Design method and preparation method of pressure sensor based on topological optimization

Abstract

The invention discloses a pressure sensor design method and preparation method based on topological optimization, and belongs to the technical field of MEMS piezoresistive pressure sensing, and the method comprises the steps: constructing a pressure film structure layer, and defining a topological region of a pressure film structure material; constructing a power exponential penalty relationship between the relative density and the elastic modulus of the pressure film structure material, and carrying out topological optimization by adopting an index weighted objective function; the method comprises the following steps: firstly, calculating a stiffness matrix of each unit grid, and assembling the stiffness matrixes of the unit grids to obtain a total stiffness matrix of the structure; calculating node displacement of each unit grid according to the total stiffness matrix; solving a target function value of the current iteration point; constructing a moving approximation subproblem of the target function to approximate an original problem, and updating upper and lower boundaries of a design variable according to an approximation value; according to the method, the MEMS pressure sensor has higher performance, and the influence of subjective experience of engineers is avoided.

Description

Technical Field

[0001] The invention relates to a design method and a preparation method of a pressure sensor based on topology optimization, and belongs to the technical field of MEMS piezoresistive pressure sensing. Background Art

[0002] MEMS piezoresistive pressure sensors are widely used in consumer electronics, aerospace, automotive electronics and other fields due to their advantages such as small size, high linearity and easy integration. With the development of society, the demand for high-precision and high-reliability pressure sensors in various fields has increased significantly. However, traditional pressure sensors face the problem of how to balance high sensitivity and high linearity. Domestic and foreign researchers usually choose to directly design new structures to balance the relationship between the two. However, the traditional structural design method relies on the designer's experience and intuition, which will further affect the standardized production and development cycle of the product. Summary of the invention

[0003] The purpose of the present invention is to provide a pressure sensor design method and preparation method based on topology optimization, avoiding the influence of engineers' subjective experience and realizing the use of algorithms to solve the problem that MEMS pressure sensors are difficult to achieve both high sensitivity and high linearity.

[0004] A first aspect: A pressure sensor design method based on topology optimization, comprising: Step 1, according to the preset pressure membrane structure, construct the pressure membrane structure layering and define the topological area of ​​the pressure membrane structure material; Step 2: In the topological region of the pressure membrane structure material, the SIMP method is used to construct the power exponential penalty relationship between the relative density x and the elastic modulus E of the pressure membrane structure material, and the objective function C of the pressure membrane structure is defined according to the minimum potential energy principle of structural mechanics. s and constraints ST; Step 3, divide the pressure membrane structure into unit grids, assign an initial relative density to each unit grid, calculate the stiffness matrix k0 of each unit grid, assemble the stiffness matrices of each unit grid, and obtain the total stiffness matrix K of the structure; Step 4: Calculate the node displacement u of each unit grid according to the total stiffness matrix i ; Step 5: Based on the unit stiffness matrix and the node displacement of the unit mesh, solve the objective function C of the current iteration point s value; Step 6: According to the MMA algorithm, a moving approximation subproblem of the objective function is constructed to approximate the original problem, and the upper and lower bounds of the design variables are updated according to the approximate values; Step 7: Determine whether the approximate solution converges. If not, jump to step 5 and iterate. If converged, determine whether the convergence tolerance value reaches the preset value. If so, output the objective function C. s Values ​​and design variable values; Step 8: Design the pressure membrane structure according to the design variable values.

[0005] Optionally, the expression of the power exponential penalty relationship between the relative density x and the elastic modulus E of the pressure membrane structure material is: (1) (2) Among them, x p is the relative density of the interpolated unit, x min is the minimum relative density defined, E0 is the Young's modulus of the material, E p is the elastic modulus after interpolation, E min Defines the minimum elastic modulus.

[0006] Optionally, the objective function C s Based on the minimum potential energy principle of structural mechanics, it is equivalent to the minimum flexibility of the pressure membrane structure, and the expression is: (3) In the formula, C s is the structural flexibility, ε ij is the strain tensor, D ij is the elastic tensor, p represents the structural volume force, is the traction force on the Newman boundary condition Г and the virtual displacement vector in the equilibrium state.

