CN115857315B - Semiconductor silicon single crystal growth model prediction control method based on event triggering - Google Patents
Semiconductor silicon single crystal growth model prediction control method based on event triggering Download PDFInfo
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Abstract
The invention discloses a semiconductor silicon single crystal growth model prediction control method based on event triggering, which is implemented according to the following steps: step 1, establishing a growth mechanism nonlinear model of a silicon single crystal in an equal diameter stage, and obtaining a linear discrete model by utilizing a small disturbance linearization theory and a zero-order retainer; step 2, designing a model predictive controller by combining the requirement of the silicon single crystal growth process; and step 3, introducing an event trigger, reducing the times of model predictive control in the rolling optimization process, and realizing real-time accurate control on the radius of the silicon single crystal and the temperature of the silicon melt. The invention solves the problems of low control precision and low quality in the growth process of large-size silicon single crystals in the prior art.
Description
Technical Field
The invention belongs to the technical field of semiconductor silicon single crystal growth process control, and particularly relates to a semiconductor silicon single crystal growth model prediction control method based on event triggering.
Background
Silicon single crystal is one of the key materials that is not alternative in the semiconductor field. With the rapid development of the integrated circuit industry, high-quality, large-sized semiconductor silicon single crystals have become a necessary trend for development. Silicon single crystal growth is a complex process in which multiple physical fields are coupled with each other, and the realization of precise control of the growth process is a necessary means for producing high-quality semiconductor silicon single crystals.
The Czochralski method is a main technical method for producing a semiconductor silicon single crystal. In the silicon single crystal growth process, the quality requirement of the high-quality semiconductor silicon single crystal is difficult to meet due to low precision of traditional PID control and difficult parameter setting. Model predictive control can timely solve the problems of model mismatch, unstable thermal field and the like in the silicon single crystal growth process through a rolling optimization strategy, and can improve the stability and robustness of the growth process, however, the method has the defects of large calculated amount, poor real-time performance and the like, so that the method has limited effect in the actual production process. Therefore, an event triggering strategy is sampled, an event triggering mechanism is constructed based on the silicon melt temperature, the silicon single crystal radius deviation and the sampling times, and under the condition of ensuring the control precision of a controlled system, the calculated amount in the control process is reduced, the control instantaneity is improved, and the high-efficiency control of the silicon single crystal growth system is realized.
Disclosure of Invention
The invention aims to provide a semiconductor silicon single crystal growth model prediction control method based on event triggering, which solves the problems of low control precision and low quality of a large-size silicon single crystal growth process in the prior art.
The technical scheme adopted by the invention is that the method for controlling the prediction of the semiconductor silicon single crystal growth model based on event triggering is implemented according to the following steps:
step 1, establishing a growth mechanism nonlinear model of a silicon single crystal in an equal diameter stage, and obtaining a linear discrete model by utilizing a small disturbance linearization theory and a zero-order retainer;
step 2, designing a model predictive controller by combining the requirement of the silicon single crystal growth process;
and step 3, introducing an event trigger, reducing the times of model predictive control in the rolling optimization process, and realizing real-time accurate control on the radius of the silicon single crystal and the temperature of the silicon melt.
The present invention is also characterized in that,
the step 1 is specifically implemented according to the following steps:
step 1.1, heater heat equation: in the state of thermal equilibrium, the heat generated and transferred by the heater is equal, and the heat equation of the heater is as follows:
Q hc ＝ε h σA hc (T h 4 -T c 4 )(2)
wherein C is h Is the product of the specific heat capacity and the mass of the heater, P is the input power of the heater, Q hc Epsilon for heat radiation to the crucible h For the emissivity of the heater, σ is Stefan-Boltzmann constant, A hc For the external surface area of the crucible, T h In order to achieve the temperature of the heater,t for temperature change of heater c The crucible temperature;
step 1.2, a crucible energy equation: in order to balance the heat of the crucible, the heat absorbed and released by the crucible reaches an equilibrium state, and the energy equation of the crucible is as follows:
Q co ＝ε c σA co (T c 4 -T o 4 )＝2πR cr (H cr -H cl )ε c σ(T c 4 -T o 4 ) (4)
wherein C is c Is the product of the specific heat capacity and the mass of the crucible, Q cl Q is the total heat transferred to the melt by the crucible through heat conduction and heat radiation co Epsilon for the heat of the crucible to be dissipated to the outside environment by thermal radiation c For emissivity of crucible A co For the inner surface area of the crucible, T c For the temperature of the crucible,t is the temperature change of the crucible o R is the temperature in the furnace cr For the radius of the crucible, H cr The height of the crucible is set to be the height of the crucible,H cl the height of the melt in the crucible is continuously reduced in the process of growing the silicon single crystal;
step 1.3, a silicon melt energy equation: according to the heat change mechanism of the silicon melt, an energy equation of the silicon melt is established as follows:
wherein c l To the specific heat capacity of the melt, M l Q is the mass of the melt in the crucible l Q is the heat conducted to the growth interface by the melt through heat lo For the heat dissipated to the environment by thermal radiation at the surface of the melt, Q is the heat released by crystallization of the silicon single crystal, ε l For emissivity of melt, T l For the temperature of the silicon melt,r is the temperature change of the silicon melt cr Is the inner radius of the crucible, R c Radius of silicon single crystal, A ls Is the cross-sectional area of the silicon single crystal;
as the silicon single crystal grows continuously, the decreasing mass of the silicon melt is equal to the increasing mass of the silicon single crystal, and the melt mass variation is dynamically expressed as:
wherein M is l For the mass of the melt in the crucible,for mass transformation of silicon melt in crucible ρ s Density, v, of silicon single crystal c Is the growth speed of the silicon single crystal;
step 1.4, in the silicon single crystal growth process, establishing a silicon single crystal mass conservation equation according to a mass conservation law:
M t ＝M l +M s +M m (10)
wherein M is t Is the total input amount of silicon in the crystal pulling process, M l Is the mass of the silicon single crystal, M s Is the mass of the silicon melt in the crucible, M m Is the mass of the meniscus;
the decrease in mass of the silicon melt is equal to the increase in mass of the silicon single crystal:
ΔM l ＝ΔM s (11)
wherein DeltaM l Is the increase in the quality of silicon single crystals, ΔM s Reduction in mass of silicon melt ρ l Is the density of the silicon melt and,is the rate of change of the melt height in the crucible.
