Three-dimensional temperature analysis method for substrate containing irregularly distributed irregular heating vias

Abstract

The application discloses a three-dimensional temperature analysis method for a substrate containing irregular heat generating through holes with random distribution, and belongs to the technical field of microelectronics, and comprises the following steps: a mathematical physical model is established; based on the mathematical physical model, a thermal conductivity expression, a volume heat capacity expression and a heat source expression of the substrate containing through holes and chips are established by using a step function; a three-dimensional heat conduction equation is established; eigen equations with unknown coefficients in each dimension direction are obtained according to the three-dimensional heat conduction equation and the heat model expression containing Fourier coefficients; the known eigen equations are solved according to the temperature boundary conditions of the substrate; and the heat model expression is updated according to the known eigen equations. The Fourier coefficients corresponding to the eigen value in each dimension direction under the combination of the known eigen equations are obtained by jointly solving the thermal conductivity expression, the volume heat capacity expression, the heat source expression of the substrate and the three-dimensional heat conduction equation, and are substituted into the updated heat model expression, so that the temperature distribution of the substrate is obtained.

Description

Technical Field

[0001] This invention belongs to the field of microelectronics technology, specifically relating to a three-dimensional temperature analysis method for substrates containing randomly distributed irregular heating vias. Background Technology

[0002] In today's rapidly developing electronic technology, 3D microsystems, with their superior integration capabilities and performance improvements, represent a revolutionary advancement in the semiconductor field. However, as 3D microsystems evolve towards miniaturization, multifunctionality, and high speed, the number of chips within these systems continues to increase, along with the power consumption of the chips themselves. If the heat within the system cannot be dissipated effectively and promptly, it can lead to a sharp rise in localized temperatures, potentially causing increased thermal stress, chip warping or deformation, and ultimately increasing the probability of electronic device failure. Therefore, designing a heat dissipation solution for 3D microsystems has become a pressing issue. Currently, the main sources of heat generation are device-level heat and the heat generated by interconnect structures during signal transmission. Through-holes, as interconnect structures, generate heat themselves and are also a primary pathway for heat conduction. Current heat dissipation methods can be categorized into active and passive cooling. While active cooling offers better heat dissipation, it occupies a larger volume and is difficult to integrate into high-density 3D microsystems. Therefore, passive cooling solutions, primarily based on through-hole arrays, are receiving more attention.

[0003] Currently, many related studies have developed thermal modeling methods for three-dimensional microsystems containing through-hole arrays: one is the finite element method, which divides the actual physical model of the three-dimensional microsystem into a finite number of simple geometric units, and establishes a corresponding mathematical model in each unit to approximate the solution of the entire physical problem; the second is the thermal resistance network model, which uses thermoelectric analogy and establishes a thermal resistance topology network method to analyze the steady-state and transient temperature distribution of the three-dimensional microsystem; the third is the thermal analytical model, which models the corresponding structure, solves the Fourier series form of the steady-state and transient temperature fields, and determines the Fourier coefficients through initial conditions and boundary conditions.

[0004] However, the above thermal analysis methods have some limitations. Method 1 usually requires dividing a large number of meshes, which is time-consuming in model building and simulation calculation. The error of Method 2 increases with the number of chips. When dealing with the case of internal vias, the thermal analysis model of Method 3 can only use the equivalent thermal conductivity method to calculate the temperature of internally uniformly distributed vias, and the calculated temperature is the interface average temperature, which cannot be applied to random via distribution. Summary of the Invention

[0005] To address the aforementioned problems in the prior art, this invention provides a three-dimensional temperature analysis method for substrates containing randomly distributed irregularly distributed heating vias. The technical problem to be solved by this invention is achieved through the following technical solution: This invention provides a three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heat-generating vias. The three-dimensional temperature analysis method includes: Extract relevant parameters of the substrate and randomly distributed vias within the substrate, and set multiple chips as planar heat sources on the substrate to establish a mathematical and physical model; Based on the mathematical physics model, and using the step function, we established the thermal conductivity expression, volumetric heat capacity expression, and heat source expression for the substrate with through holes and the chip. A three-dimensional heat conduction equation is established based on the expressions for thermal conductivity, volumetric heat capacity, and heat source. Based on the three-dimensional heat conduction equation and the thermal model expression containing Fourier coefficients, eigenvalues ​​and unknown coefficients in each dimension are obtained. Based on the temperature boundary conditions of the substrate, the eigenvalues ​​and unknown coefficients in each dimension are obtained. Based on the eigenvalues ​​and unknown coefficients in each dimension, the known eigenvalues ​​in the corresponding dimension are obtained. The thermal model expression is updated based on the known eigenvalues ​​in each dimension. By jointly solving the thermal conductivity expression, volumetric heat capacity expression, heat source expression within the substrate, and three-dimensional heat conduction equation, the Fourier coefficients under the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension direction are obtained. Substituting the Fourier coefficients into the updated thermal model expression, the temperature distribution of the substrate is obtained.