[0007] Optionally, the objective function C s Use the stiffness matrix K and displacement matrix U to convert to the finite element equation, the expression is: (4) Where F is the load vector, K is the total stiffness matrix, U is the displacement vector, B is the strain displacement matrix, n is the total number of elements, and S is e Unit area.

[0008] Optionally, the finite element equation is converted into a mathematical model for topology optimization as follows: (5) Min: (6) ST: (7) Where X is the design variable, K is the total stiffness matrix after SIMP interpolation, V is the optimized volume, and V* is the constraint volume.

[0009] Optionally, for the objective function C s To make improvements: (8) Where λ is the weighting coefficient, f, C0, C max are the objective function, minimum strain energy and maximum strain energy, respectively. , , is the average value of longitudinal and transverse stress difference, the average value of minimum stress difference, and the average value of maximum stress difference in the specified area. Assuming that the width of the black rectangular area Ω is b, the height is d, and the area is s, the average stress difference equation of the rectangular area is obtained, as shown in formula (9): (9) In the formula, is the average stress value of the black rectangular area. The average of the four black rectangular areas gives the equation: (10) (11) In the formula, is the piezoresistive domain The integral of is normalized and the stress term is kept negative. The second half of the function is divided by .

[0010] Optionally, the pressure membrane structure is a four-wing structure.

[0011] Optionally, the pressure membrane structure is divided into four layers: an upper layer, a middle layer, a lower layer and a base layer.

[0012] A method for preparing a sensor designed by the pressure sensor design method according to the first aspect, the method for preparing the sensor comprising: a: Use RCA cleaning method to clean the silicon wafer, then use thermal oxidation method to form silicon dioxide layer on the upper and lower surfaces of the silicon wafer after drying the silicon wafer; b: According to the manufactured piezoresistive layout, ions are injected after mask lithography to form the piezoresistive P-region, and after rapid annealing, a layer of Si3N4 is deposited by low-pressure chemical vapor deposition method; c: Use photolithography technology to etch windows on the silicon wafer and inject ions to form a piezoresistive P+ region. After rapid thermal annealing, the ohmic contact holes between the metal and the piezoresistive are realized. d: Oxidize the silicon wafer again to form a piezoresistive protective layer and make the micropores filled and flat; e: Photolithography is performed again to open the metal contact hole area, KOH wet etching is used to form the first-level back cavity, and photolithography and etching are performed again to form the back pattern; f: Sputtering a thin metal layer on top of the silicon wafer, then patterning and sintering to form metal leads in the ohmic contact holes; g: Photolithography again, using reactive ion etching to form the upper surface pattern; h: Bottom encapsulation by anodic bonding.

[0013] Optionally, the bonding voltage is 380 V, the temperature is heated to about 400° C., and the bonding is performed for 10 minutes.

[0014] Compared with the prior art, the present invention has the following beneficial effects: the present invention adopts an algorithm to solve the problem that MEMS pressure sensors are difficult to achieve both high sensitivity and high linearity, thereby standardizing the design of MEMS pressure sensors; The invention adopts a designed four-wing film structure to achieve both high sensitivity and high linearity. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 It is a schematic flow chart of the pressure sensor design method of the present invention; Figure 2 It is a schematic diagram of a four-wing membrane structure of a pressure sensor design method of the present invention; Figure 3 It is a schematic diagram of the layered structure of the pressure membrane structure of the present invention; Figure 4 Schematic diagram of the arrangement of the piezoresistive area of ​​(a) a square membrane structure of the present invention; (b) a schematic diagram of the arrangement of the piezoresistive area of ​​the four-wing membrane structure; Figure 5 It is a schematic diagram of optimization results of different design parameters of the present invention; Figure 6 This is the process step for preparing the pressure sensor of the present invention. DETAILED DESCRIPTION

[0016] In order to make the technical means, creative features, objectives and effects achieved by the present invention easy to understand, the present invention is further explained below in conjunction with specific implementation methods.

