Method for monitoring the growth state of semiconductor silicon single crystals based on a deep belief network

Abstract

The application discloses a kind of semiconductor silicon single crystal growth state monitoring methods based on deep belief network, first establish the model of Czochralski silicon single crystal growth process;Then based on the silicon single crystal growth model constructs the silicon single crystal growth simulation system of two inputs multiple outputs, selects the output variable of appropriate silicon single crystal growth simulation system is processed through data, finally the selected output variable is input to state monitoring model, constructs state monitoring index, completes the training of DBN network model and the construction of state monitoring model, then the silicon single crystal growth output data in running process is input to the network model trained, by comparing output state index, finally determine whether the silicon single crystal growth process running state is normal.The application solves the problem that the silicon single crystal growth process running state cannot be monitored in time in the prior art, resulting in low silicon single crystal growth efficiency and poor quality.

Description

Technical Field

[0001] This invention belongs to the field of semiconductor silicon single crystal growth process state monitoring technology, specifically relating to a semiconductor silicon single crystal growth state monitoring method based on deep belief networks. Background Technology

[0002] Silicon single crystal material is the most important fundamental material for integrated circuit chips. The continuous evolution of chip manufacturing processes has driven the development of silicon single crystal materials towards larger sizes and higher quality, thus requiring optimization and upgrades in all aspects, from crystal growth equipment to growth processes. In this process, monitoring the state of the crystal growth process is essential for achieving the growth of high-quality silicon single crystal materials. Abnormalities in melt temperature, melt height, and crystal growth rate can directly lead to dislocation defects in the crystal, reducing growth efficiency and crystal quality. Monitoring these multiple state variables is an effective way to reflect whether the crystal growth state is normal.

[0003] Silicon single crystal growth is a complex physical process. In such a multi-field, multi-phase coupled, vacuum, and high-temperature environment, how to establish a multi-state, multi-output silicon single crystal growth model and analyze the data of process parameters and variables, extract the implicit information and characteristics within the data, and realize intelligent monitoring of the crystal growth process status is a challenge to improve crystal growth efficiency and crystal quality. Summary of the Invention

[0004] The purpose of this invention is to provide a method for monitoring the growth status of semiconductor silicon single crystals based on deep belief networks, which solves the problem in the prior art that the operating status of the silicon single crystal growth process cannot be monitored in a timely manner, resulting in low silicon single crystal growth efficiency and poor quality.

[0005] The technical solution adopted in this invention is a semiconductor silicon single crystal growth state monitoring method based on deep belief networks, which is implemented according to the following steps:

[0006] Step 1: Establish a model for the Czochralski silicon single crystal growth process;

[0007] Step 2: Construct a two-input multiple-output silicon single crystal growth simulation system based on the silicon single crystal growth model. Select appropriate output variables of the silicon single crystal growth simulation system, process the data, and use them as inputs to the SM-DBN model constructed in Step 3.

[0008] Step 3: Design the SM-DBN model, input the output variables selected in Step 2 into the state monitoring model, construct state monitoring indicators, complete the training of the DBN network model and the construction of the state monitoring model, and then input the output data of the silicon single crystal growth process into the trained network model. By comparing the output state indicators, the normal operation status of the silicon single crystal growth process can be determined.

[0009] The invention is further characterized in that,

[0010] Step 1 is implemented in the following steps:

[0011] Step 1.1: Establish a silicon single crystal growth model, including a heat transfer model and a geometric model, based on the energy balance model, fluid dynamics model, and geometric model during the crystal growth process;

[0012] Step 1.2: Establish a heat transfer model. Assume that all energy released by the heater is transferred to the crucible, with no other energy loss, and that the temperature of the heater and crucible is uniform throughout the three-dimensional space. Based on the energy conservation principle of the heater, the change in heater temperature over time... The equation is:

[0013]

[0014] Wherein, the heat radiation transfer rate q from the heater to the crucible hc for:

[0015]

[0016] C h It is the heat capacity of the heater, P in It is the heater power, A c T is the surface area of ​​the crucible, σ is the Stefan-Boltzmann constant, and T is the surface area of ​​the crucible. h It is the heater temperature, T c It is the temperature of the crucible;

[0017] According to the energy conservation principle inside the crucible, the change in crucible temperature over time The equation is:

[0018]

[0019] Among them, C c It is the heat capacity of the crucible, q co It is the rate of heat radiation transfer from the crucible to the environment, q cs It is the rate of heat radiation transfer from the crucible to the melt, q cm It is the rate of heat conduction from the crucible to the melt;

[0020] Based on the conservation of melt energy and mass, the change in melt temperature over time... The equation is:

[0021]

[0022] Among them, C m It is the melt heat capacity, q so It is the rate of heat transfer from the melt to the environment, qm It is the rate of heat transfer from the melt to the meniscus. It is the rate of change of melt height, T m It is the melt temperature, H m It is the height of the melt;

[0023] At the growth interface, the crystal growth rate is determined by the heat Q of the melt flowing into the growth interface. mI Heat Q flowing into the crystal at the growth interface Ii The difference in heat is calculated, and this difference represents the heat released during crystal crystallization. Therefore, the crystal growth rate V... g According to the energy conservation equation:

[0024]

