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Measuring the Thickness of Metal Films: A Selection Guide to the Most Suitable Technique

The determination of the thickness has a fundamental importance in all the fields in which the implementation of films and coatings are required and takes a crucial role in electroplating sector. The thickness influences many aspects of the coatings such as electrical, mechanical, corrosion protection and even aesthetical properties. In the multitude of applications of thin layer coatings, the variability of thicknesses and materials is very high as well as the possible techniques that can be used to determine the characteristics of the layers of interest. The first distinction that can be made between these techniques is that which divides destructive techniques from nondestructive ones, in which however the semi or micro-destructive techniques are immediately difficult to place. Other important parameters to consider are the cost, both for the purchase of the instrumentation and for each single analysis, the difficulties in preparing and measuring the sample and data processing and obviously the detectable thickness ranges, the possible measurable materials, precision and accuracy. The purpose of this work is to compare the characteristics of the various investigation methods, with a particular focus on metal films applications, so that it will be easier to choose the most suitable technique for each purpose.


Introduction
The use of thin films has become a ubiquitous practice in many scientific and industrial sector. Coatings are widespread used to obtain a synergistic action between the characteristics of the substrate and the covering material to improve the physical, chemical and aesthetic properties and to lower the costs of the final product. For this reason, the measurement of the thickness in composite materials is mandatory both to obtain the right characteristics in the final artefact as well as to keep the costs under control. The composition of the films could be extremely vast: dielectrics (organic, such as polymers and self-assembled monolayers (SAM), or inorganic, like metal oxides), semiconductors and metals are all used in the form of films, obtaining a composite material with combined characteristics. In this work, we focused on the metal film characterization obtained through electrodeposition or vapour phase deposition but in most of the cases the same principles can be applied to films of different materials. As far as the films dimensions is concerned, the thinnest measurable thickness coincides with an atomic monolayer (ML), while the thicker layers could reach hundreds of microns in electroforming. Therefore, in this review we made an overview of all the techniques that allow to investigate in this range.

Mechanical Cross-Sectioning
The cross-section technique is very widespread and relatively simple to use even if it requires particular attention and manual skill of the operator, in fact if not executed with attention it can lead to inaccurate measurements. It consists in cutting the sample in half and observing it transversely, along the profile of the layers whose thickness must be measured. This technique therefore allows to directly measure the thickness of the film through a ruler. The entire analysis process consists of three steps: sample preparation (cutting, embedding and lapping); microscopic analysis, through optical or electronic microscopy; data processing, through dedicated software to convert the image dimensions from pixels into a unit of length through a scale.
The sample preparation process is the most critical and time-consuming step; it is during this stage that artefacts that could invalidate the subsequent analysis could be generated. Usually the sample is sawn using a disc cutting machine with the use of an abrasive resin disc or a diamond disc but, if the sample allows it, it can also be cut in other ways such as with a cutter or scissors. The cut must be made perpendicular to the surface, otherwise the thickness analysed will be overestimated due to the parallax error. In addition, the cut must neither be too fast nor overheat the sample in order to avoid damaging or detaching the film, for this reason many saws are equipped with a liquid cooling system with a direct jet on the sample.
If our interest is to accurately measure the thickness of the outermost film, and this is very thin, the cut could compromise the analysis. To avoid this problem, if the sample is conductive, it can be covered with a galvanic deposition of a few microns in order to protect the layer of interest. Alternatively, the sample can first be incorporated in resin and then cut later but in this case, it is more complex to perform a cut perpendicular to the surface. Moreover, the hardened resin tends to shrink, remaining not perfectly adherent to the sample and reducing the protection of the external film.
The incorporation can be performed with both hot melt and cold resins. The sample is placed in a housing ensuring that it is perfectly level and the resin is poured being careful not to leave air bubbles. The resin used can be both conductive and non-conductive. If the sample will be analysed under an optical microscope, there is no difference, therefore non-conductive resins are preferred because they are cheaper. On the other hand, if it will be carried out electronic microscopic analysis, a conductive resin is preferable to avoid polarization phenomena, but a non-conductive resin can also be used if it is possible to graphitize the sample.
The last step in sample preparation is lapping. Near the cut the sample will inevitably be damaged and the films will be altered: there could be both the peeling, meaning the detachment of the less adhered films, and a spreading of the softer films. The lapping process is the longest phase, it can take several hours, and consists in polishing the section of the sample with gradually finer abrasives until a mirror surface is obtained. The surface roughness and scratches must be minimized in order to accurately measure the thicknesses, as a rule of the thumb the surface roughness, and consequently the abrasive grain size, must be in the order of magnitude, of the thickness to be analysed or less, generally 1-0.3 micron. The lapping process can be carried out by hand or using special automatic lapping machines. A wet abrasion is carried out first using sandpaper, gradually finer and then with a cloth soaked in an abrasive suspension. The suspended particles can be alumina or diamond dust. Initially, with a sufficiently coarse sandpaper, all the part that was damaged during cutting is removed, about 1 mm in depth; then the mash size of the sandpaper is decreased. A fine grain size is not used immediately because otherwise it would not be possible to remove the deepest scratches. In the event of lapping by hand, the sample must not be pressed too hard and care must be taken not to consume one side more than the other to avoid introducing a parallax error. For this reason, it is advisable to rotate the sample in the same direction as the lapping machine disk and occasionally turn it of 90° with respect to its normal axis. When you think that the sample is ready, it is washed and inspected visually or, if available, under an optical microscope to check that all the scratches have been flattened, if the sample is sufficiently smooth, the next phase of microscopic analysis can be carried out. For example purposes, a microscopic analysis of the same cross-section is shown in Figure 1, where a comparison between optical and electron microscope have been carried out image. The multilayer sample is made of brass/Cu/bronze/Pd/Au/Ni. With the optical microscope it is possible to distinguish most of the layers, even the thin gold layer is visible although not quantifiable, but it is not possible to discriminate between the bronze and palladium layers. The layer with thickness over 500 nm can be measured with the optical microscope but for thinner coatings, and to distinguish brass and palladium the electron image is necessary. Figure 1. Cross section multilayer sample (brass/Cu/bronze/Pd/Au/Ni) observed with optical microscope (top) and electron microscope (bottom).