[0017] Example 1

[0018] like Figure 1-Figure 5 As shown, a pressure sensor design method based on topology optimization is disclosed, comprising: Step 1, according to the preset pressure membrane structure, construct the pressure membrane structure layering and define the topological area of ​​the pressure membrane structure material; Step 2: In the topological region of the pressure membrane structure material, the SIMP method is used to construct the power exponential penalty relationship between the relative density x and the elastic modulus E of the pressure membrane structure material, and the objective function C of the pressure membrane structure is defined according to the minimum potential energy principle of structural mechanics. sand constraints ST; Step 3, divide the pressure membrane structure into unit grids, assign an initial relative density to each unit grid, calculate the stiffness matrix k0 of each unit grid, assemble the stiffness matrices of each unit grid, and obtain the total stiffness matrix K of the structure; Step 4: Calculate the node displacement u of each unit grid according to the total stiffness matrix i ; Step 5: Based on the unit stiffness matrix and the node displacement of the unit mesh, solve the objective function C of the current iteration point s value; Step 6: According to the MMA algorithm, a moving approximation subproblem of the objective function is constructed to approximate the original problem, and the upper and lower bounds of the design variables are updated according to the approximate values; Step 7: Determine whether the approximate solution converges. If not, jump to step 5 and iterate. If converged, determine whether the convergence tolerance value reaches the preset value. If so, output the objective function C. s Values ​​and design variable values; Step 8: Design the pressure membrane structure according to the design variable values.

[0019] Further elaboration of the above technical solution: The present invention adopts the method of topological optimization to design MEMS sensors. This method is based on the principle of minimum potential energy and finds the best material distribution scheme within a given design space while satisfying the constraints. The topological optimization method avoids the influence of engineers' subjective experience to a certain extent and can find the local optimal solution under a specific load. Figure 1 This is the flow chart of the topology optimization design of the MEMS sensor.

[0020] Since the design of the pressure membrane is a crucial part of the pressure sensor design process, the structure of the membrane will directly affect the output performance of the sensor. Therefore, the structure of the membrane is predefined in the design. The most common MEMS pressure sensor is a square membrane structure because it is easy to manufacture. However, with the development of medicine, aviation and other fields, more and more high-performance pressure sensors are needed. It is necessary to introduce other membrane structures to improve sensor performance. Figure 2 As shown, this design adopts a four-wing structure, which can better concentrate the stress in the piezoresistive area and reduce the deflection of the membrane.

[0021] Next (corresponding to the third step of the flowchart), the designed sensor model is divided into four layers: upper, middle, lower and base. The thickness of the first three layers is h 1 , h 2 , h 3 ,like Figure 3shown.

[0022] According to the SIMP method, the material distribution problem of the structure is converted into the density value problem of each unit (the structure is divided into many small units). The relative density of the material of each unit is defined between 0 and 1, where 0 represents the air domain and 1 represents the solid domain. x and elastic modulus E The expression of the power exponential penalty relationship between is: (1) (2) Among them, x p is the relative density of the interpolated unit, x min is the minimum relative density defined, E0 is the Young's modulus of the material, E p is the elastic modulus after interpolation, E min Defines the minimum elastic modulus.

[0023] Among them, x p is the relative density of the interpolated unit, x min is the minimum relative density defined, E 0 is the Young's modulus of the material, E p is the elastic modulus after interpolation, E min Defines the minimum elastic modulus.

[0024] like Figure 3 As shown, the density of the upper and lower membrane units can be changed freely, and the middle part is retained. The above-mentioned four-wing membrane is a symmetrical figure including various boundary contours surrounded by regular geometric curves, such as circles, ellipses, cones, etc. Therefore, the circular membrane and the square membrane can be regarded as the simplest case of edge contours.

[0025] The minimum flexibility of the structure is the most commonly used goal of topology optimization, which can ensure that the structure has sufficient rigidity while being lightweight. Corresponding to the pressure sensor problem, it is to make the membrane deflection as small as possible when subjected to external pressure, which is equivalent to minimizing the average flexibility of the structure.

