# Compositional Metrology of Atom Probe Applied to non-Metallic Materials

Enrico Di Russo, François Vurpillot, Lorenzo Rigutti

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Enrico Di Russo, François Vurpillot, Lorenzo Rigutti. Compositional Metrology of Atom Probe Applied to non-Metallic Materials. 2026. hal-05533361

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# Compositional Metrology of Atom Probe Applied to non-Metallic Materials

Enrico Di Russo

Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy, Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Legnaro, Viale dell'Università 2, 35020 Legnaro, Italy, and CNR-ISMN, Istituto per lo Studio dei Materiali Nanostrutturati, Via Gobetti 101, 40129 Bologna, Italy

François Vurpillot, Lorenzo Rigutti*

University of Rouen Normandie, INSA Rouen Normandie, CNRS, Groupe de Physique des Matériaux UMR 6634, 76000 Rouen, France

* Corresponding author: lorenzo.rigutti@univ-rouen.fr

## Abstract

Two decades after its introduction, laser-assisted Atom Probe Tomography (La-APT) has demonstrated a unique potential for the study of the 3D distribution of atomic species in semiconductor materials and devices, and in a growing list of inorganic non-metallic solids. A crucial and often underestimated issue with APT is its accuracy in compositional measurements of non-metallic systems. This work introduces the principles of APT as an experimental technique, recalling the aspects potentially leading to compositional biases and underlining in particular the role of the surface electric field in governing the different physical-chemical phenomena that enable the measurement. It reviews the possible mechanisms of specific losses, as well as the methods for assessing a compositional bias and proposing possible correction methods. Finally, it establishes a state of the art on compositional biases in APT of non-metallic materials, on the basis of which it will be possible to conclude on specific recommendations for best practices, and the perspective of application of APT to new materials.
List of main acronyms

<table><tr><td>1D-AP</td><td>1-dimensional Atom Probe</td></tr><tr><td>aDLD</td><td>Advanced DLD</td></tr><tr><td>APT</td><td>Atom Probe Tomography</td></tr><tr><td>BIF</td><td>Best Image Field</td></tr><tr><td>CSR</td><td>Charge State Ratio</td></tr><tr><td>DE</td><td>Detection Efficiency</td></tr><tr><td>DFT</td><td>Density Functional Theory</td></tr><tr><td>DLD</td><td>Delay Line Detector</td></tr><tr><td>DUV</td><td>Deep Ultraviolet</td></tr><tr><td>EBSD</td><td>Electron Back-Scattering Diffraction</td></tr><tr><td>ECCI</td><td>Electron Channeling Contrast Imaging</td></tr><tr><td>EDS, EDX</td><td>Electron Dispersion X-ray Spectroscopy</td></tr><tr><td>EELS</td><td>Electron Energy Loss Spectroscopy</td></tr><tr><td>ERDA</td><td>Elastic Recoil Detection Analysis</td></tr><tr><td>EUV</td><td>Extreme Ultraviolet</td></tr><tr><td>FIB</td><td>Focused Ion Beam</td></tr><tr><td>FIM</td><td>Field Ion Microscopy</td></tr><tr><td>FlexTAP</td><td>Flexible Tomographic Atom Probe</td></tr><tr><td>FOV</td><td>Field of View</td></tr><tr><td>FWHM</td><td>Full Width at Half Maximum</td></tr><tr><td>ICF</td><td>Image Compression Factor</td></tr><tr><td>ISO</td><td>International Standard Organization</td></tr><tr><td>La-APT</td><td>Laser-assisted Atom Probe Tomography</td></tr><tr><td>LaWATAP</td><td>Laser-assisted Wide Angle Tomographic Atom Probe</td></tr><tr><td>LEAP</td><td>Local Electrode Atom Probe</td></tr><tr><td>LOD</td><td>Limit of Detectability</td></tr><tr><td>LOF</td><td>Length of Flight</td></tr><tr><td>LPE</td><td>Laser Pulse Energy</td></tr><tr><td>MCP</td><td>Multi-Channel Plate</td></tr><tr><td>MD</td><td>Molecular Dynamics</td></tr><tr><td>OAR</td><td>Open Area Ratio</td></tr><tr><td>PME</td><td>Probability of Multiple Events</td></tr><tr><td>RBS</td><td>Rutherford Backscattering Spectroscopy</td></tr><tr><td>SEM</td><td>Scanning Electron Microscopy</td></tr><tr><td>SIMS</td><td>Secondary Ion Mass Spectroscopy</td></tr><tr><td>(S)TEM</td><td>(Scanning) Transmission electron Microscopy</td></tr><tr><td>STM</td><td>Scanning Tunneling Microscopy</td></tr><tr><td>TKD</td><td>Transmission Kikuchi Diffraction</td></tr><tr><td>TOF</td><td>Time of Flight</td></tr><tr><td>TP-APT</td><td>Terahertz-Pulsed Atom Probe Tomography</td></tr><tr><td>VP-APT</td><td>Voltage-Pulsed Atom Probe Tomography</td></tr><tr><td>ZBEF</td><td>Zero-Barrier Electric Field</td></tr></table>
# Table of Contents