Ion Beam Cross-Sectioning
Cross sectioning procedure can be performed also by focused ion beams. By exploiting the sputtering effect of FIBs, it is possible to raster a surface producing precise trenches, useable to observe the in-depth evolution of the sample [39]. The trench dimensions can vary from mm (for a Xe or Ar plasma FIBS) to less than 1 µ m, whit a maximum depth in the range of hundreds of microns. Due to the small dimensions of the holes produced on the surface, this process (unlike mechanical cross sectioning) can be considered as semi-destructive. This process is also fast in respect to mechanical cross sectioning, because it permits to produce clear cuts from which is possible to characterize the profile of a sample in 20-60 minutes (depending on the size of the hole). For these reasons, it is particularly well suited for the characterization of thin films having thicknesses below 10 nm. The subsequent characterization step can be performed by SEM or SIM microscopy, or EDX analysis due to the small hole dimensions.
Even if the cross-sectioning process can be performed only by using a FIB for a coarse determination of thick films, films down to 10 µ m will require the use a FIB/SEM equipped with a Gas Injection System (GIS). This because, in order to avoid FIB-induced surface degradation, a thin protective layer above the surface is needed. This can be achieved only by using a beam deposition process achievable by the presence of a GIS. In FIB/SEMs the protective layer is produced by a two-step deposition process. First, a thin layer of metal is deposited on the surface using the electron beam (e-beam deposition). Then, a thicker metallic layer is deposited using the ionic beam (i-beam deposition). The e-beam deposition avoids surface degradation (and thus loss of thickness information on the topmost layer) due to direct impingement of the ionic beam on the surface [40]. The full workflow for the preparation a cross section using a FIB can be visible in Figure 2.

Angle Lapping
Angle Lapping is a sample preparation method used to increase the resolution in thin film thickness determination of the adopted microscopy characterization technique; it is based on a change in cutting geometry in respect to cross-sectioning (Cap.2.1). During traditional preparation procedures, in order to unshed the stratigraphic information from the sample, the cutting plane is perpendicular to the surface (θcut = 90°) [41]. The uncovered section gives direct stratigraphic information on the displacement and thicknesses of the layers above the substrate. Instead, in Angle lapping, the sectioning cut is performed at very low angles in respect to sample surface (θcut < 10°). This produces a "magnification effect" on the newly created section surface, on which all the layers appear stretched. Knowing the cutting angle, and a bit of trigonometry, is consequently easy to derive the film thickness. The main advantage of this method is the magnification effect, which allows to overcome resolution limits of the microscopic technique adopted for the quantification, even for very thin films. As a prerequisite, a very flat film surface is mandatory for a precise determination of the thickness of the layers underneath; moreover, particular care must be put in preparation of the surface after the cut. It is uncommon, especially for mechanically machined soft materials, to incur into thin film deformation onto the cutting surface [42]. The origin of this sample preparation technique comes from the first metallographic studies, and it is still considered as a valuable method to overcome the resolution limits of the adopted microscopic characterization methods, and its adoption has shifted to FIB cross-sectioning or lamella preparation, enabling fine characterization of ultrathin films. It can be used before both optical microscopy and SEM characterization.

Calo Tester
The Calo tester, also known as ball craterer or crater grinding, is a semi-destructive technique that is not very widespread but extremely practical in some cases, in addition it is regulated in ISO 26423 [45] (Ex EN-1071 and VDI 3198 [46]). Compared to the cross section, it has the advantage of being only locally destructive, therefore the sample, instead of having to be cut in half, is excavated in an area with a diameter of about one millimetre [47,48]. Furthermore, the whole analysis is much faster as the sample does not have to be incorporated and lapped. This technique consists in fixing the sample on a variable angle support; on it is placed a steel sphere covered by an abrasive suspension; the sphere is in contact also with a rotating cylinder that makes it roll (Figure 4). Within a few minutes, depending on the hardness of the sample and its angle of inclination, that is translated to the weight that the sphere impresses on it, a circular crater will form revealing all the layers [49]. Since the abrasion angle is very low (due to the diameter of the sphere), layers of even a few microns will have a much greater apparent size and can be appreciable under a normal optical microscope. Obviously, as in the case of the cross section, only the layers having a different colour can be distinguished. Once the diameters of the concentric circles that have been created on the sample have been measured, using a dedicated formula ( Figure 4) that takes into account the diameter of the abrasion sphere, the real thickness of the films can be obtained. Manufacturers ensure that the range of thicknesses that can be measured are between 50 and 0.1 microns.

TEM Lamella Preparation
The ultimate procedure for thickness determination of thin films is represented by the TEM lamella preparation process. It is in fact possible to extract a small portion of the sample surface, usually a 10 × 5 × 1 micron solid, containing the surface cross section. This lamella can then be thinned down to less than 100 nm in order to be observed transversally by TEM or STEM [43,44]. Due to This preparation process is important for the characterization of very thin films (below 1 µ m) using SEM and TEM. The small thickness of the lamella decreases abruptly the interaction volume, cutting down the signal coming from in-depth SE and BSE. To prepare a TEM lamella, a FIB/SEM equipped with a GIS and a nanomanipulator is required. Moreover, the lamella preparation process is quite complex, and is constituted by numerous steps that can vary depending on the load-out of the adopted machine and the geometry of the sample chamber. Generally, the workflow can be divided in three main stages: In the first stage the lamella is shaped (carved) directly onto the surface of the sample. In the second stage, called lift-out, the lamella is detached from the sample, and is mounted on a TEM support grid. In the third stage the lamella is finally thinned down to enable transmission electron analysis. This last process is crucial for the obtainment of defect-free lamellas. In Figure 5 an example of a workflow for a Tescan GAIA 3 FIB/SEM is shown: a) a raw lamella (about 10 × 5 × 1 microns) is carved on the surface, b) an undercut is performed in order to detach the lamella, c) nanomanipulator is moved on a side of the lamella and soldered to the body, then the lamella is detached from the surface, d) e) and f) lamella is moved from the surface of the sample to the TEM support, g) lamella is soldered to the support, h) nanomanipulator is detached from the lamella, i) lamella is thinned down until electrontransparence, j) in the end lamella quality is tested using the microscope in-built STEM detector.