[0026] (3) In the formula, C s is the structural flexibility, ε ij is the strain tensor, D ij is the elastic tensor, p represents the structural volume force, is the traction force on the Newman boundary condition Г, and the virtual displacement vector in the equilibrium state. It is difficult to obtain an accurate analytical equation for deflection for complex structures. Finite element analysis can be used to divide the structure into small grids, and the stiffness matrix and displacement matrix of each node can be obtained using boundary conditions. Convert the above formula into finite element equation form: (4) Where F is the load vector, K is the total stiffness matrix, U is the displacement vector, B is the strain displacement matrix, n is the total number of elements, and S is e Unit area.

[0027] The main design goal of the pressure sensor is to minimize the deformation of the film and make the structure Figure 4 The black area shown has a greater concentration of stress. The rectangular area is designated as a solid domain during topology optimization (density does not change). Due to the superior performance of the new membrane, the new membrane (four-wing type) structure and the square membrane structure designed in the previous section are used as the basic model for topology optimization.

[0028] Based on the algorithm analysis process of SIMP model, the Young's modulus of the structural unit is selected as the design variable, the minimum flexibility of the structure is taken as the goal, and the volume ratio is taken as the constraint. The mathematical model for converting the problem into topology optimization is as follows: (5) Min: (6) ST: (7) Where X is the design variable, K is the total stiffness matrix after SIMP interpolation, V is the optimized volume, and V * is the constraint volume.

[0029] In the formula, X is the design variable, K is the total stiffness matrix after SIMP interpolation, V The optimized volume, V * is the initial volume. Topology optimization problem is a special case of optimization theory. For such problems, sensitivity analysis is also very important for iterative solution and removal of numerical instability. However, conventional methods are used here and will not be described.

[0030] Since the minimum compliance topology optimization result cannot meet the design requirements of the pressure sensor, the improved objective function is considered as the design goal of this topology optimization. The most important indicators of the pressure sensor are high sensitivity and linearity, which corresponds to the mechanical problem, that is, to make the stress difference in the area where the piezoresistance is placed large enough and the film deflection small enough. Figure 4 The average of the black rectangular area shown To ensure that the structure is large and has sufficient strength, a weighted objective function is constructed as shown in Equation (8). It should be noted that the improved objective function is still a single objective function, in which the functional stress difference term is a quantity related to the rectangular area specified by the density, and does not require density interpolation penalty, and will not affect the target sensitivity, but the additional term will reduce the iteration cycle and increase the local stress, so it is very useful.

[0031] , for the objective function C s To make improvements: (8) Where λ is the weighting coefficient, f, C0, C max are the objective function, minimum strain energy and maximum strain energy, respectively. , , is the average value of longitudinal and transverse stress difference, the average value of minimum stress difference, and the average value of maximum stress difference in the specified area. Assume that the width of the black rectangular area Ω is b, the height is d, and the area is s, and the average stress difference equation of the rectangular area is obtained. The average stress difference equation of the rectangular area can be obtained, as shown in formula (9): (9) In the formula, is the average stress value of the black rectangular area. The average of the four black rectangular areas gives the equation: (10) (11) In the formula, is the piezoresistive domain The integral of the piezoresistive strips is symmetrically distributed on the x-axis and y-axis, so the above formula (11) is valid; in order to improve the sensitivity of the sensor and obtain the maximum stress in the specified area, it is necessary to make each stress average negative, so that The absolute value is the largest, and the objective function can also be minimized. To normalize and keep the stress term negative, the second half of the function is divided by .

