Wafer temperature controller, wafer temperature control method, and wafer temperature control program

Abstract

To adjust the pressure of gas to predict future wafer temperature in wafer temperature control and control the wafer temperature at target temperature.SOLUTION: A wafer temperature controller which has a temperature-adjusted plate with a wafer loaded thereon and supplies gas between the plate and the wafer to control temperature of the wafer includes: a pressure adjuster which adjusts the pressure of gas; a vicinity temperature sensor which measures vicinity temperature of the wafer; and a pressure control part which controls, based on the vicinity temperature measured by the vicinity temperature sensor and target temperature of the wafer, a pressure operation amount inputted to the pressure adjuster through model prediction control, and the pressure control part uses a model in which a heat transfer rate between the plate and the wafer obtained from the pressure of gas is variable, as a prediction model of the model prediction control.SELECTED DRAWING: Figure 2

Description

[Technical field]

[0001] The present invention relates to a wafer temperature control device, a wafer temperature control method, and a wafer temperature control program. [Background technology]

[0002] Conventionally, in a semiconductor manufacturing process such as a film formation process, a wafer to be processed is placed on a plate such as an electrostatic chuck, and the temperature of the plate such as an electrostatic chuck is adjusted to control the temperature of the wafer to a predetermined target temperature.

[0003] Since the plate on which the wafer is placed is placed in a low-pressure environment such as a vacuum, as shown in Patent Document 1, it has been considered to supply a heat transfer gas such as helium gas between the plate and the wafer in order to promote heat transfer from the temperature-regulated plate to the wafer.

[0004] Here, since the heat transfer coefficient varies depending on the pressure of the heat transfer gas supplied between the plate and the wafer, it is necessary to adjust the pressure of the heat transfer gas.

[0005] However, even if the pressure of the heat transfer gas is adjusted, it is difficult to control the temperature of the wafer itself placed on the plate due to various technical constraints. [Prior art documents] [Patent documents]

[0006] [Patent Document 1] Patent No. 4034344 Summary of the Invention [Problem to be solved by the invention]

[0007] On the other hand, the inventor of the present application is considering applying model predictive control (MPC) to the above-mentioned configuration by modeling the heat transfer from the temperature-controlled plate to the wafer. Here, in order to simplify the model and reduce the calculation load, it is considered that the state equation of the model predictive control uses constants calculated from physical property values ​​for the coefficient matrix (A matrix) of the state vector and the coefficient matrix (B matrix) of the input vector.

[0008] However, as mentioned above, since the heat transfer coefficient varies depending on the pressure of the gas supplied between the plate and the wafer, in a model in which the thermal conductivity changes over time, if a constant (fixed thermal conductivity) is used in the coefficient matrix (A matrix) of the state vector, it becomes difficult to accurately control the wafer temperature to the target temperature.

[0009] Therefore, the present invention has been made to solve the above-mentioned problems, and has an object to predict the future wafer temperature and control the wafer temperature to a target temperature in a device that controls the wafer temperature by adjusting the gas pressure. [Means for solving the problem]

[0010] That is, the wafer temperature control device of the present invention is a wafer temperature control device in which a wafer is placed on a temperature-adjusted plate and gas is supplied between the plate and the wafer to control the temperature of the wafer, and is equipped with a pressure regulator that adjusts the pressure of the gas, a nearby temperature sensor that measures the nearby temperature of the wafer, and a pressure control unit that controls a pressure manipulation amount input to the pressure regulator by model predictive control based on the nearby temperature measured by the nearby temperature sensor and a target temperature of the wafer, and is characterized in that the pressure control unit uses a model in which the heat transfer coefficient between the plate and the wafer is a variable determined from the gas pressure as a predictive model for the model predictive control.

[0011] Such a wafer temperature control device uses a model in which the heat transfer coefficient between the plate and the wafer is a variable determined from the gas pressure as a predictive model for model predictive control, so that the future wafer temperature can be predicted and the wafer temperature can be controlled to a target temperature.

