WO2025149331A1

WO2025149331A1 - Method of forecastig a drift in a parameter of interest in a semiconductor manaufacturing process

Info

Publication number: WO2025149331A1
Application number: PCT/EP2024/087002
Authority: WO
Inventors: Tariku Tessema REGASSA; Damian Andres CAMPO CAICEDO; Thiago DOS SANTOS GUZELLA
Original assignee: ASML Netherlands BV
Current assignee: ASML Netherlands BV
Priority date: 2024-01-08
Filing date: 2024-12-18
Publication date: 2025-07-17
Anticipated expiration: 2026-07-08

Abstract

Described is a method for predicting future evolution of a parameter of interest of a manufacturing process for manufacturing integrated circuits, the method comprising: obtaining metrology data relating to the parameter of interest; forecasting a first statistical characteristic of future values of the parameter of interest by applying a first parameter model to the metrology data and subsequently obtaining residuals data of the first parameter model; forecasting a second statistical characteristic of future values of the parameter of interest by applying a second parameter model to the residuals data of the first parameter model; and predicting the future evolution of the parameter of interest of the manufacturing process based on comparison of at least one of the first statistical characteristic and the second statistical characteristic to at least one reference value.

Description

METHOD OF FORECASTIG A DRIFT IN A PARAMETER OF INTEREST IN A SEMICONDUCTOR MANUFACTURING PROCESS

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority of EP application 24150631.0 which was filed on 8 lanuary 2024 which is incorporated herein in its entirety by reference.

FIELD

[0002] The present invention relates to semiconductor manufacturing processes, particularly to inspection or metrology methods in a semiconductor manufacturing process.

BACKGROUND

[0003] A lithographic apparatus is a machine constructed to apply a desired pattern onto a substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). A lithographic apparatus may, for example, project a pattern (also often referred to as “design layout” or “design”) at a patterning device (e.g., a mask) onto a layer of radiation-sensitive material (resist) provided on a substrate (e.g., a wafer).

[0004] To project a pattern on a substrate a lithographic apparatus may use electromagnetic radiation. The wavelength of this radiation determines the minimum size of features which can be formed on the substrate. Typical wavelengths currently in use are 365 nm (i-line), 248 nm, 193 nm and 13.5 nm. A lithographic apparatus, which uses extreme ultraviolet (EUV) radiation, having a wavelength within the range 4-20 nm, for example 6.7 nm or 13.5 nm, may be used to form smaller features on a substrate than a lithographic apparatus which uses, for example, radiation with a wavelength of 193 nm.

[0005] Low-ki lithography may be used to process features with dimensions smaller than the classical resolution limit of a lithographic apparatus. In such process, the resolution formula may be expressed as CD = ki xX/NA, where X is the wavelength of radiation employed, NA is the numerical aperture of the projection optics in the lithographic apparatus, CD is the “critical dimension” (generally the smallest feature size printed, but in this case half-pitch) and ki is an empirical resolution factor. In general, the smaller ki the more difficult it becomes to reproduce the pattern on the substrate that resembles the shape and dimensions planned by a circuit designer in order to achieve particular electrical functionality and performance. To overcome these difficulties, sophisticated fine-tuning steps may be applied to the lithographic projection apparatus and/or design layout. These include, for example, but not limited to, optimization of NA, customized illumination schemes, use of phase shifting patterning devices, various optimization of the design layout such as optical proximity correction (OPC, sometimes also referred to as “optical and process correction”) in the design layout, or other methods generally defined as “resolution enhancement techniques” (RET). Alternatively, tight control loops for controlling a stability of the lithographic apparatus may be used to improve reproduction of the pattern at low kl .

[0006] These tight control loops are generally based on metrology data obtained using a metrology tool measuring characteristics of the applied pattern or of metrology targets representing the applied pattern. In general, the metrology tool is based on optical measurement of the position and/or dimensions of the pattern and/or targets. It is intrinsically assumed that these optical measurements are representative for a quality of the process of manufacturing of the integrated circuits.

[0007] A process correction for an IC manufacturing process may be determined from metrology data of previously exposed wafers (the term wafer and substrate are used interchangeably and/or synonymously throughout this disclosure), so as to minimize any error in the metrology data of subsequently exposed wafers. However, it can be sometimes difficult to accurately predict the future performance (e.g., degradation in key performance indicators (KPIs)) of the manufacturing process based on the metrology data, thus resulting in higher defects and lower yields.

SUMMARY

[0008] It is an object of the inventors to address the mentioned disadvantage of the state of the art. [0009] In a first aspect of the invention, it is provided a method for predicting the future evolution of a parameter of interest of a manufacturing process for manufacturing integrated circuits, the method comprising: obtaining metrology data relating to the parameter of interest; forecasting a first statistical characteristic of future values of the parameter of interest by applying a first statistical model (hereinafter called as first parameter model) to the metrology data and subsequently obtaining residuals data of the first parameter model; forecasting a second statistical characteristic of future values of the parameter of interest by applying a second statistical model (hereinafter called as second parameter model) to the residuals data of the first parameter model; and predicting the future evolution of the parameter of interest (e.g., forecasting a drift in the parameter of interest) of the manufacturing process based on comparison of at least one of the first statistical characteristic and the second statistical characteristic to at least one reference (or baseline) value.

[00010] Also disclosed is a computer program and various apparatuses operable to perform the method of the first aspect.

BRIEF DESCRIPTION OF THE DRAWINGS

[00011] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which:

Figure 1 depicts a schematic overview of a lithographic apparatus;

Figure 2 depicts a schematic overview of a lithographic cell;

Figure 3 depicts a schematic representation of holistic lithography, representing cooperation between three key technologies to optimize semiconductor manufacturing; and Figures 4(a) and 4(b) illustrate schematically two known feedback control methods applied in a manufacturing facility.

Figure 5 is a flow diagram of a method for predicting the future evolution of a parameter of interest involved in the manufacturing of integrated circuits, in accordance with an embodiment;

Figure 6A is a plot of first time-series metrology data to which a first parameter model is applied, wherein the solid line in the figure represents the model estimation of a first statistical characteristic; Figure 6B is a plot of first residuals data obtained after applying the first parameter model on the first time-series metrology data, wherein the solid line in the figure represents the model estimation of a second statistical characteristic;

Figure 6C is a plot of second time-series metrology data to which the first parameter model is applied, wherein the solid line in the figure represents the model estimation of the first statistical characteristic; Figure 6D a plot of second residuals data obtained after applying the second parameter model on the second time-series metrology data, wherein the solid line in the figure represents the model estimation of the second statistical characteristic;

Figure 7A is a plot of reference metrology data to which the first parameter model is applied, wherein the solid line in the figure represents the model estimation of baseline (or reference) values of the first statistical characteristic; and

Figure 7B is a plot of residuals data obtained after applying the first parameter model on the reference metrology data, wherein the solid line in the figure represents the model estimation of baseline (or reference) values of the second statistical characteristic.