During the growth of the silicon single crystal, the temperature of the solid-liquid interface is kept unchanged, the heat quantity which flows into the interface is equal to the heat quantity which flows out of the interface, and the heat quantity which flows into the interface comprises the heat quantity Q transferred by the melt through heat conduction l The heat quantity flowing out of the interface is the heat quantity Q transferred into the silicon single crystal s Specific process tableThe method is shown as follows:
Q＝Q s -Q l (14)
wherein L is the latent heat of crystallization of the silicon single crystal, A is the cross-sectional area of the silicon single crystal, and k l ,k s Respectively the heat conductivity in the melt and the heat conductivity in the silicon single crystal, G l ,G s Respectively representing the temperature gradient in the melt and the temperature gradient in the silicon single crystal, so that the heat transmission at the solid-liquid interface in the constant diameter growth process must be kept stable in order to enable the silicon single crystal to grow stably;
in the process of the equal-diameter growth of the silicon single crystal, the crucible lifting speed is generally set to be k times of the crystal pulling speed, the ratio of the crucible lifting speed to the crystal pulling speed is called a crucible-to-heel ratio, and according to the growth mechanism of the silicon single crystal, the growth speed of the silicon single crystal is expressed as:
v cr ＝kv p (18)
wherein v is p For the pulling rate of the silicon single crystal, v cr The rising speed of the crucible is set;
combining the formulas (14) and (19) to obtain the silicon single crystal pulling rate v p The method comprises the following steps:
according toThe geometrical height of the meniscus gives a change in the growth radius of the silicon single crystalThe method comprises the following steps:
wherein alpha is c Is the meniscus tilt angle;
step 1.5, obtaining the heater power P and the pull-up speed v according to the above analysis c For controlling input, the temperature T of the silicon melt l And a radius R of the silicon single crystal c Nonlinear silicon single crystal growth model for control output:
wherein x is 1 Is the temperature T of the silicon melt l ，For the silicon melt temperature transformation ratio, h (x 1 ) To loss heat of silicon melt to gas x 2 For the crucible temperature +.>For crucible temperature transformation rate, x 3 For the heater temperature, +.>For heater temperature change rate, x 4 Is the radius of silicon single crystal>Radius transformation ratio of silicon single crystal, x 5 For the height of the silicon melt in the crucible, +.>The silicon melt height conversion rate is;
step 1.6, the nonlinear silicon single crystal growth model established above is expressed as:
wherein, xi (t) is the state variable of the model,the transformation rate of the state variable of the model is represented by mu (t) as the input of the system, eta (t) as the output of the system, the nonlinear functions f (ζ (t), mu (t)) and g (ζ (t), mu (t)) as the state dynamic function and the output function, respectively, and obviously, the model (23) is a complex nonlinear model which cannot be directly controlled, and the nonlinear silicon single crystal growth model is linearized to design a silicon single crystal growth controller, and one balance point (ζ) is selected in a definition domain 0 ,μ 0 ) And meet the following
Wherein eta o Is a nonlinear model at (xi) 0 ,μ 0 ) The output of the dots is provided with,is a nonlinear model at (xi) 0 ,μ 0 ) State transition rate of the point;
taylor expansion of the nonlinear model is performed, the model (23) is then at the equilibrium point (ζ 0 ,μ 0 ) The dynamic characteristics of the vicinity are:
let Δζ (t) =ζ (t) - ζ 0 ,Δμ(t)＝μ(t)-μ 0 Sum Δη (t) =η (t) - η 0 Δζ (t) represents ζ (t) and ζ 0 And Δμ (t) represents μ (t) versus μ 0 Deltaeta (t) represents eta (t) and eta 0 And (b) in combination with equation (24), the model (25) is expressed as:
wherein,
derivative of ζ (t) for function f (ζ (t), μ (t)) at ζ (ζ) 0 ,μ 0 ) Matrix of->Derivative of μ (t) at (ζ) for the function f (ζ (t), μ (t)) 0 ,μ 0 ) Matrix of->Derivative of ζ (t) for the function g (ζ (t), μ (t)) at ζ (ζ) 0 ,μ 0 ) Matrix of->Derivative of μ (t) at (ζ) for the function g (ζ (t), μ (t)) 0 ,μ 0 ) Is a matrix of (a) in the matrix.
Normalizing the system variables:
wherein Δζ 0 ,Δμ 0 And Deltaeta 0 As a reference value, x (t) is a normalized Δζ (t) value, u (t) is a normalized Δμ (t) value, and y (t) is a normalized Δη (t) value.
Combining equation (26) and equation (28), a linear model of the silicon single crystal growth process is derived as:
wherein,is the derivative of x (t).