[0006] The beneficial effects of this invention are: This invention proposes a three-dimensional temperature analysis method for substrates containing randomly distributed, irregularly shaped heat-generating vias. Based on the actual via distribution and temperature boundary conditions of the substrate, it utilizes a step function to represent the distribution of thermal conductivity, volumetric heat capacity, and heat sources. By leveraging the properties of the step function, the temperature distribution of the substrate is obtained, enabling the calculation of the temperature at any point and at any time within a substrate containing randomly distributed heat-generating vias. Compared to conventional analytical methods, this method solves the problem of its inability to calculate the three-dimensional temperature field within a substrate containing randomly distributed vias. Furthermore, it considers the influence of via heating and chip heating on the substrate temperature, thus providing a more accurate description of the temperature distribution within the substrate.

[0007] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0008] Figure 1 This is a flowchart illustrating a three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias, provided in an embodiment of the present invention. Figure 2 This is a schematic diagram of a substrate structure containing irregular through holes provided in an embodiment of the present invention; Figure 3This is a top view schematic diagram of a substrate structure containing irregular through holes provided in an embodiment of the present invention; Figures 4(a) to 4(b) This is a schematic diagram showing the value range of any point in the elliptical through hole provided in the embodiments of the present invention in the XYZ dimension directions; Figure 5 This is a schematic diagram of the XY cross-sectional range of the chip at any point in the Z-dimensional direction provided in the embodiment of the present invention; Figure 6 This is a comparative schematic diagram of the temperature field distribution of a substrate approaching a steady state when only through-hole heating is present, provided by an embodiment of the present invention. Figure 7 This is a schematic diagram comparing the substrate temperature field distribution when heating with through holes and heating with chips, which tend to reach a steady state, according to an embodiment of the present invention. Figure 8 This is a comparative schematic diagram of the temperature field distribution of the substrate containing through holes and chip heating at t=30s, provided by an embodiment of the present invention. Detailed Implementation

[0009] The present invention will be further described in detail below with reference to specific embodiments, but the implementation of the present invention is not limited thereto.

[0010] Please see Figure 1 This invention provides a three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias, specifically including the following steps: S10. Extract relevant parameters of the substrate and randomly distributed vias within the substrate, and set multiple chips as planar heat sources on the substrate to establish a mathematical and physical model.

[0011] This invention extracts relevant parameters of the substrate and its vias, and sets multiple chips as planar heat sources on the substrate to establish a mathematical physics model. This includes: extracting the structural and material parameters of the substrate, extracting the positional, structural, and material parameters of the vias, and extracting the positional and material parameters of the chips; and establishing a mathematical physics model in a three-dimensional Cartesian coordinate system. The vias randomly distributed within the substrate are all double-layered structures; among all vias, there is one elliptical cylindrical via, and the others are cylindrical vias, with the elliptical cylindrical via located at the center of the substrate. More specifically: In this embodiment of the invention, the substrate contains multiple randomly distributed irregular through-holes, including one elliptical cylindrical through-hole located at the center of the substrate, and the others randomly distributed cylindrical through-holes. Each through-hole has a double-layer structure; when the through-hole is a heat-generating through-hole, its inner cylindrical layer serves as a heat source. Multiple chips are present on the substrate, each serving as a planar heat source. With the position and size parameters of the through-holes remaining constant, the substrate temperature is calculated under two different conditions: both the chips and through-holes generate heat, and neither the chips nor the through-holes generate heat.

[0012] In a specific example, Figure 2 and Figure 3 This diagram illustrates a substrate structure containing irregular through-holes. The structure includes three chips C1, C2, and C3, and ten double-layer through-holes are disposed on the substrate. The structural parameters of the substrate include its length, width, and thickness; the material parameters include its thermal conductivity and volumetric heat capacity; the chip position parameters include the chip's coordinates in the X-axis, Y-axis, and Z-axis; and the chip's material parameters include its heating power as a planar heat source. The through-hole position parameters include the center coordinates of the through-hole in the X-axis and the through-hole's coordinates in the Y-axis (including the outer cylinder's position in the Y-axis). The range of values ​​for the inner cylinder (Y-axis) and the range of values ​​for the through-hole (Z-axis) are specified. The structural parameters of the through-hole include the radius length of the inner cylinder and the radius length of the outer cylinder for cylindrical through-holes, and the lengths of the major and minor axes for elliptical cylindrical through-holes. The material parameters of the through-hole include its thermal conductivity (including the thermal conductivity of the outer and inner cylinders), its volumetric heat capacity (including the volumetric heat capacity of the outer and inner cylinders), and its heating power. For example, using... Figure 2 Taking the substrate shown as an example, its structural parameters are shown in Table 1.

