Method of obtaining array of plurality of regions on substrate, exposure method, exposure apparatus, method of manufacturing article, non-transitory computer-readable storage medium, and information processing apparatus
A method of obtaining an array of a plurality of regions on a substrate, including obtaining position measurement data by measuring a mark assigned to each sample region among the plurality of regions on the substrate, and estimating a position of each non-measurement region excluding the sample region among the plurality of regions by using a regression model used to estimate the array from the position measurement data, wherein the regression model is a nonparametric regression model.
1 . A method of obtaining an array of a plurality of regions on a substrate, comprising:
obtaining position measurement data by measuring a mark assigned to each sample region among the plurality of regions on the substrate; and
estimating a position of each non-measurement region excluding the sample region among the plurality of regions by using a regression model used to estimate the array from the position measurement data, wherein the regression model is a nonparametric regression model;
determining, for each non-measurement region, a standard deviation of a distribution of the position of the non-measurement region estimated from the regression model; and
if the standard deviation for a non-measurement region exceeds a threshold, obtaining position measurement data by measuring a mark assigned to the non-measurement region, and obtaining the array from (i) the position measurement data of the sample region, and (ii) the position measurement data obtained for the non-measurement region for which the standard deviation exceeds the threshold.
2 . The method according to claim 1 , further comprising obtaining the array from the position measurement data and the position of the non-measurement region estimated from the regression model.
3 . The method according to claim 1 , wherein
the nonparametric regression model includes a Gaussian process regression model.
4 . The method according to claim 3 , wherein
a kernel function and a hyper parameter in the Gaussian process regression model are decided by machine learning.
5 . The method according to claim 4 , wherein
the machine learning includes one of a Markov Chain Monte Carlo method and a gradient method.
6 . An exposure method of exposing a substrate via an original, comprising:
obtaining an array of a plurality of regions on a substrate by using a method defined in claim 1 ; and
transferring a pattern of the original to each of the plurality of regions while positioning the substrate based on the array obtained in the obtaining.
7 . A method of manufacturing an article, comprising:
exposing a substrate using an exposure method defined in claim 6 ;
developing the exposed substrate; and
manufacturing the article from the developed substrate.
8 . An information processing apparatus executing a method defined in claim 1 .
9 . An exposure apparatus for exposing a substrate via an original, comprising:
a processing unit configured to obtain an array of a plurality of regions on a substrate; and
a stage configured to position the substrate based on the array obtained by the processing unit,
wherein the processing unit
obtains position measurement data by measuring a mark assigned to each sample region among the plurality of regions on the substrate, and
estimates a position of each non-measurement region excluding the sample region among the plurality of regions by using a regression model used to estimate the array from the position measurement data, and
determines, for each non-measurement region, a standard deviation of a distribution of the position of the non-measurement region estimated from the regression model, and
if the standard deviation for a non-measurement region exceeds a threshold, obtains position measurement data by measuring a mark assigned to the non-measurement region, and obtaining the array from (i) the position measurement data of the sample region, and (ii) the position measurement data obtained for the non-measurement region for which the standard deviation exceeds the threshold, and
the regression model is a nonparametric regression model.
10 . The apparatus according to claim 9 , wherein
the processing unit provides a user interface configured to display information regarding the position of the non-measurement region estimated from the regression model.
11 . The apparatus according to claim 10 , wherein
the nonparametric regression model includes a Gaussian process regression model, and
the information includes information indicating the position of the non-measurement region, and information indicating a standard deviation of a distribution of the position of the non-measurement region estimated from the regression model.
12 . The apparatus according to claim 11 , wherein
the processing unit identifiably displays, in the user interface, a non-measurement region having the standard deviation exceeding a threshold among the non-measurement regions.
13 . A non-transitory computer-readable storage medium storing a program configured to cause a computer to execute a method of obtaining an array of a plurality of regions on a substrate, the program causing the computer to execute
obtaining position measurement data by measuring a mark assigned to each sample region among the plurality of regions on the substrate, and
estimating a position of each non-measurement region excluding the sample region among the plurality of regions by using a regression model used to estimate the array from the position measurement data,
wherein the regression model is a nonparametric regression model
determining, for each non-measurement region, a standard deviation of a distribution of the position of the non-measurement region estimated from the regression model; and
if the standard deviation for a non-measurement region exceeds a threshold, obtaining position measurement data by measuring a mark assigned to the non-measurement region, and obtaining the array from (i) the position measurement data of the sample region, and (ii) the position measurement data obtained for the non-measurement region for which the standard deviation exceeds the threshold.
14 . An information processing apparatus, wherein
the apparatus is configured to
obtain position measurement data obtained by measuring a mark assigned to a sample region among a plurality of regions on a substrate,
estimate a position of a non-measurement region excluding the sample region among the plurality of regions by using a regression model used to estimate an array of the plurality of regions from the position measurement data,
determines, for each non-measurement region, a standard deviation of a distribution of the position of the non-measurement region estimated from the regression model, and
if the standard deviation for a non-measurement region exceeds a threshold, obtains position measurement data by measuring a mark assigned to the non-measurement region, and obtaining the array from (i) the position measurement data of the sample region, and (ii) the position measurement data obtained for the non-measurement region for which the standard deviation exceeds the threshold, and
provide a user interface configured to display information regarding the position of the non-measurement region estimated from the regression model, and
the regression model is a nonparametric regression model.
15 . The apparatus according to claim 14 , wherein
the nonparametric regression model includes a Gaussian process regression model, and
the information includes information indicating the position of the non-measurement region, and information indicating a standard deviation of a distribution of the position of the non-measurement region estimated from the regression model.
16 . The apparatus according to claim 15 , wherein
a non-measurement region having the standard deviation exceeding a threshold among the non-measurement regions is identifiably displayed in the user interface.