[0012] Here, the nearby temperature is, for example, the temperature of a member or space within a predetermined distance from the wafer, and includes a temperature for which a temperature model showing the relationship between the wafer temperature and the nearby temperature can be constructed. The nearby temperature also includes the temperature of a member in direct contact with the wafer, the temperature of the space or gas where the interface with the wafer exists, or the temperature of a member that exists across a gap of several μm from the wafer W. Furthermore, the nearby temperature may include the temperature of a member where heat can be conducted or transferred between the wafer and the member by at least one of conduction, convection, and radiation.

[0013] In order to accurately predict future wafer temperatures, it is desirable that the heat transfer coefficient used in the prediction model has a nonlinear or linear relationship depending on the gas pressure.

[0014] As a specific embodiment of the pressure control unit, it is considered that the pressure control unit controls the pressure manipulation amount using a state equation in which the coefficient matrix of a state vector in the model predictive control includes the pressure of the gas as a parameter.

[0015] As a specific embodiment of the pressure control unit, it is possible that the pressure control unit calculates the coefficient matrix each time a predicted trajectory for the temperature of the wafer is calculated in the model predictive control, and calculates the predicted trajectory using the calculated coefficient matrix.

[0016] Specifically, it is considered that the pressure control unit calculates the coefficient matrix using the pressure manipulation amount input to the pressure regulator or the pressure adjusted by the pressure regulator each time the predicted trajectory is calculated.

[0017] The pressure control unit may determine the pressure manipulated variable so as to minimize an evaluation function relating to a deviation between a reference trajectory for the temperature of the wafer and a predicted trajectory for the temperature of the wafer in the model predictive control. Here, the pressure control unit may use a weighted least squares method as the evaluation function. By using the weighted least squares method as the evaluation function in this manner, iterative calculations are not required, and the amount of calculations can be reduced.

[0018] The model predictive control of the present invention can be suitably used in a case where a plurality of regions in which different target temperatures for the wafer are set on the plate.

[0019] In addition, a wafer temperature control method according to the present invention is a wafer temperature control method in which a wafer is placed on a temperature-adjusted plate and gas is supplied between the plate and the wafer to control the temperature of the wafer, the method adjusting the pressure of the gas using a pressure regulator, measuring the temperature in the vicinity of the wafer using a proximity temperature sensor, and controlling the pressure manipulation amount input to the pressure regulator by model predictive control based on the proximity temperature measured by the proximity temperature sensor and a target temperature of the wafer, and is characterized in that a model in which a heat transfer coefficient between the plate and the wafer is a variable determined from the gas pressure is used as a predictive model for the model predictive control.

[0020] Furthermore, the wafer temperature control program of the present invention is a wafer temperature control program used in a wafer temperature control device that has a pressure regulator for adjusting the pressure of the gas and a nearby temperature sensor for measuring the nearby temperature of the wafer, in which a wafer is placed on a temperature-adjusted plate and gas is supplied between the plate and the wafer to control the temperature of the wafer, and is characterized in that the program provides a computer with the function of a pressure control unit that controls the pressure manipulation amount input to the pressure regulator by model predictive control based on the nearby temperature measured by the nearby temperature sensor and the target temperature of the wafer, and the pressure control unit uses a model in which the heat transfer coefficient between the plate and the wafer is a variable determined from the gas pressure as a predictive model for the model predictive control.

[0021] The wafer temperature control program may be distributed electronically, or may be recorded on a program recording medium such as a CD, a DVD, or a flash memory. Effect of the Invention

[0022] As described above, according to the present invention, in a system in which the wafer temperature is controlled by adjusting the gas pressure, it is possible to predict the future wafer temperature and control the wafer temperature to a target temperature. [Brief description of the drawings]

