Article

Modelling of the X,Y,Z positioning errors and uncertainty evaluation for the LNE's mAFM using the Monte Carlo method

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Abstract

In order to evaluate the uncertainty budget of the LNE's mAFM, a reference instrument dedicated to the calibration of nanoscale dimensional standards, a numerical model has been developed to evaluate the measurement uncertainty of the metrology loop involved in the XYZ positioning of the tip relative to the sample. The objective of this model is to overcome difficulties experienced when trying to evaluate some uncertainty components which cannot be experimentally determined and more specifically, the one linked to the geometry of the metrology loop. The model is based on object-oriented programming and developed under Matlab. It integrates one hundred parameters that allow the control of the geometry of the metrology loop without using analytical formulae. The created objects, mainly the reference and the mobile prism and their mirrors, the interferometers and their laser beams, can be moved and deformed freely to take into account several error sources. The Monte Carlo method is then used to determine the positioning uncertainty of the instrument by randomly drawing the parameters according to their associated tolerances and their probability density functions (PDFs). The whole process follows Supplement 2 to 'The Guide to the Expression of the Uncertainty in Measurement' (GUM). Some advanced statistical tools like Morris design and Sobol indices are also used to provide a sensitivity analysis by identifying the most influential parameters and quantifying their contribution to the XYZ positioning uncertainty. The approach validated in the paper shows that the actual positioning uncertainty is about 6 nm. As the final objective is to reach 1 nm, we engage in a discussion to estimate the most effective way to reduce the uncertainty.

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... Par ailleurs, certains éléments d'incertitude sont difficiles à estimer expérimentalement. C'est particulièrement le cas lorsque nous considérons les erreurs induites par les imperfections géométriques du système de mesure tels que les désalignements des interféromètres et des faisceaux, les erreurs de cosinus, la forme ou la rugosité des miroirs, la non-orthogonalité des prismes, la dilatation thermique du châssis et du prisme, les erreurs d'Abbe, les erreurs de position du prisme et l'erreur due à la variation de l'épaisseur de l'échantillon, etc. Pour contourner cette difficulté, notre modèle est développé sous Matlab par l'utilisation de la programmation orientée objet [83,84]. Il représente le plus fidèlement possible le comportement de la chaîne métrologique du mAFM. ...
... Il se pourrait que ce ne soit plus l'erreur d'Abbe qui nous pénalise mais d'autres paramètres d'entrée. Finalement, l'analyse de sensibilité des paramètres a montré que leurs indices de sensibilité n'étaient pas symétriques sur les positions X et Y , même dans le cas où les fonctions de densité de probabilité des paramètres d'entrée n'étaient pas centrées en zéro [83]. Il faudra comprendre d'où vient la dissymétrie du système en analysant une par une les composantes. ...
... Par ailleurs, un point devra être éclairci concernant la perte de symétrie du modèle constatée lors de l'analyse de sensibilité des paramètres sur les positions X et Y . En effet, malgré la symétrie de l'instrument, les indices de sensibilité évalués par le modèle ne sont pas symétriques pour certaines composantes même dans le cas où les fonctions de densité de probabilité des paramètres d'entrée ne sont pas centrées en zéro [83]. Pour élucider ce point, il sera impératif de vérifier une à une les composantes et de comprendre pourquoi le modèle de l'AFM métrologique ne répond pas aux conditions de symétrie. ...
Thesis
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... Par ailleurs, certains éléments d'incertitude sont difficiles à estimer expérimentalement. C'est particulièrement le cas lorsque nous considérons les erreurs induites par les imperfections géométriques du système de mesure tels que les désalignements des interféromètres et des faisceaux, les erreurs de cosinus, la forme ou la rugosité des miroirs, la non-orthogonalité des prismes, la dilatation thermique du châssis et du prisme, les erreurs d'Abbe, les erreurs de position du prisme et l'erreur due à la variation de l'épaisseur de l'échantillon, etc. Pour contourner cette difficulté, notre modèle est développé sous Matlab par l'utilisation de la programmation orientée objet [83,84]. Il représente le plus fidèlement possible le comportement de la chaîne métrologique du mAFM. ...
... Il se pourrait que ce ne soit plus l'erreur d'Abbe qui nous pénalise mais d'autres paramètres d'entrée. Finalement, l'analyse de sensibilité des paramètres a montré que leurs indices de sensibilité n'étaient pas symétriques sur les positions X et Y , même dans le cas où les fonctions de densité de probabilité des paramètres d'entrée n'étaient pas centrées en zéro [83]. Il faudra comprendre d'où vient la dissymétrie du système en analysant une par une les composantes. ...
