first surface S1 and the second surface S2 are the main surfaces on the front and back of the heat diffusion plate 10 that are parallel to each other and facing in opposite directions, and the distance between the first surface S1 and the second surface S2 is the plate thickness of the heat diffusion plate 10.

[0020] The heat diffusion plate 10 and the heat source 11 are arranged concentrically around the z-axis together with the control target CO. In this embodiment, the radius of the heat diffusion plate 10 is the same as the radius of the control target CO, and the radius of the heat source 11 is smaller than the radius of the heat diffusion plate 10.

[0021] The heat diffusion plate 10 is arranged on the +z side of the heat source 11 so as to be in surface contact and thermally bonded to the heat source 11 on the second surface S2. The control target CO is arranged on the +z side of the heat diffusion plate 10 so as to be in surface contact with the heat diffusion plate 10 on, for example, the first surface S1. The heat source 11 is in surface contact with the heat diffusion plate 10 in a circular region 11a on the second surface S2 (a region having the same radius as the heat source 11).

[0022] The heat source 11 heats or absorbs heat from the heat diffusion plate 10. The heat diffusion plate 10 diffuses heat in the planar direction of the first surface S1. The heat diffusion plate 10, whose temperature has been made uniform in the planar direction of the first surface S1 by the heating or heat absorption of the heat source 11, heats or absorbs heat from the control target CO.

[0023] When the heat source 11 heats the heat diffusion plate 10, the heat from the heat source 11 is applied to the heat diffusion plate 10 through the circular region 11a of the second surface S2. The heat diffusion plate 10 diffuses the heat in the planar direction of the first surface S1, i.e., in the radial direction centered on the z-axis direction. The heat diffused in the radial direction is transferred to the control target CO. The transferred heat controls the temperature of the control target CO to a target temperature.

[0024] When the heat source 11 cools the heat diffusion plate 10, the heat source 11 absorbs heat through the circular region 11a on the second surface S2, lowering the temperature of the heat diffusion plate 10. This temperature drop in the region 11a causes the refrigerant to move radially in the heat diffusion plate 10, uniformly lowering the temperature of the first surface S1. As a result, the heat diffusion plate 10 uniformly cools the control target CO across the entire first surface S1.

[0025] As shown in FIG. 2 , the heat diffusion plate 10 includes a housing 20 having a sealed internal space IS and a working fluid 21 sealed in the internal space IS as a refrigerant. In the internal space IS, liquefied and vaporized working fluids 21 coexist in equilibrium. The liquefied working fluid 21 exists directly above the heat source 11. The working fluid 21 vaporizes due to heat transferred from the heat source 11 and moves within the internal space IS toward the first surface S1, transferring (dissipating) heat to the housing 20 and gradually condensing. The condensed working fluid 21 returns to the heat source 11 due to capillary forces of capillary channels (not shown) formed in the internal space IS. In this way, the working fluid 21 circulates within the internal space IS while repeatedly vaporizing and condensing due to heat received, diffusing heat toward the first surface S1. The working fluid 21 may be, for example, pure water, but may also be an organic substance or a mixture of organic substance and water.

[0026] The housing 20 is made of a low thermal expansion material. Such a material is, for example, ceramics. Examples of ceramics include alumina, SN (silicon nitride), AIN (aluminum nitride), Y ２ O ３, zirconia, cordierite, diamond as a single crystal material, sapphire, etc. can be used, but are not limited to these. Furthermore, such materials include, for example, ceramic composite materials. Furthermore, ceramic composite materials can include, but are not limited to, SiSiC, CMC, etc. Furthermore, materials other than the above include, but are not limited to, glass, carbon, graphite, and silicon. In other words, the housing 20 is made of any of ceramics, ceramic composite materials, and inorganic materials other than metals.

[0027] The thermal expansion coefficients of each material are shown in the table below. The material used for the housing 20 is selected to have a smaller thermal expansion coefficient than metals such as copper. If a material with a small thermal expansion coefficient is used for the housing 20, deformation of the heat diffusion plate 10 can be reduced when the controlled CO or heat source 11 generates heat, and the change in the value of the overall heat transfer coefficient h can be reduced, making it easier to determine the overall heat transfer coefficient h. In this embodiment, the thermal expansion coefficient is 8.0×10 －６ It is desirable to select a material for the housing 20 that has a resistance of [1 / K] or less.