[0025] Among them, H f It is the latent heat of crystallization, ρ s It is the crystal density, q Ii It is the heat transfer rate from the solid-liquid interface to the crystal, q mI R is the heat transfer rate from the melt to the solid-liquid interface. i It is the crystal radius;

[0026] Step 1.3: Establish the geometric model:

[0027] Based on the mass conservation condition at the solid-liquid interface, the change in melt height over time... The equation is:

[0028]

[0029] Among them, V p It is the crystal pulling speed. It is the rate of change of the meniscus height, ρ m It is the melt density, R c It is the radius of the crucible;

[0030] The height H of the meniscus is expressed using the crystal radius and tilt angle. men for:

[0031]

[0032] Where 'a' is the capillary length, also known as the Laplace constant, and is related to the meniscus surface tension γ. le The value of 'a' is related to the melt density. α0 is the crystal growth angle, α c It is the crystal tilt angle;

[0033] Differentiating equation (7) yields

[0034]

[0035] The change in crystal radius over time was obtained based on the geometric relationships at the meniscus. and growth rate V g Relationship:

[0036]

[0037] Changes in the tilt angle will cause changes in the crystal diameter, and the tilt angle changes over time. Calculated using the following formula:

[0038]

[0039] in:

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] V cruc It is the crucible acceleration rate;

[0046] Based on the crystal growth rate, the crystal length changes over time. for:

[0047]

[0048] The heat transfer model and geometric model of the state variables related to the silicon single crystal growth process mentioned above together constitute the Czochralski silicon single crystal growth process model.

[0049] In step 1.3, α0 is set to 11°.

[0050] Step 2 is implemented in the following steps:

[0051] Step 2.1: Simulate the silicon single crystal growth process from the shoulder formation stage to the constant diameter growth stage. A silicon single crystal growth simulation system is built in the Matlab / Simulink environment. The simulation system consists of three subsystems. The first subsystem directly displays the total input and output of the simulation system. The input is P. in and V p The output is T. h T c T m H m R i α c V gL, H men The second layer simulates the heat flux between the melt and the crystal, as well as the crystal growth tilt angle. The third layer details the formulas and various parameter data; real-time data for each state variable can be obtained through the constructed silicon single crystal growth simulation system.

[0052] Step 2.2: Select heater temperature T h crucible temperature T c Melt temperature T m Crystal radius R i Crystal growth rate V g Melt height H m and crystal rotation speed V ir Crucible rotation speed V cr As input to the SM-DBN model established in step 3, the state of the silicon single crystal simulation system established in step 2.1 is monitored.

[0053] Step 2.3: Establish 8-dimensional sample data X, where each dimension represents a variable in the crystal growth process:

[0054]

[0055] Let n be the value of the heater temperature at the nth second of the simulation time. Let n be the crucible temperature value at the nth second of the simulation time. Let n be the value of the melt temperature at the nth second of the simulation time. The value of the crystal radius at the nth second of the simulation time. This represents the crystal growth rate at the nth second of the simulation time. Let n be the value of the crystal rotation speed at the nth second of the simulation time. Let n be the value of the crucible rotation speed at the nth second of the simulation time. Let n be the value of the melt height at the nth second of the simulation time, where n∈R;

[0056] Step 2.4: Perform L2 norm normalization on the 8-dimensional sample data X in Formula 17, and normalize the data of each row of the sample separately:

[0057]

[0058] x1,...,x n These correspond to the data values ​​of each dimension in Formula 17.

[0059] Step 3 is implemented in the following steps:

[0060] Step 3.1: The steady-state corresponding to the minimum energy function state of the DBN network, for a given set of (v,h), the specific energy function is expressed as follows:

[0061]

[0062] In the formula, v = (v1, v2, ..., v i ,…v m ) T i,m∈Z * v is the visible layer state vector. i Let be the value of the i-th node in the visual layer, and m be the number of nodes in the visual layer; h = (h1, h2, ..., h...). j ,…h n ) T j,n∈Z * h is the hidden layer state vector. j Let be the value of the j-th node in the hidden layer, and n be the number of nodes in the hidden layer; W = (w ij )i∈[1,m],j∈[1,n] is a weight matrix, a=(a1,a2,…,a i ,…a m ) T Let b be the view layer bias vector, where b = (b1, b2, ..., b i ,…b m ) T This is the hidden layer bias vector;

[0063] Step 3.2: Using the processed sample data from Step 2.4 as input to the DBN, and as the visible layer of the first RBM, the probability of the hidden layer neurons being activated is:

[0064] p(h j (0) =1|v (0) )=δ(b j +∑ i v i (0) w ij (20)

[0065] In the formula, δ is the sigmoid function, defined as δ(x) = 1 / (1+e^(x-1)). -x The superscripts of v and h indicate the state transition step k of the comparison divergence algorithm CD;

[0066] Step 3.3: Based on the probability of the hidden layer neurons being turned on, perform Gibbs sampling, sampling each neuron in the hidden layer using a binary {0,1} method to obtain the hidden layer h. (0) for:

[0067]

[0068] In the formula, r j A random number in the interval [0,1];

[0069] Step 3.4: Use hidden layer h (0) Reconstruct the visual layer v (1) Calculate the new visible layer v (1) The probability of:

