WO2020244853A1 - Inférence causale utilisant des données de série chronologique - Google Patents

Inférence causale utilisant des données de série chronologique Download PDF

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Publication number
WO2020244853A1
WO2020244853A1 PCT/EP2020/062028 EP2020062028W WO2020244853A1 WO 2020244853 A1 WO2020244853 A1 WO 2020244853A1 EP 2020062028 W EP2020062028 W EP 2020062028W WO 2020244853 A1 WO2020244853 A1 WO 2020244853A1
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Prior art keywords
parameter
parameters
series data
time series
determining
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PCT/EP2020/062028
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English (en)
Inventor
David Evert Song Kook SIGTERMANS
Marcel Richard André BRUNT
Gerardus Albertus Wilhelmus Sigbertus PREUSTING
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Asml Netherlands B.V.
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Publication of WO2020244853A1 publication Critical patent/WO2020244853A1/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N5/00—Computing arrangements using knowledge-based models
    • G06N5/04—Inference or reasoning models
    • G06N5/045—Explanation of inference; Explainable artificial intelligence [XAI]; Interpretable artificial intelligence
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
    • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
    • G03F7/70—Microphotolithographic exposure; Apparatus therefor
    • G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
    • G03F7/70491—Information management, e.g. software; Active and passive control, e.g. details of controlling exposure processes or exposure tool monitoring processes
    • G03F7/705—Modelling or simulating from physical phenomena up to complete wafer processes or whole workflow in wafer productions
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
    • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
    • G03F7/70—Microphotolithographic exposure; Apparatus therefor
    • G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
    • G03F7/70491—Information management, e.g. software; Active and passive control, e.g. details of controlling exposure processes or exposure tool monitoring processes
    • G03F7/70508—Data handling in all parts of the microlithographic apparatus, e.g. handling pattern data for addressable masks or data transfer to or from different components within the exposure apparatus
    • G—PHYSICS
    • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
    • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
    • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
    • G03F7/70—Microphotolithographic exposure; Apparatus therefor
    • G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
    • G03F7/70605—Workpiece metrology
    • G03F7/70616—Monitoring the printed patterns
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N7/00—Computing arrangements based on specific mathematical models
    • G06N7/01—Probabilistic graphical models, e.g. probabilistic networks
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B23/00—Testing or monitoring of control systems or parts thereof
    • G05B23/02—Electric testing or monitoring
    • G05B23/0205—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults
    • G05B23/0259—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterized by the response to fault detection
    • G05B23/0283—Predictive maintenance, e.g. involving the monitoring of a system and, based on the monitoring results, taking decisions on the maintenance schedule of the monitored system; Estimating remaining useful life [RUL]

Definitions

  • the present invention relates to inferring causal relationships between parameters of a system, such as an apparatus.
  • Exemplary arrangements may infer the causal relationships from time series data. More specifically, the invention may relate to determination of causal relationships between parameters using linear transformations.
  • 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).
  • 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.
  • EUV extreme ultraviolet
  • Low-ki lithography may be used to process features with dimensions smaller than the classical resolution limit of a lithographic apparatus.
  • CD 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.
  • sophisticated fine-tuning steps may be applied to the lithographic projection apparatus and/or design layout.
  • RET resolution enhancement techniques
  • Lithographic apparatus can be complex, comprising many components interacting in and with the apparatus.
  • Lithographic apparatus may be described in relation to a plurality of parameters, in which parameter values relate to operation of the apparatus.
  • the parameter values may be based on metrology data relating to the apparatus and/or to a lithographic process performed by the apparatus.
  • a method of inferring a causal relationship between first and second parameters comprises obtaining first time series data comprising a plurality of values of the first parameter at a plurality of time bins representing a plurality of points in time, and second time series data comprising a plurality of values of the second parameter at a plurality of related time bins representing a plurality of points in time.
  • the method further comprises performing quantisation of magnitudes of the plurality of values of the first time series data and the second time series data, determining a linear transformation between the first parameter and the second parameter based on the quantised magnitudes, and determining whether a causal relationship exists between the first parameter and the second parameter based on the linear transformation.
  • the quantisation may be a binary quantisation.
  • the quantisation may comprise assigning a discrete value to the plurality of values of the first time series data and the second time series data based on one or more of a comparison of the plurality of values to a threshold value, determining whether the plurality of values are a minimum or a maximum value, and determining whether the plurality of values are higher or lower than a previous value.
  • obtaining the first time series data and second time series data may comprise temporally aligning the first time series data with the second time series data, such that they comprise values of the first parameter and values of the second parameter in related time bins.
  • aligning the first time series data with the second time series data may comprise determining values of the first parameter that correspond to the plurality of time bins and/or determining values of the second parameter that correspond to the plurality of time bins.
  • determining the linear transformation may comprises generating a first set of channel capacity values indicating whether a causal relationship exists between the first and second parameters.
  • generating the first set of channel capacity values may comprise, based on a plurality of quantised magnitudes from the first time series data and the second time series data, determining a transition probability for each channel capacity value.