DETAILED DESCRIPTION

[00012] In the present document, the terms “radiation” and “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 run) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range of about 5-100 nm).

[00013] The term “reticle”, “mask” or “patterning device” as employed in this text may be broadly interpreted as referring to a generic patterning device that can be used to endow an incoming radiation beam with a patterned cross-section, corresponding to a pattern that is to be created in a target portion of the substrate; the term “light valve” can also be used in this context. Besides the classic mask (transmissive or reflective; binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include:

-a programmable mirror array. More information on such mirror arrays is given in U. S. patents US5296891A and US5523193A, which are incorporated herein by reference.

-a programmable LCD array. An example of such a construction is given in U. S. patent US5229872A, which is incorporated herein by reference. [00014] Figure 1 schematically depicts a lithographic apparatus LA. The lithographic apparatus LA includes an illumination system (also referred to as illuminator) IL configured to condition a radiation beam B (e.g., UV radiation, DUV radiation or EUV radiation), a support structure (e.g., a mask table) MT constructed to support a patterning device (e.g., a mask) MA and connected to a first positioner PM configured to accurately position the patterning device MA in accordance with certain parameters, a substrate table (e.g., a wafer table) WT constructed to hold a substrate (e.g., a resist coated wafer) W and connected to a second positioner PW configured to accurately position the substrate in accordance with certain parameters, and a projection system (e.g., a refractive projection lens system) PS configured to project a pattern imparted to the radiation beam B by patterning device MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.

[00015] In operation, the illuminator IL receives a radiation beam from a radiation source SO, e.g. via a beam delivery system BD. The illumination system IL may include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic, or other types of optical components, or any combination thereof, for directing, shaping, or controlling radiation. The illuminator IL may be used to condition the radiation beam B to have a desired spatial and angular intensity distribution in its cross section at a plane of the patterning device MA.

[00016] The term “projection system” PS used herein should be broadly interpreted as encompassing various types of projection system, including refractive, reflective, catadioptric, anamorphic, magnetic, electromagnetic and electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, or for other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term “projection lens” herein may be considered as synonymous with the more general term “projection system” PS.

[00017] The lithographic apparatus may be of a type wherein at least a portion of the substrate may be covered by a liquid having a relatively high refractive index, e.g., water, so as to fill a space between the projection system and the substrate - which is also referred to as immersion lithography. More information on immersion techniques is given in U. S. patent US6952253B2, which is incorporated herein by reference.

[00018] The lithographic apparatus LA may also be of a type having two (dual stage) or more substrate tables WT and, for example, two or more support structure MT (not shown). In such “multiple stage” machines the additional tables / structures may be used in parallel, or preparatory steps may be carried out on one or more tables while one or more other tables are being used for exposing the design layout of the patterning device MA onto the substrate W.

[00019] In operation, the radiation beam B is incident on the patterning device (e.g., mask MA), which is held on the support structure (e.g., mask table MT), and is patterned by the patterning device MA. Having traversed the mask MA, the radiation beam B passes through the projection system PS, which focuses the beam onto a target portion C of the substrate W. With the aid of the second positioner PW and position sensor IF (e.g., an interferometric device, linear encoder, 2-D encoder or capacitive sensor), the substrate table WT can be moved accurately, e.g., so as to position different target portions C in the path of the radiation beam B. Similarly, the first positioner PM and possibly another position sensor (which is not explicitly depicted in Figure 1) may be used to accurately position the mask MA with respect to the path of the radiation beam B. Mask MA and substrate W may be aligned using mask alignment marks Ml, M2 and substrate alignment marks Pl, P2. Although the substrate alignment marks as illustrated occupy dedicated target portions, they may be located in spaces between target portions (these are known as scribe-lane alignment marks).

[00020] As shown in Figure 2 the lithographic apparatus LA may form part of a lithographic cell LC, also sometimes referred to as a lithocell or (litho)cluster, which often also includes apparatus to perform pre- and post-exposure processes on a substrate W. Conventionally these include spin coaters SC to deposit resist layers, developers DE to develop exposed resist, chill plates CH and bake plates BK, e.g. for conditioning the temperature of substrates W e.g. for conditioning solvents in the resist layers. A substrate handler, or robot, RO picks up substrates W from input/output ports I/O 1 , 1/O2, moves them between the different process apparatus and delivers the substrates W to the loading bay LB of the lithographic apparatus LA. The devices in the lithocell, which are often also collectively referred to as the track, are typically under the control of a track control unit TCU that in itself may be controlled by a supervisory control system SCS, which may also control the lithographic apparatus LA, e.g. via lithography control unit LACU.

[00021] In order for the substrates W exposed by the lithographic apparatus LA to be exposed correctly and consistently, it is desirable to inspect substrates to measure properties of patterned structures, such as overlay errors between subsequent layers, line thicknesses, critical dimensions (CD), etc. For this purpose, inspection tools (not shown) may be included in the lithocell LC. If errors are detected, adjustments, for example, may be made to exposures of subsequent substrates or to other processing steps that are to be performed on the substrates W, especially if the inspection is done before other substrates W of the same batch or lot are still to be exposed or processed.

[00022] An inspection apparatus, which may also be referred to as a metrology apparatus, is used to determine properties of the substrates W, and in particular, how properties of different substrates W vary or how properties associated with different layers of the same substrate W vary from layer to layer. The inspection apparatus may alternatively be constructed to identify defects on the substrate W and may, for example, be part of the lithocell LC, or may be integrated into the lithographic apparatus LA, or may even be a stand-alone device. The inspection apparatus may measure the properties on a latent image (image in a resist layer after the exposure), or on a semi-latent image (image in a resist layer after a post-exposure bake step PEB), or on a developed resist image (in which the exposed or unexposed parts of the resist have been removed), or even on an etched image (after a pattern transfer step such as etching).