[0070] p(v i (1) =1|h (0) )=δ(a i +∑ j h j (0) w ij ) (twenty two)

[0071] Step 3.5: Following the algorithm in Step 3.3, obtain the new visible layer v. (1) for:

[0072]

[0073] Step 3.6: Following steps 3.2 and 3.3, obtain the new hidden layer h. (1) for:

[0074] p(h j (1) =1|v (1) )=δ(b j +∑ i v i (1) w ij ) (twenty four)

[0075]

[0076] Step 3.7: The above completes one step of the CD algorithm, updating and obtaining the new weight matrix and bias vector:

[0077] W←W+ε[p(h (0) =1|v (0) )v (0) Tp(h (1) =1|v (1) )v (1)T (26)

[0078] a←a+ε(v (0) -v (1) (27)

[0079] b←b+ε[p(h (0) =1|v (0) )-p(h (1) =1|v (1) (28)

[0080] In the formula, ε is the learning rate;

[0081] Step 3.8: When the state transition step k = 1, a reconstructed sample similar to the data processed in step 2.4 is obtained, achieving a better result;

[0082] Step 3.9 After the single-layer RBM network is trained, the DBN network is pre-trained layer by layer. The bottom layer RBM is trained with the original input data, and then the features extracted from the bottom RBM are used as the input of the top RBM for further training. That is, each layer of RBM is trained from bottom to top.

[0083] Step 3.10: After pre-training, the DBN network will be fine-tuned. A BP neural network will be set in the last layer of the DBN, and the output of the last RBM layer will be used as the input of the BP neural network. Supervised training will be carried out from top to bottom to optimize the parameters generated in the pre-training stage and make the prediction ability of the DBN reach its optimal level.

[0084] Step 3.11: The DBN network has been trained using the steps above. Now, we select the SM-DBN state monitoring metrics. The first state monitoring metric is the squared prediction error (SPE), which is calculated as follows:

[0085]

[0086] in

[0087]

[0088] The second condition monitoring indicator is the SM-DBN energy value, which is calculated using formula (19):

[0089]

[0090] Step 3.12: Now set the SPE control limit of the state monitoring model and compare it with the SPE value of the state monitoring model to determine the system operating state. The SPE control limit is determined by χ². 2 The distribution is approximately obtained as follows:

[0091]

[0092] In the formula,

[0093]

[0094] Finally, the SPE control limits are obtained:

[0095]

[0096] Among them, g SPE h is the weight parameter. SPEHere, SPE represents the degrees of freedom, mean(SPE) is the average value of SPE, and var(SPE) is the variance of SPE.

[0097] Step 3.13: The training of the SM-DBN model and the construction of state monitoring indicators are completed above. Based on the output data obtained from the silicon single crystal growth simulation system established in Step 2.1, after processing in Step 2.4, it is input into the DBN model. By comparing the SPE and energy value indicators with the SPE control limit, it is finally determined whether the silicon single crystal growth process is normal, thus realizing the monitoring of the operating status of the silicon single crystal growth process.

[0098] The beneficial effects of this invention are as follows: First, a two-input, multi-output silicon single crystal growth state monitoring method based on Deep Belief Network (DBN) is proposed and simulated, providing experimental data consistent with actual crystal growth and performing qualitative and quantitative analysis of the crystal growth process data. Second, a State Monitoring Model of Deep Belief Network (SM-DBN) is designed, proposing the use of a combination of energy function and squared prediction error (SPE) as an indicator to determine whether the monitored system is operating normally. SM-DBN can accurately and quickly extract the information features of crystal growth data, flexibly and accurately monitor the state under different operating conditions, playing a key role in improving crystal growth efficiency and crystal quality. Finally, experiments verify that the method proposed in this invention has high accuracy. Attached Figure Description

[0099] Figure 1 This is the overall flowchart of the present invention;

[0100] Figure 2 This is a schematic diagram of heat transfer and geometric relationships during crystal growth;

[0101] Figure 3 This is a simulation diagram of a silicon single crystal growth system;

[0102] Figure 4 This is a schematic diagram of the DBN network structure;

[0103] Figure 5 This is a comparison chart of energy results for different DBN network structures;

[0104] Figure 6 This is a comparison chart of SPE results for different DBN networks;

[0105] Figure 7 This is a diagram showing the SPE results of silicon single crystal growth under normal conditions;

[0106] Figure 8 This is a graph showing the system monitoring results under abnormal melt height values;

[0107] Figure 9 This is a graph showing the system monitoring results when the heater temperature is abnormal;

[0108] Figure 10 This is a graph showing the system monitoring results when the crystal growth rate value is abnormal. Detailed Implementation

[0109] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0110] This invention relates to a method for monitoring the growth state of semiconductor silicon single crystals based on deep belief networks. The overall process is as follows: Figure 1 As shown. Please follow these steps:

[0111] Step 1: Establish a model for the Czochralski silicon single crystal growth process;

[0112] Step 1 is implemented in the following steps:

[0113] Step 1.1: During crystal growth, multiple process variables are involved, such as heater power, crystal pulling speed, crystal growth rate, and crystal radius, and these state variables are coupled together. Based on the energy balance model, fluid dynamics, and geometric models of the crystal growth process, a silicon single crystal growth model is established, including a heat transfer model and a geometric model; for example... Figure 2 The diagram shows the heat transfer and geometric relationship during crystal growth.