Optical Microscopy
In order to measure a cross section of a sample it is necessary to use the microscope in reflection mode; additionally, to be able to recognize the different layers it is necessary that they have a sufficiently high contrast, or more simply different colours. Distinguishing different layers made, for example, of the same silvery metals, is impossible, while it is very simple to measure, for example, a silver film on copper or brass.
The lateral resolution, i.e. the minimum distance between two resolved points, is defined by the Abbe principle. The resolution is related to the wavelength of the source used, in this case the visible photons. The theoretical resolution of an optical microscope, not taking into account optical aberrations, using white light is about 0.2 microns. However, more realistically, the measurement of films with a thickness of less than one micron is difficult and with a consistent uncertainty [50][51][52][53].

Electron Microscopy
Scanning electron microscopy (SEM) is used when the thicknesses are too small to be analysed with the optical microscope or if the layers have too low contrast among them. As a general rule, if the cross section of a sample can be adequately analysed with an optical microscope, there are no valid reasons to use an electron microscope, in fact, the sample needs some characteristics in order to be measured with an SEM; moreover, SEM analysis is intrinsically more expensive than a simple optical analysis. Despite this, for most coatings optical microscopy is not enough to obtain adequate results. A sample that is analysed by SEM must be stable in high vacuum (about 10 -7 bar), must be stable when irradiated by an electron beam and must be conductive. These limitations can be circumvented by means of some tricks: there are specific environmental or low-vacuum SEM which allow analysis to be carried out with a pressure in the order of one Pascal; the acceleration potential of the electron beam can be reduced to a few kV in order not to damage the sample, but decreasing the signal-to-noise ratio; non-conductive samples can be graphitized to avoid the accumulation of surface charges that in this way are dispersed to ground. However, these problems arise when biological, organic or polymeric samples are analysed, for metallographic cross-sections there are rarely complications of this type.
Numerous advances have been made in the last few decades in the field of electron microscopy and the resolution of these instruments is quite variable and ranges from about 20 nm for older instruments with thermionic emission, up to falling below the nanometre scale for new instruments with field emission source (FEG-SEM). In SEM, the main limit to the resolution is not so much the wavelength of the probe, the electrons, as the diameter of the beam and therefore its focus and collimation. For this reason, a stratagem to improve the resolution consists in bringing the sample closer to the source, reducing the working distance and consequently the opening of the electronic cone. The SEM images can be acquired using secondary electrons (SE), backscattered electrons (BSE) or through a microanalysis map (EPMA). Because of the different nature of the signal they differ in the depth from which the signal comes (Table 1) and in their volume of interaction ( Figure 6). A higher volume of interaction results in a lower lateral resolution and translating in a less clear separation of the edges between the films that have to be analysed [1,13].  SEs are produced when a primary electron, from the beam, excites an electron of the atoms of the sample to the point of tearing it from the nucleus, these electrons are low in energy (<50 eV) and only those generated most superficially on the surface are detected. SEs carry the morphological information with them, so the contrast in the image is based on the heights of the sample. Since our sample has been levelled, the contrast produced can only be due to the different nature of the material. The SE in small part also contain information also on the composition since the quantity of electrons emitted is proportional to the atomic number of the element under the electron beam. Thus, layers of elements with significantly different atomic numbers can be distinguished [54,55]. A trick that allows to observe similar elements with the SE is to chemically etch the sample before the analysis, to slightly corrode one material compared to another, in this way morphological differences, clearly visible to the SE, are reintroduced between the different layers. However, the etching must be studied appropriately with respect to the sample that have to be analysed.
BSEs are the primary electrons which, interacting with the positive nuclei of the atoms of the sample, are back scattered towards the source, for this reason they are very energetic and penetrating. Therefore, BSE carry mainly compositional information: heavier elements generate more BSE. By sacrificing part of the lateral resolution, the image obtained will be much more contrasted by highlighting differences between the layers that can be below an atomic number unit for more modern systems, allowing in some cases to distinguish even deposits of a metal from its alloy (for example copper on brass).
Finally, the EPMA signal can be used. This is the signal with the highest volume of interaction and for this reason the lateral resolution is very poor when compared with the previous ones. In this case, to be analysed are not the electrons but the X photons that are emitted from the sample after the interaction with the primary electrons, which ones contain detailed information on the composition. With EPMA, alloys can be distinguished which vary in composition by a few percentage points. Then the different layers can be highlighted by making a map, or a linear scan perpendicular to the layers. However, given the low lateral resolution, films just below the micron are difficult to quantify.
Another trick to increase lateral resolution is to reduce the volume of interaction and this is possible by decreasing the energy of the electrons by lowering their acceleration potential. This greatly reduces the signal but with modern instruments it is possible to obtain good images using acceleration potentials of only a few kV. This strategy is very limited, however, in the case of the EPMA since, in order to have a good emission of the X photons, the electron beam must have indicatively an acceleration potential once and a half times the energy of the emission peak of interest.