[0013] Table 1

[0014]

[0015] It should be noted that the irregular through holes on the substrate are not limited to cylindrical and elliptical cylindrical shapes; for example, they can also be triangular prisms or square prisms (cubic or cubic). The embodiments of this invention provide a detailed design example using cylindrical and elliptical cylindrical shapes.

[0016] S20. Based on the mathematical physics model, and using the step function, establish the thermal conductivity expression, volumetric heat capacity expression, and heat source expression for the substrate containing through holes and the chip.

[0017] This invention, based on a mathematical physics model, uses a step function to establish expressions for the thermal conductivity and volumetric heat capacity of a substrate with vias, as well as expressions for the heat sources of a substrate with vias and chips. The process includes: Based on the properties of the step function, in a three-dimensional Cartesian coordinate system, establishing expressions for the thermal conductivity and volumetric heat capacity of vias, and expressions for the thermal conductivity and volumetric heat capacity of a substrate without vias, respectively, and obtaining the thermal conductivity expression for a substrate with vias based on the thermal conductivity expressions for vias and the substrate without vias; obtaining the volumetric heat capacity expression for a substrate with vias based on the volumetric heat capacity expressions for vias and the substrate without vias, based on the thermal conductivity expressions for vias and the substrate without vias; and based on the properties of the step function, in a three-dimensional Cartesian coordinate system, establishing expressions for heat sources using vias as heat sources, and expressions for heat sources using chips as heat sources, and obtaining the heat source expressions for a substrate with vias and chips based on the expressions for heat sources using vias and chips. More specifically: by Figure 2 A three-dimensional rectangular coordinate system is established with the lower left front corner of the substrate as the origin. A mathematical and physical model is then established within this three-dimensional rectangular coordinate system. Generally, the through-hole is assumed to be a standard cylindrical shape. However, this invention can also calculate elliptical cylindrical shapes. Both shapes can be represented by the equation of an ellipse in the three-dimensional rectangular coordinate system, and their shape expressions are as follows: ; in, For values ​​in the Z-axis direction The center coordinates of the ellipse on the XY section. , These are the lengths of the major and minor semi-axis of the ellipse, respectively. , Let be the lower and upper bounds of the elliptical cylinder along the Z-axis. When When the equation becomes the equation of the standard circle, the equation is transformed into the equation of the standard circle.

[0018] like Figures 4(a) to 4(b) As shown, the equation of the ellipse is transformed into a range expression as follows: ; The step function expression is as follows: ; Based on the properties of step functions, a single step function can represent a numerical step in a single dimension. The expression for the thermal conductivity of a single through-hole in a three-dimensional Cartesian coordinate system is as follows: ; The present invention designs the through hole as a double-layer structure. For through holes with more layers, the same method can be used to calculate the number of layers sequentially.

[0019] Furthermore, the expression for the thermal conductivity of a substrate without through holes is established as follows: ; Finally, by adding the thermal conductivity expressions for all vias to the thermal conductivity expression for the substrate without vias, we obtain the established thermal conductivity expression for the substrate with vias, as follows: ; in, This indicates the location of the substrate when the substrate contains through-holes. Thermal conductivity at that point Indicates the thermal conductivity of the substrate. Indicates the substrate position First Thermal conductivity of each through hole Indicates the location of the substrate without through holes. Thermal conductivity at that point Indicates the number of through holes in the substrate. Indicates the first Thermal conductivity of the outer cylinder with a through hole Indicates the first The thermal conductivity of the inner cylinder of a through-hole, for a cylindrical through-hole. Indicates the first The radius length of the outer cylinder of each through hole Indicates the first The radius length of the inner cylinder of a through hole, for an elliptical through hole, Indicates the first The length of the major semi-axis of the outer cylinder of each through hole Indicates the first The length of the major semi-axis of the inner cylinder of each through hole. Indicates the first The center coordinates of each through hole in the X-dimensional direction Indicates the first The lower limit coordinates of the outer cylinder of each through hole in the Y-axis direction. Indicates the first The upper limit coordinate of the outer cylinder of each through hole in the Y-axis direction. Indicates the first The lower limit coordinates of the inner cylinder of each through hole in the Y-axis direction. Indicates the first The upper limit coordinate of the inner cylinder of the through hole in the Y-axis direction. Indicates the first The lower limit coordinate of a through hole in the Z-axis direction. Indicates the first The upper limit coordinate of a through hole in the Z-axis direction. This represents the step function.

[0020] Similarly, the volumetric heat capacity expression for a single through-hole is established as follows: ; The volumetric heat capacity expression for a substrate without vias is as follows: ; Finally, by adding the volumetric heat capacity expressions for all vias to the volumetric heat capacity expression for the substrate without vias, we obtain the established volumetric heat capacity expression for the substrate with vias: ; in, This indicates the location of the substrate when the substrate contains through-holes. Volume heat capacity at that point Indicates the substrate position First The volumetric heat capacity of a through-hole Indicates the location of the substrate without through holes. Volume heat capacity at that point This indicates the volumetric heat capacity of the substrate. Indicates the first The volumetric heat capacity of the outer cylinder with a through hole Indicates the first The volumetric heat capacity of the inner cylinder with a through hole.