  • the transition probability may represent whether a quantised magnitude of the first time series data in a current time bin is predictive of a quantised magnitude of the second time series data in a related time bin.
  • determining the transition probability may further comprise determining whether a previous quantised magnitude of the first time series data in an earlier time bin is predictive of the quantised magnitude of the second time series data in the related time bin.
  • determining the transition probability comprises, for a plurality of time bins of the first time series data, and one by one treating each of the plurality of time bins as the current time bin the following steps: determining a quantised magnitude of the first time series data in the current and/or an earlier time bin, and assigning the quantised magnitude of the first time series data in the current time bin and the quantised magnitude of the second time series data in the related time bin to one of a plurality of channels based on the determined quantised magnitude of the first time series data in the current and or an earlier time bin.
  • the method may further comprise determining, for one or more of the plurality of channels, one or more channel probabilities representing whether the first time series data in the one or more channels is predictive of a the second time series data in the same channel.
  • the transition probability may be based on a plurality of channel probabilities.
  • the transition probability may comprise a weighted sum of the plurality of channel probabilities.
  • the weight of each channel probability in the weighted sum may be based on a number of quantised magnitudes from the first and second data series assigned to each channel.
  • the determined quantised magnitude may comprise a quantised magnitude in the current time bin, such that the channel capacity values represent a causal relationship between the first and second parameters at the current time bin.
  • the method may further comprise generating a first two- dimensional matrix comprising the first set of channel capacity values.
  • the method may further comprise generating a second set of channel capacity values, wherein the determined quantised magnitude comprises a quantised magnitude in the earlier time bin, such that the second set of channel capacity values represent a delayed causal relationship between the first parameter and the second parameter.
  • the method may further comprise generating a second two-dimensional matrix of the second set of channel capacity values, and concatenating the second two-dimensional matrix to the first two-dimensional matrix to form a three-dimensional matrix of channel capacity values.
  • the third dimension may represent the delayed causal relationship.
  • the method may further comprise selecting a largest channel capacity value in the three-dimensional matrix of channel capacity values; and determining, based on the selected channel capacity value, whether a causal relationship exists, and whether there is a delay in the causal relationship.
  • performing the quantisation may comprise performing a plurality of different quantisations
  • determining a linear transformation may comprise determining a linear transformation for one or more of the different quantisations.
  • the first and/or second time series data may comprise metrology data.
  • the first and second parameters may relate to an apparatus.
  • the method may further comprise determining that a causal relationship exists between the first parameter and the second parameter, and estimating, based on the causal relationship, a future behaviour of the apparatus.
  • a first computer processor may obtain the first and second time series data, and perform the quantisation.
  • the method may further comprise transmitting data representing one or more of the quantised magnitudes, the linear transformation and the causal relationship to an second processor.
  • the determining of the linear transformation may be performed by the second processor.
  • the second processor may determine a causal relationship the first and second parameters based on the transmitted data.
  • a method of inferring a causal graph between more than two parameters may comprise for a plurality of pairs of parameters, inferring a causal relationship between the pair of parameters according to the methods described above
  • a method of predicting failure of an apparatus using a causal graph comprising more than two parameters and inferred according to the method set out above.
  • the method may comprise initialising a plurality of border parameters of the more than two parameters, selecting a failure prediction parameter from non-border parameters of the more than two parameters, simulating operation of the apparatus based on the causal graph and the initialised border parameters, determining a simulated value for the failure prediction parameter as a result of the simulated operation of the apparatus, and determining a failure prediction for the apparatus based on the simulated value of the failure prediction parameter.
  • initialising the plurality of border parameters may comprise assigning values to the plurality of border parameters.
  • the assigned values comprise measured values for the border parameters from the, or a different apparatus.
  • the simulation of the operation of the apparatus may be continued until a value indicating failure is obtained for the failure prediction parameter and/or a value has been simulated for all of the more than two parameters in the causal graph.
  • the causal graph may comprise the causal relationships for the plurality of pairs of parameters.
  • the causal graph may further comprise a delay for one or more of the causal relationships.
  • Obtaining a simulated value for the failure prediction parameter may comprise propagating the initialised border parameters through the causal graph, based on the causal relationships and any related time delay.
  • the method may further comprises predicting a time corresponding to the failure prediction.
  • the method may further comprise simulating operation of the apparatus a plurality of times.
  • the method may further comprise determining one or more of the parameters as potential root causes of a failure prediction.
  • a parameter may be determined to be a potential root cause based on whether a value is obtained for a non-border parameter following the simulation of operation of the apparatus.
  • the parameter may be considered not a potential root cause if no value is obtained for that parameter.
  • the simulation of operation of the apparatus may be run a plurality of times for determining a plurality of simulated values for the parameter.
  • the method may further comprise determining a probability of no value being obtained for the parameter based on the plurality of simulated values for the parameter, and determining that the parameter is not a potential root cause if the probability is greater than a threshold.