... Par ailleurs, un point devra être éclairci concernant la perte de symétrie du modèle constatée lors de l'analyse de sensibilité des paramètres sur les positions X et Y . En effet, malgré la symétrie de l'instrument, les indices de sensibilité évalués par le modèle ne sont pas symétriques pour certaines composantes même dans le cas où les fonctions de densité de probabilité des paramètres d'entrée ne sont pas centrées en zéro [83]. Pour élucider ce point, il sera impératif de vérifier une à une les composantes et de comprendre pourquoi le modèle de l'AFM métrologique ne répond pas aux conditions de symétrie. ...
Thesis
Full-text available
À l’heure où les nanotechnologies sont en plein essor, la précision des mesures réalisées à l’échelle nanométrique devient un défi essentiel pour améliorer les performances et la qualité des produits intégrant des nano. Pour répondre aux besoins sous-jacents en nanométrologie dimensionnelle, le Laboratoire National de métrologie et d’Essais (LNE) a conçu intégralement un Microscope à Force Atomique métrologique (mAFM). Son objectif principal est d’assurer la traçabilité au mètre défini par le Système International d’unités (SI) pour les mesures à l’échelle nanométrique. Pour cela, le mAFM utilise quatre interféromètres différentiels qui mesurent en temps réel le déplacement relatif de la pointe par rapport à l’échantillon. Cet instrument de référence est destiné à l’étalonnage d’étalons de transfert couramment utilisés en microscopie à champ proche (SPM) et en microscopie électronique à balayage (SEM). Lors de ce processus, une incertitude de mesure est évaluée. Elle détermine un niveau de confiance de l’étalonnage réalisé par le mAFM. Cette incertitude est généralement évaluée grâce à des mesures expérimentales permettant de déterminer l’impact de certaines sources d’erreur qui dégradent les mesures à l’échelle du nanomètre. Pour d’autres sources d’erreur, leur évaluation reste complexe ou expérimentalement impossible. Pour surmonter cette difficulté, le travail de thèse a consisté à mettre en place un modèle numérique de l’instrument nommé « AFM virtuel ». Il permet de prévoir l’incertitude de mesure du mAFM du LNE en ciblant les sources critiques d’erreur grâce à l’utilisation d’outils statistiques tels que la Méthode de Monte Carlo (MCM), les plans de Morris et les indices de Sobol. Le modèle utilise essentiellement la programmation orientée objet afin de prendre en compte un maximum d’interactions parmi les 140 paramètres d’entrée, en intégrant des sources jusqu’ici négligées ou surestimées par manque d’informations.
... Published in 2008, the Propagation of Distributions Using a Monte Carlo Method, Supplement 1 to the guide, discusses the propagation of probability distributions through a mathematical measurement model [22,23]. The GUM method and Monte Carlo method (MCM) have been widely used in numerous fields to evaluate measurement uncertainties [24][25][26][27][28][29][30][31]. ...
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... This device is a reference instrument specifically designed to establish the traceability route dimensional measurements at the nanoscale making a direct link between the SI meter definition and AFM and SEM measurements. On this instrument, the tip/ sample relative position is measured in real time by laser interferometry and the uncertainties associated with the measurements are established through an intensive metrological qualifi- cation and modelling of the instrument [20]. The pitch and step height values of the grating resulting from the metrological AFM calibration were found to be (900.17 ...
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At this time, there is no instrument capable of measuring a nano-object along the three spatial dimensions with a controlled uncertainty. The combination of several instruments is thus necessary to metrologically characterize the dimensional properties of a nano-object. This paper proposes a new approach of hybrid metrology taking advantage of the complementary nature of atomic force microscopy (AFM) and scanning electron microscopy (SEM) techniques for measuring the main characteristic parameters of nanoparticle (NP) dimensions in 3D. The NP area equivalent, the minimal and the maximal Feret diameters are determined by SEM and the NP height is measured by AFM. In this context, a kind of new NP repositioning system consisting of a lithographed silicon substrate has been specifically developed. This device makes it possible to combine AFM and SEM size measurements performed exactly on the same set of NPs. In order to establish the proof-of-concept of this approach and assess the performance of both instruments, measurements were carried out on several samples of spherical silica NP populations ranging from 5 to 110 nm. The spherical nature of silica NPs imposes naturally the equality between their height and their lateral diameters. However, discrepancies between AFM and SEM measurements have been observed, showing significant deviation from sphericity as a function of the nanoparticle size.
... After implementing the MC simulations, the average, the uncertainty, the lower and upper limits of the coverage intervals were computed for the assessment of each iteration. The MC simulations will not be terminated until twice the standard deviation associated with it is less than the numerical tolerance associated with the standard deviation of the output quantities [34]. ...