[0021] Similarly, the heat source expression for using the through hole as a heat source is established as follows: ; Assuming the chip is extremely thin, and its heat generation is represented as a surface heat source, based on the properties of the step function, we can treat it as a distributed function and take its derivative to obtain the Dirac function, whose expression is as follows: ; The above Dirac function is expressed in There exists a point at a given location that represents 1, and all others that represent 0.

[0022] Based on the properties of the Diclave function and the step function, considering the chip as a planar heat source, we obtain the following expression for the heat source using the chip as a heat source: ; Finally, by adding the distribution of the planar heat source to the distribution of the through-hole heat source, we obtain the established expression for the heat source of the substrate containing through-holes and chips, as follows: ; in, When the substrate contains vias and chips, the location on the substrate is indicated. Heat source distribution at the location, This represents the spatial distribution of heat sources across all through-holes. This represents the spatial distribution of heat sources across all chips. Indicates the number of heating holes. Indicates the first The heating power of each heating through-hole, for a cylindrical through-hole, Indicates the first The radius length of the inner cylinder of each heating through-hole, for an elliptical through-hole, Indicates the first The length of the short semi-axis of the inner cylinder of the heating through-hole Indicates the first The center coordinates of the heating hole in the X-axis direction Indicates the first The lower limit coordinate of the inner cylinder of the heating through-hole in the Y-axis direction. Indicates the first The upper limit coordinate of the inner cylinder of the heating through-hole in the Y-axis direction. Indicates the first The lower limit coordinate of a heating through-hole in the Z-dimensional direction. Indicates the first The upper limit coordinate of a heating via in the Z-axis direction. Indicates the number of planar heat sources. Indicates the first The power of a planar heat source, Indicates the first The lower bound coordinates of a planar heat source in the X-dimensional direction. Indicates the first The upper coordinate of a planar heat source in the X-dimensional direction. Indicates the first The lower bound coordinates of a planar heat source in the Y-axis direction. Indicates the first The upper coordinate of a planar heat source in the Y-axis direction. Indicates the first The coordinates of a planar heat source in the Z-dimensional direction. Figure 5 This indicates that chip C3 is at Z= The range of values ​​for the XY section.

[0023] S30. Establish a three-dimensional heat conduction equation based on the thermal conductivity expression, volumetric heat capacity expression, and heat source expression. Obtain eigenvalues ​​and unknown coefficients in each dimension based on the three-dimensional heat conduction equation and the thermal model expression containing Fourier coefficients. Solve the eigenvalues ​​and unknown coefficients in each dimension based on the temperature boundary conditions of the substrate. Obtain the known eigenvalues ​​and unknown coefficients in the corresponding dimension based on the eigenvalues ​​and unknown coefficients in each dimension. Update the thermal model expression based on the known eigenvalues ​​and unknown coefficients in each dimension.

[0024] This invention first constructs a thermal model expression containing Fourier coefficients, as follows: ; in, Indicates the location on the substrate Place Temperature distribution over time Represents the first in the X-axis direction The eigenvalues ​​correspond to the eigenvalue equations containing unknown coefficients. Represents the first in the Y-axis direction The eigenvalues ​​correspond to the eigenvalue equations containing unknown coefficients. Represents the first in the Z-dimensional direction The eigenvalues ​​correspond to the eigenvalue equations containing unknown coefficients. express The Fourier coefficients of the combination of eigenvalue equations containing unknown coefficients corresponding to the eigenvalues ​​of temperature in each dimension at any given time, i.e. express Temperature at time X in the X-axis direction The eigenvalues, the eigenvalues ​​in the Y-dimensional direction The eigenvalues, the first eigenvalue in the Z-dimensional direction The Fourier coefficients of each eigenvalue under the combination of eigenvalue equations containing unknown coefficients.

[0025] Next, based on the expressions for thermal conductivity, volumetric heat capacity, and heat source, a three-dimensional heat conduction equation is established, as follows: ; in, Representing the thermal model The partial derivative in the X-axis direction, Representing the thermal model The partial derivative in the Y-axis direction, Representing the thermal model The partial derivative in the Z-axis direction, Representing the thermal model Regarding time The partial derivative of .

[0026] Before solving the eigenvalue equations containing unknown coefficients, we first need to define the thermal conductivity expression in the three-dimensional heat conduction equations. Volumetric heat capacity expression The values ​​are respectively taken as the thermal conductivity of the substrate. Volumetric heat capacity Internal heat source expression If the value is set to 0, and then the thermal model expression is substituted into the three-dimensional heat conduction equation processed above, the intrinsic equation expression containing unknown coefficients can be obtained as follows: ; in, Represents the first in the X-axis direction Each eigenvalue Represents the first in the Y-axis direction Each eigenvalue Represents the first in the Z-dimensional direction Each eigenvalue This represents an unknown coefficient.