  • the simulation of operation of the apparatus may be run a plurality of times for determining a plurality of true simulations, in which a failure in the failure prediction parameter was simulated, and/or a plurality of false simulations in which a failure in the failure prediction parameter was not simulated.
  • the method may further comprise determining that a parameter is a potential root cause of a failure prediction based on the variability of the parameter in a plurality of the true simulations and/or false simulations.
  • the parameter may be determined to be a potential root cause based on whether an obtained simulated value for the parameter is invariant in the true simulations and in the false simulations.
  • the parameter may be determined to be a potential root cause if the obtained simulated invariant value of the parameter in the tme simulations is different from the obtained simulated invariant value of the parameter in the false simulations.
  • determining a parameter to be invariant may comprise determining a probability that the parameter is invariant based on the obtained simulated values for the parameter, and determining that the parameter is invariant if the probability exceeds a threshold value.
  • a computer program comprising instructions which, when executed on at least one processor, cause the at least one processor to control an apparatus to carry out a method according to any of the methods described above.
  • an apparatus for inferring a causal relationship between first and second parameters comprising a processor configured to execute computer program code to undertake any of the methods set out above.
  • a lithographic apparatus comprising the apparatus for inferring a causal relationship as described above.
  • 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 a cooperation between three key technologies to optimize semiconductor manufacturing
  • Figure 4 depicts a flow diagram of a method for inferring whether a causal relationship exists between a first parameter and a second parameter
  • Figure 5 depicts a flow diagram of a method of determining linear transformations
  • Figure 6 depicts a flow diagram of a method of predicting a failure in an apparatus.
  • 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 nm) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range of about 5- 100 nm).
  • 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 and a programmable LCD array.
  • FIG. 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 mask support (e.g., a mask table) T 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 support (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 support 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.
  • the illumination system 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, and/or other types of optical components, or any combination thereof, for directing, shaping, and 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.
  • projection system PS used herein should be broadly interpreted as encompassing various types of projection system, including refractive, reflective, catadioptric, anamorphic, magnetic, electromagnetic and or electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, and 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.
  • the lithographic apparatus LA 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 PS and the substrate W - which is also referred to as immersion lithography. More information on immersion techniques is given in US6952253, which is incorporated herein by reference.
  • the lithographic apparatus LA may also be of a type having two or more substrate supports WT (also named“dual stage”).
  • the substrate supports WT may be used in parallel, and or steps in preparation of a subsequent exposure of the substrate W may be carried out on the substrate W located on one of the substrate support WT while another substrate W on the other substrate support WT is being used for exposing a pattern on the other substrate W.
  • the lithographic apparatus LA may comprise a measurement stage.
  • the measurement stage is arranged to hold a sensor and/or a cleaning device.
  • the sensor may be arranged to measure a property of the projection system PS or a property of the radiation beam B.
  • the measurement stage may hold multiple sensors.
  • the cleaning device may be arranged to clean part of the lithographic apparatus, for example a part of the projection system PS or a part of a system that provides the immersion liquid.
  • the measurement stage may move beneath the projection system PS when the substrate support WT is away from the projection system PS.
  • the radiation beam B is incident on the patterning device, e.g. mask, MA which is held on the mask support T, and is patterned by the pattern (design layout) present on 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 a position measurement system IF, the substrate support WT can be moved accurately, e.g., so as to position different target portions C in the path of the radiation beam B at a focused and aligned position.
  • the patterning device e.g. mask, MA which is held on the mask support T
  • the pattern design layout
  • the first positioner PM and possibly another position sensor may be used to accurately position the patterning device MA with respect to the path of the radiation beam B.
  • Patterning device MA and substrate W may be aligned using mask alignment marks Ml, M2 and substrate alignment marks PI, P2.
  • the substrate alignment marks PI, P2 as illustrated occupy dedicated target portions, they may be located in spaces between target portions.
  • Substrate alignment marks PI, P2 are known as scribe-lane alignment marks when these are located between the target portions C.
  • 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.
  • a lithographic cell LC also sometimes referred to as a lithocell or (litho)cluster
  • 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/Ol, 1/02, 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.
  • a supervisory control system SCS which may also control the lithographic apparatus LA, e.g. via lithography control unit LACU.
  • inspection tools 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.
  • 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 constmcted 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).
  • 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.
  • three systems may be combined in a so called“holistic” control environment as schematically depicted in Fig. 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.
  • 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 Fig. 3 by the double arrow in the first scale SCI).
  • 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) to predict whether defects may be present due to e.g. sub-optimal processing (depicted in Fig. 3 by the arrow pointing“0” in the second scale SC2).
  • 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 Fig. 3 by the multiple arrows in the third scale SC3).
  • the lithographic apparatus LA, lithographic cell LC, and metrology tool MT ah produce large amounts of data relating to a plurality of parameters, which can be used to analyse and diagnose complex issues in a system.
  • a system in this instance may comprise one or more of the whole or portions of an apparatus, such as a lithographic apparatus LA, a lithographic cell LC, or a metrology tool MT.