## 1. Introduction

a. Scope of this review

b. Relevant metrological concepts

## 2. Principles and methods of APT

a. Field Evaporation

b. Voltage vs Laser Pulsing

c. Detector in Atom Probe Tomography (APT): Principles, Metrological Limitations, and Performance

i. Detection Principle: Ion-to-Electron Conversion and Amplification

ii. Event Encoding: Spatio-Temporal Localization

iii. Event Encoding: Performance Trade-offs: MCP-DLD Combination

d. Mass spectrometry

e. 3D reconstruction

f. Correlation tables and (Saxey) diagrams

## 3. The role of surface field in APT

a. Field distribution from analytical to atomistic distribution

b. Field polarization

c. Field ionization

d. Post-ionization

e. Charge state ratio metrics (Kingham curves)

f. Field-induced stress

g. Field-induced migration

## 4. Mechanisms of loss

a. Detector losses

i. Multi-channel plate performance

ii. Dead or dazzled zone

iii. Detection efficiencies

b. Mass peak interpretation

i. Mass peak superposition

ii. Mass peak ambiguities

iii. Thermal tails and background noise

c. Physical channels

i. Neutral evaporation

ii. Preferential evaporation

iii. Correlated evaporation and multiple detection events

iv. Molecular dissociation

d. Methods for assessment and (possibly) correction of losses

i. Variation of environmental parameters

ii. Variations within a given dataset

iii. Analysis of multiple events and detector pile-up

iv. Investigation of isotopic ratios

v. Pseudo-singles and Pseudo-multiples

vi. Isotopic substitution

vii. Statistical correction of biased composition from random alloys

## 5. Review of literature data

a. Composition of the matrix

i. Arsenides, Antimonides, Phosphides

ii. III-Nitrides: GaN

iii. Other III-Nitrides: AlN, InN, AlGaN, InAlN, InGaN, ScAlN

iv. Other Nitrides

v. Oxides

vi. Group IV compounds and carbides

b. Impurities

i. Dopants in Si and SiGe

ii. Impurities in Si

iii. Point defects in oxides

iv. Dopants and point defects in nitrides

v. Dopants and impurities in GaAs

## 6. Conclusions and perspectives

a. Instrumental implementations

i. Detector Technology

ii. Terahertz pulsing

iii. Deep and Extreme UV pulsing

iv. Hybrid pulsing

v. Correlative experimental setups

b. Implementation of methods

i. Field contrast imaging of defects

ii. Cryo-APT and organic systems

iii. Automated/enhanced data mining

c. Recommendations for users

i. Good experimental practices

ii. Data mining and analysis

iii. Role of standards

iv. Promoting research in APT metrology
# 1. Introduction

Two decades after the introduction of laser-assisted Atom Probe Tomography (La-APT), this technique is today widely used for the nanoscale microscopy and microanalysis of non-metallic systems (Kelly et al., 2007; Lauhon et al., 2009; B. P. Gorman et al., 2011; Giddings et al., 2018), including minerals (Reddy et al., 2020). It has demonstrated a unique potential for the study of the 3D distribution of atomic species in semiconductor materials and devices, and in a growing list of inorganic non-metallic solids. In all these systems, the capabilities of the techniques are being pushed to their limits, in a continuous quest on impurity and interface studied within still shrinking electronic devices (Giddings et al., 2018). Thanks to the recent development of cryo-preparation techniques, its domain of application is still growing and opening to organic and biological systems (McCarroll et al., 2020; Gault et al., 2021; Meng et al., 2022a; Schwarz et al., 2025; Woods, 2025). A crucial and often underestimated issue with APT is its accuracy in compositional measurements of non-metallic phases, including those occurring in the microstructure of metals. Despite APT can detect single ions with a yield which is independent of their mass, it is potentially affected by a set of mechanisms that can significantly – sometimes severely – limit its accuracy in the measurement of chemical composition (Lefebvre-Ulrikson et al., 2016; De Geuser and Gault, 2020).

This expanding context calls for a critical review on compositional metrology in APT applied to non-metallic materials. The timeliness for such a work stems from the experience accumulated by several research groups over two decades of analyses coupled to the new challenges represented by the opening of new domains of application.

