US-12626034-B2 - Digital twin based temperature distribution estimating method and temperature distribution estimating apparatus

Abstract

The temperature distribution estimating method of the disclosure includes a building operation to build a numerical model for a form and thermal behavior of a substrate; a setting operation to set a regularization parameter to adjust noises of a temperature of the substrate measured by a temperature sensor; a generating operation to generate a sensitivity coefficient matrix that estimates a heat source received by the substrate from a plurality of heaters; a condensing operation to condense the sensitivity coefficient matrix based on a power ratio input in the heaters respectively; and estimating operation to estimate an entire temperature distribution of the substrate based on the numerical model, the regularization parameter, and the condensed sensitivity coefficient matrix, when predetermined temperature data are input.

Inventors

• JIN GYUN KIM
• Chang Uk AHN
• Ji Won Lee
• In Chan BECK
• Do Hyeong KWON

Assignees

• UNIVERSITY-INDUSTRY COOPERATION GROUP OF KYUNG HEE UNIVERSITY

Dates

Publication Date

20260512

Application Date

20220211

Priority Date

20210610

Claims (6)

1. 1 . A temperature distribution controlling method, performed by a temperature distribution estimating apparatus to measure a temperature distribution of a thermal processing target substrate, comprising: a building operation to build a virtual numerical model for a form and thermal behavior of the thermal processing target substrate using any one of a finite element method, a finite difference method, and a boundary element method as a numerical analysis method; a setting operation to set a regularization parameter a based on number and positions of temperature sensors, thermal diffusivity of the thermal processing target substrate, and performance of a chamber; a generating operation to generate a sensitivity coefficient matrix for sensor positions X ι A + 1 to estimate a heat source received by the thermal processing target substrate from a plurality of heaters, as expressed by following equation: X s i + 1 = ∂ T s i + 1 ∂ q = [ ⁠ ∂ T 1 i + 1 ∂ q 1 ∂ T 1 i + 1 ∂ q 2 … ∂ T 1 i + 1 ∂ q M ∂ T 2 i + 1 ∂ q 1 ∂ T 2 i + 1 ∂ q 2 … ∂ T 2 i + 1 ∂ q M ⋮ ⋮ ⋱ ⋮ ∂ T N i + 1 ∂ q 1 ∂ T N i + 1 ∂ q 2 … ∂ T N i + 1 ∂ q M ⁠ ] X s i + 1 = ∂ T s i + 1 ∂ q = [ ⁠ ∂ T 1 i + 1 ∂ q 1 ∂ T 1 i + 1 ∂ q 2 … ∂ T 1 i + 1 ∂ q M ∂ T 2 i + 1 ∂ q 1 ∂ T 2 i + 1 ∂ q 2 … ∂ T 2 i + 1 ∂ q M ⋮ ⋮ ⋱ ⋮ ∂ T N i + 1 ∂ q 1 ∂ T N i + 1 ∂ q 2 … ∂ T N i + 1 ∂ q M ⁠ ] where N corresponds to number of sensors used and M corresponds to number of spatially discretized heaters; wherein the heat source and an entire temperature distribution are calculated according to following equations: Δ ⁢ t = t i + 1 - t i , Δ ⁢ q i + 1 = [ ( X s i + 1 ) T ⁢ ( X s i + 1 ) + α ⁢ I ] - 1 ⁢ ( X s i + 1 ) T ⁢ ( Y i + 1 - T _ s i + 1 ) , q i + 1 = q i + Δ ⁢ q i + 1 , T i + 1 = T i + X i + 1 ⁢ Δ ⁢ q i + 1 where t i represents a past time, t i+1 represents a current time, Δt represents an interval of measurement, q i+1 represents an estimated heat source vector, Y i−1 a represents a measured temperature vector, T i+1 represents an estimated entire temperature vector, including temperatures of other points where there is no temperature sensor, and ′E represents a virtual temperature vector at locations provided with the temperature sensors when T s a is consistently heated by q i , wherein, when only one temperature sensor is provided, the virtual temperature vector becomes a constant, a condensing operation to condense the sensitivity coefficient matrix into a condensed sensitivity coefficient matrix, based on power ratio information respectively input to the heaters calculated by a heater controller, as expressed by following equations X s i + 1 = [ X s ⁢ 1 i + 1 ⁢ X s ⁢ 2 i + 1 ⁢ … ⁢ X sM i + 1 ] , X ~ s i + 1 = q 1 q 1 ⁢ X s ⁢ 1 i + 1 + q 2 q 1 + q M q 1 ⁢ X sM i + 1 , X s i + 1 = [ X s ⁢ 1 i + 1 ⁢ X s ⁢ 2 i + 1 ⁢ … ⁢ X sM i + 1 ] , X ~ s i + 1 = q 1 q 1 ⁢ X s ⁢ 1 i + 1 + q 2 q 1 ⁢ X s ⁢ 2 i + 1 + … + q M q 1 ⁢ X sM i + 1 , where X s t + 1 represents a sensitivity coefficient vector condensed using power ratios; and an estimating operation to, when predetermined temperature data are input, estimate an entire temperature distribution of the thermal processing target substrate based on the virtual numerical model, the regularization parameter, and the condensed sensitivity coefficient matrix by calculating heat sources of at least one of the plurality of heaters and restoring remaining heat sources based on the condensed sensitivity coefficient matrix; Controlling the heat sources of the plurality of heaters based on the estimated entire temperature distribution so that the temperature distribution of the thermal processing target substrate becomes uniform, to prevent warpage, cracks, or dislocations in the thermal processing target substrate.
2. 2 . The temperature distribution estimating method of claim 1 , further comprising: a monitoring operation to compare an entire temperature distribution estimated by using the predetermined temperature data that are measured in the temperature sensor with temperature data measured by using a temperature sensor for in real time to determine current status of an apparatus.
3. 3 . The temperature distribution estimating method of claim 1 , wherein the setting operation comprises: collecting temperature information of the thermal processing target substrate according to time during one cycle of processing in an initial apparatus manufacturing process; converting the collected temperature information into filtered data from which noise is removed by post-processing filtering; and setting a regularization parameter a to a value at which a least square error of the filtered data is minimized by varying the regularization parameter in following equations: Δ ⁢ t = t i + 1 - t i , Δ ⁢ q i + 1 = [ ( X s i + 1 ) T ⁢ ( X s i + 1 ) + α ⁢ I ] - 1 ⁢ ( X s i + 1 ) T ⁢ ( Y i + 1 - T _ s i + 1 ) , q i + 1 = q i + Δ ⁢ q i + 1 , T i + 1 = T i + X i + 1 ⁢ Δ ⁢ q i + 1 where t i refers to a past time, t i+1 refers to a present time, Δt refers to an interval of measurement, q i+ refers to an estimated heat source vector, Y i+1 refers to a measured temperature vector, and T i+1 refers to an estimated entire temperature vector, including temperatures of other points where there is no temperature sensor, and T _ s i + 1 refers to a virtual temperature vector of a point having a temperature sensor when T s i is consistently heated by q i .
4. 4 . A temperature distribution controlling apparatus, comprising: a plurality of heaters to supply a heat source to a thermal processing target substrate; a temperature sensor to measure temperatures of the thermal processing target substrate that is changed by the plurality of heaters; and a calculator to estimate an entire temperature distribution of the thermal processing target substrate when predetermined temperature data are input from the temperature sensor, wherein the calculator comprises: a numerical model builder to build a virtual numerical model for a form and thermal behavior of the thermal processing target substrate using any one of a finite element method, a finite difference method, and a boundary element method as a numerical analysis method; a parameter setter to set a regularization parameter a based on number and positions of temperature sensors, thermal diffusivity of the thermal processing target substrate, and performance of a chamber; a matrix generator to generate a sensitivity coefficient matrix to estimate a heat source received by the thermal processing target substrate from the plurality of heaters and to condense the sensitivity coefficient matrix based on power ratios information respectively input into the heaters calculated by a heater controller; and an estimator to, when predetermined temperature data are input, estimate an entire temperature distribution of the thermal processing target substrate based on the virtual numerical model, the regularization parameter, and the