[0114] Step 1.2: Establish a heat transfer model. Assume that all energy released by the heater is transferred to the crucible, with no other energy loss, and that the temperature of the heater and crucible is uniform throughout the three-dimensional space. Based on the energy conservation principle of the heater, the change in heater temperature over time... The equation is:

[0115]

[0116] Wherein, the heat radiation transfer rate q from the heater to the crucible hc for:

[0117]

[0118] C h It is the heat capacity of the heater, P in It is the heater power, A c T is the surface area of ​​the crucible, σ is the Stefan-Boltzmann constant, and T is the surface area of ​​the crucible. h It is the heater temperature, T c It is the temperature of the crucible;

[0119] Based on the energy conservation principle within the crucible, the change in crucible temperature over time... The equation is:

[0120]

[0121] Among them, C c It is the heat capacity of the crucible, q co It is the rate of heat radiation transfer from the crucible to the environment, q cs It is the rate of heat radiation transfer from the crucible to the melt, q cm It is the rate of heat conduction from the crucible to the melt;

[0122] Based on the conservation of melt energy and mass, the change in melt temperature over time... The equation is:

[0123]

[0124] Among them, C m It is the melt heat capacity, q so It is the rate of heat transfer from the melt to the environment, q m It is the rate of heat transfer from the melt to the meniscus. It is the rate of change of melt height, T m It is the melt temperature, H m It is the height of the melt;

[0125] At the growth interface, the crystal growth rate is determined by the heat Q of the melt flowing into the growth interface. mI Heat Q flowing into the crystal at the growth interface Ii The difference in heat is calculated, and this difference represents the heat released during crystal crystallization. Therefore, the crystal growth rate V... g According to the energy conservation equation:

[0126]

[0127] Among them, H f It is the latent heat of crystallization, ρ s It is the crystal density, q Ii It is the heat transfer rate from the solid-liquid interface to the crystal, q mI R is the heat transfer rate from the melt to the solid-liquid interface. i It is the crystal radius;

[0128] Step 1.3: Establish the geometric model:

[0129] Based on the mass conservation condition at the solid-liquid interface, the change in melt height over time... The equation is:

[0130]

[0131] Among them, V p It is the crystal pulling speed. It is the rate of change of the meniscus height, ρ m It is the melt density, R c It is the radius of the crucible;

[0132] The shape of the meniscus largely determines the diameter variation during crystal growth. The formation of the meniscus is primarily due to the surface tension at the solid-liquid interface and the silicon solution in the crucible. The height H of the meniscus is expressed as the crystal radius and tilt angle. men for:

[0133]

[0134] Where 'a' is the capillary length, also known as the Laplace constant, and is related to the meniscus surface tension γ. le The value of 'a' is related to the melt density. α0 is the crystal growth angle, α c It is the crystal tilt angle;

[0135] Differentiating equation (7) yields

[0136]

[0137] The change in crystal radius over time was obtained based on the geometric relationships at the meniscus. and growth rate V g Relationship:

[0138]

[0139] Changes in the tilt angle cause changes in the crystal diameter, and the tilt angle changes over time. Calculated using the following formula:

[0140]

[0141] in:

[0142]

[0143]

[0144]

[0145]

[0146]

[0147] V cruc It is the crucible acceleration rate;

[0148] Based on the crystal growth rate, the crystal length changes over time. for:

[0149]

[0150] The heat transfer model and geometric model of the aforementioned state variables related to the silicon single crystal growth process together constitute the Czochralski silicon single crystal growth process model. These state variables are closely coupled; simulating this process allows observation of the dynamic characteristics of each state variable over time and the export of operational data.

[0151] In step 1.3, α0 is set to 11°.

[0152] Step 2: Construct a two-input multiple-output silicon single crystal growth simulation system based on the silicon single crystal growth model. Select appropriate output variables of the silicon single crystal growth simulation system, process the data, and use them as inputs to the SM-DBN model constructed in Step 3.

[0153] Step 2 is implemented in the following steps:

[0154] Step 2.1: Based on the model established above, simulate the silicon single crystal growth process from the shoulder-forming stage to the constant-diameter growth stage. Build a silicon single crystal growth simulation system in the Matlab / Simulink environment, such as... Figure 3 As shown. This simulation system consists of three subsystems. The first subsystem directly displays the total input and output of the simulation system, with the input being P. in and V p The output is T. h T c T m H m R i α c V g L, H men The second layer primarily handles the simulation of heat flux between the melt and the crystal, as well as the crystal growth tilt angle. The third layer mainly focuses on the details of formulas and various parameter data. Real-time data for each state variable can be obtained through the constructed silicon single crystal growth simulation system.