[0032] Select Figure 2 The structural model is used as a calculation example, and the GCMMA solver of the finite element software COMSOL is used to solve it. The geometric constraints are to keep the rectangular area unchanged, the upper and lower membranes are free to change, and the volume constraint is 0.4. λ Perform a parametric sweep and use the bottom constraint as the boundary constraint, and you can get the following: Figure 5 The results are shown. Figure 5It can be seen that the output structure is very similar to the peninsula structure (an improved pressure sensor structure, which is generally considered to have excellent performance, so it also verifies from the side that the topology optimization method is feasible). λ As the flexibility decreases, the weighted flexibility becomes smaller and smaller. From the figure, it can be seen that as the flexibility decreases, the volume of the island in the middle gradually decreases, and the peninsula structure around it gradually becomes larger, which will be conducive to stress concentration and improve the sensitivity of the sensor.

[0033] Many areas of the model after topology optimization have fuzzy edges, and unpredictable stress concentration will occur during simulation. Considering the process technology and actual conditions, the exported structure is designed as a regular geometric shape with clear boundaries. Select the excellent structure after topology optimization and optimize its size (scan and compare multiple sizes, such as Figure 2 and Figure 3 All dimensions shown, etc.), and finally Figure 6 The structure shown.

[0034] Example 2

[0035] Disclosed is a method for preparing a sensor designed by the pressure sensor design method described in Example 1. The method for preparing the sensor comprises: a: Use the RCA standard cleaning method to clean n-type (100) silicon wafers, mainly to remove organic impurities, metal ions and other contaminants on the surface of the wafer. Then, after the silicon wafer is dried, a silicon dioxide layer is formed on the upper and lower surfaces of the silicon wafer using a thermal oxidation method. Then, after the silicon wafer is dried, a silicon dioxide layer is formed on the upper and lower surfaces of the silicon wafer using a thermal oxidation method; b: According to the manufactured piezoresistive layout, ions are injected after mask lithography to form the piezoresistive P-region, and after rapid annealing, a layer of Si3N4 is deposited by low-pressure chemical vapor deposition method; c: Use photolithography technology to etch windows on the silicon wafer and inject ions to form a piezoresistive P+ region. After rapid thermal annealing, the ohmic contact holes between the metal and the piezoresistive are realized. d: Oxidize the silicon wafer again to form a piezoresistive protective layer and make the micropores filled and flat; e: Photolithography is performed again to open the metal contact hole area, KOH wet etching is used to form the first-level back cavity, and photolithography and etching are performed again to form the back pattern; f: Sputtering a thin metal layer on top of the silicon wafer, then patterning and sintering to form metal leads in the ohmic contact holes; g: Photolithography again, using reactive ion etching to form the upper surface pattern; h: Bottom packaging is performed through anodic bonding, the bonding voltage is 380 V, the temperature is heated to about 400 ℃, and the bonding time is about 10 minutes.

[0036] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the technical principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A pressure sensor design method based on topology optimization, characterized in that: include: Step 1, according to the preset pressure membrane structure, construct the pressure membrane structure layering and define the topological area of ​​the pressure membrane structure material; Step 2: In the topological region of the pressure membrane structure material, the SIMP method is used to construct the power exponential penalty relationship between the relative density x and the elastic modulus E of the pressure membrane structure material, and the objective function C of the pressure membrane structure is defined according to the minimum potential energy principle of structural mechanics. s and constraints ST; Step 3, divide the pressure membrane structure into unit grids, assign an initial relative density to each unit grid, calculate the stiffness matrix k0 of each unit grid, assemble the stiffness matrices of each unit grid, and obtain the total stiffness matrix K of the structure; Step 4: Calculate the node displacement u of each unit grid according to the total stiffness matrix i ; Step 5: Based on the unit stiffness matrix and the node displacement of the unit mesh, solve the objective function C of the current iteration point s value; Step 6: According to the MMA algorithm, a moving approximation subproblem of the objective function is constructed to approximate the original problem, and the upper and lower bounds of the design variables are updated according to the approximate values; Step 7: Determine whether the approximate solution converges. If not, jump to step 5 and iterate. If converged, determine whether the convergence tolerance value reaches the preset value. If so, output the objective function C. s Values ​​and design variable values; Step 8: Design the pressure membrane structure according to the design variable values.