Step 1.7, discretizing the continuous linear model in the formula (29) by adopting a zero-order retainer to obtain a linear discrete model:
x(k+1)＝Ax(k)+Bu(k)
y(k+1)＝Cx(k)+Du(k) (30)
wherein x (k) and x (k+1) represent the values of x (t) at k and k+1, respectively, u (k) represents the values of u (t) at k and k, y (k+1) represents the values of y (t) at k+1, t is the sampling time.
The step 2 is specifically implemented according to the following steps:
step 2.1, setting a minimum value of a trigger interval of two consecutive times according to engineering practice of a silicon single crystal growth process, eliminating a Zeno phenomenon, and ensuring that the trigger time of two adjacent times meets the following conditions:
t k+1 -t k ≥βT (31)
wherein beta is a positive integer, T is a sampling period, T k And t k+1 Respectively representing the triggering moments of k and k+1;
step 2.2, the neighborhood Ω with a balance point exists, so that the system meets the control quantity constraint, namely
Step 2.3 assuming that there are three matrices P, Q, R, and Q > 0, R > 0, P > 0, satisfying the Lyapunov equation as follows
P＝(A+BK) T P(A+BK)+Q+K T RK (33)
There is a positive constant alpha such that within the neighborhood defined by (P, alpha)
Ω(P,α)＝{x∈R n |x T Px≤α} (34)
The system satisfies (A+BK) x εΩ, which represents a neighborhood range.
Step 2.4, in the systemIn consideration of a set of discrete time sequences t k K is N, N is a natural number, and the current time t is calculated k System optimization control problems. At the current time t k ，/>Representing the predicted state of the system and satisfying The prediction control is represented, U (k) represents a set of control sequences, m is a positive integer, and the current time instant is represented by m times after prediction. The model predictive controller solves the following optimization problem:
solving optimal control according to optimization problemCorresponding optimal control state-> Indicated at t k Solving a group of optimal control at any timeAnd (5) sequencing. />The first value in the control sequence is the optimal controller obtained at the current moment.
The step 3 is specifically implemented according to the following steps:
step 3.1, introducing an event triggering strategy into model predictive control, and designing event triggering conditions to obtain a group of new time sequences t k K is N, N is a natural number, the optimization problem of the system is calculated, and event triggering conditions are defined
Wherein,indicated at t k Time under the condition that the time satisfies the state trigger condition, +.>Representation model predictive computation t k The control state with optimal moment, zeta is the trigger level, < ->ρ is a positive constant, and V is an error weight matrix;
wherein t is k+1 Indicated at t k The moment satisfies the next moment of the event triggering condition.
By assuming t 0 =0, system at t 0 Triggering the event at the moment, and then strictly according to the solved trigger time point t k Triggering an event;
step 3.2, introducing an event triggering condition into the model prediction control, and solving the model prediction controller based on the event triggering into the following optimization problem:
obtaining t by calculating optimization problem k Optimal control sequence of time of dayWill->Acting in a medium silicon single crystal growth system until t meeting an event triggering condition k+1 At that point, the optimization problem is then recalculated.
The method has the beneficial effects that the method for controlling the prediction of the semiconductor silicon single crystal growth model based on event triggering realizes the real-time accurate control of the silicon single crystal growth process. And establishing a mechanism model of the growth process of the silicon single crystal by a Czochralski method according to the theory of heat transfer, dynamics and geometry in the growth process of the silicon single crystal, and obtaining a silicon single crystal growth control system model with higher precision. The model prediction control strategy is used for designing a silicon single crystal growth process controller, an event trigger mechanism is introduced, the calculated amount of model prediction in the optimization problem is greatly reduced, the instantaneity and the accuracy of control signals are enhanced, the control precision of a growth system is improved, and then the quality of silicon single crystals is improved.
Drawings
FIG. 1 is a general flow chart of the event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 2 is a diagram of the overall growth model of a single crystal in silicon based on the event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 3 is a graph showing heat transfer during growth of a single crystal in silicon based on an event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 4 is a schematic diagram of an event-triggered model predictive control in the event-triggered based semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 5 is a graph showing the relative error of the temperature of a silicon melt in the event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 6 is a graph showing relative crucible temperature errors in the event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 7 is a graph showing the relative error of heater temperature in the event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 8 is a graph of relative error in radius of a silicon single crystal in the event-triggered semiconductor silicon single crystal growth model predictive control method of the present invention;
FIG. 9 is a graph showing the relative error of the height of a silicon crucible melt in the event-triggered prediction control method of a semiconductor silicon single crystal growth model according to the present invention.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The invention discloses a semiconductor silicon single crystal growth model predictive control method based on event triggering. The process of preparing a silicon single crystal by the Czochralski method involves a number of variables, wherein the heater power and the silicon single crystal pulling rate can change the radius of the silicon single crystal and the temperature of the silicon melt, affecting the quality of the silicon single crystal. Thus, a dual-input dual-output model is established with the input of the pulling speed and the heater power and the output of the pulling speed and the heater power as the radius of the silicon single crystal and the temperature of the silicon melt. Establishing a thermal field temperature model of the silicon single crystal growth process based on energy conversion among the heater, the crucible and the silicon melt; and establishing a motion model of the silicon single crystal growth process according to dynamics and geometry theory, thereby obtaining a nonlinear system model of the silicon single crystal growth. And then linearizing the model with small disturbance, discretizing the model with a zero-order retainer, and obtaining a linear discrete model for silicon single crystal growth. Based on the linear discrete model, an event-triggered model prediction controller is designed to realize control of the silicon single crystal growth process.