  • the parameter data may be used to monitor performance of a lithographic apparatus LA, for example to determine quality or yield of patterned substrates. Monitoring behaviour may also comprise monitoring performance or degradation of one or more components of a lithographic apparatus LA, lithographic cell LC, or metrology tool MT, or to identify any issues occurring with the apparatus. Monitoring of the apparatus may be performed by monitoring parameters of and/or relating to the lithographic apparatus. Specifically, parameter metrology data obtained by metrology tools MT or the lithographic apparatus LA itself may be used to monitor performance of the apparatus.
  • Parameter data relating to a system may comprise any data that is measurable or configurable in the system.
  • parameter data my comprise, but need not be limited to, data relating to one or more components of the lithographic apparatus LA, including an illumination system IL, radiation source SO, beam delivery system BD, radiation beam B, mask support, substrate support, patterning device, substrate positioner, patterning device positioner, projection system PS, measurement stage with sensor and or cleaning device; data relating to one or more components of the lithographic cell LC, including spin coaters SC, developers DE, chill plates CH, bake plates BK, substrate handlers, control units, control systems, and inspection tools; and/or data relating to and/or obtained by a metrology tool.
  • Analysis of a system and or parameter data relating to the system can be used for failure prediction. Predicting whether a failure is likely to occur, in particular predicting which part of a system will fail and at what time, may for example be used for predictive maintenance. Predictive maintenance may reduce the amount of maintenance periods required by scheduling in downtime for maintenance on a requirement-basis based on the predictions, saving maintenance cost and costs associated with downtime of the apparatus. Predictive maintenance may further save time and cost by avoiding the amount of failures occurring.
  • Analysis of a system and/or parameter data relating to a system can also be used to determine a cause for a failure in a system. This may be a real failure, or a predicted failure. A failure may be detected for a parameter or feature, but this may be a result caused by a failure or issue in another part of the system.
  • the analysis of the parameter data may be used to determine one or more root causes or potential root causes for a real or predicted failure observed in a parameter or feature.
  • Analysis of parameter data for a system may comprise the analysis of data relating to a plurality of parameters related to that system, over time.
  • the parameters may represent characteristics and or features of and relating to a system.
  • the parameters may be described by or represented in data, which may comprise one or more values of the parameter.
  • the parameter values may comprise metrology data, calculated, predicted, or otherwise obtained data relating to the parameter. Values of a parameter may be provided linked to a moment in time, and multiple values of a parameter may be provided in a series comprising multiple moments in time. This may be referred to as time series data for a parameter.
  • parameters of and/or relating to a lithographic apparatus include temperature, pressure, flow rate, humidity, voltages, velocities, accelerations, torques forces, positioning errors, motor currents, spikes in motor currents, power spectral density in predetermined frequency bands, sudden changes in average values, sudden changes in average slope values, etc.
  • Parameters of and/or relating to the apparatus may also include recipe features and or configuration data related to processes implemented by the apparatus, wherein such features/data may be represented as a time series, for example lithographic exposure recipe features and or lithographic apparatus configuration data. The skilled person will envisage other features that may be used as parameters, depending on the system to be analysed.
  • Monitoring of the apparatus may comprise determining relationships between parameters, and may further comprise analysing these relationships over time.
  • the determination of relationships between a plurality of parameters may be used to construct a graph.
  • the graph may be a causal graph.
  • the determination of causal relationships may also be referred to as causal inference.
  • the parameters and relationships between them may be represented in multiple formats, for example as a list, as a matrix, or as a graph.
  • a large amount of parameters for example when monitoring a complex system like a lithographic apparatus LA, a large amount of relationships exist between them. This may mean that determination of the relationships between the parameters may be computationally intensive, and require large amounts of time and effort.
  • the determination may involve algorithms comprising computationally costly non-linear and or higher order calculations.
  • purpose-specific computational processing power for example a high performance cluster (HPC)
  • HPC high performance cluster
  • the determination and maintenance of up-to-date relationships may require a significant amount of time.
  • the calculation time of a causal graph may amount for hundreds of hours, where no HPC or strong computational power is provided to perform the calculations.
  • the calculation may still take hours.
  • a disadvantage of this high computational cost and resulting required time, is that the applications of causal graph inference may be limited.
  • Described herein are methods and apparatus for reducing the computational requirements for inferring a causal graph. This may be achieved through a fast approximate method for causal inference of a relationship between parameters based on time series data, as will be described in detail below.
  • Figure 4 shows a flow diagram of a method for determining whether causal relationship exists between a first parameter and a second parameter.
  • the method may be extended to incorporate additional parameters, as required.
  • the parameters may comprise parameters of an apparatus, such as a lithographic apparatus.
  • Time series data is obtained 400 for each of a plurality of parameters.
  • first and second parameters are used and first time series data is obtained that relates to the first parameter, and second time series data is obtained that relates to the second parameter.
  • exemplary methods and apparatus disclosed herein may determine causal relationships between any number of parameters.