[0028] When heating the controlled CO, the heat source 11 can be a heater. Furthermore, the heat source 11 may include an element, device, etc. that generates heat when operated, or may be a device in which a gas, liquid, etc. that generates heat convects inside. A high-frequency induction heating device, a plasma heating device, or a laser heating device may also be used as the heat source 11. In the temperature control device 1 according to this embodiment, the type of heat source 11 is not limited, and various heat sources 11 can be used.

[0029] Furthermore, when cooling the controlled CO, the heat source 11 can be a Peltier element or a cold plate. Furthermore, a water-cooled type or a type using a heat exchanger can also be used as the heat source 11. The temperature control device 1 according to this embodiment is not limited to a specific type of heat source 11, and various types of heat sources 11 can be used.

[0030] [Effects of the Configuration of the Temperature Control Device 1] When the heat source 11 is solid, the heat source 11 and the controlled object CO are indirectly connected via the thermal diffusion plate 10. This is because when the heat source 11 and the controlled object CO are directly connected, a difference in heat transfer rate occurs between the part in contact with the heat source 11 and the part that is not, resulting in non-uniformity in the temperature distribution of the controlled object CO.

[0031] It is also possible to directly connect a heat source 11 having the same radius as the controlled object CO to the controlled object CO. However, in this case, if the heat source 11 is, for example, a heater, localized changes in electrical resistance due to temperature variations within the heater make it difficult to achieve sufficient temperature smoothness. Furthermore, if the heat source 11 is, for example, a Peltier module, multiple elements with heat transfer surfaces smaller than the entire module are arranged on the surface, and each element provides or absorbs heat. However, there are gaps between the elements, and to mitigate the effects of these gaps, a small heat diffusion plate is placed on the heat transfer surface. This heat diffusion plate is much smaller than the heat diffusion plate 10. Because this heat diffusion plate is made of copper, its heat diffusion performance is insufficient. As a result, it is unable to fully mitigate the non-uniformity of the heat flux between the locations where the elements are present and those where they are not, making it difficult to achieve sufficient temperature smoothness within the allowable temperature variation limit. In this way, when the heat source 11 and the control target CO are directly connected, it becomes difficult to make the heat transfer rate uniform across the entire surface of the control target CO in order to achieve a temperature distribution that is sufficiently smooth within the tolerance for temperature unevenness. To mitigate this non-uniformity in the heat transfer rate, the temperature control device 1 is configured to connect the heat source 11 via a heat diffusion plate 10.

[0032] In addition, when the heat source 11 cools the CO to be controlled, the heat source 11 may be provided with fins (not shown).

[0033] [Design Method of Temperature Control Device] A design method of the temperature control device 1 according to this embodiment will be described. In this design method, the temperature control device 1 is designed so that the in-plane temperature variation of the first surface S1 of the thermal diffusion plate 10 that contacts the controlled object CO is within an allowable value.

[0034] [Determination of Tolerance] In this design method, first, the tolerance ΔT of the temperature variation between the control target CO and the first surface S1 of the thermal diffusion plate 10 is determined. Ｃ It is necessary to determine the tolerance ΔT Ｃ is determined based on the temperature characteristics of the CO to be controlled.

[0035] [Tolerance ΔT calculated from reaction variability Ｃ ] The control target CO is assumed to be a reaction field where substances react with each other. In a reaction field having a planar spread, it is necessary to make the reaction rate uniform within the plane, that is, to suppress the variation in the reaction rate within the plane to less than the allowable value. The reaction rate of the control target CO within the plane is v, and the allowable variation in the reaction rate v is Δv ｃ According to Arrhenius' law, the reaction rate constant v ｒ The dependence of ρ on temperature T is given by the following equation (1): Figure 3A shows a curve illustrating equation (1). Here, A ｒ is the frequency factor, and Ea [J·mol －１ ] is the activation energy, and R[J.K －１ ・mol －１ ] is the gas constant, and T [K] is the temperature of the CO to be controlled.