The method for predicting and controlling the semiconductor silicon single crystal growth model based on event triggering is shown in fig. 1, and is implemented according to the following steps:
step 1, establishing a growth mechanism nonlinear model of a silicon single crystal in an equal diameter stage, and obtaining a linear continuous model by utilizing a small disturbance linearization theory; and then a linear discrete model is obtained using a zero-order holder. The side heater is the one that powers the growth system during the constant diameter phase, while the bottom heater is not. The side heater changes the temperature of the crucible through heat radiation, and the crucible transfers heat to the silicon melt through heat conduction, so that the temperature and the temperature gradient of a growth interface are changed. The thermal field model of the growth furnace mainly comprises a heater, a crucible and a thermal transmission model of silicon melt.
Referring to fig. 2 to 4, step 1 is specifically performed as follows:
step 1.1, heater heat equation: in the state of thermal equilibrium, the heat generated and transferred by the heater is equal, and the heat equation of the heater is as follows:
wherein C is h Is the product of the specific heat capacity and the mass of the heater, P is the input power of the heater, Q hc Epsilon for heat radiation to the crucible h For the emissivity of the heater, σ is Stefan-Boltzmann constant, A hc For the external surface area of the crucible, T h In order to achieve the temperature of the heater,t for temperature change of heater c The crucible temperature;
step 1.2, a crucible energy equation: in order to balance the heat of the crucible, the heat absorbed and released by the crucible reaches an equilibrium state, and the energy equation of the crucible is as follows:
Q co ＝ε c σA co (T c 4 -T o 4 )＝2πR cr (H cr -H cl )ε c σ(T c 4 -T o 4 ) (4)
wherein C is c Is the product of the specific heat capacity and the mass of the crucible, Q cl Q is the total heat transferred to the melt by the crucible through heat conduction and heat radiation co Epsilon for the heat of the crucible to be dissipated to the outside environment by thermal radiation c For emissivity of crucible A co For the inner surface area of the crucible, T c For the temperature of the crucible,t is the temperature change of the crucible o R is the temperature in the furnace cr For the radius of the crucible, H cr For crucible height, H cl The height of the melt in the crucible is continuously reduced in the process of growing the silicon single crystal;
step 1.3, a silicon melt energy equation: according to the heat change mechanism of the silicon melt, an energy equation of the silicon melt is established as follows:
wherein c l For the specific heat capacity of the melt,M l q is the mass of the melt in the crucible l Q is the heat conducted to the growth interface by the melt through heat lo For the heat dissipated to the environment by thermal radiation at the surface of the melt, Q is the heat released by crystallization of the silicon single crystal, ε l For emissivity of melt, T l For the temperature of the silicon melt,r is the temperature change of the silicon melt cr Is the inner radius of the crucible, R c Radius of silicon single crystal, A ls Is the cross-sectional area of the silicon single crystal;
as the silicon single crystal grows continuously, the decreasing mass of the silicon melt is equal to the increasing mass of the silicon single crystal, and the melt mass variation is dynamically expressed as:
wherein M is l For the mass of the melt in the crucible,for mass transformation of silicon melt in crucible ρ s Density, v, of silicon single crystal c Is the growth speed of the silicon single crystal;
step 1.4, in the silicon single crystal growth process, establishing a silicon single crystal mass conservation equation according to a mass conservation law:
M t ＝M l +M s +M m (10)
wherein M is t Is the total input amount of silicon in the crystal pulling process, M l Is the mass of the silicon single crystal, M s Is the mass of the silicon melt in the crucible, M m Is the mass of the meniscus;
in the modeling process, on the premise of ensuring the model accuracy, the meniscus quality is assumed to be unchanged in the whole process in the silicon single crystal growth process. Therefore, the decrease in mass of the silicon melt is equal to the increase in mass of the silicon single crystal:
ΔM l ＝ΔM s (11)
wherein DeltaM l Is the increase in the quality of silicon single crystals, ΔM s Reduction in mass of silicon melt ρ l Is the density of the silicon melt and,is the rate of change of the melt height in the crucible.
During the growth of the silicon single crystal, the temperature of the solid-liquid interface is kept unchanged, the heat quantity which flows into the interface is equal to the heat quantity which flows out of the interface, and the heat quantity which flows into the interface comprises the heat quantity Q transferred by the melt through heat conduction l The heat quantity flowing out of the interface is the heat quantity Q transferred into the silicon single crystal s The specific process is as follows:
Q＝Q s -Q l (14)
wherein L is the latent heat of crystallization of the silicon single crystal, A is the cross-sectional area of the silicon single crystal, and k l ,k s Respectively the heat conductivity in the melt and the heat conductivity in the silicon single crystal, G l ,G s Respectively representing the temperature gradient in the melt and the temperature gradient in the silicon single crystal, so that the heat transmission at the solid-liquid interface in the constant diameter growth process must be kept stable in order to enable the silicon single crystal to grow stably;
in the process of growing a silicon single crystal in equal diameter, the crucible raising speed is generally set to be k times the crystal pulling speed, and the ratio of the crucible raising speed to the crystal pulling speed is called the crucible-to-heel ratio. According to the silicon single crystal growth mechanism, the silicon single crystal growth rate is expressed as:
v cr ＝kv p (18)
wherein v is p For the pulling rate of the silicon single crystal, v cr The rising speed of the crucible is set;
combining the formulas (14) and (19) to obtain the silicon single crystal pulling rate v p The method comprises the following steps:
from the geometrical height of the meniscus, the radius of growth of the silicon single crystal is transformedThe method comprises the following steps:
wherein alpha is c Is the meniscus tilt angle;
step 1.5, obtaining the heater power P and the pull-up speed v according to the above analysis c For controlling input, the temperature T of the silicon melt l And a radius R of the silicon single crystal c Nonlinear silicon single crystal growth model for control output:
wherein x is 1 Is the temperature T of the silicon melt l ，For the silicon melt temperature transformation ratio, h (x 1 ) To loss heat of silicon melt to gas x 2 For the crucible temperature +.>For crucible temperature transformation rate, x 3 For the heater temperature, +.>For heater temperature change rate, x 4 Is the radius of silicon single crystal>Radius transformation ratio of silicon single crystal, x 5 For the height of the silicon melt in the crucible, +.>The silicon melt height conversion rate is;
step 1.6, the nonlinear silicon single crystal growth model established above is expressed as:
wherein, xi (t) is the state variable of the model,the transformation rate of the state variable of the model is represented by mu (t) as the input of the system, eta (t) as the output of the system, the nonlinear functions f (ζ (t), mu (t)) and g (ζ (t), mu (t)) as the state dynamic function and the output function, respectively, and obviously, the model (23) is a complex nonlinear model which cannot be directly controlled, and the nonlinear silicon single crystal growth model is linearized to design a silicon single crystal growth controller, and one balance point (ζ) is selected in a definition domain 0 ,μ 0 ) And meet the following
Wherein eta o Is a nonlinear model at (xi) 0 ,μ 0 ) The output of the dots is provided with,is a nonlinear model at (xi) 0 ,μ 0 ) State transition rate of the point;
taylor expansion of the nonlinear model is performed, the model (23) is then at the equilibrium point (ζ 0 ,μ 0 ) The dynamic characteristics of the vicinity are:
let Δζ (t) =ζ (t) - ζ 0 ,Δμ(t)＝μ(t)-μ 0 Sum Δη (t) =η (t) - η 0 Δζ (t) represents ζ (t) and ζ 0 And Δμ (t) represents μ (t) versus μ 0 Deltaeta (t) represents eta (t) and eta 0 And (b) in combination with equation (24), the model (25) is expressed as:
wherein,
derivative of ζ (t) for function f (ζ (t), μ (t)) at ζ (ζ) 0 ,μ 0 ) Matrix of->Derivative of μ (t) at (ζ) for the function f (ζ (t), μ (t)) 0 ,μ 0 ) Matrix of->Derivative of ζ (t) for the function g (ζ (t), μ (t)) at ζ (ζ) 0 ,μ 0 ) Matrix of->Derivative of μ (t) at (ζ) for the function g (ζ (t), μ (t)) 0 ,μ 0 ) Is a matrix of (a) in the matrix.
Due to different physical dimensions in the system, the normalization processing is carried out on the system variables before the controller is designed:
wherein Δζ 0 ,Δμ 0 And Deltaeta 0 For reference values, they are selected according to the physical quantity of the single crystal furnace. x (t) is the normalized Δζ (t), u (t) is the normalized Δμ (t), and y (t) is the normalized Δη (t). Combining equation (26) and equation (28), a linear model of the silicon single crystal growth process is derived as:
wherein,is the derivative of x (t).
Step 1.7, discretizing the continuous linear model in the formula (29) by adopting a zero-order retainer to obtain a linear discrete model
x(k+1)＝Ax(k)+Bu(k)
y(k+1)＝Cx(k)+Du(k) (30)
Wherein x (k) and x (k+1) represent the values of x (t) at k and k+1, respectively, u (k) represents the values of u (t) at k and k, y (k+1) represents the values of y (t) at k+1, t is the sampling time.
Step 2, designing a model predictive controller by combining the requirement of the silicon single crystal growth process;
the step 2 is specifically implemented according to the following steps:
and 2.1, eliminating the Zeno phenomenon, and preventing the controller from performing infinite triggering control actions within a limited time. Too small a trigger interval will cause the system to not operate properly, e.g., the controller cannot meet the high sampling requirements. Therefore, according to the engineering practice of the silicon single crystal growth process, the minimum value of the triggering interval of two continuous times is set, the Zeno phenomenon is eliminated, and the triggering time of two adjacent times meets the following conditions:
t k+1 -t k ≥βT (31)
wherein beta is a positive integer, T is a sampling period, T k And t k+1 The k and k+1 trigger times are indicated, respectively.
The model prediction control method is based on a silicon single crystal growth discrete model, and the triggering time interval sigma adopts integer times of the sampling period T, so that obviously, the requirement of the above formula can be ensured. Therefore, the model predictive control based on the discrete system does not have the Zeno phenomenon.
Step 2.2, for a linear system, if (a, B) is controllable, a state feedback u=kx must be present to make the system progressively stable. From the continuity of the function, it is known that there must be a neighborhood Ω of a balance point so that the system satisfies the control quantity constraint, i.e
Step 2.3 assuming that there are three matrices P, Q, R, and Q > 0, R > 0, P > 0, satisfying the Lyapunov equation as follows
P＝(A+BK) T P(A+BK)+Q+K T RK (33)
There is a positive constant alpha such that within the neighborhood defined by (P, alpha)
Ω(P,α)＝{x∈R n |x T Px≤α} (34)
The system satisfies (A+BK) x εΩ, which represents a neighborhood range.
And 2.4, designing a silicon single crystal growth process controller by a model predictive control method. In the systemIn consideration of a set of discrete time sequences t k K is N, N is a natural number, and the current time t is calculated k System optimization control problems. At the current time t k ，/>Representing the predicted state of the system and satisfying +.> The prediction control is represented, U (k) represents a set of control sequences, m is a positive integer, and the current time instant is represented by m times after prediction. The model predictive controller solves the following optimization problem:
solving optimal control according to optimization problemCorresponding optimal control state-> Indicated at t k And solving a group of optimal control sequences at the moment. />The first value in the control sequence is the optimal controller obtained at the current moment.