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Thesis
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Avec l’émergence des nanosciences et nanotechnologies ces dernières années, l’étude et la caractérisation des propriétés dimensionnelles et physicochimiques sur des structures ayant des dimensions inférieures à 100 nm sont devenues indispensables. Cela nécessite la mise au point de techniques de mesures et le développement d’instruments adaptées aux échelles nanométriques. Depuis les années 90, les laboratoires nationaux de métrologie ont relevé le défi du développement d'une nouvelle activité de métrologie de référence destinée à satisfaire les besoins de la mesure dimensionnelle à l'échelle nanométrique. Cela a conduit à l’émergence d’une nouvelle science appelée « nanométrologie » qui est définit comme étant la science de la mesure à l’échelle du nanomètre (gamme allant de 1 nm à 100 nm) et à l’estimation des incertitudes de mesure associées. Cette science suscite un intérêt croissant dans la recherche fondamentale et dans l’industrie. A titre d’exemple, la mesure de paramètres géométriques (taille et morphologie) d’un nano-objet est incontournable pour l’investigation de ses propriétés physicochimiques. Ces paramètres se retrouvent au coeur des préoccupations métrologiques des industriels (ex. : microélectronique) et des études sur la toxicité éventuelle des nano produits. En effet, depuis les travaux de l’organisation internationale de normalisation (ISO), et plus particulièrement de son comité technique en charge de la normalisation des nanomatériaux (TC229), la taille et la forme d’un nanoobjet sont reconnus comme un des paramètres indispensables pour son identification. De plus, depuis l’entrée en vigueur le premier janvier 2013 du décret français n◦ 2012-232 concernant la déclaration des substances à l’état nano-particulaire, les activités liées à la caractérisation des nanomatériaux sont en forte croissance. Le développement de ces activités et le fort couplage existant entre propriétés dimensionnelles et propriétés physico-chimique des nanomatériaux, pousse à l’amélioration de la fiabilité et de la comparabilité des mesures à l’échelle nanométrique. Cela génère un réel besoin d’étalonnage et de mise à disposition d’étalons de transferts. Ces étalons, permettent d’étalonner les instruments utilisés pour la mesure des nanomatériaux et d’y associer des incertitudes de mesure nanométriques. L’état actuel de l’instrumentation susceptible d’être utilisée dans ce cadre montre que les microscopes à sonde locale (SPM pour Scanning Probe Microscope) et les microscopes électroniques à balayage (SEM pour Scanning Electron Microscope) représentent des outils puissants pour caractériser des échantillons à l’échelle du nanomètre. Ces instruments équipent la plupart des laboratoires de recherche académiques et industriels. 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Cependant, malgré les très bonnes performances atteintes par l’instrument en termes de stabilité thermique et mécanique (sans tenir compte de la tête AFM), son incertitude de mesure est pénalisée par l’utilisation d’une tête AFM commerciale mal adaptée à la discipline métrologique. Ces pour ces raisons qu’a été initié le développement d’une tête AFM spécifiquement conçue pour les besoin de nanométrologie. Un des objectifs principaux de la thèse a consisté à mener un important travail de développement instrumental afin de poursuivre la conception et l’optimisation des performances du mAFM en l’équipant d’une tête AFM métrologique dans le but de minimiser l’incertitude de mesure globale de l’instrument. Cette tête AFM comporte un système original de mesure des déflexions du levier nécessaire à la détection des forces s’exerçant à l’extrémité de la pointe. Parallèlement à ce développement, le projet a aussi porté sur la caractérisation fine de l’instrument afin d’établir un bilan d’incertitude ainsi que l’optimisation de l’architecture du contrôleur dans le but d’améliorer la vitesse de balayage des échantillons. Le travail présenté dans ce manuscrit est structuré comme suit : Dans un premier temps, le premier chapitre introduit le principe de la microscopie à force atomique. Les notions de traçabilité et d’étalonnage sont abordées et leur mise en pratique est illustrée sur le mAFM. Dans une seconde partie, et suite à la description du mAFM, les limites de l’instrument avec l’ancienne tête AFM sont abordées. La fin du chapitre présente un cahier des charges pour la conception de la nouvelle tête AFM. Le chapitre deux représente une étude bibliographique des principaux systèmes de mesure de déflexions du levier. Les avantages et les inconvénients de chaque système sont présentés et leur éventuelle intégration sur le mAFM est discutée. 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A computational model is a representation of some physical or other system of interest, first expressed mathematically and then implemented in the form of a computer program; it may be viewed as a function of inputs that, when evaluated, produces outputs. Motivation for this article comes from computational models that are deterministic, complicated enough to make classical mathematical analysis impractical and that have a moderate-to-large number of inputs. The problem of designing computational experiments to determine which inputs have important effects on an output is considered. The proposed experimental plans are composed of individually randomized one-factor-at-a-time designs, and data analysis is based on the resulting random sample of observed elementary effects, those changes in an output due solely to changes in a particular input. Advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs, monotonicity of outputs with respect to inputs, or ad...