[0027] The temperature boundary condition of the substrate is that it is thermally adiabatic on all sides and has vertical convection, and its expression is as follows: ; in, The convection coefficients are the upper and lower surfaces of the substrate.

[0028] Substituting the eigenvalue equation containing unknown coefficients into the temperature boundary conditions, we obtain multiple sets of eigenvalues ​​and values ​​of unknown coefficients, expressed as follows: .

[0029] Furthermore, by solving for multiple sets of eigenvalues ​​and unknown coefficients, we obtain the known eigenvalue equations, which are denoted as follows: , Represents the first in the X-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Y-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Z-dimensional direction The known eigenvalues ​​correspond to known eigenvalue equations. Substituting these known eigenvalue equations into the thermal model expression updates the thermal model expression as follows: ; in, express Fourier coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of temperature in each dimension at any given time.

[0030] S40. By jointly solving the thermal conductivity expression, volumetric heat capacity expression, heat source expression within the substrate, and three-dimensional heat conduction equation, the Fourier coefficients under the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension direction are obtained. The Fourier coefficients are then substituted into the updated thermal model expression to obtain the temperature distribution of the substrate.

[0031] This invention provides embodiments that, by jointly solving the thermal conductivity expression, volumetric heat capacity expression, heat source expression within the substrate, and three-dimensional heat conduction equation, obtains the Fourier coefficients under the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension. This includes: calculating the modulus in the corresponding dimension based on the known eigenvalue equations in each dimension; calculating the coefficients of the total heat source under the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension based on the heat source expression and the modulus in each dimension; substituting the coefficients of the known eigenvalue equation combination, the thermal conductivity expression, the heat capacity expression, and the heat source expression within the substrate into the three-dimensional heat conduction equation to obtain a new three-dimensional heat conduction equation; multiplying both sides of the new three-dimensional heat conduction equation by the known eigenvalue equations in each dimension and then integrating to obtain an equation expression containing unknown Fourier coefficients; arranging the equation expression containing unknown Fourier coefficients into a matrix form according to row and column indices to obtain a first-order ordinary differential matrix equation; and solving the first-order ordinary differential matrix equation to obtain the Fourier coefficients under the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension. More specifically: The modulus in each dimension is obtained by taking the norm from the known eigenvalue equations, and its calculation expression is as follows: .

[0032] Next, the coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of the total heat source in each dimension are calculated, and the formula is expressed as: ; in, This represents the coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of the total heat source in each dimension. When the substrate contains vias and chips, the location on the substrate is indicated. Heat source distribution at the location, Represents the first in the X-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Y-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Z-dimensional direction The known eigenvalue equations corresponding to each eigenvalue. This represents the modulus in the X-axis direction. This represents the modulus in the Y-axis direction. This represents the modulus in the Z-axis direction. Indicates the length of the substrate. Indicates the width of the substrate. This indicates the thickness of the substrate.

[0033] Substituting the coefficients, thermal conductivity expressions, heat capacity expressions, and substrate heat source expressions from the known eigenvalue combinations corresponding to the eigenvalues ​​in each dimension into the three-dimensional heat conduction equation, we obtain a new three-dimensional heat conduction equation, expressed as:

[0034] in, This indicates the thermal conductivity of the substrate containing through-holes. Indicates thermal conductivity The partial derivative in the X-dimensional direction, Indicates thermal conductivity The partial derivative in the Y-dimensional direction, Indicates thermal conductivity The partial derivative in the Z-dimensional direction, This indicates the volumetric heat capacity of the substrate containing through-holes. express The Fourier coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of temperature in each dimension at any given time. express The first derivative, Represents the first in the X-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Y-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Z-dimensional direction The known eigenvalue equations corresponding to each eigenvalue. , They represent The first derivative and the second derivative, , They represent The first derivative and the second derivative, , They represent The first derivative and the second derivative, This represents the coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of the total heat source in each dimension.

[0035] Based on the properties of the step function, the partial derivative of thermal conductivity is expressed as follows: ; in, Represents the Dirac function, express The first derivative, express The first derivative.

[0036] ; .

[0037] Multiplying both sides of the new three-dimensional heat conduction equation by the known intrinsic equations in the three dimensions and then integrating, we obtain the following expression for the equation containing unknown Fourier coefficients: ; In the equation expression after integration Arranged in matrix form using row and column indices, the expression for the first-order ordinary differential matrix equation is as follows: ; in, To correct the coefficient matrix of thermal conductivity, , To correct the coefficient matrix of volumetric heat capacity, , For the heat source coefficient vector, , This is the calculated value of the corresponding triple integral.

[0038] ; in, express The first derivative, express The first derivative.

[0039] ; ; ; ; .