[0023] [Figure 1] 1 is a perspective view showing a schematic configuration of a wafer temperature control apparatus according to an embodiment of the present invention; [Diagram 2] 11 is a diagram showing the relationship between the pressure of the heat transfer gas supplied between the wafer and the suction plate and the heat transfer coefficient between the wafer and the suction plate. FIG. [Diagram 3] FIG. 2 is a diagram illustrating a mechanism of model predictive control in the embodiment. [Figure 4] FIG. 2 is a diagram showing a control object model used in model predictive control in the embodiment. [Diagram 5]4 is a diagram showing a reference trajectory and a predicted trajectory in which two different regions are not distinguished from each other in the model predictive control of the embodiment. FIG. [Figure 6] FIG. 2 is a diagram illustrating a state equation and a reference trajectory of the model predictive control according to the embodiment. [Figure 7] 4A to 4C are diagrams illustrating predicted values ​​based on a free response, predicted values ​​based on a step response, and predicted trajectories of the model predictive control of the embodiment. [Figure 8] FIG. 4 is a diagram illustrating an evaluation function of the model predictive control according to the embodiment. [Figure 9] FIG. 11 is a diagram showing simulation results of a case where the wafer temperature is controlled using the wafer temperature control device of the same embodiment, and a case where the wafer temperature is controlled by fixing the pressure of the coefficient matrix (A matrix) at the initial pressure in model predictive control (comparative example). DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0024] <One embodiment of the present invention> DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of a wafer temperature control device according to the present invention will be described below with reference to the drawings. In addition, in any of the drawings shown below, for the purpose of easy understanding, some parts are omitted or exaggerated in schematic form as appropriate. The same components are denoted by the same reference numerals and the description thereof is omitted as appropriate.

[0025] <1. Basic configuration of wafer temperature control device> The wafer temperature control device 100 of this embodiment is used in a semiconductor manufacturing device that performs a semiconductor manufacturing process such as a film formation process, and is configured to electrostatically chuck the back surface of a wafer W in a vacuum chamber, for example.

[0026] Specifically, the wafer temperature control device 100 includes an adsorption plate 2 on the upper surface of which a wafer W is placed, and a temperature regulator 3 that adjusts the temperature of the adsorption plate 2, as shown in FIG.

[0027] The suction plate 2 constitutes a so-called electrostatic chuck that holds the wafer W by electrostatic suction force. The suction plate 2 in this embodiment is a ceramic plate having a substantially circular shape, and the upper surface thereof serves as an suction surface 2a that suctions the wafer W. An electrostatic electrode (not shown) for generating an electrostatic force between the suction plate 2 and the wafer W is provided inside the suction plate 2.

[0028] The temperature regulator 3 regulates the temperature of the suction plate 2 to a preset temperature, and includes a heater 31 for heating the suction plate 2 and a cooler 32 for cooling the suction plate 2. The temperature regulator 3 may not necessarily include the heater 31.

[0029] The heater 31 is provided inside the suction plate 2, and has a plurality of heater electrodes 31a for heating the suction plate 2. The power supplied to the plurality of heater electrodes 31a is controlled by a heating control unit 11a of the control device CTL, and each of the heater electrodes 31a is independently controlled according to a heating operation amount set by a user.

[0030] The cooler 32 is provided in contact with the lower surface of the suction plate 2, and includes a substantially disk-shaped base plate 32a and a cooling passage 32b formed within the base plate 32a.

[0031] The cooling flow passage 32b is formed in a spiral shape in a plan view inside the base plate 32a. An inlet flow passage 32c and an outlet flow passage 32d, which are connected to a cooling source (not shown) such as a chiller, are connected to the cooling flow passage 32b. A control valve 32e for controlling the flow rate of the refrigerant is provided in the flow passage connected to the cooling flow passage 32b, and the valve opening of the control valve 32e is controlled by the cooling control unit 11b of the control device CTL.

[0032] The wafer temperature control apparatus 100 of this embodiment also includes a gas supply mechanism 4 for supplying a gas that transfers heat (hereinafter, referred to as a heat transfer gas), such as helium gas or argon gas, between the suction plate 2 and the wafer W.

[0033] The gas supply mechanism 4 supplies a heat transfer gas at a predetermined pressure between the suction surface 2a of the suction plate 2 and the rear surface of the wafer W that is suctioned.