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Tactile ultra-precise coordinate measuring machines (CMMs) are very attractive for accurately measuring optical components with high slopes, such as aspheres. The METAS µ-CMM, which exhibits a single point measurement repeatability of a few nanometres, is routinely used for measurement services of microparts, including optical lenses. However, estimating the measurement uncertainty is very demanding. Because of the many combined influencing factors, an analytic determination of the uncertainty of parameters that are obtained by numerical fitting of the measured surface points is almost impossible. The application of numerical simulation (Monte Carlo methods) using a parametric fitting algorithm coupled with a virtual CMM based on a realistic model of the machine errors offers an ideal solution to this complex problem: to each measurement data point, a simulated measurement variation calculated from the numerical model of the METAS µ-CMM is added. Repeated several hundred times, these virtual measurements deliver the statistical data for calculating the probability density function, and thus the measurement uncertainty for each parameter. Additionally, the eventual cross-correlation between parameters can be analyzed. This method can be applied for the calibration and uncertainty estimation of any parameter of the equation representing a geometric element. In this article, we present the numerical simulation model of the METAS µ-CMM and the application of a Monte Carlo method for the uncertainty estimation of measured asphere parameters.
Article
This article describes the context of the development and the implementation of a metrological atomic force microscope. This is a reference instrument traceable to the International System of Units and dedicated to the practice of dimensional nanometrology. Its specific design allows a control of the measurement uncertainty. It is mainly used for the calibration of standards usually employed in the field of scanning probe microscopy or scanning electron microscopy.
Article
In order to obtain the high accuracy required for a metrological atomic force microscope, the sample approach mechanism meets strict specifications. The design presented in this paper offers a stiff construction, which limits the influences of floor vibrations on the measurement. Next to this, thermal considerations in the design decrease the uncertainties introduced by temperature variations of the environment. Uncertainties can also be caused by misalignment of the sample holder with respect to the measurement system of three interferometers. To limit these uncertainties, the approach mechanism provides sufficient alignment possibilities. The performance of the sample approach mechanism was evaluated by means of a finite element simulation of its dynamic stiffness. A series of experiments provide the unknown parameters to the simulation model. The dynamic stiffness lies around 395 Hz, which is sufficiently high to provide accurate measurements.
Article
A proprietary metrological scanning probe microscope (SPM) with an interferometer, developed by the Institute of Process Measurement and Sensor Technology at the Ilmenau University of Technology (IPMS), is used as a stationary probe system in the nanomeasuring and nanopositioning machine (NPMM). Due to the movements of the NPMM, the total microscope measuring range is 25mm × 25mm × 5mm with a positioning resolution of less than 0.1nm. Examples for specimens are step height standards and one-dimensional gratings. The repeatability has been determined at less than 0.5nm for measurements on calibrated step height standards and less than 0.2nm for the gratings. The measurement results of these samples are always directly related to the corresponding measurement uncertainty, which can be calculated using an uncertainty budget. A new traceable method has been developed using a vectorial modular model. With this approach, it is possible to quickly insert new sub-models and to individually analyze their effects on the total measurement uncertainty. The analysis of these effects with regard to their uncertainties is done by Monte Carlo Simulation (MCS), because some models have partially or fully nonlinear character of which one example is the interferometer model of the metrological SPM. The complete development and analysis of these models is presented for one specific measurement task. The measurement results and the corresponding measurement uncertainty were obtained by Monte Carlo Simulation. Comparisons with the GUM have shown that the proposed procedure is a good alternative to achieve reasonable measurement results with uncertainty estimation.
Article
A scalar model output Y is assumed to depend deterministically on a set of stochastically independent input vectors of different dimensions. The composition of the variance of Y is considered; variance components of particular relevance for uncertainty analysis are identified. Several analysis of variance designs for estimation of these variance components are discussed. Classical normal-model theory can suggest optimal designs. The designs can be implemented with various sampling methods: ordinary random sampling, latin hypercube sampling and scrambled quasi-random sampling. Some combinations of design and sampling method are compared in two small-scale numerical experiments.
Article
The theorem that an integrable function can be decomposed into summands of different dimensions is proved. The Monte Carlo algorithm is proposed for estimating the sensitivity of a function with respect to arbitrary groups of variables.
Article
A computational model is a representation of some physical or other system of interest, first expressed mathematically and then implemented in the form of a computer program; it may be viewed as a function of inputs that, when evaluated, produces outputs. Motivation for this article comes from computational models that are deterministic, complicated enough to make classical mathematical analysis impractical and that have a moderate-to-large number of inputs. The problem of designing computational experiments to determine which inputs have important effects on an output is considered. The proposed experimental plans are composed of individually randomized one-factor-at-a-time designs, and data analysis is based on the resulting random sample of observed elementary effects, those changes in an output due solely to changes in a particular input. Advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs, monotonicity of outputs with respect to inputs, or adequacy of a low-order polynomial as an approximation to the computational model.