condensed sensitivity coefficient matrix by calculating heat sources of at least one of the plurality of heaters and restoring remaining heat sources based on the condensed sensitivity coefficient matrix; controlling the heat sources of the plurality of heaters based on the estimated entire temperature distribution so that the temperature distribution of the thermal processing target substrate becomes uniform, to prevent warpage, cracks, or dislocations in the thermal processing target substrate, wherein the sensitivity coefficient matrix X s t + 1 is: ∂ T s i + 1 ∂ q = [ ⁠ ∂ T 1 i + 1 ∂ q 1 ∂ T 1 i + 1 ∂ q 2 … ∂ T 1 i + 1 ∂ q M ∂ T 2 i + 1 ∂ q 1 ∂ T 2 i + 1 ∂ q 2 … ∂ T 2 i + 1 ∂ q M ⋮ ⋮ ⋱ ⋮ ∂ T N i + 1 ∂ q 1 ∂ T N i + 1 ∂ q 2 … ∂ T N i + 1 ∂ q M ⁠ ] r at ∂ T s i + 1 ∂ q = [ ⁠ ∂ T 1 i + 1 ∂ q 1 ∂ T 1 i + 1 ∂ q 2 … ∂ T 1 i + 1 ∂ q M ∂ T 2 i + 1 ∂ q 1 ∂ T 2 i + 1 ∂ q 2 … ∂ T 2 i + 1 ∂ q M ⋮ ⋮ ⋱ ⋮ ∂ T N i + 1 ∂ q 1 ∂ T N i + 1 ∂ q 2 … ∂ T N i + 1 ∂ q M ⁠ ] , and the condensed sensitivity coefficient matrix is: X s i + 1 = [ X s ⁢ 1 i + 1 ⁢ X s ⁢ 2 i + 1 ⁢ … ⁢ X sM i + 1 ] , X ~ s i + 1 = q 1 q 1 ⁢ X s ⁢ 1 i + 1 + q 2 q 1 ⁢ X s ⁢ 2 i + 1 + … + q M q 1 ⁢ X sM i + 1 , X s i + 1 = [ X s ⁢ 1 i + 1 ⁢ X s ⁢ 2 i + 1 ⁢ … ⁢ X sM i + 1 ] , X ~ s i + 1 = q 1 q 1 ⁢ X s ⁢ 1 i + 1 + q 2 q 1 ⁢ X s ⁢ 2 i + 1 + … + q M q 1 ⁢ X sM i + 1 , where N corresponds to number of sensors used and M corresponds to number of spatially discretized heaters, and X s t + 1 represents a sensitivity coefficient vector condensed using power ratios, wherein the heat source and the entire temperature distribution are calculated according to following equations g s , T _ s t + 1 ⁢ Υ z + 1 t + 1 ⁢ Δ ⁢ ts t + 1 ⁢ t s where t i represents a past time, t i+1 represents a current time, Δt represents an interval of measurement, q i+ represents an estimated heat source vector, Y i+1 represents a measured temperature vector, T i+1 represents an estimated entire temperature vector, including temperatures of other points where there is no temperature sensor, and T′″ represents a virtual temperature vector at locations provided with the temperature sensors when ′ is consistently heated by q 1 , wherein, when only one temperature sensor is provided, the virtual temperature vector becomes a constant.
5. 5 . The temperature distribution estimating apparatus of claim 4 , wherein the temperature sensor comprises a temperature sensor for measurement and a temperature sensor for verification, wherein a monitor is further included to compare an entire temperature distribution estimated by using the predetermined temperature data that are measured in the temperature sensor for measurement with temperature data measured by using the temperature sensor for verification in real time to determine current status of an apparatus.
6. 6 . The temperature distribution estimating apparatus of claim 4 , wherein the parameter setter is configured to collect temperature information of the thermal processing target substrate according to time during one cycle of processing in an initial apparatus manufacturing process, convert the collected temperature information into filtered data from which noise is removed by post-processing filtering; and set a regularization parameter α to a value at which a least square error of the filtered data is minimized by varying the regularization parameter in following equations: Δ ⁢ t = t i + 1 - t i , Δ ⁢ q i + 1 = [ ( X s i + 1 ) T ⁢ ( X s i + 1 ) + α ⁢ I ] - 1 ⁢ ( X s i + 1 ) T ⁢ ( Y i + 1 - T _ s i + 1 ) , q i + 1 = q i + Δ ⁢ q i + 1 , T i + 1 = T i + X i + 1 ⁢ Δ ⁢ q i + 1 where t 1 refers to a past time, t i+1 refers to a present time, Δt refers to an interval of measurement, q i+1 refers to an estimated heat source vector, Y i+1 refers to a measured temperature vector, and T i+1 refers to an estimated entire temperature vector, including temperatures of other points where there is no temperature sensor, and T _ s i + 1 refers to a virtual temperature vector of a point having a temperature sensor when T s i is consistently heated by q i .