[0034] Specifically, the gas supply mechanism 4 has a gas circulation groove 41 formed on the suction surface 2a of the suction plate 2, a gas supply path 42 for supplying a heat transfer gas to the gas circulation groove 41, and a pressure regulator 43 for adjusting the pressure of the heat transfer gas supplied to the gas circulation groove 41. The heat transfer gas supplied to the gas circulation groove 41 flows from the gas circulation groove 41 into the gap between the suction surface 2a of the suction plate 2 and the back surface of the wafer W being suctioned.

[0035] The gas flow grooves 41 include, for example, a plurality of linear grooves formed radially from the central axis of the suction plate 2, and a plurality of circular grooves formed circularly from the central axis of the suction plate 2. The gas supply passage 42 is formed along the central axis of the suction plate 2, and is connected to a heat transfer gas source (not shown).

[0036] Moreover, the pressure regulator 43 can change the heat transfer rate from the suction plate 2 to the wafer W (the heat transfer coefficient between the wafer W and the suction plate 2) by adjusting the pressure of the heat transfer gas. Note that Fig. 2 shows the relationship between the pressure of the heat transfer gas supplied between the wafer W and the suction plate 2 and the heat transfer coefficient between the wafer W and the suction plate 2. In this embodiment, the heat transfer coefficient has a nonlinear relationship depending on the pressure of the heat transfer gas, but may have a linear relationship.

[0037] Specifically, the pressure regulator 43 has a pressure sensor and a pressure control valve, and the valve opening degree of the pressure control valve is controlled by the pressure control section 12 of the control device CTL.

[0038] The wafer temperature control device 100 also includes a proximity temperature sensor 5 that measures the temperature in the vicinity of the wafer W. The proximity temperature sensor 5 is provided on the rear surface side of the base plate 32a, and measures the temperature of the base plate 32a as the temperature in the vicinity of the wafer W. The proximity temperature sensor 5 in this embodiment is, for example, an infrared sensor such as a radiation thermometer. The proximity temperature sensor 5 may also measure the temperature of the suction plate 2 as the temperature in the vicinity of the wafer W.

[0039] <2. Wafer temperature control system> Furthermore, the wafer temperature control apparatus 100 includes a control device CTL that controls the operations of at least the temperature regulator 3 and the pressure regulator 43 .

[0040] The control device CTL is a so-called computer equipped with a CPU, a memory, an A / D converter, a D / A converter, and various input / output devices. A wafer temperature control program stored in the memory is executed, and the control device CTL functions as a temperature control unit 11 that controls the operation of the temperature regulator 3 and a pressure control unit 12 that controls the operation of the pressure regulator 43 by the cooperation of various devices.

[0041] The temperature control unit 11 of this embodiment has a heating control unit 11a and a cooling control unit 11b, and supplies fixed power to each heater electrode 31a of the heater 31 by the heating control unit 11a, and controls the valve opening degree of the control valve 32e of the cooler 32 to be constant by the cooling control unit 11b. That is, the temperature control unit 11 fixes the heating operation amount and the cooling operation amount of the heater 31 and the cooler 32 during operation, and controls so that the temperature adjustment amount per unit time by the temperature regulator 3 is constant. Note that in the heating control unit 11a and the cooling control unit 11b, the heating operation amount and the cooling operation amount do not become variables of other physical property values ​​(for example, heat transfer coefficient, etc.) and are not included in the coefficient matrix (A matrix) described later, so the prediction model control performed by the pressure control unit 12 described below is not performed.

[0042] The pressure control unit 12 controls the pressure P of the heat transfer gas by model predictive control based on the nearby temperature measured by the nearby temperature sensor 5 and the target temperature of the wafer W.HG This controls the pressure regulator 43 that adjusts the pressure.

[0043] Here, as shown in Fig. 3, model predictive control is a control method for performing optimization while predicting a future response at each time, and predicts future behavior of the controlled object over a finite interval from the current time by having a prediction model (controlled object model) inside pressure control unit 12. In Fig. 3, target command r(t) is the target temperature of wafer W, control input u(t) is the pressure manipulation amount input to pressure regulator 43, and control output y(t) is the nearby temperature measured by nearby temperature sensor 5. The nearby temperature value measured from this controlled object is applied to the prediction model inside pressure control unit 12, and a new control input u(t) is determined in an optimizer so as to minimize the tracking error from the current time to a predetermined time.