And step 3, introducing an event trigger, reducing the times of model predictive control in the rolling optimization process, and realizing real-time accurate control on the radius of the silicon single crystal and the temperature of the silicon melt.
The step 3 is specifically implemented according to the following steps:
step 3.1, introducing an event triggering strategy into model predictive control, and designing event triggering conditions to obtain a group of new time sequences t k K is N, N is a natural number, the optimization problem of the system is calculated, and event triggering conditions are defined
Wherein,indicated at t k Time under the condition that the time satisfies the state trigger condition, +.>Representation model predictive computation t k The control state with optimal moment, zeta is the trigger level, < ->ρ is a positive constant, and V is an error weight matrix;
wherein t is k+1 Indicated at t k The moment satisfies the next moment of the event triggering condition.
By assuming t 0 =0, system at t 0 Triggering the event at the moment, and then strictly according to the solved trigger time point t k Event touch is performedAnd (5) hair growing.
Step 3.2, introducing an event triggering condition into the model prediction control, and solving the model prediction controller based on the event triggering into the following optimization problem:
obtaining t by calculating optimization problem k Optimal control sequence of time of dayWill->Acting in a medium silicon single crystal growth system until t meeting an event triggering condition k+1 At that point, the optimization problem is then recalculated.
Model predictive control based on event triggering is relative to model predictive control, at t k Solving the optimal control sequence at any timeThe model predictive control takes only the first value, while the event-triggered model predictive control will get t according to the event-triggered constraint k+1 The time of day determines how many values to take. The event-triggered model predictive control can reduce the number of calculations of the optimization problem at each sampling point.
The invention will be further described with reference to the accompanying drawings and a specific example.
According to the process, based on the actual equipment parameters and experimental data of the silicon single crystal growth furnace, the linear discrete model is obtained as
Wherein,
according to formula (39), performing event-triggered model predictive control optimization problem calculation, and initializing state x 0 ＝[155-41.5]，m＝20，α＝0.0003，R＝diag(0.1,0.1)，Q＝diag(2,2,2,2,2)，ζ＝0.03，
Equation (35) is a silicon single crystal growth model predictive controller equation, equation (38) is an event-triggered silicon single crystal growth model predictive controller equation, and controller parameters are obtained by solving equation (35) and equation (38) respectively, so that control over a silicon single crystal growth process is realized. In equation (35) the scroll optimization calculation is required for each sample, whereas in equation (38) the trigger conditions are compared first and the optimization problem calculation is performed only if the trigger conditions are reached.
The simulation diagram 5 is a graph of the relative error of the temperature of the silicon melt, the error is almost zero after the 9 th sampling, the model prediction control is still performed at each sampling point, the point is an event trigger point, the event trigger is performed every time in the first 4 sampling because the relative error is larger than the trigger condition, the relative error is smaller, and the event trigger is performed only when the sampling interval times are larger than the trigger condition.
The simulation diagram 6 is a relative error diagram of crucible temperature, the error is nearly zero after the 5 th sampling, the model prediction control still calculates at each sampling point, the model prediction control is an event trigger point, the event trigger is carried out each time in the first 4 sampling because the relative error is larger than the trigger condition, the relative error is smaller, and the event trigger is carried out only when the sampling interval times are larger than the trigger condition.
The simulation diagram 7 is a relative error diagram of the heater temperature, the error is almost zero after the 4 th sampling, the model prediction control still calculates at each sampling point, the o is an event trigger point, the event trigger is performed every time in the previous 4 sampling because the relative error is larger than the trigger condition, the relative error is smaller, and the event trigger is performed only when the sampling interval times are larger than the trigger condition.
The simulation diagram 8 is a graph of the relative radius error of the silicon single crystal, the error is almost zero after the 5 th sampling, the model prediction control is still calculated at each sampling point, the graph is an event trigger point, the event trigger is carried out each time in the first 4 sampling because the relative error is larger than the trigger condition, the relative error is smaller, and the event trigger is carried out only when the sampling interval times are larger than the trigger condition.
The simulation diagram 9 is a relative error diagram of the height of the crucible melt, the error is almost zero after the 6 th sampling, the model prediction control is still calculated at each sampling point, the point is an event trigger point, the event trigger is carried out every time in the previous 4 sampling because the relative error is larger than the trigger condition, the relative error is smaller, and the event trigger is carried out only when the sampling interval times are larger than the trigger condition.
In summary, it can be obtained that the model predictive control needs to be calculated at each sampling point, and the model predictive control based on event triggering only performs calculation when the current state deviation value is greater than the set deviation value in the first 4 times. In the 5 th and 6 th times, since the control sampling time is longer than the set sampling condition, the calculation is performed. It is apparent that adding event triggers can greatly reduce the computational effort of model predictive control.