[0155] Step 2.2: The heat transfer and thermal equilibrium processes in the system have a crucial impact on the crystal radius, crystal growth rate, solid-liquid interface shape, and crystal defects. The heater temperature, crucible temperature, and melt temperature reflect the internal temperature conditions of the system to some extent. Melt height is an important parameter that affects crystal quality. Monitoring and effectively controlling the melt level is a fundamental condition for growing high-quality crystals. Controlling the crystal diameter can improve crystal utilization and reduce dislocation defects within the crystal. The crystal growth rate directly affects the efficiency and defects of crystal growth and also has a destructive impact on the periodicity and symmetry of the crystal. Considering all these factors, the heater temperature T is selected. h crucible temperature T c Melt temperature T m Crystal radius R i Crystal growth rate V g Melt height H m and crystal rotation speed V ir Crucible rotation speed V cr As input to the SM-DBN model established in step 3, the state of the silicon single crystal simulation system established in step 2.1 is monitored.

[0156] Step 2.3: Establish 8-dimensional sample data X, where each dimension represents a variable in the crystal growth process:

[0157]

[0158] Let n be the value of the heater temperature at the nth second of the simulation time. Let n be the crucible temperature value at the nth second of the simulation time. Let n be the value of the melt temperature at the nth second of the simulation time. The value of the crystal radius at the nth second of the simulation time. This represents the crystal growth rate at the nth second of the simulation time. Let n be the value of the crystal rotation speed at the nth second of the simulation time. Let n be the value of the crucible rotation speed at the nth second of the simulation time. Let be the value of the melt height at the nth second of the simulation time, where n∈R.

[0159] Step 2.4: Perform L2 norm normalization on the 8-dimensional sample data X in Formula 17, and normalize the data of each row of the sample separately:

[0160]

[0161] x1,...,x n These correspond to the data values ​​of each dimension in Formula 17.

[0162] Step 3: Design the SM-DBN model, input the output variables selected in Step 2 into the state monitoring model, construct state monitoring indicators, complete the training of the DBN network model and the construction of the state monitoring model, and then input the output data of the silicon single crystal growth process into the trained network model. By comparing the output state indicators, the normal operation status of the silicon single crystal growth process can be determined.

[0163] Step 3 is implemented in the following steps:

[0164] Step 3.1: The core of the DBN network is several layers of Restricted Boltzmann Machines (RBMs), whose structure is as follows: Figure 4 As shown, the output of the bottom-level RBM serves as the input of the top-level RBM, and the output of the top-level RBM serves as the input of a BP neural network layer for backpropagation. RBM is an energy-based model; the steady state of the DBN network corresponds to the state with the lowest energy function value. For a given set of (v, h), the specific energy function is expressed as follows:

[0165]

[0166] In the formula, v = (v1, v2, ..., v i ,…v m ) T i,m∈Z * v is the visible layer state vector. i Let be the value of the i-th node in the visual layer, and m be the number of nodes in the visual layer; h = (h1, h2, ..., h...). j ,…h n ) T j,n∈Z * h is the hidden layer state vector. j Let be the value of the j-th node in the hidden layer, and n be the number of nodes in the hidden layer; W = (w ij Let i∈[1,m],j∈[1,n] be the weight matrix, and a=(a1,a2,…,a…) i ,…a m ) T Let b be the view layer bias vector, where b = (b1, b2, ..., b i ,…b m ) T This is the hidden layer bias vector;

[0167] Step 3.2: Using the processed sample data from Step 2.4 as input to the DBN, and as the visible layer of the first RBM, the probability of the hidden layer neurons being activated is:

[0168] p(h j (0)=1|v (0) )=δ(b j +∑ i v i (0) w ij (20)

[0169] In the formula, δ is the sigmoid function, defined as δ(x) = 1 / (1+e^(-1 / x)). -x The superscripts of v and h indicate the state transition step k of the Contrastive Divergence (CD) algorithm.

[0170] Step 3.3: Based on the probability of the hidden layer neurons being turned on, perform Gibbs sampling, sampling each neuron in the hidden layer using a binary {0,1} method to obtain the hidden layer h. (0) for:

[0171]

[0172] In the formula, r j A random number in the interval [0,1];

[0173] Step 3.4: Use hidden layer h (0) Reconstruct the visual layer v (1) Calculate the new visible layer v (1) The probability of:

[0174] p(v i (1) =1|h (0) )=δ(a i +∑ j h j (0) w ij ) (twenty two)

[0175] Step 3.5: Following the algorithm in Step 3.3, obtain the new visible layer v. (1) for:

[0176]

[0177] Step 3.6: Following steps 3.2 and 3.3, obtain the new hidden layer h. (1) for:

[0178] p(h j (1) =1|v (1) )=δ(b j +∑ i v i (1) w ij ) (twenty four)

[0179]

[0180] Step 3.7: The above completes one step of the CD algorithm, updating and obtaining the new weight matrix and bias vector:

[0181] W←W+ε[p(h (0) =1|v (0) )v (0)T -p(h (1) =1|v (1) )v (1)T (26)

[0182] a←a+ε(v (0) -v (1) (27)

[0183] b←b+ε[p(h (0) =1|v (0) )-p(h (1) =1|v (1) (28)

[0184] In the formula, ε is the learning rate;

[0185] Step 3.8: When the state transition step k = 1, a reconstructed sample similar to the data processed in step 2.4 is obtained, achieving a better result;

[0186] Step 3.9 After the single-layer RBM network is trained, the DBN network is pre-trained layer by layer. The bottom layer RBM is trained with the original input data, and then the features extracted from the bottom RBM are used as the input of the top RBM for further training. This process is repeated to train as many RBM layers as possible, that is, to train each layer of RBM from bottom to top.