[00023] Typically the patterning process in a lithographic apparatus LA is one of the most critical steps in the processing which requires high accuracy of dimensioning and placement of structures on the substrate W. To ensure this high accuracy, three systems may be combined in a so called “holistic” control environment as schematically depicted in Figure 3. One of these systems is the lithographic apparatus LA which is (virtually) connected to a metrology tool MT (a second system) and to a computer system CL (a third system). The key of such “holistic” environment is to optimize the cooperation between these three systems to enhance the overall process window and provide tight control loops to ensure that the patterning performed by the lithographic apparatus LA stays within a process window. The process window defines a range of process parameters (e.g. dose, focus, overlay) within which a specific manufacturing process yields a defined result (e.g. a functional semiconductor device) - typically within which the process parameters in the lithographic process or patterning process are allowed to vary.

[00024] The computer system CL may use (part of) the design layout to be patterned to predict which resolution enhancement techniques to use and to perform computational lithography simulations and calculations to determine which mask layout and lithographic apparatus settings achieve the largest overall process window of the patterning process (depicted in Figure 3 by the double white arrow in the first scale SCI). Typically, the resolution enhancement techniques are arranged to match the patterning possibilities of the lithographic apparatus LA. The computer system CL may also be used to detect where within the process window the lithographic apparatus LA is currently operating (e.g. using input from the metrology tool MT), so as to predict whether defects may be present due to e.g. sub-optimal processing (depicted in Figure 3 by the arrow pointing “0” in the second scale SC2). [00025] The metrology tool MT may provide input to the computer system CL to enable accurate simulations and predictions, and may provide feedback to the lithographic apparatus LA to identify possible drifts, e.g. in a calibration status of the lithographic apparatus LA (depicted in Figure 3 by the multiple arrows in the third scale SC3).

[00026] The lithographic apparatus LA is configured to accurately reproduce the pattern onto the substrate. The positions and dimensions of the applied features need to be within certain tolerances. Position errors may occur due to an overlay error (often referred to as “overlay”). The overlay is the error in placing a first feature during a first exposure relative to a second feature during a second exposure. The lithographic apparatus minimizes the overlay errors by aligning each wafer accurately to a reference prior to patterning. This is done by measuring positions of alignment marks on the substrate using an alignment sensor. More information on the alignment procedure can be found in U.S. patent application publication US20100214550A1, which is incorporated herein by reference. Pattern dimensioning (e.g., CD) errors may, for example, occur when the substrate is not positioned correctly with respect to a focal plane of the lithographic apparatus. These focal position errors may be associated with un-flatness of a substrate surface. The lithographic apparatus minimizes these focal positon errors by measuring the substrate surface topography prior to patterning using a level sensor. Substrate height corrections are applied during subsequent patterning to assure correct imaging (focusing) of the patterning device onto the substrate. More information on the level sensor system can be found in U.S. patent application publication US20070085991A1, which is incorporated herein by reference.

[00027] Besides the lithographic apparatus LA and the metrology apparatus MT other processing apparatus may be used during IC production as well. An etching station (not shown) processes the substrates after exposure of the pattern into the resist. The etch station transfers the pattern from the resist into one or more layers underlying the resist layer. Typically etching is based on application of a plasma medium. Local etching characteristics may, for example, be controlled using temperature control of the substrate or directing the plasma medium using a voltage controlled ring. More information on etching control can be found in international patent application publication WO2011081645A2 and U.S. patent application publication US20060016561A1 which are incorporated herein by reference.

[00028] During the manufacturing of the ICs, it is of great importance that the process conditions for processing substrates using processing apparatus, such as the lithographic apparatus or etching station, remain stable such that properties of the features remain within certain control limits. Stability of the process is of particular importance for features of the functional parts of the IC, i.e., the product features. To guarantee stable processing, process control capabilities need to be in place. Process control involves monitoring of processing data and implementation of means for process correction, e.g. control the processing apparatus based on characteristics of the processing data. Process control may be based on periodic measurement by the metrology apparatus MT, often referred to as “Advanced Process Control” (further also referenced to as APC). More information on APC can be found in U.S. patent application publication US20120008127A1, which is incorporated herein by reference. A typical APC implementation involves periodic measurements on metrology features on the substrates to monitor and correct drifts associated with one or more processing apparatus. The metrology features reflect the response to process variations of the product features.

[00029] In US20120008127A1, a lithographic apparatus is calibrated by reference to a primary reference substrate. Using an apparatus which need not be the same as the one being calibrated, there is obtained an apparatus-specific fingerprint of the primary reference substrate. Using the same set-up there is then obtained an apparatus-specific fingerprint of a secondary reference substrate. The apparatus-specific fingerprint of the primary reference substrate is subtracted from the apparatus specific fingerprint of the secondary reference substrate to obtain and store an apparatus -independent fingerprint of the secondary reference substrate. The secondary reference substrate and stored apparatus-independent fingerprint are subsequently used together in place of the primary reference substrate as a reference for the calibration of the lithographic apparatus to be calibrated. Initial set-up for a cluster of lithographic tools can be performed with less use of the costly primary reference substrate, and with less interruption to normal production. The initial set-up can be integrated with ongoing monitoring and re -calibration of the apparatuses. [00030] The term fingerprint may refer to a main (systematic) contributor (“latent factor”) of a measured signal, and in particular a contributor connected to the performance impact on-wafer or to previous processing steps. Such a fingerprint can refer to substrate (grid) patterns (e.g. from alignment, leveling, overlay, focus, CD), field patterns (e.g., from intrafield alignment, leveling, overlay, focus, CD), substrate zone patterns (e.g., outermost radius of wafer measurements) or even patterns in scanner measurements related to wafer exposure (e.g., heating signature through-lot from reticle alignment measurements, temperature/pressure/servo profiles, etc.). Fingerprints may be comprised within a fingerprint collection, and may be encoded homogenously or heterogeneously therein.

[00031] As such, APC identifies correctable variation in a performance parameter such as overlay, and applies one set of corrections to a lot (batch) of wafers. In determining these corrections, corrections from previous lots are taken into account in order to avoid overcorrecting the noise in the measurements. For adequate smoothing of current corrections with previous ones, the history of corrections taken into account may match the context of the current lot. “Context” in this regard encompasses any parameters that identify variants arising within the same overall industrial process. The layer ID, layer type, product ID, product type, reticle ID and so forth are all context parameters that may lead to different fingerprints in the finished performance. In addition to the individual scanners that may be used in a high-volume manufacturing (HVM) facility, the individual tools used for each of the coating, etching, and other steps involved in semiconductor manufacturing can also vary from lot to lot or wafer to wafer. Each of these tools can impose a particular error “fingerprint” on the products. Outside the field of semiconductor manufacturing, similar situations may arise in any industrial process.