Focused Ion Beam Methods
Another series of techniques enabling fast characterization of thin metallic films relies on the use of focused beams of ions. Ionic probes, display different beam-matter interaction in respect to electrons, resulting in different effects of the impinging beam on the surface [56][57][58][59][60][61]. When a focussed ion beam (FIB) hits a surface, it can generate a series of different effect in respect to an electron beam; all these effects are a consequence of elastic or inelastic collisions between the charged particles and the atoms forming the surface. The most important interactions can be listed: a) Surface sputtering and secondary ion emission, b) deformation of the surface reticule, c) ion implantation, d) emission of electrons, e) emission of electromagnetic waves and f) heating. All these effects manifest at the same time, but their ratio is strongly dependent on the physicochemical nature of both the probe and targeted surface, and on the apparatus set-up. For a certain ion source, parameters such beam energy, beam current, and impact angle can be tailored to favour one phenomenon in respect to the others. Among all the ion beam-matter interactions, two are widespread used for the characterization of thin metallic films: a) the production of electrons [62] and c) the sputtering and secondary ion emission [63] . The first one exploits a signal which is similar to the one responsible for SEM images, while the second permits to perform subtractive manufacturing onto the surface. FIB apparatus could in fact perform both sample preparation and analysis in a single workflow, by preparing the surface cross section and acquiring its images, though not all the FIB columns are fitted to perform well both the processes. As already mentioned, different elements could be used as ion sources; their charge/mass ratio could modify deeply the ratios at high sputtering and SE production manifest. Today, the main commercial FIB machines are engineered to work using He, Ga, Ar or Xe ions [64]. Smaller ions, like He, are well suited for imaging purposes, while bigger ions like Ar and Xe are used for fast sputtering; Ga FIB machines are considered a good trade-off between the two effects and are still considered as the best choice for all purpose tasks.
From a technical point of view, FIB machines are very similar to SEMs. In both of them, we have a column that can be divided in: a) a top part composed by the source, responsible for the ion/electron generation and b) a bottom part, the focusing apparatus, a series of electrostatic/electromagnetic lenses, condensers and apertures to focalize and control the beam [57]. In FIBs, the working principle exploited for ion emission varies depending on the physicochemical properties of the element of choice; we could divide the sources in three main branches: Gaseous Field Ionization Sources (GFISs) for He [65], Liquid Metal Ion Sources (LMISs) for Ga [57] and Plasma sources for Ar and Xe [66,67].
Despite SEMs, in FIBs the focusing apparatus is composed by electrostatic elements (lenses, condensers, etc.) rather than electromagnetic. Ions in fact suffer weakly by Lorentz forces due to their slower travelling speed (in respect to electrons) in the column, meaning also that these instruments are less prone to suffer from stray external magnetic fields. Ion sources are not interchangeable; the focusing apparatus must be finely engineered to suit for the particular physical properties of the source element. This means also that the choice between different FIB machines, exploiting different sources, must be carefully planned in respect of the need. Historically, industrial FIB columns were intended as standalone instruments, these devices were diffused especially in the semiconductor/ electronic industry, where their use both for quality check and prototyping falls back from the '70s. Today FIBs can be found easily when paired whit a SEM column [68]. This double column configuration allows a vast array of different procedures for the characterization of materials in the range of mm down to the nm for all R&D fields. FIB/SEMs can be equipped with a vast array of accessories and sensors to enable different characterization techniques. Among the available accessories, the gas injection system (GIS) has a particular relevance, because permits additive manufacturing in the range of tens of nm using both the ionic and the electronic beams. The GIS is constituted by a series of external reservoirs containing the precursors of the elements we want to deposit onto surfaces, which are heated to produce a reactive gas. This gas is injected by a hollow needle in the vicinity of the surface of the sample, where it forms a cloud between the beam and the surface. The beam particles hitting on the precursor produce the degradation of the molecules, followed by precipitation of the elements onto the surface. It is possible to use both the electron beam (e-beam deposition) or the ionic beam (i-beam deposition) [69] to achieve deposition. In the following paragraphs the main techniques which can be exploited for metallic layer thickness determination will be presented: an analytical technique (Scanning Ion Microscopy) and two preparation techniques (TEM lamella preparation and angle lapping). All the FIB methods are well suited for the characterization of metallic films of thicknesses ranging from 50 µ m to 10 nm.

Scanning Ion Microscopy
Scanning Ion Microscopy (SIM) can be used as an alternative to SEM for the study of cross sections obtained by mechanical methods. The main advantage in using charged atoms instead of electrons is related to the smaller ion mean free path inside the matter [65,70]. This effect is responsible to a smaller beam-surface interaction volume, to a higher number of surface SEs, and thus to crispier images. Moreover, the number of SEs produced per impacting Ion is much bigger in respect to the number of SEs produced by the impact of an electron, greatly enhancing the brilliance of the signal [71]. Ions are also particularly sensitive to crystal orientation; images acquired by this method tend to possess strong crystalline contrast due to enhanced channeling effect [72]. Ionic imaging can be performed using the same SEM SE detector, whit a lesser lateral resolution in respect to electrons.

Data Analysis
Once the image is obtained with a microscopic technique, the pixels must be converted into a unit of length. Most of the software that allow to acquire microscopic images (both optical and electronic) commonly have a tool for the extraction of this information, otherwise there are free or paid software that allow to do the same job as ImageJ, Gimp and Adobe Photoshop to name the most known. In order to convert the pixels to length on the image, a reference scale must be printed on it, through that scale we can get all the dimensions of interest in the image. The thickness of the films can therefore be measured, being careful to carry out the measurement perpendicular to the film. By making the measurement in several points, a statistic of the thicknesses can also be carried out. However, in the event that the edges of the film in the image are not very defined, finding the limits by eye could be complex, as showed in the blur image of a Au film reported in Figure 7. In these cases, instead of analysing the image, it is more practical to observe the profile graph in which the greyscale values are reported. If the different layers have contrast between them, they also have a different grey value; the delimitation between one layer and another can be defined as the point where the value is intermediate between the two layers or, more rigorously, at the inflection points of the graph. In this case the spatial resolution, and therefore the uncertainty of the measurement, is defined as the distance between the points where the variation of the grey value is in the range 20%-80% [73], as required by ISO 18516 [74]. On the other hand, an incorrect method to make the separation of the layers more defined is to digitally postprocessing the images by acting on brightness and contrast. In fact, contrary to varying these parameters during the measurement, by performing software alterations of the image, the edge of the film can move as the different shades of grey are processed, ad demonstrated in Figure 7.