[0187] Step 3.10: After pre-training, the DBN network will be fine-tuned. A BP (Back Propagation) neural network will be set in the last layer of the DBN. The output of the last RBM layer will be used as the input of the BP neural network. Supervised training will be carried out from top to bottom to optimize the parameters generated in the pre-training stage and make the prediction ability of the DBN reach its optimal level.

[0188] Step 3.11: The DBN network has been trained using the steps above. Now, we select the SM-DBN state monitoring metrics. The first state monitoring metric is the squared prediction error (SPE), which is calculated as follows:

[0189]

[0190] in

[0191]

[0192] The second condition monitoring indicator is the SM-DBN energy value, which is calculated using formula (19):

[0193]

[0194] Step 3.12: Now set the SPE control limit of the state monitoring model and compare it with the SPE value of the state monitoring model to determine the system operating state. The SPE control limit is determined by χ. 2 The distribution is approximately obtained as follows:

[0195]

[0196] In the formula,

[0197]

[0198] Finally, the SPE control limits are obtained:

[0199]

[0200] Among them, g SPE h is the weight parameter. SPE Here are the degrees of freedom, mean(SPE) is the average value of SPE, and var(SPE) is the variance of SPE; generally, a confidence level of 1-α is chosen to be 0.99.

[0201] Step 3.13: The training of the SM-DBN model and the construction of state monitoring indicators are completed above. Based on the output data obtained from the silicon single crystal growth simulation system established in Step 2.1, after processing in Step 2.4, it is input into the DBN model. By comparing the SPE and energy value indicators with the SPE control limit, it is finally determined whether the silicon single crystal growth process is normal, thus realizing the monitoring of the operating status of the silicon single crystal growth process.

[0202] Example

[0203] First, a suitable DBN network structure is selected. Sample data is generated using steps 2.3 and 2.4. To improve computation speed, the simulation duration is set to 12000s with a fixed step size of 0.6s, resulting in 20001 data points. The last data point is removed, leaving 20000 sample data points. 14000 sets are selected as training data, and 6000 sets as test data. The training data is batch-processed, divided into multiple mini-batch datasets of 200 samples each, with 200 iterations per batch. Six network structures are provided for training, with the structure [8,8,9,10] having the lowest energy value (-771949). Figure 5 Comparison of energy values ​​for six different network structures;

[0204] SPE values ​​of six types of networks as follows Figure 6 As shown in the figure, the structure labeled [8,8,9,10] has the smallest value, 0.1401. Considering the overall network energy value, the DBN structure with network structure [8,8,9,10] is finally selected, as this network structure can more accurately uncover the hidden features in the performance parameters. Based on this network structure, an SM-DBN state monitoring model is constructed.

[0205] The new silicon single crystal growth sample under normal conditions is now input into the condition monitoring model, and the SPE result diagram is shown below. Figure 7 As shown, the control limit value of SPE is 0.06639, which is above the SPE monitoring curve, indicating that the system is operating normally.

[0206] An abnormal operating condition is then created for the crystal growth process. Melt height is a parameter related to crystal quality, heater temperature can reflect the internal temperature of the system to some extent, and crystal growth rate is related to crystal growth efficiency and defects. Therefore, certain degrees of disturbance are applied to these three process variables to verify the effectiveness of SM-DBN. The specific procedures and results are as follows:

[0207] To simulate an anomaly or external disturbance at a certain point during crystal growth, the output H of the silicon single crystal simulation system is given. m A step disturbance with an amplitude of 0.006 is added, and the data from this abnormal condition is then input into the SM-DBN to obtain the system status monitoring results as follows. Figure 8 As shown, the SPE value is above the control limit, indicating a system anomaly; similarly, test several more sets of operating conditions and assign T... h Adding a ramp increment with a slope of 0.5, the system state monitoring results are as follows: Figure 9 As shown, the SPE value is above the control limit, indicating a system malfunction; V g Adding a step jump with an amplitude of 0.0005, the system state monitoring results are as follows: Figure 10 As shown, the SPE value is above the control limit, indicating a system anomaly. Experimental results demonstrate that this state monitoring model can effectively monitor whether the system is operating normally.

[0208] The parameters of the silicon single crystal growth process were adjusted to obtain more sets of silicon single crystal growth data. Then, different perturbations were added to the sample data to form different abnormal operating conditions. These conditions were then input into the state monitoring model to determine whether the system state was abnormal or normal. Based on all experimental results, the accuracy rate was 96.8%, which verified the effectiveness and accuracy of the SM-DBN monitoring model applied to the silicon single crystal growth process.

[0209] Figure 5The horizontal axis represents the number of DBN network training iterations, and the vertical axis represents the network energy value. There are 6 energy curves representing different DBN structures. Among them, the curve for the network structure [8,8,9,10] shows a downward trend and has the smallest energy value, -771949.

[0210] Figure 6 The horizontal axis represents the number of DBN network training iterations, and the vertical axis represents the squared prediction error. There are 6 curves representing the squared prediction error of different DBN structures. The curve with network structure [8,8,9,10] has the smallest value, 0.1401.