  • the first and second time series data comprises a plurality of values of the respective parameter, each magnitude corresponding to a specific time bin.
  • a time bin corresponds to a particular moment in time.
  • Obtaining the time series data may comprise measuring the respective parameter, which may be done during operation of the system or apparatus to be monitored. In alternative arrangements, one or more of the first and second time series data may have been obtained previously. Obtaining the time series data may therefore be by transmission or other means of transference. In some implementations, time series data may be enriched with context information. Context information may be used to determine the validity of parameter data at the moment it was generated or sensed. For example, context information may comprise a machine state of the apparatus, indicating whether a machine was running or whether it was down.
  • the values of the second time series data correspond to time bins related to the time bins of the first time series data.
  • the related time bins may be the same time bins or a later time bin.
  • obtaining the first and second time series data may include some temporal alignment 402 of the values thereof. This may be necessary because the time bins of measured or otherwise obtained time series data may not be synchronous.
  • values of one or both of the first and second time series data that correspond to specific time bins, having a specific spacing may be determined. This may be done using interpolation and/or decimation algorithms.
  • the magnitudes of the plurality of values of the first and or second time series data are quantised 404.
  • the quantisation may be a binary quantisation, such that the quantised magnitudes are represented by either a“1” or a“0”.
  • Quantisation may comprise assigning a discrete value to value to the plurality of values of the time series data by comparing each of the plurality of values to one or more threshold values. If a value of the time series data is above the threshold, it may be assigned a first value (e.g. a“1”) and if it is below the threshold, it may be assigned a second value (e.g.
  • quantisation may comprise assigning a discrete value to value to the plurality of values of the time series data by determining whether a value of the time series data is a minimum or maximum value (local or global) within its respective time series. For example, if a value of the time series data is a local or global maximum or minimum value it may be assigned a first value (e.g. a“1”), and if it is a not a maximum or minimum value, it may be assigned a second value (e.g.
  • quantisation may comprise assigning a discrete value to value to the plurality of values of the time series data by determining whether one of the plurality of values is higher or lower than a previous value in the respective time series data. For example, if a value of the time series data is higher than a previous value, it may be assigned a first value (e.g. a“1”) and if it is less than a previous value, it may be assigned a second value (e.g. a“0”) ⁇
  • a first value e.g. a“1”
  • a second value e.g. a“0”
  • a plurality of quantisations of the time series data may be undertaken. This may result in a plurality of sets of quantised magnitudes for one or more of the time series.
  • the processes for determining a causal relationship between the parameters may be undertaken using one or more of the plurality of quantised magnitudes.
  • a linear transformation is determined 406 between the first and second parameters. It is determined 408 whether a causal relationship exists between the first and second parameters, based on the linear transformation. More detail on an exemplary process for determining the linear transformation and the causal relationship is provided below.
  • the determined causal relationships may be used in some exemplary arrangements to generate a causal graph.
  • the term“causal graph” may encompass any representation of a set of causal relationships between a plurality of parameters within a system or apparatus.
  • a causal graph may comprise a matrix of values in which rows and columns represent the parameters and each value in a cell of the matrix represents a probability of a causal relationship between the parameters identified with the row and column.
  • the matrix may also comprise a third dimension representing time, as explained below. Such matrices, are therefore able to show if the causal relationship has an associated time delay.
  • Causal graphs may also comprise a graphical representation of causal relationships between parameters in a system.
  • a parameter may be represented by a node and an arrow from one node to another may represent a causal relationship, wherein the direction and optionally the thickness of the arrow identifies the direction and magnitude of the causal relationship.
  • Figure 5 shows a flow diagram of an exemplary method for determining causal relationships between parameters.
  • the exemplary method of Figure 5 uses linear transformations to determine the causal relationships. All or part of the process described below may be substituted in Figure 4 for steps 406 and/or 408.
  • the quantised magnitudes, for example those determined as set out above, of the first and second time series data are used to determine channel capacity values.
  • the channel capacity values indicate whether a causal relationship exists between the first and second parameters.
  • Channel capacity values may encompass other metrics that may be determined, such as values providing an indication whether a causal relationship exists between the first and second parameters, for example transfer entropy values.
  • the quantised magnitudes are split into a plurality of channels.
  • the example given below is based on a binary quantisation and uses only a single quantisation method. However, the method may be expanded to cover more quantisation levels than two. Further, the method may be expanded to incorporate a plurality of sets of quantised magnitudes that have been determined using different quantisation methods.
  • the quantised magnitudes for the first and second parameters are collated together 500, as shown below, wherein the first and second parameter data is temporally aligned.
  • a current and/or previous value of the quantised magnitude of the first parameter is then determined 502 and the current quantised magnitudes of the first and second parameters may be assigned to one of two channels 504 based on that determination.
  • the current quantised magnitudes may be assigned to a channel based on whether an earlier value for the first and or second parameter is a“0” or a“1”.
  • the current quantised magnitudes are placed into a first channel (channel 1) if a value of the first parameter in an earlier time bin is a“0”, and placed into a second channel (channel 2) if a value of the first parameter in an earlier time bin is a“1”. This is shown below in Table 2.