Article
This paper describes modelling of an integrated AFM - CMM instrument, its calibration, and estimation of measurement uncertainty. Positioning errors were seen to limit the instrument performance. Software for offline stitching of single AFM scans was developed and verified, which allows compensation of such errors. A geometrical model of the instrument was produced, describing the interaction between AFM and CMM systematic errors. The model parameters were quantified through calibration, and the model used for establishing an optimised measurement procedure for surface mapping. A maximum uncertainty of 0.8% was achieved for the case of surface mapping of 1.2×1.2 mm2 consisting of 49 single AFM scanned areas.
Article
An atomic force microscope with a high-resolution three-axis laser interferometer for real-time correction of distorted topographic images has been constructed and investigated. With this apparatus, standard samples for a scanning probe microscope can be directly calibrated on the basis of stabilized wavelength of He–Ne lasers. The scanner includes a three-sided mirror block as a mobile target mirror for the interferometer, which realizes a rectangular coordinate system. The position coordinates of the sample is independently and simultaneously acquired with high-resolution (0.04 nm) X/Y/Z interferometer units and fed back for XY servo scanning and height image construction. The probe is placed on the sample surface at the intersection of the three optical axes of the interferometer with good reproducibility, so that the Abbe error caused by the rotation of the scanner is minimized. Image distortion in the XY plane and vertical overshoot/undershoot due to nonlinear motion of piezo devices, hysteresis, and creep are eliminated. The optical properties of the interferometers, mechanical characteristics of the scanner, and system performances in dimensional measurements for calibration standards are demonstrated. © 1999 American Institute of Physics.
Article
A metrological atomic force microscope (mAFM) has been developed at LNE. It will be dedicated to traceable dimensional measurements and calibrations of transfer standards with a maximum size of 25 mm × 25 mm × 7 mm. The displacement range is 60 µm for the X and Y axes and 15 µm for the Z axis. The instrument uses four laser differential interferometers in an original Abbe-compliant arrangement to measure the position of the tip relative to the sample and to be directly traceable to the SI. The expected uncertainty for the measurement of the tip/sample relative position is 1 nm for the whole range without taking into account the tip contribution. To fulfill this specification, the design of the instrument has been optimized to minimize Abbe errors, to reduce the metrology loop length, to limit the drifts due to thermal dilatation and to improve the stability of interferometric position measurement in ambient air. To limit Abbe errors, a dedicated three-axis flexure stage has been developed to reduce parasitic rotational motion at the level of 1 µrad for the whole range. This stage is driven by piezo-actuators. The instrument controller is based on a FPGA combined with an embedded PXI controller to perform real-time control of the XYZ position. We present the design of the instrument and the very first results.
Article
A long-range atomic force microscope (AFM) profiler system was built based on a commercial metrology AFM and a home-made linear sample displacement stage. The AFM head includes a parallelogram-type scanner with capacitive position sensors for all three axes. A reference cube located close to the tip acts as the counter electrode for the capacitive sensors. Below this metrology AFM head we placed a linear sample displacement stage, consisting of monolithic flexures forming a double parallelogram. This piezo actuated stage provides a highly linear motion over m. Its displacement is simultaneously measured by a capacitive position sensor and a differential double-pass plane mirror interferometer; both measuring systems have subnanometre resolution capability. For the measurement of periodical structures two operating modes are possible: a direct scanning mode, in which the position of the displacement stage is increased point by point while the AFM head measures the height, and a combined scanning mode where the displacement stage produces offsets which are multiples of the pitch to be measured while the AFM head is simultaneously scanning to locate an edge or a line centre position. Construction details, system characteristics and results from first pitch measurements are presented. The estimated relative combined uncertainties for pitch values on different standards are in the range to . Laser diffraction measurements of comparable uncertainty were performed on the same standards and show a very good agreement.
Article
An atomic force microscope (AFM) head designed for nanometrology is accomplished in this study. It is the sensing component of the nano-measuring machine, a nanometrological instrument with a working range of 50 mm × 50 mm × 2 mm, as well as a part of the metrological system of the instrument. Three reference mirrors are mounted on the head and arranged without Abbe error. Relative displacement of the AFM head and the specimen is measured by interferometers and results are traceable. The optical beam deflection method is used to detect the atomic force. The laser beam is introduced through a single-mode polarization-maintaining optical fibre from an external laser diode. With a compact design, a 100 mm optical lever is realized inside the AFM head that is less than 20 mm in thickness and 200 g in weight. A force–distance curve is obtained using a gauge block in a test. Furthermore, online tests of the measurement of a step scale have been made. According to the calculation and experimental verification, the resolution of our AFM head reaches 0.05 nm.
