A global equation-based semiconductor heater power density design simulation acceleration method

Abstract

This invention discloses an accelerated simulation method for power density design of semiconductor heaters based on global equations. The invention uses the areal power density of each heating zone of the semiconductor heater as the design variable to be determined, the area-average temperature of the target heated inner surface as the observation, and the deviation between the area-average temperature and the preset target temperature as the constraint residual. An augmented nonlinear equation system is constructed, including state variables and power density variables related to the target temperature constraint. By simultaneously solving the augmented nonlinear equation system, the power density distribution of the zones that satisfies the target temperature constraint is directly obtained. Compared with manual parameter tuning, this invention reduces repeated trial calculations. Compared with common gradient-based outer-layer optimization methods, derivative-free outer-layer search methods, and swarm search-based outer-layer optimization methods, this invention requires no additional iterations or parameter adjustments, and features a short computational path, high convergence efficiency, and small target temperature control error.

Description

Technical Field

[0001] This application relates to the field of precision temperature control technology for semiconductor devices, specifically to a simulation acceleration method for power density design of semiconductor heaters based on global equations. Background Technology

[0002] In semiconductor manufacturing processes, core components such as gas heating modules, process reaction chambers, hot plates, electrostatic chucks, and wafer carrier stages all rely on heaters to provide stable and controllable heat input. The temperature control accuracy and temperature field uniformity of the heaters not only determine the system's thermal field distribution but also affect wafer deposition rate, etching uniformity, thin film thickness consistency, and process window stability. Therefore, they are important factors affecting semiconductor manufacturing yield.

[0003] In existing technologies, heater power density design largely relies on manual parameter tuning. The basic process involves: presetting initial power density values ​​for each heating zone, performing numerical simulations to obtain the temperature field distribution, and then repeatedly adjusting the power density parameters based on the deviation between the simulated temperature and the target temperature. When there are many heater zones, strong inter-zone thermal coupling, and differentiated target temperature requirements across multiple temperature zones, this method requires extensive trial and error, resulting in long design cycles, high simulation costs, poor stability of design results, and reliance on manual experience.

[0004] Another existing approach employs common gradient-based outer-layer optimization methods, derivative-free outer-layer search methods, and population search-based outer-layer optimization methods. These methods construct the target temperature deviation as the objective function and achieve automatic power density adjustment through outer-layer iterative search. While this approach can achieve automatic power density correction, it still requires setting optimization objectives, variable boundaries, initial values, step size control, or convergence criteria. Furthermore, it necessitates constructing a two-layer iterative architecture of "outer-layer optimization search - inner-layer physics field solution," resulting in high computational resource consumption, long convergence paths, low simulation efficiency, and limitations on the target temperature residual due to search accuracy.

[0005] Therefore, there is an urgent need for a design method that can directly construct constraint relationships around a preset target temperature and simultaneously correct the flow field, thermal field state variables, and power density design variables within a unified coupled equation framework, in order to break through the efficiency bottleneck of existing technologies and achieve rapid, high-precision reverse calculation and simulation acceleration of the power density of semiconductor heaters. Summary of the Invention

[0006] The purpose of this application is to provide a simulation acceleration method for the design of semiconductor heater power density based on global equations, in order to solve the problems of low efficiency of manual parameter tuning, long simulation cycle, redundant calculation path of outer layer optimization algorithm, and difficulty in further compressing the target temperature control error in the prior art.

[0007] In a first aspect, the present invention provides a method for accelerating the simulation of semiconductor heater power density design based on global equations, comprising the following steps:

[0008] S1: Obtain the geometric and physical property parameters, operating conditions and boundary conditions of the heater and related structures in the semiconductor device;

[0009] S2: Divide the heated inner surface of the heater into multiple independent heating zones, and construct the surface power density of each heating zone as a vector of design variables to be determined;

[0010] S3: Construct a state residual operator based on the actual coupling relationship of the target object. The state residual operator includes at least the heat conduction control relationship and can further incorporate flow and heat transfer coupling relationships based on the actual coupling relationship.

[0011] S4: Define the corresponding area average temperature operator for each target heated surface;

[0012] S5: Construct a global constraint residual based on the deviation between the area average temperature of the target heated surface and the preset target temperature;

[0013] S6: Integrate the state residual operator and the global constraint residual into an augmented nonlinear equation system. By solving the augmented nonlinear equation system simultaneously, the power density of each heating zone that meets the preset target temperature requirement can be directly obtained.

[0014] Furthermore, the state variable vector corresponding to the state residual operator includes the fluid velocity field, the fluid pressure field, and the system temperature field.