  • a stepwise progression through the quantised magnitudes of Table 1 is undertaken, wherein the time bin at each step is considered to be the “current” time bin. For that current time bin, it is determined whether an earlier quantised magnitude (i.e. in an earlier time bin) of the first parameter is a“0” or a“1” and the quantised magnitude in the current time bin for the first and second parameter is assigned to channel 1 or channel 2 based on that determination. It is noted here that the“related” time bin of the second parameter time series is coincident with the current time bin. However, in other exemplary methods and apparatus, the related time bin may be later than the current time bin.
  • the quantised magnitude in the current time bin for the first and second parameter is assigned to channel 1 if the quantised magnitude of the first parameter in the earlier time bin is a“0”, and is assigned to channel 2 if the quantised magnitude of the first parameter in the earlier time bin is a“1”.
  • the next latest time bin is then considered to be the“current” time bin and the process is repeated.
  • a different time bin i.e. separated from the current time bin by more than one previous time bin
  • a plurality of earlier time bins may be used, e.g. if the quantised magnitudes of the two immediately previous time bins are“0” and “0” then the current quantised magnitudes may be assigned to channel 1, and so on. The number of channels may therefore change accordingly.
  • no earlier time bins may be used when assigning the current quantised magnitude to a channel and the value of the quantised magnitude(s) in the current time bin may be used in isolation.
  • the quantised magnitudes can be used to generate 506 a channel probability for each channel.
  • the channel probability is a conditional probability, e.g. conditional on previous values in each channel.
  • a channel may consist of inputs and outputs, which in the case of Table 2 are the 1 st and 2 nd parameters and current, future and/or previous values thereof.
  • the number of inputs may represent the number of unique values the first parameter can be.
  • the number of outputs may reflect the number of unique values the second parameter can be.
  • a channel probability may be calculated.
  • the channel probability for a specific channel input and channel output combination may represent a measure of probability that a causal relationship exists between these specific values.
  • channel probabilities may be represented in a matrix. Each matrix element may represent a probability for a specific input value and output value combination.
  • Table 2 where both the first parameter and the second parameter are binary quantised values, there are 4 probability matrix elements.
  • the determination of a channel probability may be performed by noting the number of instances for each channel that a specific quantised value of the first parameter resulted in a particular quantised value for the second parameter as a proportion of the total number of time bins in the channel. This is shown for channel 1 below in Table 3.
  • the greyed out cells of the matrix shown in Table 3 comprise the channel probabilities for channel 1 for each combination of quantised values for the first and second parameter. The same process may be undertaken for the remaining channels.
  • Transition probabilities may be determined 508 based on the determined channel probabilities for that channel.
  • the transition probability determination may be based on a weighted sum of a plurality of the channel probabilities of the channel.
  • the weight applied to each channel probability in the weighted sum may be based on the number of time bins of quantised magnitudes assigned to respective channels.
  • the channel capacity value for a channel based on the transition probability of the channel may provide a quantification of the causal relationship of whether the first parameter is causal on the second parameter.
  • a channel capacity value may be determined based on the Arimoto-Blahut algorithm. Based on one or both of the transition probabilities and the channel capacity values, a classification (e.g. 1/0, yes/no, causal/not-causal) may be determined on whether the first parameter is causal on the second parameter.
  • a two-dimensional matrix may be generated comprising the channel capacity values relating to causal relationships between the parameters, wherein each column and row of the two-dimensional matrix relates to one of the parameters.
  • the two dimensional matrix mentioned above may be determined using no historical data from the collated quantised magnitudes. That is, the quantised magnitudes of the first and second parameters in the current time bin may be assigned to a channel based on the value of one or both of those quantised values. The process may then be repeated to determine a second two-dimensional matrix using historical data, as in the method described above. That is, the collated quantised magnitude data may be assigned to channels based on a value of an earlier quantised magnitude of the first parameter, for example the quantised magnitude in the immediately preceding time bin. This process may be repeated to form further two-dimensional matrices using quantised magnitudes in earlier and earlier time bins. After a plurality of two-dimensional matrices have been generated, they may be concatenated to form a three dimensional matrix. The third dimension of the matrix represents time or a delayed causal relationship between the parameters.
  • causal relationship and optionally any associated delay may be used to predict failures or other behaviours of a system or apparatus.
  • the methods disclosed herein do not require large amounts of computing power to process. Therefore, in exemplary arrangements, the methods may be undertaken using the processor and computing power present on the apparatus itself, which may be a lithographic apparatus, such as that disclosed herein. This means that sensitive recipe information need not be transmitted away from the apparatus for fault detection and/or prediction of apparatus behaviour. If the quantised data is exported to an external processor, the quantised values may obscure a significant amount of the information contained in the original data, maintaining and protecting confidentiality (e.g.