[0040] Solving the obtained ordinary differential equation yields the following expression for the transient Fourier coefficients: ; in, This indicates the inverse operation. When At this point, the expression for the Fourier coefficients becomes the steady-state expression, as follows: .

[0041] Therefore, by substituting the Fourier coefficients of the known eigenvalues ​​in each dimension into the updated thermal model expression, the analytical solution of the temperature of the substrate containing irregular through holes under steady-state and transient conditions is completed.

[0042] To verify the effectiveness of the three-dimensional temperature analysis method for substrates containing randomly distributed irregular heating vias provided in this embodiment of the invention, the following experiments were conducted for verification.

[0043] by Figure 2 and Figure 3 Taking the substrate shown as an example, the structural parameters of the substrate are consistent with those in Table 1. The thermal conductivity of the substrate is set to 100 W / (m·K), and the volumetric heat capacity is 700 W / (m·K). 2300 J / (k·m 3 The chip is designed to not generate heat, while the inner cylinders of all vias generate heat, except for vias 1 and 3. The outer cylinders of the vias have a thermal conductivity of 200 W / (m·K) and a volumetric heat capacity of 700 W / (m·K). 2300 J / (k·m 3 The thermal conductivity of the inner cylinder with the through-hole is 400 W / (m·K), and the volumetric heat capacity is 400 W / (m·K). 8930 J / (k·m 3 The heating power of the heating through-hole is 250,000 kW / m. 3 . Figure 6 This is a comparative schematic diagram showing the temperature distribution of a substrate approaching steady state when only through-holes generate heat. Figure 6 From top to bottom, the diagrams show the substrate temperature distribution as analyzed in this invention, the substrate temperature distribution as analyzed by the finite element method, and the error distribution between the two. Figure 6 It can be seen that when only the heating of the through-hole is considered, the thermal analysis method proposed in this invention can describe the temperature distribution inside the substrate more accurately.

[0044] Similarly, with Figure 2 and Figure 3 Taking the substrate shown as an example, the structural parameters of the substrate are consistent with those in Table 1, wherein the heat dissipation power applied to chips C1, C2, and C3 on the substrate is 250,000 W / m. 2 The heat dissipation power of the through-hole in the substrate is 100,000 kW / m 3 Among them, through holes 1 and 3 do not generate heat. The thermal conductivity of the outer cylinder of the through holes is 1.5 W / (m·K), and the volumetric heat capacity is 730. 2200J / (k·m 3 The thermal conductivity of the inner cylinder with the through-hole is 400 W / (m·K), and the volumetric heat capacity is 400 W / (m·K). 8930J / (k·m 3 Considering the transient temperature distribution of a substrate containing irregular through-holes, Figure 7 This is a comparison of the substrate temperature distribution when the chip, containing vias, approaches steady state and when the chip heats up. Figure 8 This is a comparative diagram showing the temperature distribution of the substrate containing vias and chip heating at the 30th second of calculation. Similar to... Figure 6 , Figure 7 , Figure 8 From top to bottom, the diagrams show the substrate temperature distribution as analyzed in this invention, the substrate temperature distribution as analyzed by the finite element method, and the error distribution between the two. Figure 7 , 8 It can be seen that when both via heating and chip heating are considered, the thermal analysis method proposed in this invention can still describe the temperature distribution inside the substrate relatively accurately.

[0045] The thermal analysis model of this invention is compared with the finite element model. The comparison results are shown in Table 2. The comparison data are the substrate temperature distribution of each model at t=30s and the substrate temperature distribution when approaching steady state.

[0046] Table 2 Comparison results of the thermal desorption model and the finite element model of the present invention

[0047] As shown in Table 2, the thermal desorption temperature calculated by the thermal desorption model proposed in this invention at 30s has an average absolute error of 0.3041K compared to the finite element simulation; the average relative error at 30s is 0.2596%; the average absolute error at steady state is 0.0343K; and the average relative error at steady state is 0.0207%. The calculation time is 62.3s, which is significantly higher than the 435s of the finite element simulation, demonstrating a substantial improvement in computational efficiency.

[0048] In summary, the three-dimensional temperature analysis method for substrates containing randomly distributed irregular heat-generating vias proposed in this invention utilizes a step function to represent the distribution of thermal conductivity, volumetric heat capacity, and heat sources based on the actual via distribution and temperature boundary conditions. By leveraging the properties of the step function, the temperature distribution of the substrate is obtained, enabling temperature calculation at any point and time within a substrate containing randomly distributed heat-generating vias. Compared to conventional analytical methods, this method solves the problem of its inability to calculate the three-dimensional temperature field within a substrate containing randomly distributed vias. Furthermore, it considers the influence of via heating and chip heating on the substrate temperature, thus providing a more accurate description of the temperature distribution within the substrate.