[0036] Frequency Factor A ｒ The units of change depend on the order of the reaction. For example, if the assumed reaction is a first-order reaction, the frequency factor A ｒ The unit is [S －１ In this case, the concentration of the reactant is C Ａ [mol m －３ ], the reaction rate v is expressed by the following formula: From this, the variation Δv in the reaction rate v is expressed by the following formula: Here, <v ｒ >, <C Ａ > are the reaction rate constants v at radius r, respectively. ｒ and concentration C Ａ The second-order minute quantity Δv is the average value within the plane. ｒ , ΔC Ａis small compared to the others and is small enough to be ignored, the following equation is obtained from equation (3):

[0037] The tolerance for the variation in the reaction rate in the reaction field is Δη. In this case, the variation Δv in the reaction rate v must satisfy the condition expressed by the following formula: t ｐ is the reaction time. The variation Δv of the reaction rate v that satisfies this conditional expression is the tolerance value Δv of the variation of the reaction rate v in the above expression (4). ｃ Therefore, the following equation is obtained from the above equation (4): This formula is transformed into the following: Average concentration <C Ａ >In-plane concentration non-uniformity ΔC Ａ If is sufficiently small, the following equation is obtained from equation (5):

[0038] Here, the minimum temperature in the surface of the thermal diffusion plate 10 is T ｍｉｎ [K], and the maximum value is T ｍａｘ [K], as shown in FIG. 3A, ΔT=T ｍａｘ -T ｍｉｎ and from equation (1) Therefore, the condition that the variation ΔT of the temperature T within the surface must satisfy is given by the following equation: Here, the upper limit of ΔT that satisfies the formula (8) is the allowable temperature variation ΔT ｒ The range of ΔT that satisfies equation (8) can be determined using a solution method such as the bisection method. Therefore, by the above procedure, the allowable temperature variation ΔT when the controlled CO is the reaction field can be determined. ｒ can be determined.

[0039] Furthermore, as shown in FIG. 3B, the physical property value Φ ｐｃ．ｉ The temperature dependence of (pc.i=1, 2, 3, . . . , n: n is an arbitrary integer) is given by the following equation (9). where f ｐｃ．ｉ (T) is a function of temperature T [K]. ｐｃ．ｉAn example of (T) is shown in FIG. ｐｉ The tolerance for variation (minor amount) within the surface is ΔΦ ｐｃ．ｉ Then, ΔΦ ｐｃ．ｉ The temperature variation tolerance ΔT to satisfy ｐｃ．ｉ must satisfy the following conditions: Therefore, the tolerance Δv, which is a small amount of the variation Δv in the reaction rate v, ｒ The temperature variation tolerance ΔT is determined from ｒ and the temperature variation tolerance ΔT determined from the tolerance of the physical property variation. ｐｃ．ｉ The smallest value among (pc.i=1, 2, 3, . . . , n) is the tolerance ΔT for the variation in the in-surface temperature T of the first surface S1 of the control object CO. ｃ Here, the tolerance for the variation in the in-surface temperature T of the first surface S1 is ΔT ｃ can be determined as:

[0040] The temperature T [K] shown in the above formulas (1) to (10) is the temperature of the CO2 to be controlled. When the thickness of the CO2 to be controlled is thin enough to be ignored, the temperature of the CO2 to be controlled and the temperature of the first surface S1 of the heat diffusion plate 10 are almost the same. When the thickness of the CO2 to be controlled is not negligible, the heat transfer coefficient of the CO2 to the surface opposite to the heat diffusion plate 10 is expressed as U ｔｏｐ The temperature of the heat diffusion plate 10 is T, and the temperature of the CO2 on the side opposite to the heat diffusion plate 10 is T ｔｏｐ and the thermal conductivity of the control target CO in the thickness direction is k ｔ , the thickness of the CO to be controlled is δ ｔ Then, when the field of heat conduction within the thermal diffusion plate 10 and the field of heat transfer to the control object CO are coupled, the following relationship holds: Therefore, for example, T>T ｔｏｐ And T ｔｏｐ >T ∞ In this case, T ｔｏｐ For , the following equation is obtained: This formula is transformed as follows: When the thickness of the control target CO cannot be ignored, T[K] in the above formulas (1) to (10) is ｔｏｐ It is necessary to calculate it as corresponding to

[0041] [Design parameters of the temperature control device] Tolerance ΔT of variation in the in-plane temperature T of the first surface S1 Ｃ The design parameters relating to the size and heat transfer of the temperature control device 1 that satisfy the above can be determined, for example, as follows.