Article
As part of the development of a traceable scanning probe microscope (SPM) within the Dutch standards laboratory (NMi Van Swinden Laboratorium) novel methods have been developed for the calibration of this instrument. The SPM has been constructed using a commercial atomic force microscope (AFM) head that has been embedded in a metrology frame using an accurate 3D translation stage. The position of the AFM probe relative to the sample is determined along 3 orthogonal measurement axes by 3 individual laser interferometers. Since the AFM probe position is determined in 3D accurate calibration of the angles between the measurement axes is important if the measurement error is to be minimized to several nanometers. Different calibration setups to accurately determine these angles will be presented. Due to the properties of the 3D translation stage the Abbe offset between the probe position and the measurement axes should remain below 0.1 mm. Since the laser interferometers use a four pass optical configuration the measurement axis is defined by the virtual centre of the four positions of the laserspots onto the plane retro mirror. In order to accurately determine this virtual centre and therefore the position of the measurement axis a device has been designed to measure the four laserspot positions of each axis within the SPM. A method for the alignment of the optical axes of the laser interferometers with respect to the end mirrors of the interferometers will be discussed.
Article
Two-dimensional (2D) gratings are widely used for calibrating the xy-plane of nearly all kinds of microscopes. The mean pitch, orthogonality and local pitch uniformity of the 2D gratings have to be calibrated prior to usage. In this paper, a method of accurate calibration of 2D gratings using a metrological large-range scanning force microscope is presented. A new measurement strategy is proposed, where the 2D gratings are measured in two narrow rectangular areas for determining all desired measurands and a small square area for viewing the grids in detail. The proposed strategy greatly shortens the measurement time, reduces the drift and eases the data processing procedure. Different data evaluation methods are introduced. Several 2D gratings with mean pitches from 100 nm to 10 µm have been calibrated, the results agree well with the values determined by an optical diffractometer. The expanded uncertainty (k = 2) of the mean pitch was approximately 15 pm for a 2D grating with a nominal mean pitch of 1000 nm, i.e. a relative uncertainty of 1.5 × 10−5.
Article
Nanometre accuracy and resolution metrology over technically relevant areas is becoming a necessity for the progress of nanomanufacturing. At the National Institute of Standards and Technology, we are developing the Molecular Measuring Machine, a scanned probe microscope (SPM) and Michelson interferometer based metrology instrument, designed to achieve nanometre measurement uncertainty for point-to-point measurements over a 50 mm by 50 mm working area. The salient design features are described, along with example measurements that demonstrate the measurement capabilities so far achieved. Both long-range measurements of sub-micrometre pitch gratings over 10 mm, and short-range, high-resolution measurements of a molecular crystal lattice have been accomplished. The estimated relative measurement uncertainty so far attained for pitch measurements is 6 × 10−5, coverage factor k = 2. We have also used this instrument and scanning probe oxidation lithography for creating some simple nanometre dimension patterns that could serve as prototype calibration standards, utilizing the SPM probe tip positioning accuracy.
Article
A long-range scanning probe microscope (SPM) designed for the measurement of micro- and nanoscale forms, roughness and surface defects was constructed. It is based on commercial crossed roller bearing stages combined with piezoceramic actuators used to compensate the imperfections of the bearing mechanism. Three interferometers are used for all three-axis translation monitoring and feedback. For stage rotation monitoring (axis normal to the sample surface), an autocollimator is used. For nonplanarity compensation and two more axis rotation compensations (axes parallel to sample surface), an optical quality reference plane and a set of tunneling current sensors are used. The developed system enables us to perform large-scale measurements of the surface form with no influence of positioning system non-planarities and piezoceramic component hysteresis. In contrast to specialized metrology systems, e.g. using a six-axis interferometer for stage motion monitoring and feedback, this approach enables a more compact and much cheaper metrology SPM construction.
Article
A virtual scanning force microscope (virtual SFM) is established to assess the measurement uncertainty of SFMs, especially due to the SFM scanning apparatus and its non-ideal properties, to meet the increasing demand in industry and metrology. It builds numerical models for the individual errors from the instrument, the artefact or the environment, sequences the error models to simulate the measurement process, imitates the change of the measurement data during the whole measurement process to reveal the influences of these error sources, and calculates the measurement uncertainty from the statistical distribution of simulation results. The Monte Carlo method is applied to generate parameters within a given range of assumed values for the errors of the instrument, the sample and the set-up. In this paper, the principle and structure of the virtual SFM are introduced, its construction and simulation procedure are described, and a verification experiment is performed choosing a well-investigated instrument. The experiment result indicates that measurements can very well be simulated in the virtual SFM to obtain the measurement uncertainty if important parameters are determined.