  • a set of parameters may be quantised to a set of discrete values, e.g.“1” and“0”)- Accordingly, exemplary methods and apparatus may be configured to transmit data representing one or more of the quantised magnitudes, the linear transformation and the causal relationship to a second processor, which may be on a separate device.
  • the separate device may, for example, be a fault diagnosis tool or a laptop used by an engineer. If the quantised magnitudes and and/or linear transformations are transmitted to the second processor then the second processor may be configured to determine the causal graph, which may include any associated delay.
  • Figure 6 shows a flow diagram of a method for predicting failure of an apparatus using a causal graph.
  • a plurality of border parameters may be determined and initialised with values. If a graph is a directed graph with parentless nodes, also referred to as roots, these parentless nodes may be borders of the graph for the target node. If the graph has no parentless nodes, borders of the graph may be determined as nodes that are furthest away from the target node in terms of causal delay. That is to say, a node is a border node if none of its neighbouring nodes has a longer causal delay away from the graph than that node.
  • the initialisation values may be quantised magnitudes. The initialisation values may have been measured in the apparatus under test and may be received therefrom. Alternatively, the initialisation parameters may have been measured in a different apparatus to the one under test. In some exemplary arrangements, the causal graph may have been determined based on a different apparatus to the one under test.
  • Border parameters may be determined based on whether the causal graph has roots.
  • a root is a node in a directed graph which has no incoming edge (the effect node in a causal relationship), that is to say, it is not caused by anything in the graph. If the causal graph has roots then the roots may be determined to be the border parameters. If the causal graph does not have roots then the border parameters may be those which fit one or more of the following criteria: the parameter is the furthest away from the failure prediction parameter compared to its immediate neighbours; there is no direct path between the parameter and another border parameter; and there is a direct path to all other parameters in the causal graph.
  • the failure prediction parameter may represent a phenomenon the reason why the prediction method is run, represented by a phenomenon to be explained by the failure prediction, for example represent a fatal error in the machine, or an excursion of a control limit of a specific parameter.
  • the operation of the apparatus is then simulated based on the initialisation of the border parameters. That is, data in each of a plurality of time bins for each of the border parameters is stepped through in succession and values for the remaining parameters are populated in dependence on the probabilities and optionally the delays provided by the causal graph. The simulation may be ended when the failure prediction parameter is populated with a value indicating a failure.
  • the simulation may be ended when all parameters except the failure prediction parameter are populated (not empty), irrespective of whether the failure prediction parameter is populated.
  • the simulation may be repeated a plurality of time. Each simulation may be based on the same initialisation of the border parameters.
  • a simulated value of the failure prediction parameter may be determined 606. In cases where there are multiple simulations, the simulated value of the failure prediction parameter will be determined for each simulation.
  • the simulated value(s) of the failure prediction parameter may be used to determine a likelihood that the apparatus will fail (a failure prediction) 608.
  • Running the simulation a plurality of times may permit the determination of probability data relating to whether a failure of the apparatus is likely to occur.
  • the probability data relating to whether a failure of the apparatus is likely to occur may be expressed in a probability density function based on time from the start of the simulation. If the probability of a failure at any given time exceeds a threshold then maintenance of the apparatus may be scheduled to prevent that failure.
  • failure prediction could be implemented offline or online.
  • the simulations could be performed offline (e.g. at local offices or locally in the fab). These simulations could be executed in a scheduled fashion (e.g. every day at midnight or based on certain warning indicators).
  • a scheduled fashion e.g. every day at midnight or based on certain warning indicators.
  • both offline and online simulation is possible.
  • the apparatus itself may perform, as part of a daily test, simulation runs based on the latest data to predict a service need for the future.
  • a further advantage may be that specific causal graphs may be determined per apparatus. These causal graphs may be determined at regular intervals. This allows apparatus specific predictions, mitigating the need to include the configuration of the apparatus.
  • one or more potential root causes of a predicted failure may be determined.
  • determination of root causes may be based on whether or not a parameter is populated with a value during a simulation.
  • a plurality of simulations may be run and a probability may be determined 610 of whether a particular parameter will not be populated with a value during a simulation.
  • the determined probability it may be determined 612 whether the parameter is a root cause. If the determined probability that the parameter will not be populated is greater than a threshold value then it may be determined that the parameter is not a root cause of a predicted failure. This is because the data relating to that parameter is not propagated through the apparatus sufficiently quickly.
  • a true simulation encompasses one in which the failure prediction parameter is populated with a value and indicates a failure.
  • a false simulation encompasses one in which there is no indication of a failure, for example based on the failure prediction parameter.
  • the variability of a particular parameter in one or both of the true and false simulations may be indicative of whether the parameter is a root cause of a failure. For example, a parameter may be determined as invariant in true and false simulations.
  • the term invariant encompasses a value for a parameter that meets the same criteria in a plurality, possibly all, true and/or false simulations.
  • the same criteria may comprise having the same value (within a range) or being above or below a threshold value.
  • the criteria might relate to a probability that one or more criteria is met. If a parameter is invariant in both a true network and a false network, this may be determined to be a root cause of a failure. Further, if the parameter is invariant in both a true network and a false network and has a different value in the true network to the value in the false network, the parameter may be determined to be a root cause of the failure.