[0044] The controlled object model used in this model predictive control is shown in Fig. 4. In this embodiment, the controlled object model has two concentric heating zones, a circular heating zone (Zone 1) located in the center and an annular heating zone (Zone 2) located on the periphery. In this suction plate 2, the target temperature of the central heating zone is T W1_SET and the target temperature of the outer heating area is T W2_SET In addition, in FIG. 5, the reference trajectory and the predicted trajectory are shown without distinguishing between the two different regions.

[0045] In the control model shown in Figure 4, the wafer temperature T W1 Time change (T W1 (variables showing dots on top of each other), and the wafer temperature T W2 Time change (T W2 The thermal radiation shown in term A is proportional to the fourth power of the temperature, but since term A itself is a small value, it is expediently set to the first power here to simplify the calculation.

[0046] Specifically, as described above, the pressure control unit 12 calculates the heat transfer coefficient between the suction plate 2 and the wafer W as a prediction model (control target model) of the model predictive control using the pressure P HG More specifically, the pressure control unit 12 uses a model in which the variables are calculated from the vicinity temperature T P1 , T P2 and the target temperature T of the wafer W W1_SET , T W2_SET Based on this, the pressure manipulation amount input to the pressure regulator 43 is controlled by model predictive control.

[0047] Here, the pressure control unit 12 controls the pressure manipulation amount using a state equation in which the coefficient matrix (A matrix) of the state vector includes the pressures P1 and P2 of the heat transfer gas as parameters in the model predictive control. Here, the reason why the coefficient matrix (A matrix) of the state vector includes the pressures P1 and P2 of the heat transfer gas is that the heat transfer coefficients α1 (P1) and α2 (P2) between the wafer and the chucking plate change depending on the pressures P1 and P2 of the heat transfer gas.

[0048] Specifically, the pressure control unit 12 uses the state equation shown in FIG. In this state equation, the coefficient matrix (A matrix) multiplied by the state vector is a matrix indicating the thermal conductivity and heat capacity of each divided area (zone), and includes the heat transfer coefficients α1 (P1) and α2 (P2) between the wafer W and the suction plate 2, which are determined from the pressures P1 and P2 of the heat transfer gas. In addition, the coefficient matrix (B matrix) multiplied by the input vector in the state equation is a matrix indicating the coefficients for converting the heat transfer coefficients α1 (P1) and α2 (P2) between the wafer and the suction plate, and the steady heat supplies q1 and q2 from the outside air into temperature changes. Furthermore, the reference trajectory in the model predictive control indicates an ideal trajectory that exponentially approaches the set temperature from the current temperature, as shown in Figure 6.

[0049] Then, the pressure control unit 12 calculates a coefficient matrix (A matrix) every time a predicted trajectory regarding the temperature of the wafer W is calculated in the model predictive control, and calculates the next predicted trajectory using the coefficient matrix (A matrix) updated by the calculation.

[0050] Specifically, the pressure control unit 12 calculates a predicted trajectory obtained from a state equation as shown in Figures 5 and 6, and more specifically, calculates the predicted trajectory using a predicted value based on a free response and a predicted value based on a step response.

[0051] The predicted value based on the free response indicates the trajectory for the next Np steps if the current (time k) inputs P1(k) and P2(k) remain unchanged.

[0052] The predicted value by the step response is the trajectory for the future Np steps and the current temperature T W1 (k), T W2 This shows the difference (change) from (k).

[0053] Here, since the inputs P1(k) and P2(k) are updated sequentially, they must be calculated every hour. The inputs P1(k) and P2(k) used for the predicted value based on this step response are the pressure manipulated variable input to the pressure regulator 43 or the pressure adjusted by the pressure regulator 43.