X-Ray Fluorescence Spectroscopy
X-ray fluorescence spectroscopy (XRF) is an analysis tool widely used for the elemental analysis and chemical analysis of materials. When materials are exposed to high-energy X-rays, ionization of their component atoms may take place exciting them, during the relaxation process characteristic X photons are emitted and detected for analysis. Due to incident high-energy X-rays the inner shell (K, L, M, etc.) transition phenomena occurs within 100 fs producing a characteristic fluorescence radiation. Ionization consists of the ejection of one or more electrons from the atom and may occur if the atom is exposed to radiation with energy greater than its ionization energy. X-rays and gamma rays can be energetic enough to eject tightly held electrons from the inner orbitals of the atom. The removal of an electron in this way makes the electronic structure of the atom unstable, and electrons in higher orbitals "fall" into the lower orbital to fill the holes left behind. In falling, energy is released in the form of photons with an amount of energy equal to the difference between the two orbitals involved. Thus, materials emit radiations of the characteristic energies of the present atoms. A variety of samples in different states, such as solids, powders, and liquids, can be analysed using this technique. It can also be used to measure the composition, thickness of coating and layers. The characteristic photons of the sample are collected by a detector that uses the same working principle of EPMA. Both the source photons as well the emitted ones could pass through an analysing crystal that act as monochromator differentiating between energy dispersive (ED) XRF without analysing crystal; wavelength dispersive (WD) XRF, in which the emitted photons are selected with a monochromator; monochromatic wavelength dispersive (MWD) XRF, in which two optics are used: one for the source and one for the emitted photons. For reasons of cost and ease of use, energy dispersion instruments are the most used. The incoming high-energy beam is very penetrating Figure  8, for this reason the maximum detectable thickness is related to the energy to the emitted X-rays. XRF is the most common instrument used by industries for the film thickness investigation the since it is fast, non-destructive and relatively simple to use, making it perfect for the quality control of the products [75,76], and for this reason there are present also standard procedures to perform the measurement like ISO 3497 [77] and ASTM B568 [78]. Commercial instruments can measure easily the thickness of almost every material (with some restriction for lighter elements), whether conductive or not, in the range from 10 nm to 100 µ m [79,80]; nevertheless, depending on the materials under investigation and the instrumental settings, the limits of measurement could be extended from less than 1 nm [81] to a few centimetres [82]. The lateral resolution of XRF very low and spot size commonly ranges from 0.1 to 15 mm. The relative intensity (normalized respect to bulk element) of an emitted from a film follow an exponential trend [18], but could be approximated to a second order curve for small far from the saturation thickness [7]. The emission of a gold film on Cu substrate, as function of the thickness, is reported in Figure 9 using a log-log scale. In this case the range of thicknesses between the relative intensities 0.9 (semi-infinite thickness [83]) and 0.1 (infinitely small) gold is between 0.2 µ m and 50 µ m. The output of the instrument is a spectrum in which the position of the peaks corresponds to the spectroscopic emission of the elements present in the sample while the intensity is correlated to the sample composition in the volume of interaction of the incident beam. For this reason, there is no direct information on the thicknesses, but the intensity of the peaks in the spectra will be function of thickness. In fact, a sample with a thicker coating will emit more photons from the film and less from the substrate than a thinner one. Since no information about the thickness can be extracted a priori from the spectra, only with the right assumptions on the nature of the sample and the use of an adequate calibration curve, the thickness information can be deconvoluted: this complication could bring to high uncertainties or even wrong results.
Deriving the coating's thickness from the X-ray spectrum requires an experimental calibration curve that employs standards; however, due to the large dependence of the X-ray spectrum on the nature of the coating and the substrate, standards are not always available. The variability of thickness, layer composition, multilayer architectures and substrate chemical nature create difficulties in producing certified standards. This issue is critical in industrial applications, indeed the determination of precious metal coatings in the fashion industry is a major one where the products are made with many coatings and substrates, with extreme variability in the system. Calibration curve obtained with standards of known thickness was used to measure vanadium(V) oxide nanometric films on glass with portable XRF measuring the attenuation of Ca emission [5]. Hamann [84] was able to detect fraction, up to 1 %, of a monolayer of over 20 samples without the use of standards or models, combining WD-XRF and XRD measurement to obtain the proportionality constant between X-ray emitted intensity and the number of atoms per unit area.
Nowadays, the most common approach is the use of the fundamental parameter (FP) method [6,79,85,86]. FP relies on theoretical equations that consider the composition and thickness of the sample to evaluate the XRF intensity. Practically, the FP method is combined with a few pure element empirical standards to correct unpredicted deviations due to matrix effects [87,88]. With the FP method, it is possible to determine the film thickness of single and even multilayer samples if the structure and the composition are known exactly; nevertheless, the error correlated to the measurement is significant. Typical accuracy for single layer samples is ±5 %, while for multiple-layer samples this value grows to ±10 % for the upper layer and ±37 % for the first underlayer [89][90][91] due to inaccuracy in the method for complex samples. Additionally, very often the thickness and composition of the underlying layers in multilayer architectures are not exactly known and they are introduced in the measurement software using an initial estimation [92]. The FP method was investigated by many authors for multi-layered samples in the micron range (Au/Ni/Cu [89]) as well as in the nanometres range (Ni/Cu/Si [93]). This method is very useful when it is difficult to obtain accurate certified reference materials for layer thickness calibration, such as in the case of semiconductor research [94]. Vrilink [90] showed a good correlation between FP and SEM and profilometry measurement of multilayer samples with different composition (Rh, Ta, W, Ti, Pd, Pt, Ni, Au, Cr) between 20 and 250 nm, considering the density variation for thin film. Ager [92] highlights discrepancy between SEM and non-destructive techniques like RBS and XRF measurement due to differences in density between bulk metal and thin film due to porosity; in the paper comparison between references and electroplated samples were performed to prove the hypothesis for ancient gildings but his considerations are valid in many other fields. Exploiting the FP both the emission line of the top layer as well the reduction in intensity of the underling layer can be used for the thickness determination, as shown in a study of 2017 in which the results of ALD oxides samples are tested [83].
An alternative to the use of standards and FP method, consists in a semiquantitative approaches based on calibration curves obtained with a simulation software using Monte Carlo (MC) algorithms. During the simulation, when the materials and the architecture to simulate are chosen, it is also possible to specify the density of the materials; in this way the user can decide to simulate materials that have a porosity different from the nominal one due to the deposition method, as for example happens during electroplating in which the density of the coatings is often lower than that of the bulk material. Moreover, the MC method simulates X-ray spectra using a statistical approach that counts the photon interactions in the sample. With this approach, inhomogeneities of the sample, spectral and spatial distribution of the beam, polarization effects, photo-absorption, multiple fluorescence, and scattering effects, which are difficult to model with the FP method, can be considered. The simulation approach is not very common, probably because the FP was preferred for many years since it was computationally favourable, but with the last technological development even a personal computer could arrive to obtain a good simulation in relatively small amount of time. The two main software that provide a simulated spectra with MC approach are: XRMC [95] and XMI-MSIM [96]. Both codes use the Xraylib database [97,98]. XRMC is generally used for complex 3D geometries while XMI-MSIM can only simulate samples composed of parallel layers, but for simple geometries XMI-MSIM is currently superior to XRMC in simulating XRF experiments [99]. Thickness evaluation using the MC method is diffuse in the field of cultural heritage applications. Schiavon [100] use XRMC code to obtain the thickness and composition of Nuragic manufacts comparing the simulations with the experimental measurements to confirm hypotheses based on bulk chemical composition, structural observations, and historical information. A similar approach was used by Brunetti [101] ad Bottaini [102] for Peruvian and Portuguese manufacts: a MC simulation is performed defining the experimental setup and the sample, then the simulate spectra is compared to the measured one visually and with the chi-squared test. if differences are found, the model is corrected until the two spectra matches determining both the composition and structures. Beside the comparative method MC simulation can be also employed to obtain a calibration curve based on simulated standard. XMI-MSIM has been successfully used for this purpose for electroplated samples, normalizing the result respect to semi-infinite bulk element, with even better results than with the FP semi-empirical method [7]. A similar approach was used by Pessanha [8] for cultural heritage gildings on Pb using PENELOPE code, exploiting the ratio between two lines of the same element for normalization, as if it were an internal standard. This latter method of data processing has been widely used by Cesareo in the last decade [103][104][105][106][107][108] exploiting the differential attenuation (or self-attenuation) of the substrate (or coating) of two lines of the same element. The curves of the X lines ratios over the thickness can be obtained by knowing the value of this ratios for an infinitely thin layer and a semi-infinite one, these values are tabulated.
XRF are not commonly used for organic films since they don't give fluorescent radiation detectable in air Porcinai [109] used the X-rays attenuation of the substrate, calculating the ratio between two emission lines of the same element, to evaluate the thickness of polymers for protective purposes; empirical, semi-empirical and analytical (FP) methods were compared. Recently De Almeida [110] used a multivariate approach to evaluate multiple regions in the XRF spectra and obtain the thickness of polymeric films. To keep up with all the innovations in this field, every year a review [111][112][113] on the last advances in XRF group techniques is published to highlights the last developments in instrumentation, methodologies and data handling of this world.