Article
Microscope calibration standards in nanometrology were calibrated using a metrological atomic force microscope (metrological AFM) and the validity of calibrated values was shown. The metrological AFM was developed through the modification of a commercial AFM, which replaced the PZT tube scanner with flexure hinge scanners and displacement sensors. These modifications improved the traceability of measured values to metrological primary standards. The grating pitch and step height specimens, which are typical standard artefacts for the calibration of lateral and vertical magnifications of microscopes, were measured using the metrological AFM. The expanded uncertainties (k = 2) of calibrated values were estimated considering the characteristics of the calibration process and were less than 1 nm. The measurement results were compared with those obtained by other metrological methods or the certified values and their consistency was verified by checking the En numbers. These experimental results show that the metrological AFM can be used effectively for the measurements of microscope calibration standards in nanometrology.
Article
The National Physical Laboratory has recently developed a traceable measuring instrument that uses a stylus probe to measure topography in three dimensions. This paper presents the uncertainty evaluation associated with the new instrument. A sound mathematical model has been developed to describe the instrument geometry and functionality. Based on this model the uncertainty associated with the point co-ordinate measurement has been evaluated using a Monte Carlo technique, similar to the “virtual CMM” method. A short description of the instrument is presented along with the mathematical model and uncertainty calculation results.
Article
a b s t r a c t A new metrological AFM is developed for the Dutch standards laboratory. This instrument consists of a translation stage with a stroke of 1 Â 1 Â 1 mm and a custom-designed AFM measurement head. Here the design of the translation stage, consisting of elastic straight guides, Lorentz-actuators with weight and stiffness compensation and interferometric translation measurement systems, will be discussed. Some preliminary results on the performance of the actuation system are presented.
Article
The design of a large measurement-volume metrological atomic force microscope (AFM) is presented. The translation of the sample is accomplished with multiple stages which allow for separate 'coarse' and 'fine' motion. Interferometers and autocollimators are used to measure the position and orientation of the sample. The instrument does not attempt to control position via feedback from the interferometers, thereby allowing use of readily available commercial translation stages and controllers.
Article
The uncertainty associated with a value of some quantity is widely recognized throughout scientific disciplines as a quantitative measure of the reliability of that value. In addition, measurement uncertainty is increasingly seen as essential in quality assurance for industry. The Guide to the Expression of Uncertainty in Measurement (GUM) provides internationally agreed recommendations for the evaluation of uncertainties. This paper outlines the current situation of uncertainty evaluation in the context of international norms and arrangements. It describes the basic ideas and concepts that underlie the GUM and serves as a brief tutorial on methods for evaluating measurement uncertainty in a manner consistent with the GUM. It recommends an approach to evaluating measurement uncertainty based on the propagation of distributions using Monte Carlo simulation. An example is presented to illustrate Monte Carlo simulation.
Article
A Metrological Atomic Force Microscope (MAFM) has been constructed for the traceable calibration of transfer standards for scanning probe microscopy. It uses optical interferometry to generate image scales with direct traceability to the national standard of length. Three interferometers monitor the relative displacements of the AFM tip and sample in the x, y and z directions and the interferometer data is used directly to construct 3D images of sample surfaces. Traceable dimensional measurement of surface features may then be derived from the image data. This paper describes the MAFM instrument and presents a measurement uncertainty budget. Examples are given of measurements of pitch and step height on calibration transfer standards for scanning probe microscopy.
Article
An interferometrically traceable metrological atomic force microscope (IT-MAFM) has been developed at MIKES. It can be used for traceable atomic force microscope (AFM) measurements and for calibration of transfer standards of scanning probe microscopes (SPMs). Sample position is measured online by 3 axes of laser interferometers. A novel and simple method for detection and online correction of the interferometer nonlinearity was developed. Effect of the nonlinearity in measurements is demonstrated. In the design, special attention has been paid to elimination of external disturbances like electric noise, acoustic noise, ambient temperature variations and vibrations. The instrument has been carefully characterized. The largest uncertainty components are caused by Abbe errors, orthogonality errors, drifts and noise. Noise level in Z direction was 0.25 nm, and in X and Y directions 0.36 nm and 0.31 nm, respectively. Standard uncertainties for X, Y and Z coordinates are ucx = q[0.48; 0.04x; 0.17y; 1.7z; 2 time] nm, ucy = q[0.45; 0.31x; 0.07y; 0.14z; 4 time] nm and ucz = q[0.42; 3x; 7.2y; 0.18z; 2 time] nm where x, y, z are in μm and time in h. Standard uncertainty for 300 nm pitch is 0.023 nm,and for 7 nm step height measurement is 0.35 nm. Uncertainty estimates are supported by an international comparison.