[0042] As shown in Figure 4, consider a minute section between position r and position r + Δr in the radial direction of the heat diffusion plate 10, with the center point as the reference. This minute section forms a double cylindrical shell. The thickness of the heat diffusion plate 10 is assumed to be d. In the direction of the radius r, this shell transmits heat from position r to the center point by thermal conduction with a quantity of heat q. ｒ is entered, and the heat quantity q ｒ The amount of heat input from the heat source 11 is q ｉｎ The amount of heat from the heat diffusion plate 10 to the control target CO is q ｏｕｔ In addition, in the shell, heat is transferred in the thickness direction from the section in contact with the heat source 11, and heat is released from the section in contact with the controlled object CO. In this case, the governing equation for thermal energy conservation in the shell in a steady state can be defined as follows: Dividing both sides of the above equation by 2πrdΔr and making Δr approach 0 gives the following equation: According to Fourier's law of heat conduction, q ｒ [W m －２ ] is as follows: k ｒ is the radial thermal conductivity. According to Newton's law, q ｏｕｔ [W m －２ ] becomes as follows:

[0043] h [W m －２ ・K －１ ] is the overall heat transfer coefficient based on the first surface S1 of the thermal diffusion plate 10. ∞ is the reference temperature that is the standard for thermal diffusion. If there is a CO to be controlled, the reference temperature T ∞ The ambient temperature around the controlled object CO can be set as the thermal conductivity k ｒ [W m －１ ・K－１ ] and the overall heat transfer coefficient h [W m －２ ・K －１ ] are constant within the heat diffusion plate 10, the above formula (12) becomes as follows:

[0044] Here, the surface temperature at the radius r [m] on the first surface S1 of the control object CO is T ＊ (r) [K]. Surface temperature T ＊ (r) [K] is defined as follows: where T(r) [K] is the temperature of the heat diffusion plate 10 at the radius r, and q ｉｎ [W m －２ ] is the heat flux at the heat input to the heat diffusion plate 10.

[0045] Ambient temperature T around the controlled CO ∞ [K] and q ｒ [W m －２ ] is also constant, the following equation is obtained from equation (15): The temperature at the end of the heat source 11 (r=R) is T ＊ R [K], and T ＊ R The dimensionless temperature θ[-] based on [K] is defined by the following equation. In addition, the radius R [m] of the heat source 11 is used as a reference, and the dimensionless radius r ＊ is defined by the following formula:

[0046] Using equations (18A) and (18B), the following equation is obtained from equation (16). In addition, the Biot number in the radial direction, Bi ｒ is defined by the following formula: Also, this Biot number Bi ｒ From this, the design parameter a is defined by the following equation:

[0047] Here, R [m] is the radius of the heat source 11. d [m] is the thickness of the heat diffusion plate 10, as described above. Also, α is the ratio of the radius r of the heat diffusion plate 10 to the radius R of the heat source 11. The following equation is obtained as a general solution of the above equation (22). Here, I ０ is the zeroth-order modified Bessel function of the first kind. ０ is the modified Bessel function of the second kind of order 0.

[0048] When the above equation (22) is differentiated, the following equation is obtained. Here, I １ is the first-order modified Bessel function of the first kind. １ is a first-order modified Bessel function of the second kind.