Electron Probe Microanalysis
The electron probe microanalysis (EPMA) was first developed in 1951 by Casting [114]. EPMA permits to analyse the composition of homogeneous materials in a region of few microns from the surface. The EPMA can be conducted using two different approaches: wavelength dispersive X-ray spectroscopy (WDS) [115] or energy dispersive X-ray spectroscopy (EDS) [116][117][118][119][120]. WDS is generally considered an excellent method for microanalysis because is more sensitive and has a higher resolution than EDS but it is more expensive and needs a dedicated device. EDS, on the other hand, can be conducted by simply coupling a detector to SEM, a widespread instrument in the academic and industrial sphere, especially since the recent spread of inexpensive benchtop instrument.
EPMA was mentioned before in the destructive section for mapping the cross-sectioned samples, here it is used as a non-destructive technique measuring the sample perpendicularly to the surface [121]. This technique interprets every sample as homogeneous, since the output information is a spectrum. For this reason, there is no direct information on the thicknesses but the intensity of the peaks in the spectra are function of thickness as well as it is for XRF. EPMA is not much known for thickness measurement but it is an attractive candidate because it enables fast, quantitative [122,123] and non-destructive [124] analysis with the additional benefit of having a lateral resolution in the micron range [125]. In addition to that, the probe (electrons) is not very penetrating ( Figure 10) and for this reason it is possible (by adjusting the beam energy) to analyse ultrathin films or just the top layer to obtain its composition [126][127][128][129][130]. The EPMA detector is present as an upgrade of conventional SEM, but most of the instruments comes at least with the EDS detector by default. The electron bombardment of the beam excites the atoms in the sample knocking out the electrons from of the inner shells. Such state is unstable, and the resulting electron hole is immediately filled by a higher-energy electron from a higher atomic orbital. The energy difference is released in the form of an X-ray quantum. The resulting X-ray radiation is characteristic of the transition and the atom. For a single element, different transitions are allowed, depending on which shell the higher-energy electron comes from and which shell the hole must be filled in. This results in X-ray quanta, which are marked with Kα, Kβ, Lα, etc. The energy of an X-ray lines (position of the lines in the spectrum) is an indicator of which element is under investigation. The intensity of the line depends on the concentration of the element within the sample. Furthermore, the electrons, slowing down in the electric field of the atomic nuclei, generate an X-ray braking radiation, called bremsstrahlung, which constitutes the continuous background of the EPMA spectrum. The EPMA detector exploits the energy interaction between X-rays and a suitable material, generally represented by a silicon single crystal doped with lithium, coated at both ends with a gold conductive layer, at a temperature of -192 °C with liquid nitrogen. Other variants are the high purity germanium detectors and silicon drift detector (SSD) with Peltier cooling. When an X-ray photon is absorbed in the sensitive area of the detector, then electron-hole pairs are produced, this cause the production of an electric current, which is then sensitively amplified. In WDS instruments is present a diffracting crystal that select the photons to be send to the detector, which measure only the number of pulses, i.e. photons. In EDS system there is not a photon selector thus the signal of each photon is processed to obtain its energy value; during this time the system reject every other signal resulting in a dead time. High dead time produce high spectra with high resolution but low signal because many photons are rejected, on the other hand low dead time produce high signal but wide peaks. A longer process time is needed for quantitative analysis where spectral resolution is important, whereas if maximizing the number of X-rays in a spectrum or map is most important a shorter process time can be used. Si (Li) detectors operate at count rates of about 1 to 20 kCPS with optimal dead times of 20 -30 %. The reason why SDD are now preferred to Si (Li) detectors is that they can handle much higher count rates of >100 kCPS and dead times of 50 %. The count rate can be optimized by adjusting the beam current (probe current or spot size) and the process time. It is important to select a process time and beam current that will give an acceptable X-ray count rate and detector dead time for analysis, as well as the desired spectral resolution. The typical energy resolution of an EDS detector is 130 -140 eV while it is of a about only 10 eV for WDS systems. Moreover, EDS systems have a much lower count rates and poor reproducibility, generally a factor of ten respect to WDS detectors. The Beam energy can be varied to increase sensitivity for thinner of thicker coatings ( Figure 11).
The film thickness can be obtained from the measured spectra through various approaches. The calibration curve can be obtained using standards of known thickness [131] or Monte Carlo simulations [19,20]. In terms of the quantification method, the quantity used in the calibration curves, multiple alternatives were evaluated: the K-ratio [127,[132][133][134], which is the ratio between the intensity in the sample and the intensity in a standard with known composition, commonly used in que quantification analysis; the ratio of intensities [135] and the atomic ratio [131], obtained performing the ZAF correction algorithm on the K-ratios. Both the absolute thickness [136,137] as well the mass thickness [138][139][140][141] was taken into consideration for the quantification. In the last fifty years many software were written to simulate EDS spectra [142]; many of them are written by researcher and some were commercial: MAGIC [143,144], STRATAGEM [145][146][147], GMRFILM [122], Electron Flight Simulator [148,149], ThinFilmID [150] and LayerProbe [150,151], pyPENELOPE [152,153], Win X-Ray [154,155] and MC X-Ray [154,156], XFilms [157], CASINO [124,[158][159][160][161], CalcZAF [162,163] and DTSA-II [164][165][166]. Many of these software exploits the PENEPMA algorithm [153]. PENEPMA is a simplified version dedicated to EPMA, written to perform simulation of X-ray spectra and calculates different quantities of interest, of another algorithm called PENELOPE. PENELOPE (Penetration and ENErgy LOss of Positrons and Electrons) is a general-purpose Monte Carlo code system for the simulation of coupled electron-photon transport in arbitrary materials. PENELOPE covers the energy range from 1 GeV down to, nominally, 50 eV. The physical interaction models implemented in the code are based on the most reliable information available at present, limited only by the required generality of the code. These models combine results from first-principles calculations, semi-empirical models and evaluated data bases. It should be borne in mind that although PENELOPE can run particles down to 50 eV, the interaction cross sections for energies below 1 keV may be affected by sizeable uncertainties; the results for these energies should be considered as semi-quantitative. PENELOPE incorporates a flexible geometry package called PENGEOM that permits automatic tracking of particles in complex geometries consisting of homogeneous bodies limited by quadratic surfaces. The PENELOPE code system is distributed by the OECD/NEA Data Bank. The distribution package includes a report [167] that provides detailed information on the physical models and random sampling algorithms adopted in PENELOPE, on the PENGEOM geometry package, and on the structure and operation of the simulation routines. PENELOPE is coded as a set of FORTRAN subroutines, which perform the random sampling of interactions and the tracking of particles (either electrons, positrons or photons).