## a. Scope of this review

The structure of the review (Fig. [1]) includes the following key chapters: (1) This introduction, including a historical overview and the position of APT in the context of composition measurement in microanalysis; (2) Principles of APT as an experimental technique, underlining the aspects that may intervene in compositional biases; (3) A focus on the role of the surface electric field in APT and how it governs the different physical-chemical phenomena that enable the measurement; this section also includes a review of different quantities interpreted as “surface field” and a criticism of the use of the notion of “evaporation field”; (4) A review of mechanisms of loss, i.e. what are the physical or instrumental channels which lead to an error in the measurement of composition; based on these mechanisms, the methods for assessing a compositional bias will be summarized, including possible methods for the correction of ascertained biases; (5) A review of literature data on compositional biases in APT of non-metallic materials; (6) A final chapter will contain the perspectives, specific recommendations for best practices, and the main conclusions of this work. The main advantages and limitations of APT as a microanalytical tool will be clearly set out.

This review develops the problem of compositional biases in APT on the basis of the actual knowledges in the domain of field ion evaporation. It does not exclusively focus on the experimental technique and related methods, but also delves into the fundamental physical processes on which the technique relies. It calls to an expanding readership spanning from theoretical/computational materials scientists interested in surface physics of non-metallic materials to routine APT users that face the everyday problem of the soundness of their APT data. Furthermore, it may be a reference for a broader panel of scientists facing the problem of measuring composition in nanoscale systems and of selecting the most adapted technique and methods.

This article focuses on the application of APT to the measurement of composition of non-metallic inorganic systems. Among these, the most represented are semiconductors and other dielectrics such as oxides, nitrides and carbides. Compositional biases in non-metallic minerals share many common points
with the aforementioned systems, along with specific problems related to the variety of impurity distributions. For specific cases concerning minerals, the reader is referred to the comprehensive review of Cappelli et al. (Cappelli et al., 2021). Organic systems are discussed in perspective.

<figure><img src="image_2.png" /><figcaption>Figure [1]. The structure of this review at a glance.</figcaption></figure>

b. Relevant metrological concepts

Within the above-described scope, we remind here the main metrological concepts and quantities that will be dealt with. The spatial metrology issues and concepts related to the technique have been extensively reviewed in the past (Larson et al., 2013; Vurpillot and Oberdorfer, 2015; De Geuser and Gault, 2020; Gault et al., 2026) and will be only rapidly reminded here.

*Measurement error.* The error is meant as the measured quantity value minus a reference quantity value. Note that the measurement error may be known or not, depending on the existence of the reference quantity. In APT the reference quantity for the composition is often not known, as APT is in many cases the only technique that can determine it.

*Accuracy.* The accuracy is defined as the closeness of agreement between a measured quantity value and a true quantity value of a measurand. The term 'measurement accuracy' is rather a concept than a quantity and is not given a numerical quantity value. A measurement is said to be more accurate when it offers a smaller measurement error. In composition measurements by APT, the accuracy should thus be meant as the closeness of agreement between the composition measured by APT within a specified region of a dataset and the reference value of the composition, determined either as a standard value or by an independent technique.