[0049] As shown in FIG. 4, the first surface S1 can be divided into the following two regions. Zone I (0≦r≦R): a central region where heat is transferred from the bottom surface Zone II (R≦r): a peripheral region where heat is not transferred from the bottom surface In Zone I, r ＊ As approaches +0, the second term of equation (22) diverges to +∞. Ｉ Since (r) is a bounded function, C ２ must be 0. Then, the second term of the equation (22) in region I becomes 0. Also, Θ Ｉ (r ＊ ) is r ＊ = 1, which gives 1, so the following equation is obtained.

[0050] Furthermore, the edge of the heat diffusion plate 10 (r ＊ = α), the boundary condition of r ＊ When = 1, Θ ＩＩ From various boundary conditions such as Θ = 1, ＩＩ (r ＊ ) is derived. As shown in FIG. 5, the highest temperature point on the surface of the heat diffusion plate 10 is the center in contact with the center of the heat source 11, and Θ Ｉ The lowest temperature point is the end of the heat diffusion plate 10 (r=αR), and Θ ＩＩ Therefore, the dimensionless temperature variation value ΔΘ within the surface is given by the following formula: Θ Ｉ (0), Θ ＩＩ (α) can be determined from equations (25) and (26), respectively.

[0051] Θ Ｉ (0), Θ ＩＩ After (α) is determined, the r of the first surface S1 at each point is calculated from the formula (18A). ＊ = 0, r ＊ = Surface temperature T at α ＊ (r ＊ ) [K] are as follows: The value ΔT of the temperature unevenness within the first surface S1 is given by the following formula. Here, the target temperature of the CO2 to be controlled is set as the boundary condition T(R) of the temperature of the thermal diffusion plate 10, and the ambient temperature around the CO2 to be controlled is set as T ∞ and the heat input flux is q ｉｎ These values ​​can be set as given design conditions. Ｉ (0), Θ ＩＩ (α) is a, α, and Biot number Bi ｒ is determined when the radii of the heat diffusion plate 10 and the heat source 11 are determined, and a is determined when d, R, and Biot number Bi ｒ Once it's decided, it's decided.

[0052] Therefore, in this design method, T(R), T ∞ , q ｉｎ Assuming that the value of is a given design condition, and further determining the values ​​of the design parameters d, R, and α relating to the sizes of the heat diffusion plate 10 and the heat source 11, as shown in FIG. 5, from equation (29), ΔT = ΔT ｃ The Biot number Bi that satisfies ｒ Upper limit Bi ｒ．ＭＡＸ The value of can be determined.

[0053] Biot number Bi ｒ Upper limit Bi ｒ．ＭＡＸ Once the value of is obtained, the thermal conductivity k of the heat diffusion plate 10 in the surface direction is calculated so as to satisfy the following equation (30). ｒ By determining the overall heat transfer coefficient h, the design conditions of the temperature control device 1 can be determined. Therefore, the equations (29) and (30) are expressed as the ambient temperature T ∞ , the target temperature T(R) of the controlled CO, and the amount of heat input q from the heat source 11 to the heat diffusion plate 10.ｉｎ the values ​​of design parameters d, R, and α relating to the sizes of the heat diffusion plate 10 and the heat source 11, and the thermal conductivity k of the heat diffusion plate 10. ｒ and the tolerance ΔT for the variation in the relationship between the overall heat transfer coefficient h and the temperature T [K] on the first surface S1. ｃ Using the formula (29) and the formula (30), the design parameters d, R, α, k ｒ , h can be determined.

[0054] [Design Flow of Temperature Control Device 1] As shown in Fig. 6A, the design of the temperature control device 1 is executed by an information processing device 100. The information processing device 100 is realized in a computer HW having a CPU 60, a memory 61, an external storage device 62, an operation unit 63, a display unit 64, and an internal bus 65, by the CPU 60 executing a software program loaded from the external storage device 62 into the memory 61 in accordance with an operation via the operation unit 63. The results of the design process of the temperature control device 1 executed by the CPU 60 are displayed on, for example, the display unit 64. The information processing device 100, whose functions are realized by the computer executing this program, performs the design process (design method) of the temperature control device 1 shown in Fig. 6B.