Article
The practical realization of the definition of the metre, including recommended radiations of other optical frequency standards (2001) was presented in the article. The CIPM acknowledged the considerable effort put into its preparation by the Consultative Committee for Length (CCL) through its Working Group on the mise en pratique that included representatives of national metrology institutes and the BIPM. The list of recommended radiations given by the CIPM in 1997 be replaced by the list of radiations including, updated frequency values for cold Ca atom, H atom and the trapped Sr+ ion.
Article
This paper presents the state of the art in scanning force microscopy for dimensional metrology. A detailed description is given of the important factors affecting the major components of a scanning force microscope from the metrological point of view. Both instrument design and calibration are discussed together with an overview of industrial applications. Recent achievements by national metrology institutes and others to improve calibration procedures, traceability, reduce measurement uncertainty, ensure consistancy of measurement and broaden the range of applications are described.
Article
Our prototype atomic force microscope (AFM) uses a piezoelectric tube to scan a probe tip over a sample surface, while a PC-based digital controller maintains a constant tip–sample separation based on feedback from a quartz tuning fork proximity sensor. We have successfully run the AFM in both the shear and tapping modes, using optical fibers as probe tips. The AFM utilizes a set of capacitance sensors with a spherical target for direct measurement of probe tip displacements. We have used this system to image localized surface topography of a square wave silicon calibration grating with localized step height accuracy of 1 nm in the vertical direction and vertical RMS noise less than 4 nm. The lateral accuracy is on the order of 100 nm, and our largest lateral measurement scans have been performed on regions with step heights of 26.5 nm and a modest scan speed of .
Article
Variance based methods have assessed themselves as versatile and effective among the various available techniques for sensitivity analysis of model output. Practitioners can in principle describe the sensitivity pattern of a model Y=f(X1,X2,…,Xk) with k uncertain input factors via a full decomposition of the variance V of Y into terms depending on the factors and their interactions. More often practitioners are satisfied with computing just k first order effects and k total effects, the latter describing synthetically interactions among input factors. In sensitivity analysis a key concern is the computational cost of the analysis, defined in terms of number of evaluations of f(X1,X2,…,Xk) needed to complete the analysis, as f(X1,X2,…,Xk) is often in the form of a numerical model which may take long processing time. While the computational cost is relatively cheap and weakly dependent on k for estimating first order effects, it remains expensive and strictly k-dependent for total effect indices. In the present note we compare existing and new practices for this index and offer recommendations on which to use.
Article
A sample scanning device operating in a working volume of 30 x 30 x 18 microm with interferometer and capacitance-based controls of displacements, is described. The xy-stage uses plane mirror linear interferometers and fast phase-meters for control of displacements of precise ball-bearing stages driven by piezo flexure actuators. The stage operates with a full range bandwidth of 200 Hz, and an estimated accuracy (k = 2) of 3 nm + 1 x 10(-3) L, where L is the lateral displacement. A novel z-stage based on a kinematic coupling between two plates, the upper one being moved by three bimorph plates and the distance being measured by three capacitive sensor, is described. The tilt of the z-stage is kept within fractions of a microrad, leading to a full range estimated accuracy of 2 nm + 2 x 10(-3) h, where h is the vertical displacement. The control bandwidth is of about 1 kHz, thus allowing fast and accurate step-height measurements. In order to test the device used in a scanning probe microscope, micrometric patterned surfaces made using high resolution e-beam lithography and precise metal deposition on silicon are imaged. Results of pitch measurements are discussed and compared with those obtained using optical diffractometry.
Conception d'un microscope à force atomique métrologique PhD Thesis Ecole doctorale Société du Future
  • B Poyet
Poyet B 2010 Conception d'un microscope à force atomique métrologique PhD Thesis Ecole doctorale Société du Future, Versailles
The sensitivity Package R package version 1
  • G Pujol
  • B Iooss
  • A Janon
Pujol G, Iooss B and Janon A 2015 The sensitivity Package R package version 1.11.1 (https://cran.r-project.org/web/ packages/sensitivity/index.html)
  • G Dai
  • F Pohlenz
  • H-U Danzebrink
  • M Xu
  • K Hasche
  • G Wilkening
Dai G, Pohlenz F, Danzebrink H-U, Xu M, Hasche K and Wilkening G 2004 Metrological large range scanning probe microscope Rev. Sci. Instrum. 75 962–9
Evaluation of Measurement Data— Supplement 2 to the 'Guide to the Expression of Uncertainty in Measurement'—Extension to
JCGM 102 2011 Evaluation of Measurement Data— Supplement 2 to the 'Guide to the Expression of Uncertainty in Measurement'—Extension to Any Number of Output Quantities (BIPM)