[00032] To ensure accurate feedback control appropriate to the particular context, different lots (batches) of product units can be treated as separate “threads” in the APC algorithms. Context data can be used to assign each product unit to the correct thread. In the case of a manufacturing plant producing usually high volumes of only a few types of products by the same process steps, the number of different contexts may be relatively small, and the number of product units in each thread will be ample to allow smoothing of noise. All of the lots having a common context can be assigned to their own thread to optimize the feedback correction and the ultimate performance. In case of a foundry producing many different types of product in very small production runs, the context may change more often, and the number of lots having exactly the same context data may be quite small. Using only the context data to assign lots to different APC ‘threads’ may then result in a large number of threads, with a small number of lots per thread. Complexity of the feedback control increases, and the ability to improve performance for low-volume products is reduced. Combing different lots into the same threads without sufficient regard to their different contexts will cause loss of performance also. [00033] Figure 4 (a) illustrates schematically the operation of one type of control method implemented by APC system 250. Historic performance data PDAT is received from the storage 252, having been obtained by metrology apparatus 240 or other means from wafers 220 that have been processed by lithographic apparatus 200 and associated apparatuses of the lithocell. A feedback controller 300 analyzes the performance parameters represented in the performance data for recent lots, and calculates process corrections PC which are fed to the lithographic apparatus 200. These process corrections are added to the wafer-specific corrections derived from the alignment sensors and other sensors of the lithographic apparatus, to obtain a combined correction for the processing of each new lot.

[00034] Figure 4 (b) illustrates schematically the operation of another type of control method implemented by a known APC system 250. As can be seen, the general form of the feedback control method is the same as that shown in Figure 4 (a), but in this example, context data relating to the historic wafers and context data CTX relating to the current wafer is used to provide a more selective use of the performance data PDAT. Specifically, while in the earlier example the performance data for all historic wafers was combined in a single stream 302, and the modified method, context data from storage 256 used to assign the performance data for each historic lot to one of several threads 304. These threads are processed effectively in parallel feedback loops by feedback controller 300, yielding multiple process corrections 306, each based on the historic performance data of wafers in one of the threads 304. Then, when new lots are received for processing, their individual context data CTX can be used to select which of the threads provides the appropriate context data 306 for the current wafer. [00035] There are a number of alternative methods for temporal processing and/or filtering of (feedback) control data (e.g. overlay fingerprints or EPE fingerprints). Such methods include using a low pass temporal filtering method or moving average processing method; e.g., a weighted moving average or Exponentially Weighted Moving Average EWMA. Other methods include a machine learning model such as a Neural Network (NN). For example, an advanced NN filtering method may be trained to learn the appropriate response to temporal behavior based on historic control parameter data and provide a (feedback) correction prediction for the next lot, e.g. in an APC control loop. [00036] A disadvantage of these existing methods is that the temporal processor (e.g., NN or EWMA filter) “learns” based only on the behavior of the control parameters, as can be determined from metrology data of exposed structures. Measured parameter values (e.g., overlay, edge placement error, critical dimension, focus) in the metrology data vary with changes in behavior overtime of these control parameters (i.e., a different output is measured for the same control input). Control parameters may be any input parameters of the scanner or other tool used in IC manufacture (e.g., etch chamber, deposition chamber, bonding tool etc.) which control the manufacturing process (e.g., exposure process, etch process, deposition process, bonding process etc.). Therefore, control of the output process, more specifically the formation, configuration and/or positioning of the exposed and/or etched structures, is dependent on these control parameters. As such, it is these control parameters which may be controlled to correct or compensate for any measured error in the metrology data, either in a feedback loop to correct future wafers/lots or as a feedforward correction to correct a present wafer/lot. Note than “learning” in this context includes averaging in a moving average example, as the moving average output effectively “learns” (in a loose sense) to respond to inputted data by averaging the data (its output response changes overtime based on the previous few inputs).

[00037] The APC control loop described above has a primary task of monitoring drift in metrology data indicative of drift of the control parameter behavior and determining appropriate corrections for the control parameters to address this drift and maintain the measured metrology parameter values within specification (i.e., within a certain acceptable tolerance or “process window”) within which the IC device can be expected to function with good probability.

[00038] As one type of APC methods, statistical process control (SPC) is currently being used to monitor on-product performance (OPP). SPC is a quality control method that monitors the OPP of a process and helps to ensure that the process operates efficiently and within the expected range of variation. SPC can also detect changes in the OPP over time and indicates when it drifts from a normal pattern. SPC is limited in its ability to identify drifts as it only detects a drift when it has occurred in historical data and cannot predict or forecast if the drift will continue into the future. Furthermore, gradual or subtle deviations (or drifts) from the normal pattern are less likely to be detected by SPC, e.g., because the gradual or subtle deviations may still fall within the specified tolerance range of the SPC model. As such, SPC cannot detect drifts in metrology data in a reliable and timely manner. This disadvantage leads to a delay in detecting a potential drift, which is highly unwanted in the semiconductor industry. This issue has been identified in other industries, which has led to works that focus on acknowledging the autocorrelation of OPP signals.

[00039] In the semiconductor industry, the mean and variance of a representative performance parameter (or KPI) are two commonly used statistical parameters employed to characterize a given process. A drift in said mean and/or variance can indicate a decline in performance, potentially caused by various factors in the process; and SPC can proactively detect anomalies in these statistical parameters based on historical data. However, since current implementations of SPC cannot reliably and timely identify and address the underlying causes of a drift, performance degradation can result in lost production and increased costs.

[00040] It is thus an objective of the present disclosure to propose a method that is capable of predicting and notifying potential drifts in at least one parameter of interest before such drifts actually occur by using statistical models. Accordingly, it is provided a method for predicting the future evolution (or performance) of a parameter of interest of a manufacturing process for manufacturing integrated circuits. The proposed method not only can estimate historical performance data in relation to the parameter of interest, but also can forecast or predict the future evolution of such a parameter.

[00041] In an embodiment, the proposed method may be in the form of a computer program, which may be stored in a non-transient computer program carrier. In an embodiment, the computer program carrier may be comprised in a processing system which may be operable to execute the computer program so as to perform the proposed method.