[0055] As shown in FIG. 6B, first, the information processing device 100 calculates the ambient temperature T ∞ , the heat input q from the heat source 11 ｉｎ、 Given design conditions such as the target temperature T(R) of the controlled CO, and the values ​​of design parameters d, R, and α relating to the sizes of the heat diffusion plate 10 and the heat source 11 are input (step S1).

[0056] Next, the information processing device 100 calculates the variation Δr of the reaction value based on the relationships shown in equations (1) to (8). ｃ The temperature variation tolerance ΔT corresponding to ｒ is determined, and f is calculated based on the equations (9) and (10). ｐｃ．ｉ (T) to Δφ ｐｃ，ｉ ΔT that satisfies ｐｃ，ｉ Calculate ΔT ｒ、 ΔT ｐｃ．ｉ The smallest one is the temperature variation tolerance ΔT ｃThat is, the information processing device 100 sets the allowable value for the variation in the temperature of the control target CO within the plane based on the temperature characteristics of the physical property values ​​of the control target CO or the characteristics of the reaction rate within the plane when the control target CO is a reaction field.

[0057] Next, the information processing device 100 uses the above formula (29) to obtain ΔT=ΔT ｃ The maximum Biot number Bi ｒ．ＭＡＸ (Step S3). The maximum Biot number Bi ｒ．ＭＡＸ , ΔT = ΔT ｃ The maximum Biot number Bi is the Biot number Bi when ｒ．ＭＡＸ For example, the bisection method is used to search for the above.

[0058] In the bisection method, F=ΔT−ΔTc is defined as an evaluation function. In this case, F is a monotonically increasing function of Bi. The bisection method is performed in the following procedure.

[0059] Step 1) Biot number Bi ｒ As the initial values ​​of both ends of the solution interval, a sufficiently small appropriate Biot number Bir1 (for example, 10 －６ ) and a sufficiently large appropriate Biot number Bi ｒ２ (For example, 10 ６ ) and the Biot number Bi ｒ１ The value of F, that is, F(Bi ｒ１ ) is not a negative value, F(Bi ｒ１ ) becomes negative. ｒ１ Continue dividing the value of F(Bi ｒ２ ) is negative, F(Bi ｒ２ ) becomes positive. ｒ２ Continue doubling the value of

[0060] Step 2) Bi ｒ１ and Bi ｒ２ The intermediate value Bi ｒｍ = (Bi ｒ１ +Bi ｒ２ ) / 2, that is, F(Bi ｒｍ ) is calculated using equation (29). ｒｍ Depending on whether the value of φ(Biｒｍ ) < 0, Bi ｒｍ New Bi ｒ１ Let φ(Bi ｒｍ ) ≧0, Bi ｒｍ New Bi ｒ２ Step 3) The newly determined Bi ｒ１ , Bi ｒ２ From Bi ｒｍ Ask for.

[0061] Step 4) Bi ｒ２ -Bi ｒ１ is the threshold (e.g., 10 ―６ Repeat steps 2 and 3 until the final Bi ｒｍ The value of the maximum Biot number Bi ｒ．ＭＡＸ The convergence value of the maximum Biot number Bi ｒ．ＭＡＸ End the search for

[0062] Next, the information processing device calculates the maximum Biot number Bi in equation (30). ｒ．ＭＡＸ Enter the value of the thermal conductivity k that satisfies equation (30). ｒ and the overall heat transfer coefficient h are determined (step S4). ｒ is calculated by using the temperature characteristics shown in FIG. 7, for example, to obtain the thermal conductivity k ｒ It can be said that:

[0063] In this embodiment, the Biot number Bi ｒ was searched for, and also, based on the temperature characteristics shown in FIG. 7, the thermal conductivity k corresponding to the target temperature T(R) of the controlled CO ｒ is determined first, and the overall heat transfer coefficient h is searched for as described above to obtain the maximum Biot number Bi ｒ．ＭＡＸ Alternatively, the Biot number Bi ｒ In addition, any one of the radius R of the heat source 11, the thickness d of the heat diffusion plate 10, and the ratio α of the radius of the heat diffusion plate 10 to the heat source 11 may be searched for.