*Bias and systematic error.* The systematic error is the component of measurement error that in replicate measurements remains constant or varies in a predictable manner. A bias is the estimate of a systematic measurement error. In APT, these terms are often used as synonyms. A compositional bias refers thus to a systematic error in the measurement of composition.

*Precision.* The precision is the closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions. In compositional measurements by APT, the precision can thus refer (i) to repeated analyses performed on similar objects (specimens with similar shape and similar composition) under the same conditions (i.e.
within the same instrument and with the same set of experimental parameters: base temperature, laser intensity per pulse, DC voltage, etc.) or (ii) to repeated analyses extracted from different subvolumes of a given dataset from a homogeneous sample, acquired under uniform experimental conditions. In APT, the compositional precision is primarily related to the counting error which is introduced by the limitations in the detection efficiency, while the spatial precision – often confused with the resolution – can be considered as the closeness of agreement between the positioning of an atom after reconstruction and its effective position within the material (within the lattice if the material is crystalline).

**Spatial resolution.** Spatial resolution is a key quantity in microscopy, meaning means the smallest feature (or separation between features) that can be distinguished and measured with specified uncertainty under defined conditions. This quantity has different specific origins in different microscopy techniques. While in optical (photonic or electronic) microscopy it is related to the full width at half maximum (FWHM) of the illuminating beam point spread function (PSF), its definition in APT depends on both the instrument and the feature being quantified (lattice plane, defect, cluster, local order/disorder, etc.). The related term *resolvable feature size* (Gault et al., 2021) is in APT a task-specific measurand: it is the smallest size/spacing for which a chosen measurement procedure (APT experiment + reconstruction + analysis) yields estimates of a feature’s properties (e.g., diameter, composition, concentration profile) within acceptable error bounds. This is conceptually analogous to ISO’s PSF-based resolution, for APT the equivalent “PSF” is the broadening of reconstructed features and depends on material, microstructure and analysis workflow, so one must report resolution in relation to the feature class (interfaces, clusters, particles) rather than as a universal instrument constant (De Geuser and Gault, 2020; Gault et al., 2026).

**Limit of detection.** The limit of detection (LOD) is the measured quantity value, obtained by a given measurement procedure, for which the probability of falsely claiming the absence of a component in a material is $\beta$, given a probability $\alpha$ of falsely claiming its presence (IUPAC recommends $\alpha = \beta = 0.05$). The term “sensitivity” or “analytical sensitivity” is often used as a synonym of the LOD (Reddy et al., 2020; Kelly et al., 2022), but this use is discouraged by ISO, as sensitivity is rather the quotient of the change in measurement indication and the corresponding change in value of a quantity being measured. Another used synonym for this term in radiometry and in APT is the term “detectability” (Currie, 1968; Baptiste Gault et al., 2012). In APT, the LOD is related to the ability to detect a given atomic species, requiring both identification of a peak in the mass spectrum and accurate counting of ions. It is first limited by counting statistics, which also include the effect of detector efficiency: for instance, measuring 200 ppm with 50 ppm precision and 100% detection efficiency requires more than $10^5$ detected atoms, which becomes problematic for very dilute species. It is also limited by mass resolution and by the signal-to-background ratio, since a peak must rise above background noise to be distinguished (Currie, 1968). Because mass resolution, background level, and other experimental factors vary between datasets, the limit of detection in APT depends in a complex manner on all these factors and cannot be indicated in a close, simple form.