[0015] Furthermore, the area-averaged temperature operator is obtained by performing an area-weighted integral on the temperature field at the boundary of the target heated surface and dividing it by the total area of ​​the target heated surface.

[0016] Furthermore, the global constraint residual is the difference between the area average temperature and the preset target temperature of the corresponding target heated surface.

[0017] Furthermore, the augmented nonlinear equation system is constructed by stacking the state residual operator and the global constraint residual into a unified residual vector according to the variable dimension, and combining the state variables and the design variables into augmented solution variables.

[0018] Furthermore, the augmented nonlinear equations are solved iteratively using the block Jacobi linearization correction form to obtain the state variable correction and the power density design variable correction, respectively; and an equivalent temperature constraint sensitivity matrix is ​​constructed based on the block elimination method to characterize the comprehensive impact of power density changes on the target temperature residual.

[0019] Furthermore, the power density design variable is directionally corrected based on the equivalent temperature constraint sensitivity matrix, thereby eliminating the dependence of the design variable correction process on the outer search algorithm.

[0020] Furthermore, the state residual operator also includes at least one of the following: fluid flow conservation control relationship, fluid energy transfer control relationship, and boundary heat transfer coupling relationship.

[0021] In a second aspect, the present invention provides a computer device, the computer device comprising:

[0022] One or more processors;

[0023] Memory, used to store one or more programs;

[0024] When the one or more programs are executed by the one or more processors, the one or more processors perform the steps in the method described above.

[0025] Thirdly, the present invention provides a computer-readable storage medium storing computer instructions that, when executed by one or more processors, cause the one or more processors to perform the steps of the method described above.

[0026] Compared with the prior art, this application has at least the following beneficial effects:

[0027] Firstly, by directly constructing global constraint equations with a preset target temperature and embedding them into the solution framework, the reliance on manual parameter tuning experience and optimization of objective function weight settings is reduced.

[0028] Secondly, the flow field and thermal field state variables and power density design variables are coupled and updated synchronously within the same equation system framework, eliminating the need for two-level iterations and improving solution efficiency.

[0029] Third, by combining the design of global constraint residuals and equivalent temperature constraint sensitivity matrices, it is beneficial to reduce the target temperature control error. Attached Figure Description

[0030] Figure 1 Flowchart for constructing physical field and temperature constraints for global equations;

[0031] Figure 2 This is an internal structural diagram of the computer device in this embodiment. Detailed Implementation

[0032] The present application will be further described in detail below with reference to specific embodiments. However, the scope of protection of the present application is not limited to the following embodiments.

[0033] This application proposes a method for constructing an augmented equation system centered on state residual operators and global constraint residuals. The key technical point of this application is not to require all physical processes to be uniformly incorporated into the same equation system at once, but rather to uniformly write the state residuals and power density design variables related to the target temperature constraint into the augmented equation system for simultaneous solution, thereby reducing the efficiency loss caused by the two-layer iterative architecture.

[0034] Specifically, this application first constructs a state variable vector x and a design variable vector q, where the state variable vector x is used to characterize the relevant state field of the system, and the design variable vector q is used to characterize the surface power density distribution of each heating zone of the multi-zone heater; then, based on the actual coupling relationship of the target object, relevant control equations are selected to construct the state residual operator. The governing equations include at least heat conduction and may further incorporate flow and heat transfer coupling relationships; an area-average temperature operator is introduced for the i-th target heated surface. And based on this operator, a global constraint residual is constructed. .

[0035] When all target heated surfaces meet the preset design temperature requirements, the global constraint residuals are satisfied. Based on this, construct the overall residual vector. It is then integrated with the state residual operator into an augmented nonlinear equation system F(X)=0 to achieve co-firing of the physical field solution and the temperature target constraint within the same framework. For different engineering objects, the specific composition of the state residual operator can be adjusted according to the actual coupling relationship, but it does not change the unified augmented solution framework of this application.

[0036] Furthermore, to improve the stability and convergence of the equation system, this application employs a block Jacobi linearization method to iteratively correct the augmented nonlinear equation system, and constructs an equivalent temperature constraint sensitivity matrix S using block elimination to assess the comprehensive sensitivity of power density changes to the target temperature residual. Using this sensitivity matrix, the directional correction of the power density design variable can be achieved without introducing an outer search algorithm, thereby reducing the convergence path and the number of iterations.

[0037] In one specific embodiment, the target semiconductor device includes a fluid flow channel, a multi-zone heater shell structure, and a solid entity structure thermally coupled to the heater shell structure; the known operating conditions include fluid inlet flow rate, inlet temperature, outlet boundary conditions, and outer boundary heat dissipation conditions; the heated inner surface of the heater is divided into multiple independent heating zones, each heating zone corresponding to a surface power density parameter to be solved.