[00042] With reference to Figure 5, in an embodiment, the proposed method may comprise the following four main steps.

[00043] Step 510: obtaining metrology data relating to the parameter of interest. The parameter of interest may be any key performance indicator (KPI) associated with the manufacturing process, such as wafer alignment readout data, wafer alignment quality parameter(s), Critical Dimension (CD) data, Edge Placement Error (EPE) data, wafer overlay, exposure focus, etc.

[00044] In an embodiment, the proposed method may further comprise routinely measuring substrates to obtain said metrology data. Substrates may be measured on a metrology device, such as a scatterometry based metrology device (e.g., MT as shown in Figure 3) which may be part of a lithographic cell (e.g., LC as shown in Figure 2).

[00045] In an embodiment, the metrology data may comprise time -series metrology data obtained in a previous period of time. Here, the term “previous” refers to any event occurring or data collected before the application of the method 500. The previous period of time may end on the day before or on the same day when the method 500 is applied to predict the future evolution of the parameter of interest of the manufacturing process.

[00046] Step 520: forecasting a first statistical characteristic of future values of the parameter of interest by applying a first parameter model to the metrology data and subsequently obtaining residuals data of the first parameter model. The metrology data may comprise a plurality of historical (or previously measured) values of the parameter of interest.

[00047] In an embodiment, the first statistical characteristic may be a forecasted mean of future values of the parameter of interest in a specified future period. The future period may start from the time immediately following the application of the method 500 for predicting the future evolution of the parameter of interest of the manufacturing process. In some cases, the gap between future and historical data may be treated as a missing observation. In other cases, the model may be informed that there is a gap. The duration of the gap may depend on the type of the model and the nature and size of data. It may be that the narrower the gap the better the forecasting performance of the model.

[00048] In an embodiment, the first parameter model may comprise a time-series model configured for analyzing a sequence of data points collected over time (i.e. time series data) and predicting or forecasting events based on the collected data points. The first parameter model may comprise for example one of, inter alia-, a state space model (e.g., a local level model), a time varying regression model (e.g., autoregressive integrated moving average (ARIMA)), or a long short-term memory (LSTM) network. A state space model is a general model that provides a mathematical representation of time-series data by using observed and hidden states related by differential or stochastic equations. More details about state space modelling can be found in the book by Durbin, J., & Koopman, S.J., “Time Series Analysis by State Space Methods ” , published by Oxford University Press on 3 May 2012, which is incorporated herein by reference. Depending on the nature of the metrology data and/or the type of the parameter of interest, other suitable statistical models may be used as the first parameter model.

[00049] In an example implementation, the first parameter model is a local level model which is a special case of a state space model. The local level model models the hidden stochastic trend and irregular components of time-series data and can be expressed by: where £_t is the measurements error at time t, 8_t is the process noise accounting for unobserved factors affecting local level transition from time tto time t + 1, a_t (or a_t+1) denote unobserved components of the model (also called hidden states) at time t (or t + 1) which encode the dynamical processes the system (e.g., forecasted values of the parameter), y_t denote the observed performance metrics (e.g., measured values of the parameter)!:, af and erf are the model’s unknown parameters that are required to be estimated. The unobserved components and unknown parameters in the model can be estimated for example by using a Kalman filter and maximum likelihood estimation, respectively. The Kalman filter algorithm can be used to predict future values of the unobserved components with a high degree of precision. A well-trained local level model (e.g., the model trained with sufficient metrology data) can provide accurate predictions, which can help determine the likelihood of observing a drift in the future. Further details about the local level model and Kalman filter can be found in the presentation by Koopman, S.J., “Local Level Model”, CREATES, Aarhus University, October 2016, which is incorprated herein by reference.

[00050] In an embodiment, the first parameter model (e.g., Eq.(l) and Eq.(2)) may be fitted to the metrology data to forecast a plurality of mean values a_t of future values of the parameter of interest at a plurality of time instances t. Each of the plurality of mean values a_t is forecasted for corresponding future values of the parameter of interest at a respective one of the plurality of time instances t. Subsequently, the future mean Mf of future values of the parameter of interest may be determined by summing the plurality of forecasted mean values a_t and dividing the sum by the total number n of the plurality of forecasted mean values a_t, as expressed by:

Eq. (3)

[00049] Step 530: forecasting a second statistical characteristic of future values of the parameter of interest by applying a second parameter model to the residuals data of the first parameter model. [00050] In an embodiment, the second statistical characteristic may be a forecasted variance of future values of the parameter of interest in a specified future period. The future period may start from the time immediately following the application of the method 500 for predicting the future evolution of the parameter of interest of the manufacturing process.

[00051] The second parameter model may comprise for example an auto-regressive conditional heteroskedasticity (ARCH) model or a generalized auto-regressive conditional heteroskedasticity (GARCH) model. Depending on the nature of the metrology data and/or the type of the parameter of interest, other suitable statistical models may be used as the second parameter model.

[00052] Referred back to the above example implementation where the first parameter model is a local level model, the second parameter model may be a GARCH model which is used to model conditional changes in variance of the parameter of interest over time. The GARCH model models the variance as a function of both past samples and conditional variances, thus allowing for persistent and complex variability patterns to be modeled. Similar to the local level model, the GARCH model can be represented in the form of a state space model. Specifically, the simplified version of the GARCH(1,1) model can be expressed by: where £_t is the residual value at a time instance t obtained from the local level model as expressed by is the residual value of the GARCH(1,1) model at the time instance t, o_t denotes the estimated standard deviation of the residual at the time instance t (cr² denotes the corresponding estimated variance of the residual at the time instance t). The and y are the model’s unknown parameters that are required to be estimated. The unknown parameters in the GARCH(1,1) model can be estimated for example either by quasi-maximum likelihood estimation or an extended Kalman filter.

[00053] As shown in Eq.[4] and Eq.[5], the second parameter model (in this case the GARCH model) may be fitted to the residuals data of the first parameter model (in this case the local level model) to account for drifts in variance. In the case where the manufacturing process has a stable performance (i.e. no drifts in the parameter of interest), the residuals data of the local level model are assumed to have a zero mean and a constant variance. Whereas, in the case where the manufacturing process consists of a drift, the variance of the residuals data from the local level model changes conditionally over time. If the manufacturing process drifts only in the mean of historical values the parameter of interest, the variance of the residuals data is expected to be constant. Whereas, if the process drifts only in the variance or in both the variance and the mean, the variance of the residuals data from the local level model is expected to be non-constant and the GARCH model can capture this variability.