The main quantities that usually define composition as measured by APT are the following:

**Atomic fraction.** In APT, the atomic fraction $X_i$ of a component $i$ is given by the expression

$$X_i = \frac{n_i}{\sum_j n_j} = \frac{n_i}{N_{tot}} \quad (1)$$

Where $n_i$ is the number of ions, atoms or detected events of the component $i$, while $j$ runs over all the components defined for the given analysis, so that the sum in the denominator eventually yields the total number of atoms $N_{tot}$ defined as identifiable signal (i.e. excluding the noise) in the set of detected events. The atomic fraction can be expressed as a percentage, and is sometimes confused with the term *concentration*. Notice that the component $i$ can be an element, a charge state of a given element appearing in the mass/charge histogram, an isotope of an element (possibly with a given charge state), a molecular ion species.
**Site fraction.** The site fraction (or *sublattice occupancy*) of element *i* on a given sublattice (which should be explicitly specified) is

$$x_i = \frac{N_i}{N_{\text{sites}}} \qquad (2)$$

where $N_i$ is the number of atoms of element *i* on those sites and $N_{\text{sites}}$ is the total number of crystallographic sites in that sublattice. In the context of APT and materials analysis, *site fraction* refers to the fraction of crystallographic sites (or lattice positions) in a crystal structure that are occupied by atoms of a specific element or type, relative to the total number of available sites of that sublattice. This is a dimensionless quantity distinct from atomic fraction, as it considers the structural arrangement of atoms within a crystal. Its use is common in the analysis of oxides or semiconductor compounds in which the lattice structure is known, as for instance in AlₓGa₁₋ₓN, where *x* is the III-site fraction of Al within the AlGaN lattice (notice that in this example the V-sites are all occupied by N).

**Isotopic abundance.** The isotopic abundance is the proportion of a specific isotope of an element relative to the total amount of that element in a sample. Denoted as $a(i)$ for isotope *i*, it is given by

$$a(i) = \frac{n(i)}{\sum_k n(k)} \qquad (3)$$

where $n(i)$ is the number of detected ions of isotope *i* and the sum is over all isotopes *k* of the element. In APT, it can be defined for a given charge state and compared with the natural isotopic abundance, which can be useful for peak decomposition (i.e. identifying contributions from different elements within a given mass/charge peak, for instance ¹⁶O₂⁺ and ⁶⁴Zn²⁺ both occurring at 32 m/z) or for assessing specific losses related to detector limitations (Gopon et al., 2022; Ndiaye et al., 2024).

**Relative abundance or relative fraction.** This quantity is the equivalent of a fraction of a component *i* calculated over a specific subset of events and not on the total number of events. It can be useful in order to track spatial dependences of species in the detector or reconstructed space. For instance, the distribution of the relative abundances of the hydrogen species H⁺, H₂⁺, H₃⁺ over the sum of these three components can be tracked in order to disclose their correlation with the equivalent electric field (Rigutti et al., 2021).

**Volume concentration.** Or more shortly, *concentration*. refers to the amount-of-substance concentration

$$C_i = \frac{n_i}{V} \qquad (4)$$

where $n_i$ is the amount of substance of component *i* and *V* is the volume in which this amount is measured, with SI units mol/m³ or, far more frequently in semiconductor science, cm⁻³. In APT, volume concentration of solute *i* should consider the limited detector efficiency $DE_{MCP}$ (~50-80%, section 4.a), which is applied as a correction factor. Furthermore, the volume *V* is determined upon reconstruction, with all cautions that this operation requires and with all associated uncertainties.

For other metrologically relevant quantities the reader may refer to online resources (“ISO/IEC Guide 99:2007(en), International vocabulary of metrology — Basic and general concepts and associated terms (VIM),” n.d.). Last but not least, the Charge State Ratio, a paramount quantity for APT field metrology, will be introduced and discussed in section 3.e.
### c. APT vs other microanalysis techniques

APT may be compared with similar microanalysis techniques such as (Scanning) Transmission Electron Microscopy ((S)TEM) including Energy Dispersive X-ray Spectroscopy (EDS) and Electron Energy Loss Spectroscopy (EELS), as well as with Secondary Ion Mass Spectrometry ((Nano-)SIMS). Most APT users are well acquainted with SEM based techniques (EDS, Electron Backscattering Scanning Diffraction – EBSD, Transmission Kikuchi Diffraction – TKD, Electron Channeling Contrast Imaging - ECCI) as they are currently applied to APT sample preparation protocols (Saxey et al., 2007; I. Blum et al., 2016; Breen et al., 2017; Yang et al., 2025). These techniques are based on different physical principles, instrumental implementation and detection schemes. APT delivers 3D maps of positions and chemical identity of individual atoms with sub-nanometer precision. APT may thus resolve features as small as 1 nm—while simultaneously achieving limits of detection down to parts-per-million (ppm) levels. This combination allows APT to analyze nanoscale features like precipitates, grain boundaries, and segregation layers with quantitative atomic precision, even for light elements such as hydrogen, carbon, and lithium (Gault et al., 2021; Lawitzki et al., 2021).