[0038] For the aforementioned system, the control relationships related to the target temperature constraint can be uniformly represented as state residual equations. To facilitate the explanation of the technical implementation path of this application, the standard equations used for heat conduction and, in one embodiment, fluid, solid, shell heat transfer, and global constraint solving are listed below:

[0039] (1) Continuity equation:

[0040]

[0041] In the formula, Here, is the divergence operator, and u is the fluid velocity vector.

[0042] (2) Navier-Stokes momentum equation:

[0043]

[0044] In the formula, ρ is the fluid density, p is the fluid pressure, μ is the dynamic viscosity, and F is the volume force term.

[0045] (3) Fluid energy equation:

[0046]

[0047] In the formula, C p For the specific heat capacity at constant pressure, T f k represents the fluid temperature. f Q is the thermal conductivity of the fluid. f It serves as a heat source for the fluid domain.

[0048] (4) Solid heat conduction equation:

[0049]

[0050] In the formula, T s For solid temperature, k s Q is the thermal conductivity of a solid. s It is a solid-state heat source.

[0051] (5) Shell heat transfer equation:

[0052]

[0053] In the formula, the relevant parameters represent the shell tangential gradient operator, shell thickness, shell equivalent thermal conductivity, shell temperature, partition surface power density, and shell additional surface heat source term, respectively.

[0054] (6) Interfacial thermal continuity condition:

[0055]

[0056]

[0057] In the formula, T1 and T2 are the temperatures on both sides of the interface, k1 and k2 are the thermal conductivity on both sides of the interface, and n is the interface normal vector.

[0058] (7) Convection heat transfer boundary conditions:

[0059]

[0060] In the formula, h is the convective heat transfer coefficient, and T ∞ The ambient temperature.

[0061] (8) The state variable vector and the design variable vector to be determined corresponding to the state residual operator:

[0062]

[0063]

[0064] In the formula, x is the state variable vector, q is the design variable vector, u is the fluid velocity field, p is the fluid pressure field, and T is the system temperature field. Let n be the surface power density of the i-th heating zone, and n be the total number of heating zones.

[0065] (9) Area-average temperature operator:

[0066]

[0067] In the formula, Denotes the boundary of the i-th target heated surface. This represents the heated surface area of ​​the i-th target.

[0068] (10) Global constraint residual equation:

[0069]

[0070] In the formula, the residual represents the deviation between the area average temperature of the i-th target heated surface and the preset target temperature.

[0071] (11) Augmented nonlinear equations:

[0072]

[0073]

[0074]

[0075] In the formula, For the global constraint residual vector, For the augmented nonlinear system of equations, X is the augmented solution variable.

[0076] (12) Block Jacobi modified equation:

[0077]

[0078] In the formula, Δx and Δq are the partial derivative block matrices of the state residual and the global constraint residual with respect to the state variables and design variables, respectively. Δx is the state variable correction amount, and Δq is the power density correction amount.

[0079] (13) Equivalent temperature constraint sensitivity matrix:

[0080]

[0081] In the formula, S is the equivalent temperature constraint sensitivity matrix, which is used to characterize the comprehensive impact of power density changes on the target temperature residual.

[0082] (14) Variable update relationship:

[0083]

[0084] In the formula, the state residual norm is recalculated after the update. With global constraint residual norm Until the convergence tolerance is satisfied simultaneously.

[0085] Based on the aforementioned physical field equations, an area-average temperature observation is constructed for each target heated surface. And establish the corresponding global temperature constraint residual. This transforms the temperature matching problem, which originally relied on manual trial and error or outer optimization search, into a unified simultaneous solution problem under augmented variable X, thereby simplifying the solution process.

[0086] When the heater has N heating zones, a corresponding design variable vector is constructed. Simultaneously, a global temperature constraint equation consistent with the number of partitions is established; if different target temperature zones have different preset target temperatures, the target temperature parameters can be assigned to the corresponding target temperature zones. This allows for the synchronous solution of parameters across multiple temperature zones within the same coupled solution framework.

[0087] In practical calculations, the state residual R and global constraint residual H can be calculated first based on the state variables and power density variables of the current iteration step. Then, the state variable correction Δx and power density correction Δq can be obtained by solving the block Jacobian matrix. Furthermore, the power density correction path can be compressed and characterized by the equivalent temperature constraint sensitivity matrix S to improve the convergence stability of the parameter correction process under multi-partition thermal coupling conditions.