[0054] Then, as shown in FIG. 8, the pressure control unit 12 determines the pressure manipulation amount so as to minimize the evaluation function using the weighted least squares method for the deviation between the reference trajectory for the temperature of the wafer W and the predicted trajectory for the temperature of the wafer W in the model predictive control. In addition, in order to prevent hunting from occurring in the evaluation function, terms r1 and r2 that reduce the pressure are added, and these terms r1 and r2 are made as small as possible. By using the weighted least squares method as the evaluation function in this way, iterative calculation is not required, and the amount of calculation can be reduced. In addition, when the evaluation function cannot be analytically solved due to constraints such as constraint conditions, a method for exploratory solution such as the shooting method, gradient descent method, or Newton's method may be used.

[0055] <3. Simulation results> Next, FIG. 9 shows simulation results of a case where the temperature of the wafer W is controlled using the wafer temperature control device 100 of this embodiment, and a case where the temperature of the wafer W is controlled by fixing the pressure of the coefficient matrix (matrix A) at the initial pressure in model predictive control (comparison example).

[0056] In both simulations, the target temperature (℃) is (T W1_SET ,T W2_SET )=(14.0,12.7), initial temperature (℃) is (T W1 ,T W2 )=(15.5,14.0), and the initial pressure (P1,P2)=(3.0,3.0). Also, the sampling period T S is 1 second, and the reference orbit time constant T ref is 15 seconds, the prediction horizon score Np is 30, and the match score Na is 15.

[0057] As can be seen from FIG. 9A, when the temperature of the wafer W is controlled using the wafer temperature control device 100 of this embodiment, the temperature T W1 ,T W2 The target temperature T W1_SET ,T W2_SET On the other hand, as can be seen from FIG. 9(b), when the pressure of the coefficient matrix (matrix A) is fixed at the initial pressure in the model predictive control and the temperature of the wafer is controlled, the temperature T W1 ,T W2 The target temperature T W1_SET ,T W2_SET It is not possible to control the temperature accurately.

[0058] <4. Effects of this embodiment> As described above, according to the wafer temperature control device 100 of this embodiment, a model in which the heat transfer coefficient between the suction plate 2 and the wafer W, which is determined from the pressure of the heat transfer gas, is used as a predictive model for model predictive control, so that the future wafer temperature can be predicted and the wafer temperature can be controlled to a target temperature. Specifically, in the model predictive control, a state equation in which the pressure of the heat transfer gas is included as a parameter in the coefficient matrix (A matrix) of the state vector is used to control the pressure manipulation amount, so that the wafer temperature can be controlled to a target temperature with high accuracy.

[0059] <5. Other embodiments> For example, in the above embodiment, the suction plate 2 has multiple regions with different target temperatures for the wafer W set by varying the pressure of the heat transfer gas across the surface of the suction plate 2, but it is also possible to set a single target temperature across the entire suction plate 2 without varying the pressure of the heat transfer gas across the surface of the suction plate 2.

[0060] Moreover, the heater 31 of the embodiment can form a temperature distribution on the suction plate 2 by using a plurality of heater electrodes 31a. For example, the heater 31 can make the heating amount different between the center and the outer periphery of the suction plate 2, and can also make the heating amount different between a large region that is roughly C-shaped on the outer periphery and the remaining small region. That is, three heating regions are set on the suction plate 2, and a temperature distribution is formed on the suction plate 2. The cooler 32 may also be configured to form three cooling regions on the surface of the base plate 32a in correspondence with the three heating regions of the suction plate 2. In addition, various setting methods are possible, such as setting a plurality of heating regions by radially dividing the outer periphery.

[0061] The heating or cooling region of the wafer W and the suction plate 2 is not limited to being divided into three regions, but may be divided into more regions, or may be divided into two regions. Also, the entire wafer W or the suction plate 2 may be treated as one temperature without setting regions.

[0062] Furthermore, the suction plate 2 may not have a suction function and may simply be a plate on which the wafer W is placed.

[0063] The configuration of the cooler or heater is not limited to those described above. For example, the cooler may be configured using a Peltier element, and the heater is not limited to a heater electrode, but may be configured to heat the wafer by light irradiation, or may be configured to heat the wafer by plasma.