[00054] Figures 6A and 6C respectively show a plot of first time-series metrology data y_t and a plot of second time-series metrology data to which the first parameter model is applied. Figures 6B and 6D respectively show a plot of first residuals data £_t after applying the first parameter model to the first time-series metrology data and a plot of second residuals data £_t' after applying the second parameter model to the second time-series metrology data. Note that the first time-series metrology data and the second time-series metrology data are not real measurement data but virtual data generated for demonstration of the concept of the method.

[00055] With reference to Figure 6A, after applying to the first time-series metrology data y_t at a time instance of t₀, the first parameter model not only estimates the first past mean a_t-p (the mean for the period before the time instance of t₀) of the metrology data but also forecasts the first future mean a_t-f (the mean for the period after the time instance of t₀) of the metrology data. The resultant first residuals data £_t of the first parameter model is shown in Figure 6B. With reference to Figure 6B, after applying to the first residuals data £_t, the second parameter model not only estimates the first past variance c_t ²-p (the variance for the period before the time instance of t₀) of the metrology data but also predicts the first future variance Of -f (the variance for the period after the time instance of t₀) of the metrology data. In this example case, the method 500 predicts that the future mean a_t-f will continue to drift but the future variance o'²- will remain substantially stable or constant.

[00056] With reference to Figure 6C, after applying to the second time-series metrology data y_t' at a time instance of t₀. the first parameter model not only estimates the second past mean a_t-p^r (the mean for the period before the time instance of t₀) of the metrology data but also predicts the second future mean a_t-f^r (the mean for the period after the time instance of t₀) of the metrology data. The resultant second residuals data £_t ^r of the first parameter model is shown in Figure 6D. With reference to Figure 6D, after applying to the second residuals data £_t ^r, the second parameter model not only estimates the second past variance Ot -p' (the variance for the period before the time instance of t₀) of the metrology data but also predicts the second future variance Ot -f' (the variance for the period after the time instance of t₀) of the metrology data. In this example case, the method 500 predicts that the future mean a_t-f^r will remain substantially stable or constant, but the future variance c -T will continue to drift in the future.

[00057] In an embodiment, the second parameter model (e.g., Eq.(4) and Eq.(5)) may be fitted to the residuals data £_t of the first parameter model (i .e . the remaining values after applying the first parameter model to the metrology data) to forecast a plurality of variance values o'² of the parameter of interest at the plurality of time instances t . Each of the plurality of variance values o'² is forecasted for corresponding future values of the parameter of interest at a respective one of the plurality of time instances t . Subsequently, the future variance V of future values of the parameter of interest is determined by summing the plurality of forecasted variance values <7_t ² and dividing the sum by the total number n of the plurality of forecasted variance values a². as expressed by:

17 = ^. Eq. (6)

[00058] Step 540: predicting the future evolution (or variations) of the parameter of interest of the manufacturing process based on comparison of at least one of the first statistical characteristic and the second statistical characteristic to at least one reference value. In particular, this prediction may be based on comparison of each of the first statistical characteristic and the second statistical characteristic to a respective reference value; e.g., to assess whether at least one of the first statistical characteristic and the second statistical characteristic differs from its respective reference value by a respective threshold value.

[00059] In an embodiment, the at least one reference value may be user defined e.g., based on previous experience in carrying out the manufacturing process. The at least one reference value directly input to the model.

[00060] In an alternative embodiment, the at least one reference value may be determined by applying the first parameter model and the second parameter model to reference metrology data relating to the parameter of interest, said reference metrology data comprising historical (or previously measured) values of the parameter of interest which represent a stable performance of the parameter of interest (e.g., no drifts in the parameter of interest) in the manufacturing process. The reference metrology data may be generated in a previous period of time.

[00061] In an embodiment, the at least one reference value may comprise a reference mean of historical values of the parameter of interest and a reference variance of historical values of the parameter of interest, each determined based on the reference metrology data.

[00062] In an embodiment, the determining of the reference mean of historical (or previously measured) values of the parameter of interest may comprise: determining a plurality of mean values of historical values of the parameter of interest at a plurality of time instances by applying the first parameter model to the reference metrology data and subsequently obtaining residuals data of the first parameter model, each of the plurality of mean values being determined for corresponding historical values of the parameter of interest at a respective one of the plurality of time instances; and determining the reference mean of historical values of the parameter of interest by summing the plurality of determined mean values and dividing the sum by the total number of the plurality of determined mean values.

[00063] In an embodiment, the determining of the reference variance of the parameter of interest may comprise: determining a plurality of variance values of the parameter of interest at the plurality of time instances by applying the second parameter model to the residuals data of the first parameter model, each of the plurality of variance values being determined for corresponding historical values of the parameter of interest at a respective one of the plurality of time instances; and determining the reference variance of historical values of the parameter of interest by summing the square of the plurality of determined variance values and dividing the sum by the total number of the plurality of determined variance values.

[00064] In an example implementation, the reference mean and reference variance may be determined using historic metrology data with stable measurements of the parameter of interest (e.g., historical overlay data that is known to be stable during a certain manufacturing process). This historic metrology data is assumed to represent the “normal” or “accepted” performance of the manufacturing process; i.e., the historic metrology data may be representative or indicative of normal or baseline performance of the parameter of interest. To estimate the reference mean and reference variance, a two-step statistical modeling approach may be used.

[00065] With reference Figure 7A, in the first step, the first parameter model (e.g., Eq.(l) and Eq. (2)) is fitted to the reference metrology data y_t to determine a plurality of mean values a_t (not shown) of historical (or previously measured) values of the parameter of interest at a plurality of time instances t. Each of the plurality of mean values a_t is determined for corresponding historical values of the parameter of interest at a respective one of the plurality of time instances t. Subsequently, the reference mean M_b of historical values of the parameter of interest is determined by summing the plurality of determined mean values a_t and dividing the sum by the total number n of the plurality of determined mean values a_t, as expressed by:

M_b = ^. Eq. (7)

[00066] With reference to Figure 7B, in the second step, the second parameter model (e.g., Eq. (4) and Eq. (5)) is fitted to the residuals data £_t of the first parameter model (i.e. the remaining values after applying the first parameter model to the reference metrology data) to determine a plurality of variance values o'² (not shown) of historical values of the parameter of interest at the plurality of time instances t. Each of the plurality of variance values o'² is determined for corresponding historical values of the parameter of interest at a respective one of the plurality of time instances t. Subsequently, the reference variance V_b of historical values of the parameter of interest is determined by summing the plurality of variance values o'² and dividing the sum by the total number n of the plurality of variance values o'², as expressed by:

F_b = ^. Eq. (8) [00067] In an embodiment, the predicting of the future evolution of the parameter of interest of the manufacturing process may comprise: determining a first difference between the forecasted mean of future values of the parameter of interest in the specified future period and the determined reference mean; determining a second difference between the forecasted variance of future values of the parameter of interest in the specified future period and the determined reference variance; comparing the first difference to a first threshold; comparing the second difference to a second threshold; and predicting a future drift in the parameter of interest the manufacturing process if one or both of the first difference and the second difference exceed the first threshold and the second threshold, respectively. [00068] In an example implementation, the first difference (i.e. the difference between the predicted mean and the reference mean) and the second difference (i.e. the difference between the predicted variance and the reference variance) may be expressed as: where threshold_mean is the first threshold and threshold_var is the second threshold. In an embodiment, the first threshold and the second threshold may be user defined.

[00069] In an embodiment, the method 500 may further comprise determining a correction for the manufacturing process from said prediction of the future evolution of the parameter of interest. The determining of a correction may comprise determining a correction for a setting of an apparatus used in said manufacturing process.

[00070] In an embodiment, the method 500 may further comprise manufacturing further integrated circuits using said correction.

[00071] Further embodiments of the invention are disclosed in the list of numbered clauses below:

1. A method for predicting the future evolution of a parameter of interest of a manufacturing process for manufacturing integrated circuits, the method comprising: obtaining metrology data relating to the parameter of interest; forecasting a first statistical characteristic of future values of the parameter of interest by applying a first parameter model to the metrology data and subsequently obtaining residuals data of the first parameter model; forecasting a second statistical characteristic of future values of the parameter of interest by applying a second parameter model to the residuals data of the first parameter model; and predicting the future evolution of the parameter of interest of the manufacturing process based on comparison of at least one of the first statistical characteristic and the second statistical characteristic to at least one reference value.

2. A method according to clause 1, wherein the metrology data comprises time-series metrology data obtained in a previous period of time. 3. A method according to clause 2, wherein the previous period of time ends on a day immediately preceding the day on which the first parameter model and the second parameter model are applied to predict the future evolution of the parameter of interest.

4. A method according to any preceding clause, wherein the first parameter model comprises a time-series model configured for analyzing a sequence of data points collected over time and forecasting events based on the collected data points.

5. A method according to any preceding clause, wherein the first parameter model comprises one of: a state space model, a time varying regression model, or a long short-term memory (LSTM) network.

6. A method according to any preceding clause, wherein the first parameter model comprises an autoregressive integrated moving average (ARIMA) model.

7. A method according to any preceding clause, wherein the first parameter model comprises a local level model.

8. A method according to any preceding clause, wherein the forecasting of the first statistical characteristic of future values of the parameter of interest comprises estimating one or more unobserved components and unknown parameters of the first parameter model using a Kalman filter and a maximum likelihood estimation method, respectively.

9. A method according to any preceding clause, wherein the second parameter model comprises an auto-regressive conditional heteroskedasticity (ARCH) model.

10. A method according to any of clauses 1 to 8, wherein the second parameter model comprises a generalized auto-regressive conditional heteroskedasticity (GARCH) model.

11. A method according to any preceding clause, wherein the forecasting of the second statistical characteristic of future values of the parameter of interest comprises estimating unknown parameters of the second parameter model using a quasi -maximum likelihood estimation method or an extended Kalman filter.

12. A method according to any preceding clause, further comprising applying the first parameter model and the second parameter model to reference metrology data relating to the parameter of interest to determine the at least one reference value, said reference metrology data comprising historical values of the parameter of interest which represent a stable performance of the parameter of interest in the manufacturing process.

13. A method according to clause 12, further comprising simulating at least one aspect of a lithographic process to obtain synthetic metrology data, said reference metrology data comprising at least in part said synthetic metrology data.

14. A method according to any preceding clause, wherein the first statistical characteristic is a forecasted mean of future values of the parameter of interest in a specified future period.

15. A method according to any preceding clause, wherein the second statistical characteristic is a forecasted variance of future values of the parameter of interest in a specified future period. 16. A method according to clause 14 or 15, wherein the forecasting of the first statistical characteristic of future values of the parameter of interest comprises: forecasting a plurality of mean values of future values of the parameter of interest at a plurality of time instances by applying the first parameter model to the metrology data and subsequently obtaining residuals data of the first parameter model, each of the plurality of mean values being forecasted for corresponding future values of the parameter of interest at a respective one of the plurality of time instances; and determining the forecasted mean of future values of the parameter of interest from said plurality of forecasted mean values.

17. A method according to clause 16, wherein said determining the forecasted mean of future values of the parameter of interest from said plurality of forecasted mean values comprises summing the plurality of forecasted mean values and dividing the sum by the total number of the plurality of forecasted mean values.

18. A method according to clause 16 or 17, wherein the forecasting of the second statistical characteristic of future values of the parameter of interest comprises: forecasting a plurality of variance values of future values of the parameter of interest at the plurality of time instances by applying the second parameter model to the residuals data of the first parameter model, each of the plurality of variance values being forecasted for corresponding future values of the parameter of interest at a respective one of the plurality of time instances; and determining the forecasted variance of future values of the parameter of interest from said plurality of forecasted variance values.

19. A method according to clause 18, wherein said determining the forecasted variance of future values of the parameter of interest from said plurality of forecasted variance values comprises determining the forecasted variance of future values of the parameter of interest by summing the plurality of forecasted variance values and dividing the sum by the total number of the plurality of forecasted variance values.

20. A method according to any of clauses 14 to 19, wherein the at least one reference value comprises a reference mean of historical values of the parameter of interest and a reference variance of historical values of the parameter of interest, each determined based on the reference metrology data.

21. A method according to clause 20, wherein the determining of the reference mean of historical values of the parameter of interest comprises: determining a plurality of mean values of historical values of the parameter of interest at a plurality of time instances by applying the first parameter model to the reference metrology data and subsequently obtaining residuals data of the first parameter model, each of the plurality of mean values being determined for corresponding historical values of the parameter of interest at a respective one of the plurality of time instances; and determining the reference mean of historical values of the parameter of interest from said plurality of determined mean values. 22. A method according to clause 21, wherein said determining the reference mean of historical values of the parameter of interest from said plurality of determined mean values comprises summing the plurality of determined mean values and dividing the sum by the total number of the plurality of determined mean values.