[0088] Combination Figure 1 The simulation acceleration method for semiconductor heater power density design based on global equations in this application, as shown in the flowchart, can be summarized as follows:

[0089] Parameter import and variable definition: Import the heater's geometric parameters, material thermal properties, operating conditions, and boundary conditions, and define the state variable vector x and design variable vector q based on the known conditions;

[0090] State residual operator construction: Constructing state residual operators using the state variable vector and design variable vector as independent variables. The operator selects relevant control equations based on the actual coupling relationship of the target object, at least characterizing the heat conduction control relationship, and can further characterize the fluid flow conservation equation, energy conservation equation, and boundary heat transfer coupling relationship; under the given current power density distribution q, the corresponding relevant state field is obtained by solving;

[0091] Construction of the average temperature operator: for the i-th target heated surface Construct an area-average temperature operator The distributed temperature field parameters are mapped to a surface average temperature scalar that directly corresponds to semiconductor process control.

[0092] Global constraint residual construction: obtaining area-averaged temperature Then, construct the i-th global constraint residual. And construct a global constraint residual vector based on the constraint residuals of all target heated surfaces. ;

[0093] Construction of augmented equation system: Combine the global constraint residual vector with the state residual operator to form an augmented nonlinear equation system. ;

[0094] Linearization and Correction Calculation: Local linearization is performed on the augmented nonlinear equation system to obtain the block Jacobian correction equations, and the state variable corrections are solved. and power density design variable correction amount Furthermore, an equivalent temperature constraint sensitivity matrix S is constructed using the block elimination method to clarify the regulatory relationship between power density and target temperature deviation.

[0095] Iterative Update and Convergence Judgment: Based on the corrected values ​​obtained from the solution, iterative updates are performed on the state variables and power density variables. The state residual operator is re-solved and the global constraint residual is calculated. The residual norm is then determined. and Whether both are less than the preset convergence tolerance. When both meet the convergence criteria, the iteration stops and the final power density distribution of each partition and the corresponding temperature field results are output; when either index fails to meet the convergence requirement, the solution step returns to continue iterating.

[0096] The above method is applicable to the coordinated design of power density and temperature field of semiconductor process cavities, gas heating boxes, hot plates, wafer carrier stages and other multi-zone semiconductor heaters, so as to reduce the adverse effects of temperature deviation on the stability of semiconductor processes and product yield, such as thin film deposition, etching, annealing, epitaxy, crystal growth and reaction gas heating.

[0097] Based on the same concept as the above method, this application also provides a computer device, the internal structure of which can be shown in the following diagram. Figure 2 As shown, the computer device includes a processor, memory, input / output (I / O) interfaces, and a communication interface. The processor, memory, and I / O interfaces are connected via a system bus, and the communication interface is also connected to the system bus via the I / O interfaces. The processor provides computing and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs stored in the non-volatile storage media. The database stores data related to the embodiments of this application, such as binding relationships. The I / O interfaces are used for exchanging information between the processor and external devices. The communication interface is used for communicating with external terminals via a network connection. When the computer program is executed by the processor, it implements the steps in the simulation acceleration method for semiconductor heater power density design based on global equations.

[0098] Those skilled in the art will understand that Figure 2 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0099] Based on the same concept as the above method, embodiments of this application also provide a computer-readable storage medium storing computer instructions, which, when executed by one or more processors, cause the one or more processors to perform steps in the simulation acceleration method for semiconductor heater power density design based on global equations.

[0100] To further illustrate the simulation acceleration effect of this application, the following illustrative example is constructed: the heated surface of the semiconductor heater is divided into 50 independent heating zones, and the 50 surface power density design variables are synchronously inversely calculated; at the same time, a uniform accuracy requirement is set, that is, the maximum absolute value of the deviation between the area average temperature of each target heated surface and the preset target temperature does not exceed 1℃, or the global temperature constraint residual satisfies the same preset convergence tolerance.

[0101] To ensure fairness and comparability in the comparison, all methods to be compared use the exact same geometric model, mesh size, physical property parameters and boundary conditions, and the time taken for a single complete flow-thermal field coupled simulation is uniformly denoted as τ.

[0102] Under the aforementioned accuracy requirements, common gradient-based outer-layer optimization methods, derivative-free outer-layer search methods, and swarm search-based outer-layer optimization methods typically require a complete flow-thermal field coupled simulation. Therefore, the total time consumption can be expressed as follows: ,in The number of simulations required to achieve the same accuracy requirement. For the problem of simultaneous design of 50 partitions, due to the high dimensionality of design variables and significant thermal coupling between partitions, the outer optimization method often requires many iterations to gradually approach the target temperature distribution, resulting in a significant increase in the total time consumption.