[0211] Figure 7 The horizontal axis represents the number of DBN network training iterations, and the vertical axis represents the squared prediction error value. The solid line represents the monitored value of the squared prediction error obtained after training the silicon single crystal growth sample data with SM-DBN under normal conditions, and the dashed line represents the control limit value of the squared prediction error. The control limit value of the squared prediction error is 0.06639. The control limit is above the squared prediction error monitoring curve, indicating that the silicon single crystal growth system is operating normally.

[0212] Figure 8 The horizontal axis represents the number of DBN network training iterations, and the vertical axis represents the squared prediction error. The solid line represents the monitored value of the squared prediction error obtained after training with SM-DBN on silicon single crystal growth sample data with abnormal melt height data, and the dashed line represents the control limit of the squared prediction error. A squared prediction error value above the control limit indicates that the silicon single crystal growth system is malfunctioning.

[0213] Figure 9 The horizontal axis represents the number of DBN network training iterations, and the vertical axis represents the squared prediction error. The solid line represents the monitored value of the squared prediction error obtained after training with SM-DBN on silicon single crystal growth sample data with abnormal heater temperature data, and the dashed line represents the control limit of the squared prediction error. A squared prediction error value above the control limit indicates that the silicon single crystal growth system is malfunctioning.

[0214] Figure 10 The horizontal axis represents the number of DBN network training iterations, and the vertical axis represents the squared prediction error. The solid line represents the monitored value of the squared prediction error obtained after training with SM-DBN on silicon single crystal growth sample data with abnormal crystal growth rate data, and the dashed line represents the control limit of the squared prediction error. A squared prediction error value above the control limit indicates that the silicon single crystal growth system is malfunctioning.