Description

CROSS-REFERENCE TO RELATED APPLICATION This application claims the benefit under 35 U.S.C. § 119 of Korean Patent Application No. 10-2021-0075546 filed on Jun. 10, 2021, in the Korean Intellectual Property Office, the entire disclosure of which is incorporated herein by reference for all purposes. BACKGROUND 1. Field The following description relates to a temperature distribution estimating method and a temperature distribution estimating apparatus during a rapid thermal processing when manufacturing a semiconductor device, more specifically, a temperature distribution estimating method and a temperature distribution estimating apparatus that are able to estimate a temperature distribution of a wafer only by small data of a point measurement temperature. 2. Description of Related Art In manufacturing a semiconductor device, a temperature of a wafer is rapidly increased in a rapid thermal processing (hereinafter, ‘RTP’). In this case, when a temperature distribution of a wafer is not maintained uniformly, there may be problems such as a warpage, crack, and dislocation, etc. Therefore, it is important to maintain a temperature distribution uniformly in a RTP operation, and accordingly, a temperature distribution measurement technology for a wafer is desired. A contact temperature sensor, thermocouple, and a non-contact temperature sensor, pyrometer are mainly used to measure a temperature of a wafer. In measuring a temperature with a thermocouple, numerous sensors are needed to measure the entire temperature of a wafer, which may cause an error due to increasing a thermal resistance of the thermocouple. Also, the error may become larger by keeping using it. Therefore, measuring a temperature with using a pyrometer is the main method in a typical RTP apparatus. In measuring the entire temperature distribution with a pyrometer, an amount of photons is measured from a point or points of a wafer. However, since a temperature is changed dramatically in a wafer during a RTP operation, an emissivity of an object may be changed. This makes it difficult to precisely measure a temperature with using a pyrometer. The reason why a pyrometer is mainly used to measure a temperature of a RTP apparatus even though it is relatively expensive compared with a thermocouple is that a thermocouple needs a number of sensors to measure a temperature. Therefore, a production cost may rise with when using a thermocouple in a mass production of semiconductor devices. Consequently, it is really difficult to measure the entire temperature distribution of a wafer only with current hardware apparatus, and it leads to another problem to calculate a uniformity of a wafer. A calculation formula for a thin film uniformity may vary according to an apparatus or a company, but the calculation is conducted mainly by using max/min value such as (max−min)/(2*avg) and (max−min)/(max+min) because measuring data are not enough. When measuring data are enough, a precise calculation for a uniformity may be possible by calculating a standard deviation of a space. Korean Patent Publication No. 10-2021-0000731 (Title: Virtual sensor for spatially resolved wafer temperature control, Applicant: Applied Materials, Inc.) relates to methods and apparatus for temperature sensing and control during substrate processing. To solve a technical issue that it is difficult to measure a substrate temperature directly during a substrate processing, the publication establishes models to estimate a temperature change of a substrate according to changes in power output by building models via machine learning techniques, and the models are used to adjust heating device setpoints for future processing operations. However, the invention disclosed in Korean Patent Publication No. 10-2021-0000731 has two problems. One is that massive data are needed before establishing models. The other is that maintaining heating device setpoints do not guarantee maintaining a substrate temperature at a target substrate temperature because a change in a heat transfer may occur between a substrate and a chamber due to a deterioration of a process chamber as time goes by. SUMMARY This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. In a general aspect, a temperature distribution estimating method may include a building operation to build a numerical model for a form and thermal behavior of a thermal processing target substrate; a setting operation to set a regularization parameter to adjust noises of a temperature of the substrate measured by a temperature sensor; a generating operation to generate a sensitivity coefficient matrix that estimates a heat source received by the substrate from a plurality of hea