23. A method according to clause 21 or 22, wherein the determining of the reference variance of historical values of the parameter of interest comprises: determining a plurality of variance values of historical values of the parameter of interest at the plurality of time instances by applying the second parameter model to the residuals data of the first parameter model, each of the plurality of variance values being determined for corresponding historical values of the parameter of interest at a respective one of the plurality of time instances; and determining the reference variance of historical values of the parameter of interest from said plurality of determined variance values.

24. A method according to clause 23, wherein said determining the reference variance of historical values of the parameter of interest from said plurality of determined variance values comprises summing the plurality of determined variance values and dividing the sum by the total number of the plurality of determined variance values.

25. A method according to any of clauses 20 to 24, wherein the predicting of the future evolution of the parameter of interest of the manufacturing process comprises: determining a first difference between the forecasted mean of future values of the parameter of interest in the specified future period and the determined reference mean; determining a second difference between the forecasted variance of future values of the parameter of interest in the specified future period and the determined reference variance; comparing the first difference to a first threshold; comparing the second difference to a second threshold; and predicting a future drift in the parameter of interest the manufacturing process if the first difference exceeds the first threshold and/or if the second difference exceeds the second threshold.

26. A method according to any preceding clause, further comprising determining a correction for the manufacturing process from said prediction of the future evolution of the parameter of interest.

27. A method according to clause 26, wherein said determining of a correction comprises determining a correction for a setting of an apparatus used in said manufacturing process.

28. A method according to clause 26 or 27, further comprising manufacturing further integrated circuits using said correction.

29. A method according to any preceding clause, further comprising scheduling a maintenance action and/or calibration action based on said prediction of the future evolution of the parameter of interest.

30. A method according to any preceding clause, comprising routinely measuring substrates to obtain at least some of said metrology data. 31. A computer program comprising program instructions operable to perform the method of any of clauses 1 to 30, when run on a suitable apparatus.

32. A non-transient computer program carrier comprising the computer program of clause 31.

33. A processing system comprising a processor and a storage device comprising the computer program carrier of clause 32.

34. A lithographic apparatus arrangement comprising: a lithographic exposure apparatus; and the processing system of clause 33.

35. A lithographic cell comprising: the lithographic apparatus arrangement of clause 34; and a metrology device comprising the processing system of clause 33 and further operable to perform the method of clause 30.

[00072] It should be appreciated that the metrology data used in the methods described herein may comprise synthetic metrology data, non-synthetic metrology data (e.g., as measured from one or more physical wafers) or a combination of synthetic metrology data and non-synthetic metrology data.

Synthetic metrology data, for example, may be obtained via computational lithography techniques which simulate one or more steps of a semiconductor manufacturing process.

[00073] Although specific reference may be made in this text to the use of lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications. Possible other applications include the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat-panel displays, liquidcrystal displays (LCDs), thin-film magnetic heads, etc.

[00074] Although specific reference may be made in this text to embodiments of the invention in the context of a lithographic apparatus, embodiments of the invention may be used in other apparatus. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus that measures or processes an object such as a wafer (or other substrate) or mask (or other patterning device). These apparatuses may be generally referred to as lithographic tools. Such a lithographic tool may use vacuum conditions or ambient (non-vacuum) conditions.

[00075] Although specific reference may have been made above to the use of embodiments of the invention in the context of optical lithography, it will be appreciated that the invention, where the context allows, is not limited to optical lithography and may be used in other applications, for example imprint lithography.

[00076] While specific embodiments of the invention have been described above, it will be appreciated that the invention may be practiced otherwise than as described. The descriptions above are intended to be illustrative, not limiting. Thus, it will be apparent to one skilled in the art that modifications may be made to the invention as described without departing from the scope of the claims set out below.

Claims

2. A method as claimed in claim 1, wherein the metrology data comprises time-series metrology data obtained in a previous period of time.

3. A method as claimed in claim 1, wherein the first parameter model comprises a time-series model configured for analyzing a sequence of data points collected over time and forecasting events based on the collected data points.

4. A method as claimed in claim 1, wherein the first parameter model comprises one of: a state space model, a time varying regression model, or a long short-term memory (LSTM) network.

5. A method as claimed in claim 1, wherein the first parameter model comprises an autoregressive integrated moving average (ARIMA) model.

6. A method as claimed in claim 1, wherein the forecasting of the first statistical characteristic of future values of the parameter of interest comprises estimating one or more unobserved components and unknown parameters of the first parameter model using a Kalman filter and a maximum likelihood estimation method, respectively.

7. A method as claimed in claim 1, further comprising applying the first parameter model and the second parameter model to reference metrology data relating to the parameter of interest to determine the at least one reference value, said reference metrology data comprising historical values of the parameter of interest which represent a stable performance of the parameter of interest in the manufacturing process.

8. A method as claimed in claim 7, further comprising simulating at least one aspect of a lithographic process to obtain synthetic metrology data, said reference metrology data comprising at least in part said synthetic metrology data.

9. A method as claimed in claim 1, wherein the first statistical characteristic is a forecasted mean of future values of the parameter of interest in a specified future period.

10. A method as claimed in claim 1, wherein the second statistical characteristic is a forecasted variance of future values of the parameter of interest in a specified future period.

11. A method as claimed in claim 9, wherein the forecasting of the first statistical characteristic of future values of the parameter of interest comprises: forecasting a plurality of mean values of future values of the parameter of interest at a plurality of time instances by applying the first parameter model to the metrology data and subsequently obtaining residuals data of the first parameter model, each of the plurality of mean values being forecasted for corresponding future values of the parameter of interest at a respective one of the plurality of time instances; and determining the forecasted mean of future values of the parameter of interest from said plurality of forecasted mean values.

12. A method as claimed in claim 9, wherein the at least one reference value comprises a reference mean of historical values of the parameter of interest and a reference variance of historical values of the parameter of interest, each determined based on the reference metrology data.

13. A method as claimed in claim 1, further comprising determining a correction for the manufacturing process from said prediction of the future evolution of the parameter of interest.

14. A method as claimed in claim 1, further comprising scheduling a maintenance action and/or calibration action based on said prediction of the future evolution of the parameter of interest.

15. A computer program comprising program instructions operable to perform the method of any of claims 1 to 6, when run on a suitable apparatus.