Claims

1. A method for monitoring the growth state of semiconductor silicon single crystals based on deep belief networks, characterized in that, The specific steps are as follows: Step 1: Establish a model for the Czochralski silicon single crystal growth process; Step 1 is implemented in the following steps: Step 1.1: Establish a model including a heat transfer model and a Czochralski silicon single crystal growth process model based on the energy balance model, fluid dynamics and geometric model of the crystal growth process; Step 1.2: Establish a heat transfer model. Assume that all energy released by the heater is transferred to the crucible, with no other energy loss, and that the temperature of the heater and crucible is uniform throughout the three-dimensional space. Based on the energy conservation principle of the heater, the change in heater temperature over time... The equation is: （1） Among them, the heat radiation transfer rate from the heater to the crucible for: （2） It is the heat capacity of the heater. It is the heater power. It is the surface area of ​​the crucible. It is the Stefan-Boltmann constant. It is the heater temperature. It is the temperature of the crucible; According to the energy conservation principle inside the crucible, the change in crucible temperature over time The equation is: （3） in, It is the heat capacity of the crucible. It is the rate of heat radiation transfer from the crucible to the environment. It is the rate of heat radiation transfer from the crucible to the melt. It is the rate of heat conduction from the crucible to the melt; Based on the conservation of melt energy and mass, the change in melt temperature over time... The equation is: （4） in, It is the heat capacity of the melt. It is the rate of heat transfer from the melt to the environment. It is the rate of heat transfer from the melt to the meniscus. It is the rate of change of melt height. It is the melt temperature. It is the height of the melt; At the growth interface, the crystal growth rate depends on the heat generated by the melt flowing into the growth interface. Heat flowing into the crystal at the growth interface The difference in heat is calculated, and this difference represents the heat released during crystal crystallization. Therefore, the crystal growth rate... According to the energy conservation equation: （5） in, It is the latent heat of crystallization. It is the crystal density. It is the heat transfer rate from the solid-liquid interface to the crystal. It is the heat transfer rate from the melt to the solid-liquid interface. It is the crystal radius; Step 1.3: Establish the geometric model: Based on the mass conservation condition at the solid-liquid interface, the change in melt height over time... The equation is: （6） in, It is the crystal pulling speed. It is the rate of change of the meniscus height. It is the melt density. It is the radius of the crucible; The height of the meniscus is expressed using the crystal radius and tilt angle. for: （7） in, The capillary length, also known as the Laplace constant, is related to the meniscus surface tension. It is related to the melt density. The value is , It is the crystal growth angle. It is the crystal tilt angle; Differentiating equation (7) yields （8） The change in crystal radius over time was obtained based on the geometric relationships at the meniscus. and growth rate Relationship: （9） Changes in the tilt angle cause changes in the crystal diameter, and the tilt angle changes over time. Calculated using the following formula: （10） in: （11） （12） （13） （14） （15） It is the crucible acceleration rate; Based on the crystal growth rate, the crystal length changes over time. for: （16） The heat transfer model and geometric model of the state variables related to the silicon single crystal growth process mentioned above together constitute the Czochralski silicon single crystal growth process model; Step 2: Construct a two-input, multiple-output silicon single crystal growth simulation system based on the Czochralski silicon single crystal growth process model. Select appropriate output variables of the silicon single crystal growth simulation system, process the data, and use them as inputs to the SM-DBN model constructed in Step 3. Step 2 is implemented in the following steps: Step 2.1: Simulate the silicon single crystal growth process from the shoulder-forming stage to the constant-diameter growth stage. A silicon single crystal growth simulation system is built in the Matlab / Simulink environment. The simulation system consists of three subsystems. The first subsystem directly displays the total input and output of the simulation system. The input is... and The output is , , , , , , , , The second layer simulates the heat flux between the melt and the crystal, as well as the crystal growth tilt angle; the third layer details the formulas and various parameter data. Real-time data of each state variable can be obtained through the well-built silicon single crystal growth simulation system. Step 2.2: Select heater temperature crucible temperature Melt temperature Crystal radius Crystal growth rate Melt height and crystal rotation speed Crucible rotation speed As input to the SM-DBN model established in step 3, the state of the silicon single crystal simulation system established in step 2.1 is monitored. Step 2.3: Establish 8-dimensional sample data X, where each dimension represents a variable in the crystal growth process: （17） Let n be the value of the heater temperature at the nth second of the simulation time. Let n be the crucible temperature value at the nth second of the simulation time. Let n be the value of the melt temperature at the nth second of the simulation time. The value of the crystal radius at the nth second of the simulation time. This represents the crystal growth rate at the nth second of the simulation time. Let n be the value of the crystal rotation speed at the nth second of the simulation time. Let n be the value of the crucible rotation speed at the nth second of the simulation time. Let n be the value of the melt height at the nth second of the simulation time. ; Step 2.4: Perform L2 norm normalization on the 8-dimensional sample data X in Formula 17, and normalize the data of each row of the sample separately: （18） These correspond to the data values ​​of each dimension in Formula 17; Step 3: Design the SM-DBN model, input the output variables selected in Step 2 into the state monitoring model, construct state monitoring indicators, complete the training of the DBN network model and the construction of the state monitoring model, and then input the output data of the silicon single crystal growth process into the trained network model. By comparing the output state indicators, it is finally determined whether the silicon single crystal growth process is operating normally. Step 3 is implemented in the following steps: Step 3.1: The steady-state corresponding to the minimum energy function state of the DBN network, for a given set of... The specific energy function is expressed as follows: （19） In the formula This is the visible layer state vector. For the visible layer The value of each node, The number of nodes in the visualization layer; This represents the hidden layer state vector. For the hidden layer The value of each node, This represents the number of hidden layer nodes; The weight matrix, This is the view layer bias vector. This is the hidden layer bias vector; Step 3.2: Using the processed sample data from Step 2.4 as input to the DBN, and as the visible layer of the first RBM, the probability of the hidden layer neurons being activated is: （20） In the formula, It is the sigmoid function, defined as , and The superscript indicates the number of state transition steps in the Comparative Divergence (CD) algorithm. k ; Step 3.3: Based on the probability of the hidden layer neurons being turned on, perform Gibbs sampling and perform binary sampling on each neuron in the hidden layer. Sampling to obtain the hidden layer for: （21） In the formula, A random number in the interval [0,1]; Step 3.4: Use hidden layers Reconstruct the visual layer Calculate the new visible layer The probability of: （22） Step 3.5: Following the algorithm in Step 3.3, obtain the new visible layer. for: （23） Step 3.6: Following steps 3.2 and 3.3, obtain the new hidden layer. for: （24） （25） Step 3.7: The above completes one step of the CD algorithm, updating and obtaining the new weight matrix and bias vector: （26） （27） （28） In the formula, The learning rate; Step 3.8, State Transition Steps k When =1, a reconstructed sample similar to the data processed in step 2.4 is obtained, achieving a better result; Step 3.9 After the single-layer RBM network is trained, the DBN network is pre-trained layer by layer. The bottom layer RBM is trained with the original input data, and then the features extracted from the bottom RBM are used as the input of the top RBM for further training. That is, each layer of RBM is trained from bottom to top. Step 3.10: After pre-training, the DBN network will be fine-tuned. A BP neural network will be set in the last layer of the DBN, and the output of the last RBM layer will be used as the input of the BP neural network. Supervised training will be carried out from top to bottom to optimize the parameters generated in the pre-training stage so that the prediction ability of the DBN can reach the optimal level. Step 3.11: The DBN network has been trained using the steps above. Now, we select the SM-DBN state monitoring metrics. The first state monitoring metric is the squared prediction error (SPE), which is calculated as follows: （29） in （30） The second condition monitoring indicator is the SM-DBN energy value, which is calculated using formula (19): （31） Step 3.12: Now set the SPE control limit of the state monitoring model and compare it with the SPE value of the state monitoring model to determine the system operating state. The SPE control limit is set by... The distribution is approximately obtained as follows: （32） In the formula, （33） Finally, the SPE control limits are obtained: （34） in, For weight parameters, Here are the degrees of freedom, mean(SPE) is the average of SPE, and var(SPE) is the variance of SPE. The confidence level is selected. It is 0.99; Step 3.13: The training of the SM-DBN model and the construction of state monitoring indicators are completed above. Based on the output data obtained from the silicon single crystal growth simulation system established in Step 2.1, after processing in Step 2.4, the data is input into the DBN model. By comparing the SPE and energy value indicators with the SPE control limit, it is finally determined whether the silicon single crystal growth process is normal, thus realizing the monitoring of the operating status of the silicon single crystal growth process.

2. The method for monitoring the growth state of semiconductor silicon single crystals based on deep belief networks according to claim 1, characterized in that, In step 1.3 The value is 11°.

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