[0064] The location where the proximity temperature sensor measures is not limited to the above-mentioned location, but may be other locations. In short, any temperature that is likely to have some correlation or relationship with the wafer temperature may be measured as the proximity temperature. Furthermore, the proximity temperature sensor is not limited to an infrared temperature sensor, but may be, for example, a thermocouple or the like provided in the plate.

[0065] In addition, various modifications and combinations of the embodiments may be made as long as they do not go against the spirit of the present invention. [Explanation of symbols]

[0066] 100 Wafer temperature control device W: Wafer 2. Adsorption plate (plate) 3...Temperature regulator 43 Pressure regulator 5. Near-field temperature sensor 12 Pressure control section

Claims

1. A wafer temperature control device in which a wafer is placed on a temperature-adjusted plate and a gas is supplied between the plate and the wafer to control the temperature of the wafer, a pressure regulator for adjusting the pressure of the gas; a proximity temperature sensor for measuring a proximity temperature of the wafer; a pressure control unit that controls a pressure manipulation amount input to the pressure regulator by model predictive control based on the nearby temperature measured by the nearby temperature sensor and a target temperature of the wafer, The wafer temperature control device, wherein the pressure control unit uses a model in which a heat transfer coefficient between the plate and the wafer is a variable determined from the pressure of the gas as a predictive model for the model predictive control.

2. The wafer temperature control apparatus according to claim 1 , wherein the heat transfer coefficient used in the prediction model has a nonlinear or linear relationship depending on the pressure of the gas.

3. 3. The wafer temperature control apparatus according to claim 1, wherein the pressure control unit controls the pressure manipulated variable using a state equation in which the pressure of the gas is included as a parameter in a coefficient matrix of a state vector in the model predictive control.

4. 4. The wafer temperature control device according to claim 3, wherein the pressure control unit calculates the coefficient matrix each time a predicted trajectory related to the temperature of the wafer is calculated in the model predictive control, and calculates the predicted trajectory using the calculated coefficient matrix.

5. 5. The wafer temperature control device according to claim 4, wherein the pressure control unit calculates the coefficient matrix using a pressure manipulation amount input to the pressure regulator or a pressure adjusted by the pressure regulator each time the predicted trajectory is calculated.

6. 3. The wafer temperature control device according to claim 1, wherein the pressure control unit determines the pressure manipulation amount so as to minimize an evaluation function related to a deviation between a reference trajectory related to the temperature of the wafer and a predicted trajectory related to the temperature of the wafer in the model predictive control.

7. 7. The wafer temperature control apparatus according to claim 6, wherein the pressure control unit uses a weighted least squares method as the evaluation function.

8. 3. The wafer temperature control apparatus according to claim 1, wherein the plate has a plurality of regions each having a different target temperature for the wafer.

9. 1. A wafer temperature control method comprising: placing a wafer on a temperature-adjusted plate; and supplying a gas between the plate and the wafer to control the temperature of the wafer, the method comprising: The pressure of the gas is adjusted by a pressure regulator; measuring the temperature near the wafer with a proximity temperature sensor; a method for controlling a pressure manipulation amount input to the pressure regulator by model predictive control based on the nearby temperature measured by the nearby temperature sensor and a target temperature of the wafer, a model in which a heat transfer coefficient between the plate and the wafer is a variable determined from the pressure of the gas, is used as a predictive model for the model predictive control;

10. A wafer temperature control program used in a wafer temperature control device that controls the temperature of a wafer by supplying gas between a temperature-adjusted plate and the plate, the wafer temperature control program including a pressure regulator that adjusts the pressure of the gas and a proximity temperature sensor that measures the temperature near the wafer, the program comprising: a computer is provided with a function as a pressure control unit that controls a pressure manipulation amount input to the pressure regulator by model predictive control based on the nearby temperature measured by the nearby temperature sensor and the target temperature of the wafer, the pressure control unit uses a model in which a heat transfer coefficient between the plate and the wafer is a variable determined from the pressure of the gas as a predictive model for the model predictive control.

Images