APT's ability to directly quantify atomic fractions in small volumes—with well-defined statistical uncertainties—makes it uniquely powerful for studying nanoscale phase separation, clustering, and solute distribution in advanced materials. APT also offers extremely high detection limits and isotopic sensitivity, achieving single-ion detection with 50–80% efficiency. It quantifies trace elements (ppm levels), including light elements like hydrogen, carbon, and lithium, which are challenging for EDS or EELS. Time-of-flight mass spectrometry further distinguishes isotopes, supporting tracer and isotope-enriched studies. Finally, APT provides direct number-fraction quantification in small volumes, converting counted ions into atomic fractions with clear statistical uncertainties. This is especially useful for analyzing solutes in nanoscale features like precipitates, Guinier-Preston zones, or segregation at defects, often yielding lower compositional standard deviations than (S)TEM-EDS for comparable volumes.

<table><caption>Table [I] Comparison of microanalysis techniques by some of their typical metrological figures of merit.</caption><thead><tr><td>Technique</td><td>Resolvable Feature Size</td><td>Limit of detection</td><td>Notes</td></tr></thead><tbody><tr><td>SEM (EDS, EBSD, TKD, ECCI)</td><td>10 nm – 1 µm</td><td>1% - 1‰</td><td>Surface and bulk analysis; limited sensitivity for trace elements.</td></tr><tr><td>(S)TEM (EDS, EELS)</td><td>1 nm – 10 nm</td><td>1%- 1‰</td><td>High spatial resolution; limited to electron-transparent samples.</td></tr><tr><td>(Nano-)SIMS</td><td>10 nm – 100 nm</td><td>1 ppm - 1 ppb</td><td>Excellent isotopic sensitivity; lower spatial resolution compared to APT.</td></tr><tr><td>APT</td><td>0.1 nm – 10 nm</td><td>1‰ - 1 ppm</td><td>Near-atomic resolution; quantifies trace elements and isotopes in nanoscale volumes.</td></tr></tbody></table>

Techniques like (Scanning) Transmission Electron Microscopy ((S)TEM) and Secondary Ion Mass Spectrometry ((Nano-)SIMS) can be considered as complementary to APT (Eswara et al., 2019).

**APT vs (S)TEM-EDS/EELS.** APT delivers 3D atom-by-atom composition mapping, while STEM-EDS/EELS offer 2D nanoscale projections with thickness-integrated signals and potential delocalization, though TEM excels at linking composition with crystallography, defects, strain and phase contrast—areas where APT provides only indirect insights. For quantification, STEM-EDS/EELS face challenges like X-ray absorption, fluorescence, and thickness effects, requiring standards or modeling for accuracy, especially in complex systems (Nilsen and van Helvoort, 2022). APT avoids absorption/fluorescence issues but introduces a number of artifacts (see parts 3 and 4). However, APT uniquely detects and maps dilute solutes and light elements at trace levels within tiny volumes, a task
difficult for EDS (De Geuser and Gault, 2020; Lawitzki et al., 2021). In terms of materials systems, STEM-based techniques can address 2D materials, a domain still inaccessible to APT.

APT vs SIMS. These techniques differ significantly in resolution, quantification, and application scope. APT delivers sub-nanometer 3D spatial resolution, but only for small, geometrically constrained volumes, while SIMS excels in depth profiling over micrometer-thick films, though with depth resolution of a few nanometers and lateral resolution typically exceeding 50–100 nm (Eswara et al., 2019). For quantification, SIMS is highly matrix- and analyst protocol-dependent, requiring matched standards for accurate concentration measurements. APT, while less affected by classical matrix effects, faces material-specific biases due to field evaporation, often needing empirical corrections or cross-validation for reliable stoichiometry, especially in multi-component systems. The detection limit of APT can be as good as ppm-level, which can be outperformed by SIMS (down to ppb level) as it covers larger surface areas and can thus rely on larger statistics (McPhail, 2006).