[0049] In the description of this invention, it should be understood that the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0050] Although the invention has been described herein in conjunction with various embodiments, those skilled in the art, by reviewing the specification and accompanying drawings, will understand and implement other variations of the disclosed embodiments in carrying out the claimed invention. In the specification, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude a plurality. While certain measures are described in different embodiments, this does not mean that these measures cannot be combined to produce good results.

[0051] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A three-dimensional temperature analysis method for a substrate containing randomly distributed irregular heat generating vias, characterized by, The three-dimensional temperature analysis method includes: Extract relevant parameters of the substrate and randomly distributed vias within the substrate, and set multiple chips as planar heat sources on the substrate to establish a mathematical and physical model; Based on the mathematical physics model, and using the step function, we established the thermal conductivity expression, volumetric heat capacity expression, and heat source expression for the substrate with through holes and the chip. A three-dimensional heat conduction equation is established based on the expressions for thermal conductivity, volumetric heat capacity, and heat source. Based on the three-dimensional heat conduction equation and the thermal model expression containing Fourier coefficients, eigenvalues ​​and unknown coefficients in each dimension are obtained. Based on the temperature boundary conditions of the substrate, the eigenvalues ​​and unknown coefficients in each dimension are obtained. Based on the eigenvalues ​​and unknown coefficients in each dimension, the known eigenvalues ​​in the corresponding dimension are obtained. The thermal model expression is updated based on the known eigenvalues ​​in each dimension. By jointly solving the thermal conductivity expression, the volumetric heat capacity expression, the heat source expression within the substrate, and the three-dimensional heat conduction equation, the Fourier coefficients of the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension direction are obtained. Substituting these Fourier coefficients into the updated thermal model expression, the temperature distribution of the substrate is obtained. Among them, based on a mathematical physics model, and using a step function, expressions are established for the thermal conductivity, volumetric heat capacity, and heat source of the substrate containing vias and the chip, including: Based on the properties of the step function, in a three-dimensional rectangular coordinate system, expressions for the thermal conductivity and volumetric heat capacity of through holes are established based on mathematical physics models. Expressions for the thermal conductivity and volumetric heat capacity of substrates without through holes are also established. Based on the thermal conductivity expressions for through holes and substrates without through holes, expressions for the thermal conductivity of substrates with through holes are obtained. Based on the volumetric heat capacity expressions for through holes and substrates without through holes, expressions for the volumetric heat capacity of substrates with through holes are obtained. Based on the properties of the step function, in a three-dimensional rectangular coordinate system, a heat source expression is established based on a mathematical physics model, which uses vias as heat sources, and a heat source expression is established based on chips as heat sources. Based on the heat source expressions using vias and chips as heat sources, the heat source expression for a substrate containing vias and chips is obtained. The established expression for the thermal conductivity of a substrate containing through-holes is: ； in, This indicates the location of the substrate when the substrate contains through-holes. Thermal conductivity at that point Indicates the thermal conductivity of the substrate. Indicates the number of through holes in the substrate. Indicates the first Thermal conductivity of the outer cylinder with a through hole Indicates the first The thermal conductivity of the inner cylinder of a through-hole, for a cylindrical through-hole. Indicates the first The radius length of the outer cylinder of each through hole Indicates the first The radius length of the inner cylinder of a through hole, for an elliptical through hole, Indicates the first The length of the major semi-axis of the outer cylinder of each through hole Indicates the first The length of the major semi-axis of the inner cylinder of each through hole. Indicates the first The center coordinates of each through hole in the X-dimensional direction Indicates the first The lower limit coordinates of the outer cylinder of each through hole in the Y-axis direction. Indicates the first The upper limit coordinate of the outer cylinder of each through hole in the Y-axis direction. Indicates the first The lower limit coordinates of the inner cylinder of each through hole in the Y-axis direction. Indicates the first The upper limit coordinate of the inner cylinder of the through hole in the Y-axis direction. Indicates the first The lower limit coordinate of a through hole in the Z-axis direction. Indicates the first The upper limit coordinate of a through hole in the Z-axis direction. This represents the step function.

2. The method according to claim 1, wherein the three-dimensional temperature analysis method of a substrate containing randomly distributed irregular heat generating vias, characterized by, Extract relevant parameters of the substrate and its internal vias, and place multiple chips on the substrate as planar heat sources to establish a mathematical and physical model, including: The structural and material parameters of the substrate, the positional, structural, and material parameters of the vias within the substrate, and the positional and material parameters of the chip are extracted. A mathematical and physical model is established in a three-dimensional Cartesian coordinate system. The vias randomly distributed within the substrate are all double-layer structures. Among all the vias, there is one elliptical cylindrical via, and the others are cylindrical vias. The elliptical cylindrical via is located at the center of the substrate.