In principle, the user should provide a main steering program to follow the particle histories through the material structure and to keep score of quantities of interest. In PENEPMA photon interactions are simulated in chronological succession, allowing the calculation of X-ray fluorescence in complex geometries. PENEPMA makes extensive use of interaction forcing (a variance-reduction technique which artificially increases the probability of occurrence of relevant interactions) to improve the efficiency. CalcZAF [162] simulation software is based on PENEPMA and is a general-purpose software package for simulation of both relativistic and sub relativistic electron interactions with matter. Even in this case, the characteristics and the geometry of the detector are not taken into account and the output consists in a lines-like unconvoluted spectrum. DTSA-II [164] shares many physical models with PENEPMA but was designed exclusively for simulation of X-ray spectra generated by sub relativistic electrons. DTSA2 uses variance reduction techniques unsuited to general purpose code. These optimizations help the program to be orders of magnitude more computationally efficient while retaining the detector position sensitivity. Simulations are executed in minutes rather than hours and differences that result from varying the detector position can be modelled. It is possible to insert the characteristics and the geometry of the detector in DTSA2, which is capable of handling complex sample geometries. The primary and secondary bremsstrahlung and fluorescence can be calculated. The outputs consist in a real-looking spectrum since it is deconvoluted considering the detector resolution; even the electron trajectories can ben visualized. The CASINO [158] is a single scattering Monte Carlo simulation software of electron trajectory in solid specifically designed for low beam interaction in a bulk and thin foil. This software can be used to generate many of the recorded signals (X-rays and backscattered electrons) in a scanning electron microscope. This program can also be efficiently used for all the accelerated voltage found on a field emission scanning electron microscope (0.1 to 30 keV). The characteristics and the geometry of the detector are not taken into account and the output is not a spectrum but the characteristic emission lines intensity as function of the depth.
The X-ray depth distribution of the emissions is described by the φ(ρz) curve, which can be used for the determination of thin film thicknesses [12]. The thickness of the film can vary between two extremes relatively to the curve: extremely thin or extremely thick [137]. In the first case, the emission corresponds to a bulk sample with the composition of the substrate, in the second case, to a bulk with the composition of the film. In the intermediate cases the φ (ρz) curves vary between these two extremes. The maximum thickness that can be analysed with the EPMA method is about some microns: this is determined by the acceleration potential of the electrons together with the atomic number of the elements in the sample [168]. On the other hand, the minimum detectable thickness (lower detection limit) is given by the combination of the X-ray energy characteristics of the elements in the sample and the properties of the detector and can be as low as a few monolayer or less [169].
Exploiting STRATAGEM software Kühn [146] was capable to obtain both the elemental composition and thickness of a thin film ternary alloy Pd-Ni-Co co-deposited via magnetron sputtering on silicon wafer by ED-EPMA in the range of 50 to 250 nm. The results were confirmed by AES and XPS measurement, for the composition, and by SEM imaging for the thickness. The volume of interanion was confirmed by CASINO simulations. A similar approach was used for the determination of electrodeposited Ni, Pd and Au on Cu comparing the results of CASINO, CalcZAF and DTSA [18]. A comparation between GRMfilm, DTSA-II and PENEPMA was performed for very thin films (5 -20 m) of Al an Cu on Bi, in this study also the variation in the film density respect to the bulk material was evaluated [170]. Ultra-thin film of Ge, Sn, Ag and Au on Si wafer was evaluated also by Campos performing multiple analysis with different beam energy [171]. DTSA-II was used also to determine the sputter coater deposition of Ti and Ag on Si for medical applications [172]. Osada [136] developed it's a new MC simulation software to evaluate the thickness of aluminium oxide on aluminium sheets in in the range 5 nm to 50 nm. Recently, in 2018, Darznek Performed thickness measurement tilting the sample off to the normal incidence angle to increase the signal of the superficial coatings, specifically to determine the thickness of chromium film on a silicon substrate. With this approach he was able to determine up to 10 14 atoms per square centimetre with an error less than 10 % exploiting K-ratio measurement with MC simulation.
In 2016 Sokolov [124] measured the thickness of Silicon dioxide and silicon nitride thin films using EDS varying the penetration depth of the analysis changing the acceleration voltage of the beam and correlating the thickness of the film with the signal of the substrate elements to the collected noise. Stanford [173] in 2020 measured the oxide layer formation on Pu from 35 nm to 400 nm using measured standards to build the k-ratio calibration curves of oxygen through FIM-SEM analysis. Previously Bastin made a massive work collecting the K-ratio of Al [174] and Pd [175] of films from 10 to 320 nm in thickness at various beam energies between 3 kV and 30 kV on many substrates between Be and Bi.
Even the thickness of multi-layered samples can be measured using EPMA measurements [176,177]. In 2019 Pazzaglia [178] developed a new model for the standardless determination of mass thickness and composition using EDS for multilayer samples with an accuracy of 10 μg/cm 2 . Previously Lesch used a sputtering method with EPMA, the signal deconvolution with max entropy algorithm provide the thickness of Ti/Al/Ti layers deposited on Si.
EPMA was used by some authors to evaluate important information about layered samples besides their thickness: Christien [119] used the EDS measurement to determine the interdiffusion coefficient between thin films of miscible metals. Using various annealing temperature and the Fick's diffusion equations he was able to estimate the coefficients for a Ni film on Pd. Darznek [179] proposed a method to evaluate thickness uniformity of nanofilms by means MC simulations correlating the peaks intensities in the EDS spectrum to the film thickness. In 2016 Ortel [116] developed a technique combining EPMA measurement for mass deposition determination and SEM analysis for thickness determination to obtain the change in density of the films respect to the bulk materials and consequently extrapolate the porosity of the coatings.