2. The pressure sensor design method based on topology optimization according to claim 1 is characterized in that: The expression of the power exponential penalty relationship between the relative density x and the elastic modulus E of the pressure membrane structure material is: (1) (2) Among them, x p is the relative density of the interpolated unit, x min is the minimum relative density defined, E0 is the Young's modulus of the material, E p is the elastic modulus after interpolation, E min Defines the minimum elastic modulus.

3. The pressure sensor design method based on topology optimization according to claim 1 is characterized in that: The objective function C s Based on the minimum potential energy principle of structural mechanics, it is equivalent to the minimum flexibility of the pressure membrane structure, and the expression is: (3) In the formula, C s is the structural flexibility, ε ij is the strain tensor, D ij is the elastic tensor, p represents the structural volume force, is the traction force on the Newman boundary condition Г and the virtual displacement vector in the equilibrium state.

4. The pressure sensor design method based on topology optimization according to claim 3 is characterized in that: The objective function C s Use the stiffness matrix K and displacement matrix U to convert to the finite element equation, the expression is: (4) Where F is the load vector, K is the total stiffness matrix, U is the displacement vector, B is the strain displacement matrix, n is the total number of elements, and S is e Unit area.

5. The pressure sensor design method based on topology optimization according to claim 3 is characterized in that: The finite element equation is transformed into a mathematical model for topology optimization as follows: (5) Min: (6) ST: (7) Where X is the design variable, K is the total stiffness matrix after SIMP interpolation, V is the optimized volume, and V * is the constraint volume.

6. The pressure sensor design method based on topology optimization according to claim 1 is characterized in that: For the objective function C s To make improvements: (8) Where λ is the weighting coefficient, f, C0, C max are the objective function, minimum strain energy and maximum strain energy, respectively. , , is the average value of longitudinal and transverse stress difference, the average value of minimum stress difference, and the average value of maximum stress difference in the specified area. Assuming that the width of the black rectangular area Ω is b, the height is d, and the area is s, the average stress difference equation of the rectangular area is obtained, as shown in formula (9): (9) In the formula, is the average stress value of the black rectangular area. The average of the four black rectangular areas gives the equation: (10) (11) In the formula, is the piezoresistive domain The integral of is normalized and the stress term is kept negative. The second half of the function is divided by .

7. The pressure sensor design method based on topology optimization according to claim 1 is characterized in that: The pressure membrane structure is a four-wing structure.

8. The pressure sensor design method based on topology optimization according to claim 1, characterized in that: The pressure membrane structure is divided into four layers: an upper layer, a middle layer, a lower layer and a base layer.

9. A method for preparing a sensor designed by the pressure sensor design method according to any one of claims 1 to 8, characterized in that: The method for preparing the sensor comprises: a: Use RCA cleaning method to clean the silicon wafer, then use thermal oxidation method to form silicon dioxide layer on the upper and lower surfaces of the silicon wafer after drying the silicon wafer; b: According to the manufactured piezoresistive layout, ions are injected after mask lithography to form the piezoresistive P-region, and after rapid annealing, a layer of Si3N4 is deposited by low-pressure chemical vapor deposition method; c: Use photolithography technology to etch windows on the silicon wafer and inject ions to form a piezoresistive P+ region. After rapid thermal annealing, the ohmic contact holes between the metal and the piezoresistive are realized. d: Oxidize the silicon wafer again to form a piezoresistive protective layer and make the micropores filled and flat; e: Photolithography is performed again to open the metal contact hole area, KOH wet etching is used to form the first-level back cavity, and photolithography and etching are performed again to form the back pattern; f: Sputtering a thin metal layer on top of the silicon wafer, then patterning and sintering to form metal leads in the ohmic contact holes; g: Photolithography again, using reactive ion etching to form the upper surface pattern; h: Bottom encapsulation by anodic bonding.

10. The method for preparing a pressure sensor according to claim 9, characterized in that: Including, the bonding voltage is 380 V, the temperature is heated to about 400° C., and the bonding is performed for 10 minutes.

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