[0103] In contrast, this application integrates 50 power density design variables and physical field state variables into an augmented nonlinear equation system, directly achieving synchronous variable correction through block Jacobian correction and an equivalent temperature constraint sensitivity matrix. Under the condition of meeting the same temperature accuracy requirements, the total time consumption of this application can be expressed as k×τ, where k is the number of global simultaneous iterations required for the augmented equation system to converge.

[0104] Taking the design of 50 heaters as an example, the single simulation time τ = 30 minutes. Gradient-based optimization methods require more than 50 iterations, derivative-free outer-layer search methods require more than 80 iterations, and swarm search outer-layer optimization methods require more than 120 iterations. This application, however, only requires one complete global simultaneous iteration calculation. Based on this calculation, the total computation time for the four methods is approximately 1500 minutes, 2400 minutes, 3600 minutes, and 30 minutes, respectively, as shown in the table below:

[0105]

[0106] Therefore, this application can significantly reduce the number of repeated solutions to the physical field, greatly shorten the calculation process, and effectively improve the design accuracy of the target temperature, highlighting significant technical advantages.

[0107] For semiconductor manufacturing equipment, higher target temperature control accuracy can reduce the probability of problems such as uneven heating of wafer surface, film thickness deviation, etching rate fluctuation and reaction gas heating deviation, thereby reducing process drift, expanding the stable process window, and reducing the adverse effects of temperature deviation on product yield.

[0108] In summary, the global equation-based simulation acceleration method for semiconductor heater power density design described in this application balances solution efficiency, temperature control accuracy, and engineering feasibility. By coupling the solution to the global equations, the overhead of double-layer iterations in traditional methods can be reduced, improving simulation efficiency and making it suitable for power density design of multi-zone precision heating systems in semiconductor devices.

Claims

1. A method for accelerating the simulation of the power density design of a semiconductor heater based on global equations, characterized in that, Includes the following steps: S1: Obtain the geometric and physical property parameters, operating conditions and boundary conditions of the heater and related structures in the semiconductor device; S2: Divide the heated inner surface of the heater into multiple independent heating zones, and construct the surface power density of each heating zone as a vector of design variables to be determined; S3: Construct a state residual operator based on the actual coupling relationship of the target object. The state residual operator includes at least the heat conduction control relationship and can further incorporate flow and heat transfer coupling relationships based on the actual coupling relationship. S4: Define the corresponding area average temperature operator for each target heated surface; S5: Construct a global constraint residual based on the deviation between the area average temperature of the target heated surface and the preset target temperature; S6: Integrate the state residual operator and the global constraint residual into an augmented nonlinear equation system. By solving the augmented nonlinear equation system simultaneously, the power density of each heating zone that meets the preset target temperature requirement can be directly obtained.

2. The global equation based semiconductor heater power density design emulation acceleration method of claim 1, wherein, The state variable vector corresponding to the state residual operator includes the fluid velocity field, the fluid pressure field, and the system temperature field.

3. The method of claim 1, wherein, The area-averaged temperature operator is obtained by performing an area-weighted integral on the temperature field at the boundary of the target heated surface and dividing it by the total area of ​​the target heated surface.

4. The method of claim 3, wherein, The global constraint residual is the difference between the area average temperature and the preset target temperature of the corresponding target heated surface.

5. The method according to claim 1, characterized in that, The augmented nonlinear equation system is constructed by stacking the state residual operator and the global constraint residual into a unified residual vector according to the variable dimension, and combining the state variables and the design variables into augmented solution variables.

6. The method according to claim 5, characterized in that, The augmented nonlinear equations are solved iteratively using the block Jacobi linearization correction form to obtain the state variable correction and the power density design variable correction, respectively. An equivalent temperature constraint sensitivity matrix is ​​constructed based on the block elimination method to characterize the comprehensive impact of power density changes on the target temperature residual.

7. The method according to claim 6, characterized in that, The power density design variable is directionally corrected based on the equivalent temperature constraint sensitivity matrix, thereby eliminating the dependence of the design variable correction process on the outer search algorithm.

8. The method according to claim 1, characterized in that, The state residual operator also includes at least one of the following: fluid flow conservation control relationship, fluid energy transfer control relationship, and boundary heat transfer coupling relationship.

9. A computer device, characterized in that, The computer device includes: One or more processors; Memory, used to store one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors perform the steps in the method as described in any one of claims 1-8.

10. A computer-readable storage medium storing computer instructions, characterized in that, When the computer instructions are executed by one or more processors, the one or more processors cause the processors to perform the steps of the method described in any one of claims 1-8.

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