Conclusions
Thickness measurement is a challenge that affects many scientists and companies. The composite materials on which a coating is present are ubiquitous and allow to obtain properties that a single element would not have. Thickness is decisive for obtaining these properties, and thus its control and measurement. In this review we describe the main techniques, both of preparation and analysis, which are used both in research and in the industrial sector for this purpose. In fact, there is no perfect technique suitable for any type of sample, but the most appropriate route must be chosen for every need. At the end of this work, it seems appropriate to report a roughly comparison between the various methods as regards the range of thicknesses that can be analyzed ( Figure 12) and the times required to carry out the analysis ( Figure 13). Obviously, the costs are also an important parameter to consider and they are very variable: from a general point of view mechanical and optical techniques are cheaper than the electronic, ionic and spectroscopic ones. It should also be taken into account that some instruments are supplied with multiple combined analyzers such as in the case of SE, BSE and EDS or for FIB techniques.
Talking about the measurable ranges of thickness, microscopic techniques tend to have only a lower limit, dictated by the aberrations that the beam undergoes under certain dimensions, while it is possible to lower the magnifications until observing shapes above the millimeter. Spectroscopic techniques instead suffer from the attenuation of the signal inside the sample and therefore cannot measure coatings beyond a certain size which appear as infinitely thick. Furthermore, for those analysis techniques that require sample preparation, the range that can be analyzed is the intersection of the ranges of the individual techniques. Considering now the preparation and measurement time, it is highly dependent on the presence of automated systems and on the experience and manual skills of the operator, as well as on the degree of accuracy required for the result. In general, the XRF, in addition to being extremely versatile, is the fastest technique, not even requiring sample preparation. On the other hand, microscopic techniques coupled to cross-sectioning are extremely widespread as they allow to obtain a result in which the thickness is directly visible. Furthermore, cross sectioning, although being time consuming, is a procedure that is generally automated. Some of the illustrated techniques are well established and have not undergone many innovations in recent years, if not an engineering optimization of performance and costs; for others however, research is still very active, as we have shown in this work, therefore they must be followed with interest in order to make the best use of them as a powerful analytical tool as they are.

Conflicts of Interest:
The authors declare no conflict of interest.