[0064] (Design Example) For example, it is assumed that the design conditions are set as follows: Diameter 2R of heat source 11: 55 mm Diameter (2αR) of heat diffusion plate 10 and controlled object CO: 30 mm Allowable value ΔT of in-plane temperature unevenness Ｃ: 0.1K Target temperature of CO to be controlled T (R): 350K Environmental temperature (ambient temperature) T ∞ : 298K When this condition is input into equation (29), the maximum Biot number Bi ｒ．ＭＡＸ The value of is 2.321 x 10 －３２ Furthermore, this maximum Biot number Bi ｒ．ＭＡＸ The thermal conductivity k that satisfies the condition ｒ The minimum value of is 1.185 × 10 ３ [W m －１ ・K －１ The thermal conductivity k of the ceramic heat diffusion plate 10 is ｒ The temperature dependence of kr. is given in Figure 7. In Figure 7, kr. exp is the experimental value, and kr. fitted is the fitting curve showing the characteristics. Here, the temperature T ｂｔｍ Thermal conductivity k at 350[K] ｒ The value is 5.987 x 10 ３ [W m －１ ・K －１ This value is k ｒ The minimum value that must be satisfied (maximum Biot number Bi ｒ．ＭＡＸ Since the in-plane temperature unevenness of the control target CO at this time is 0.0201 [K], it is clear that the thermal diffusion plate 10 can achieve a temperature unevenness of the control target CO below the allowable value.

[0065] As described above in detail, according to this embodiment, the maximum temperature T on the first surface S1 that comes into contact with the control target CO ＊ (0) and the minimum temperature T ＊ The radius R of the contact area between the heat source 11 and the heat diffusion plate 10, the radius ratio α between the heat diffusion plate 10 and the circular region 11a, the plate thickness d of the heat diffusion plate 10, and the thermal conductivity k of the heat diffusion plate 10 are set so that the difference between (α) is within the allowable value ΔTc. ｒ and the overall heat transfer coefficient h can be determined, so that the variation in the temperature of the CO to be controlled can be suppressed within an allowable value.

[0066] [Variations of Configuration] The configuration of the temperature control device 1 is not limited to that shown in Fig. 1. For example, as shown in Fig. 8A, the heat source 11 and the heat diffusion plate 10 may be integrated at the second surface S2. Specifically, the housing 20 of the heat source 11 may be formed from the same material as the housing 20 of the heat diffusion plate 10, and the heat source 11 and the housing 20 of the heat diffusion plate 10 may be directly joined at the second surface S2 to form an integrated unit. Alternatively, the heat source 11 may be formed by screen-printing a heater circuit on a ceramic substrate, and the heat diffusion plate 10 and the heat source 11 may be integrally formed by simultaneous sintering.

[0067] The heat source 11 may also be incorporated into the heat diffusion plate 10. In this case, a heater layer serving as the heat source 11 is incorporated into the housing 20 of the heat diffusion plate 10, as shown in FIG.

[0068] [Substrate Holding Device] The configuration of the temperature control device 1 according to this embodiment can be applied to electrostatic or vacuum suction type substrate holding devices. As shown in FIG. 9 , a substrate holding device 50 includes the temperature control device 1. The temperature control device 1 controls the temperature of the wafer W and also has a function of attracting the wafer W by electrostatic or vacuum suction. This allows the in-plane temperature T of the wafer W to be kept within a tolerable range during exposure, development, etching, and other processes performed on the wafer W. The temperature control device 1 can also be incorporated into substrate holding devices that hold liquid crystal substrates or other substrates other than the wafer W.

[0069] Although the temperature control device 1 according to the present embodiment is disk-shaped, the present invention is not limited to this and may be, for example, a polygonal flat plate. Furthermore, the temperature control device 1 is not limited to a planar shape. For example, as shown in FIG. 10 , the temperature control device 1 may be cylindrical. In this case, the heat source 11 is cylindrical, and a cylindrical heat diffusion plate 10 is formed so as to contact the side of the heat source 11. The side of the heat diffusion plate 10 contacts the CO2 to control the temperature of the CO2. When the temperature control device 1 is polygonal flat, consider a plane that passes through the center of the heat source 11 and includes the Z axis. When this plane is rotated around the Z axis, the length of the line segment intersecting the heat source 11 changes with the rotation. In such a temperature control device 1, the heat flux flowing into the heat diffusion plate 10 is maximized in the plane where the length of the line segment intersecting the heat source 11 is shortest. Therefore, the above equation can be applied to the plane where the heat flux is maximized, and thermal design and safety evaluation can be performed.