Claims (5)
1. The prediction control method of the semiconductor silicon single crystal growth model based on event triggering is characterized by comprising the following steps:
step 1, establishing a growth mechanism nonlinear model of a silicon single crystal in an equal diameter stage, and obtaining a linear discrete model by utilizing a small disturbance linearization theory and a zero-order retainer;
the step 1 is specifically implemented according to the following steps:
step 1.1, heater heat equation: in the state of thermal equilibrium, the heat generated and transferred by the heater is equal, and the heat equation of the heater is as follows:
wherein C is h Is the product of the specific heat capacity and the mass of the heater, P is the input power of the heater, Q hc Epsilon for heat radiation to the crucible h For the emissivity of the heater, σ is Stefan-Boltzmann constant, A hc For the external surface area of the crucible, T h In order to achieve the temperature of the heater,t for temperature change of heater c The crucible temperature;
step 1.2, a crucible energy equation: in order to balance the heat of the crucible, the heat absorbed and released by the crucible reaches an equilibrium state, and the energy equation of the crucible is as follows:
wherein C is c Is the product of the specific heat capacity and the mass of the crucible, Q cl Q is the total heat transferred to the melt by the crucible through heat conduction and heat radiation co Epsilon for the heat of the crucible to be dissipated to the outside environment by thermal radiation c For emissivity of crucible A co For the inner surface area of the crucible, T c For the temperature of the crucible,t is the temperature change of the crucible o R is the temperature in the furnace cr For the radius of the crucible, H cr For crucible height, H cl The height of the melt in the crucible is continuously reduced in the process of growing the silicon single crystal;
step 1.3, a silicon melt energy equation: according to the heat change mechanism of the silicon melt, an energy equation of the silicon melt is established as follows:
wherein c l To the specific heat capacity of the melt, M l Q is the mass of the melt in the crucible l Q is the heat conducted to the growth interface by the melt through heat lo For the heat dissipated to the environment by thermal radiation at the surface of the melt, Q is the heat released by crystallization of the silicon single crystal, ε l For emissivity of melt, T l For the temperature of the silicon melt,r is the temperature change of the silicon melt cr Is the inner radius of the crucible, R c Radius of silicon single crystal, A ls Is the cross-sectional area of the silicon single crystal;
as the silicon single crystal grows continuously, the decreasing mass of the silicon melt is equal to the increasing mass of the silicon single crystal, and the melt mass variation is dynamically expressed as:
wherein M is l For the mass of the melt in the crucible,for mass transformation of silicon melt in crucible ρ s Density, v, of silicon single crystal c Is the growth speed of the silicon single crystal;
step 1.4, establishing a silicon single crystal mass conservation equation according to a mass conservation law in the silicon single crystal growth process;
step 1.5, obtaining a nonlinear silicon single crystal growth model according to the analysis;
step 1.6, deforming the established nonlinear silicon single crystal growth model;
step 1.7, obtaining a linear discrete model by adopting a zero-order retainer;
step 2, designing a model predictive controller by combining the requirement of the silicon single crystal growth process;
the step 2 is specifically implemented according to the following steps:
step 2.1, setting a minimum value of a trigger interval of two consecutive times according to engineering practice of a silicon single crystal growth process, eliminating a Zeno phenomenon, and ensuring that the trigger time of two adjacent times meets the following conditions:
t k+1 -t k ≥βT (31)
wherein beta is a positive integer, T is a sampling period, T k And t k+1 Respectively representing the triggering moments of k and k+1;
step 2.2, the neighborhood Ω with a balance point exists, so that the system meets the control quantity constraint, namely
Step 2.3 assuming that there are three matrices P, Q, R, and Q > 0, R > 0, P > 0, satisfying the Lyapunov equation as follows
P＝(A+BK) T P(A+BK)+Q+K T RK (33)
There is a positive constant alpha such that within the neighborhood defined by (P, alpha)
Ω(P,α)＝{x∈R n x T Px≤α} (34)
The system satisfies (A+BK) x epsilon omega, wherein omega represents a neighborhood range;
step 2.4, in the systemIn consideration of a set of discrete time sequences t k K is N, N is a natural number, and the current time t is calculated k System optimization control problem at current time t k ，/>Representing the predicted state of the system and satisfying The prediction control is represented, U (k) represents a group of control sequences, m is a positive integer, m times after the current time is predicted are represented, and the model prediction controller solves the following optimization problem:
solving optimal control according to optimization problemCorresponding optimal control state-> Indicated at t k Solving a group of optimal control sequences at any time; />The first value in the control sequence is the optimal controller obtained at the current moment;
step 3, introducing an event trigger, reducing the times of model predictive control in the rolling optimization process, and realizing real-time accurate control on the radius of the silicon single crystal and the temperature of the silicon melt;
the step 3 is specifically implemented according to the following steps:
step 3.1, introducing an event triggering strategy into model predictive control, and designing event triggering conditions to obtain a group of new time sequences t k K is N, N is a natural number, the optimization problem of the system is calculated, and event triggering conditions are defined
Wherein,indicated at t k Time under the condition that the time satisfies the state trigger condition, +.>Representation model predictive computation t k The control state with optimal moment, zeta isTrigger level, & lt>ρ is a positive constant, and V is an error weight matrix;
wherein t is k+1 Indicated at t k The next moment when the moment meets the event triggering condition;
by assuming t 0 =0, system at t 0 Triggering the event at the moment, and then strictly according to the solved trigger time point t k Triggering an event;
step 3.2, introducing an event triggering condition into the model prediction control, and solving the model prediction controller based on the event triggering into the following optimization problem:
obtaining t by calculating optimization problem k Optimal control sequence of time of dayWill->Acting in a medium silicon single crystal growth system until t meeting an event triggering condition k+1 At that point, the optimization problem is then recalculated.