3. The three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias according to claim 1, characterized in that, The established expression for the volumetric heat capacity of a substrate containing through-holes is: ； in, This indicates the location of the substrate when the substrate contains through-holes. Volume heat capacity at that point This indicates the volumetric heat capacity of the substrate. Indicates the first The volumetric heat capacity of the outer cylinder with a through hole Indicates the first Volumetric heat capacity of the inner cylinder with a through hole; The established expression for the heat source of the substrate containing vias and chips is: ； in, When the substrate contains vias and chips, the location on the substrate is indicated. Heat source distribution at the location, Indicates the number of heating holes. Indicates the first The heating power of each heating through-hole, for a cylindrical through-hole, Indicates the first The radius length of the inner cylinder of each heating through-hole, for an elliptical through-hole, Indicates the first The length of the short semi-axis of the inner cylinder of the heating through-hole Indicates the first The center coordinates of the heating hole in the X-axis direction Indicates the first The lower limit coordinate of the inner cylinder of the heating through-hole in the Y-axis direction. Indicates the first The upper limit coordinate of the inner cylinder of the heating through-hole in the Y-axis direction. Indicates the first The lower limit coordinate of a heating through-hole in the Z-dimensional direction. Indicates the first The upper limit coordinate of a heating via in the Z-axis direction. Indicates the number of planar heat sources. Indicates the first The heating power of a planar heat source Indicates the first The lower bound coordinates of a planar heat source in the X-dimensional direction. Indicates the first The upper coordinate of a planar heat source in the X-dimensional direction. Indicates the first The lower bound coordinates of a planar heat source in the Y-axis direction. Indicates the first The upper coordinate of a planar heat source in the Y-axis direction. Indicates the first The coordinates of a planar heat source in the Z-dimensional direction.

4. The three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias according to claim 1, characterized in that, The updated thermal model expression is: ； in, Indicates the location on the substrate Place Temperature distribution over time Represents the first in the X-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Y-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Z-dimensional direction The known eigenvalue equations corresponding to each eigenvalue. express Fourier coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of temperature in each dimension at any given time.

5. The three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias according to claim 1, characterized in that, The temperature boundary conditions of the substrate are thermal insulation on all sides and convection on the top and bottom.

6. The three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias according to claim 1, characterized in that, By jointly solving the thermal conductivity expression, the volumetric heat capacity expression, the heat source expression within the substrate, and the three-dimensional heat conduction equation, the Fourier coefficients under the known eigenvalue equation combination corresponding to the eigenvalues ​​in each dimension are obtained, including: Calculate the modulus in the corresponding dimension direction based on the known intrinsic equations in each dimension direction; Calculate the coefficients of the total heat source under the combination of known eigenvalue equations corresponding to the eigenvalues ​​in each dimension based on the heat source expression and the modulus in each dimension. Substituting the coefficients, thermal conductivity expression, heat capacity expression, and substrate heat source expression under the combination of known eigenvalues ​​in each dimension into the three-dimensional heat conduction equation, a new three-dimensional heat conduction equation is obtained. By multiplying both sides of the new three-dimensional heat conduction equation by the known intrinsic equations in each dimension and then integrating, we obtain an equation expression containing unknown Fourier coefficients. Arrange the equation expressions containing unknown Fourier coefficients into matrix form according to row and column indices to obtain the first-order ordinary differential matrix equation. Solve the first-order ordinary differential matrix equation to obtain the Fourier coefficients under the combination of known eigenvalue equations corresponding to the eigenvalues ​​in each dimension.

7. The three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias according to claim 6, characterized in that, The coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of the total heat source in each dimension are calculated using the following formula: ； in, This represents the coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of the total heat source in each dimension. When the substrate contains vias and chips, the location on the substrate is indicated. Heat source distribution at the location, Represents the first in the X-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Y-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Z-dimensional direction The known eigenvalue equations corresponding to each eigenvalue. This represents the modulus in the X-axis direction. This represents the modulus in the Y-axis direction. This represents the modulus in the Z-axis direction. Indicates the length of the substrate. Indicates the width of the substrate. This indicates the thickness of the substrate.

8. The three-dimensional temperature analysis method for a substrate containing randomly distributed irregularly distributed heating vias according to claim 6, characterized in that, The new three-dimensional heat conduction equation is expressed as follows: in, This indicates the thermal conductivity of the substrate containing through-holes. Indicates thermal conductivity The partial derivative in the X-dimensional direction, Indicates thermal conductivity The partial derivative in the Y-dimensional direction, Indicates thermal conductivity The partial derivative in the Z-dimensional direction, This indicates the volumetric heat capacity of the substrate containing through-holes. express The Fourier coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of temperature in each dimension at any given time. express The first derivative, Represents the first in the X-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Y-axis direction The known eigenvalue equations corresponding to each eigenvalue. Represents the first in the Z-dimensional direction The known eigenvalue equations corresponding to each eigenvalue. , They represent The first derivative and the second derivative, , They represent The first derivative and the second derivative, , They represent The first derivative and the second derivative, This represents the coefficients of the known eigenvalue equations corresponding to the eigenvalues ​​of the total heat source in each dimension.

Images