  • 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 apparatus may be generally referred to as lithographic tools. Such a lithographic tool may use vacuum conditions or ambient (non- vacuum) conditions.
  • a method of inferring a causal relationship between first and second parameters comprising:
  • first time series data comprising a plurality of values of the first parameter at a plurality of time bins representing a plurality of points in time
  • second time series data comprising a plurality of values of the second parameter at a plurality of related time bins representing a plurality of points in time
  • the quantisation comprises assigning a discrete value to the plurality of values of the first time series data and the second time series data based on one or more of: a comparison of the plurality of values to a threshold value, determining whether the plurality of values are a minimum or a maximum value, and determining whether the plurality of values are higher or lower than a previous value.
  • obtaining the first time series data and second time series data comprises temporally aligning the first time series data with the second time series data, such that they comprise values of the first parameter and values of the second parameter in related time bins.
  • aligning the first time series data with the second time series data comprises determining values of the first parameter that correspond to the plurality of time bins and/or determining values of the second parameter that correspond to the plurality of time bins.
  • determining the linear transformation comprises:
  • generating the first set of channel capacity values comprises, based on a plurality of quantised magnitudes from the first time series data and the second time series data, determining a transition probability for each channel capacity value, the transition probability representing whether a quantised magnitude of the first time series data in a current time bin is predictive of a quantised magnitude of the second time series data in a related time bin.
  • determining the transition probability further comprises determining whether a previous quantised magnitude of the first time series data in an earlier time bin is predictive of the quantised magnitude of the second time series data in the related time bin.
  • determining the transition probability comprises, for a plurality of time bins of the first time series data, and one by one treating each of the plurality of time bins as the current time bin:
  • the method further comprising determining, for one or more of the plurality of channels, one or more channel probabilities representing whether the first time series data in the one or more channels is predictive of a the second time series data in the same channel.
  • transition probability is based on a plurality of channel probabilities.
  • transition probability comprises a weighted sum of the plurality of channel probabilities.
  • the method further comprising generating a first two-dimensional matrix comprising the first set of channel capacity values.
  • the method further comprising generating a second two-dimensional matrix of the second set of channel capacity values, and concatenating the second two-dimensional matrix to the first two- dimensional matrix to form a three-dimensional matrix of channel capacity values, the third dimension representing the delayed causal relationship.
  • performing the quantisation comprises performing a plurality of different quantisations
  • determining a linear transformation comprises determining a linear transformation for one or more of the different quantisations.
  • first and/or second time series data comprises metrology data.
  • the method further comprising:
  • the determining of the linear transformation is performed by the second processor; and wherein the second processor determines a causal relationship the first and second parameters based on the transmitted data.
  • a method of inferring a causal graph between more than two parameters comprising, for a plurality of pairs of parameters, inferring a causal relationship between the pair of parameters according to the method any of clauses 1 - 19.
  • initialising the plurality of border parameters comprises assigning values to the plurality of border parameters.
  • determining border parameters comprises:
  • the border parameters to be parameters in the causal graph that meet all of the following criteria:
  • the parameter is the furthest away from the failure prediction parameter compared to its immediate neighbours
  • causal graph comprises the causal relationships for the plurality of pairs of parameters, and further comprises a delay for one or more of the causal relationships, and wherein obtaining a simulated value for the failure prediction parameter comprises:
  • determining that a parameter is a potential root cause of a failure prediction based on the variability of the parameter in a plurality of the true simulations and or false simulations.
  • a computer program comprising instructions which, when executed on at least one processor, cause the at least one processor to control an apparatus to carry out a method according to any of clauses 1 to 36.
  • An apparatus for inferring a causal relationship between first and second parameters comprising a processor configured to execute computer program code to undertake the method as set out in any of clauses 1 to 36.
  • a lithographic apparatus comprising the apparatus of clause 38.

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Abstract

La présente invention concerne un procédé de déduction d'une relation causale entre des premier et second paramètres, le procédé comprenant les étapes consistant à obtenir des premières données de série chronologique comprenant une pluralité de valeurs du premier paramètre à une pluralité de cellules temporelles représentant une pluralité de points dans le temps et des secondes données de série chronologique comprenant une pluralité de valeurs du second paramètre à une pluralité de cellules temporelles associées représentant une pluralité de points dans le temps. Le procédé comprend en outre l'étape consistant à effectuer une quantification des amplitudes de la pluralité de valeurs des premières données de série chronologique et des secondes données de série chronologique. Le procédé comprend en outre les étapes consistant à déterminer une transformation linéaire entre le premier paramètre et le second paramètre sur la base des amplitudes quantifiées et à déterminer si une relation causale existe entre le premier paramètre et le second paramètre sur la base de la transformation linéaire.
PCT/EP2020/062028 2019-06-03 2020-04-30 Inférence causale utilisant des données de série chronologique WO2020244853A1 (fr)

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