[0070] This invention allows various embodiments and modifications without departing from the broad spirit and scope of this invention. Furthermore, the above-described embodiments are intended to explain this invention and do not limit the scope of this invention. That is, the scope of this invention is defined by the claims, not the embodiments. Various modifications made within the scope of the claims and the meaning of the invention equivalent thereto are considered to be within the scope of this invention.

[0071] In addition, this application claims priority based on Japanese Patent Application No. 2022-189258 filed on November 28, 2022, and the specification, claims, and drawings of Japanese Patent Application No. 2022-189258 are incorporated herein by reference.

[0072] The present invention can be applied to uniformize the temperature of a substance or reaction field that has an extension in the plane direction.

[0073] 1 Temperature control device, 10 Heat diffusion plate, 11 Heat source, 11a Area, 20 Housing, 21 Working fluid, 50 Substrate holding device, 60 CPU, 61 Memory, 62 External storage device, 63 Operation unit, 64 Display unit, 65 Internal bus, 100 Information processing device, CO Control target, IS Internal space, S1 First surface, S2 Second surface, HW Computer, W Wafer

Claims

1. A method for designing a temperature control device, wherein the temperature control device comprises: a heat diffusion plate, which is a member provided with a first surface facing a control target and a second surface parallel to and facing opposite to the first surface, diffusing heat in the planar direction of the first surface; and a heat source, which is thermally joined to the heat diffusion plate at the second surface and heats or absorbs heat from the heat diffusion plate; and the size of the heat diffusion plate and the heat source, the thermal conductivity and overall heat transfer coefficient of the heat diffusion plate are determined based on a calculation formula showing the relationship between the ambient temperature around the control target, the target temperature of the control target, the amount of heat input from the heat source to the heat diffusion plate, the sizes of the heat diffusion plate and the heat source, the thermal conductivity and overall heat transfer coefficient of the heat diffusion plate, and the temperature variation on the first surface, so that the difference between the maximum and minimum temperatures of the portion of the first surface that comes into contact with the control target when the ambient temperature, the target temperature, and the heat input amount are given design conditions, falls within an allowable value for the in-plane variation of the temperature of the control target. How to design a temperature control device.

2. The method for designing a temperature control device according to claim 1, wherein the heat diffusion plate comprises: a housing having an enclosed internal space and made of any of ceramics, ceramic composites, and inorganic materials excluding metals; and a working fluid located in the internal space, which circulates in the internal space along the surface direction of the first surface while repeatedly evaporating due to heat reception and condensing due to heat release, and which diffuses heat in the surface direction of the first surface.

3. The thermal expansion coefficient of the material of the housing is 8.0 x 10 －６ The method for designing a temperature control device according to claim 2, wherein the temperature difference is [1 / K] or less.

4. A method for designing a temperature control device as described in claim 1, wherein a tolerance for the variation in the temperature of the controlled object within a surface is set based on the temperature characteristics of the physical property values ​​of the controlled object or the characteristics of the reaction rate within the surface when the controlled object is a reaction field.

5. The method for designing a temperature control device according to claim 1, wherein the heat source is one of a heater, a Peltier element, and a cold plate.

6. The design method for a temperature control device according to claim 1, wherein the heat source and the heat diffusion plate are integrated on the second surface.

7. The method for designing a temperature control device according to claim 1, wherein the heat source is incorporated into the heat diffusion plate.

8. The method for designing a temperature control device according to claim 1, wherein the heat source and the heat diffusion plate each have a disk-shaped outer shape and are arranged concentrically with each other, and the radius of the heat source is smaller than the radius of the heat diffusion plate.

9. A temperature control device designed using the temperature control device design method according to any one of claims 1 to 8.

Images