2. The method for controlling prediction of a semiconductor silicon single crystal growth model based on event triggering according to claim 1, wherein the step 1.4 of establishing a conservation equation of mass of the silicon single crystal is specifically implemented according to the following steps:
M t ＝M l +M s +M m (10)
wherein M is t Is the total input amount of silicon in the crystal pulling process, M l Is the mass of the silicon single crystal, M s Is the mass of the silicon melt in the crucible, M m Is the mass of the meniscus;
the decrease in mass of the silicon melt is equal to the increase in mass of the silicon single crystal:
ΔM l ＝ΔM s (11)
wherein DeltaM l Is the increase in the quality of silicon single crystals, ΔM s Reduction in mass of silicon melt ρ l Is the density of the silicon melt and,is the rate of change of the melt height in the crucible;
during the growth of the silicon single crystal, the temperature of the solid-liquid interface is kept unchanged, the heat quantity which flows into the interface is equal to the heat quantity which flows out of the interface, and the heat quantity which flows into the interface comprises the heat quantity Q transferred by the melt through heat conduction l The heat quantity flowing out of the interface is the heat quantity Q transferred into the silicon single crystal s The specific process is as follows:
Q＝Q s -Q l (14)
wherein L is the latent heat of crystallization of the silicon single crystal, A is the cross-sectional area of the silicon single crystal, and k l ,k s Respectively the heat conductivity in the melt and the heat conductivity in the silicon single crystal, G l ,G s Respectively representing the temperature gradient in the melt and the temperature gradient in the silicon single crystal, so that the heat transmission at the solid-liquid interface in the constant diameter growth process must be kept stable in order to enable the silicon single crystal to grow stably;
in the process of the equal-diameter growth of the silicon single crystal, the crucible lifting speed is generally set to be k times of the crystal pulling speed, the ratio of the crucible lifting speed to the crystal pulling speed is called a crucible-to-heel ratio, and according to the growth mechanism of the silicon single crystal, the growth speed of the silicon single crystal is expressed as:
v cr ＝kv p (18)
wherein v is p For the pulling rate of the silicon single crystal, v cr The rising speed of the crucible is set;
combining the formulas (14) and (19) to obtain the silicon single crystal pulling rate v p The method comprises the following steps:
from the geometrical height of the meniscus, the radius of growth of the silicon single crystal is transformedThe method comprises the following steps:
wherein alpha is c Is the meniscus tilt angle.
3. The method for controlling prediction of a semiconductor silicon single crystal growth model based on event triggering as set forth in claim 2, wherein the step 1.5 is specifically as follows:
from the above analysis, the heater power P and the pull-up speed v were obtained c For controlling input, the temperature T of the silicon melt l And a radius R of the silicon single crystal c Nonlinear silicon single crystal growth model for control output:
wherein x is 1 Is the temperature T of the silicon melt l ，For the silicon melt temperature transformation ratio, h (x 1 ) To loss heat of silicon melt to gas x 2 For the crucible temperature +.>For crucible temperature transformation rate, x 3 For the heater temperature, +.>For heater temperature change rate, x 4 Is the radius of silicon single crystal>Radius transformation ratio of silicon single crystal, x 5 For the height of the silicon melt in the crucible, +.>Is the silicon melt height conversion rate.
4. The method for controlling prediction of a semiconductor silicon single crystal growth model based on event triggering as set forth in claim 3, wherein the step 1.6 is specifically as follows:
the established nonlinear silicon single crystal growth model is expressed as:
wherein, xi (t) is the state variable of the model,the transformation rate of the state variable of the model is represented by mu (t) which is the input of the system and eta (t) which is the input of the systemThe nonlinear functions f (ζ (t), μ (t)) and g (ζ (t), μ (t)) are respectively a state dynamic function and an output function, and it is apparent that the model (23) is a complex nonlinear model that cannot be directly controlled, and in order to design a silicon single crystal growth controller, the nonlinear silicon single crystal growth model is linearized, and one balance point (ζ) is selected in a definition domain 0 ,μ 0 ) And meet the following
η 0 ＝g(ξ 0 ,μ 0 ) (24)
Wherein eta o Is a nonlinear model at (xi) 0 ,μ 0 ) The output of the dots is provided with,is a nonlinear model at (xi) 0 ,μ 0 ) State transition rate of the point;
taylor expansion of the nonlinear model is performed, the model (23) is then at the equilibrium point (ζ 0 ,μ 0 ) The dynamic characteristics of the vicinity are:
let Δζ (t) =ζ (t) - ζ 0 ,Δμ(t)＝μ(t)-μ 0 Sum Δη (t) =η (t) - η 0 Δζ (t) represents ζ (t) and ζ 0 And Δμ (t) represents μ (t) versus μ 0 Deltaeta (t) represents eta (t) and eta 0 And (b) in combination with equation (24), the model (25) is expressed as:
wherein,
derivative of ζ (t) for function f (ζ (t), μ (t)) at ζ (ζ) 0 ,μ 0 ) Matrix of->Derivative of μ (t) at (ζ) for the function f (ζ (t), μ (t)) 0 ,μ 0 ) Matrix of->Derivative of ζ (t) for the function g (ζ (t), μ (t)) at ζ (ζ) 0 ,μ 0 ) Matrix of->Derivative of μ (t) at (ζ) for the function g (ζ (t), μ (t)) 0 ,μ 0 ) Is a matrix of (a);
normalizing the system variables:
wherein Δζ 0 ,Δμ 0 And Deltaeta 0 For reference value, x (t) is normalized deltaThe value of ζ (t), u (t) is the normalized value of Δμ (t), and y (t) is the normalized value of Δη (t);
combining equation (26) and equation (28), a linear model of the silicon single crystal growth process is derived as:
wherein,is the derivative of x (t).
5. The method for controlling prediction of a semiconductor silicon single crystal growth model based on event triggering as set forth in claim 4, wherein the step 1.7 is specifically as follows:
discretizing the continuous linear model in the formula (29) by using a zero-order retainer to obtain a linear discrete model
x(k+1)＝Ax(k)+Bu(k)
y(k+1)＝Cx(k)+Du(k) (30)
Wherein x (k) and x (k+1) represent the values of x (t) at k and k+1, respectively, u (k) represents the values of u (t) at k and k, y (k+1) represents the values of y (t) at k+1, t is the sampling time.
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