Finally, compositional measurements by APT can be compared to and completed with those from other techniques, such as Rutherford Backscattering (RBS), nanoscale and standard X-ray diffraction (XRD), laser-induced breakdown spectroscopy (LIBS), optical, capacitive or thermal spectroscopies.

## 2. Principles and methods of APT

In this part the basic principles and methods in APT will be introduced, with an accent on the subjects relevant to composition metrology.

### a. Field Evaporation

The basic theories of field evaporation describe the escape process as thermodynamically determined, thermally activated, with only one escape charge-state involved (Müller, 1941, 1956; Gomer, 1959). The emission and detection rates (measured in ions/s, over the surface of analysis) are then assumed to be proportional to the evaporation rate-constant $K_{EV}$ (measured in s⁻<fnref n="1" />, on a single atomic site of emission) given by an Arrhenius-type equation:

$$K_{EV} = K_0 \exp \left( - \frac{Q(F)}{k_B T} \right) \quad (5)$$

where $k_B$ is the usual Boltzmann factor, Q is the activation energy for field evaporation, and $K_0$ is the field-evaporation pre-exponential that is also $K_{EV}$ for Q=0. Note that experimental measurements validate this expression over large range of evaporation rates (Ernst, 1979; Kellogg, 1984). In the simplest approximation, pre-factors are taken as slowly varying with T and F<fnref n="1" />, so that the control of the evaporation rate is made by manipulating the applied voltage (and hence F the surface microscopic field in the electrostatic approximation) and/or the temperature, T of the specimen. Later theories assumed that the atom might initially escape into a charge-state higher than 1, and might then be subject to field-induced post-ionization (see section 3.d). The activation energy, Q, is a microscopic energetic parameter that integrates the energy required by an atom to be liberated, expelled, and detected as an ion. In the early ages of APT analysis, it was found that Q was extremely sensitive to the applied microscopic field. For metals, a variation of few percent in the field can change the evaporation rate constant by several

<footnote n="1">¹ The validity of a constant pre-factor can be debated significantly, since it depends on the variability of the jump frequency over the barrier that is dependant of the strength of the binding interaction of the atom with surface, and of the statistics of vibration frequency that are temperature dependant. Experimentally, Kellogg measured for instance several ranges of order of variation for tungsten atoms as a function of the surface field.</footnote>
orders of magnitude (Tsong, 1971). It was also found that each element has its own magnitude of field required to cause significant field evaporation from the surface. Empirically, Q has been described via two simple properties: the microscopic field F<sub>EV</sub> for which the barrier vanishes or zero barrier evaporation field ZBEF and one (or two) parameter(s) (described below) describing the "field-sensitivity" of Q, i.e., how Q depends empirically on F (Müller, 1941, 1956; Tsong, 1978a; Tsong and Müller, 1969; Forbes, 1995; Forbes et al., 1982; Brandon, 1966; Larson et al., 2013; Miller and Forbes, 2014a). The electrostatic field F (microscopic) applied experimentally is normally always slightly lower than F<sub>EV</sub>.

F<sub>EV</sub> is one of the most fundamental parameters influencing data analysis. For a given element, some F<sub>EV</sub> tables are given in literature (Tsong, 1978b) based on either experimental observations or some analytical models. Experimental values accuracy is strongly discussed, and is generally about ±15%. The table is limited to a selection of pure materials, because measurement was made mostly in field ion microscopy mode on pure metals for an evaporation rate value not equal to K<sub>0</sub> (much lower). Analytical models gave also values that must be taken carefully due to the roughness of approximations made to calculate them. These tables should therefore be taken with caution. In particular for non-metallic compounds, it is not generally true that the evaporation field is defined for a given species: evaporation fields and activation energy barriers depend not only on the element, but also on the type of bond it builds with its nearest neighbors (Yamaguchi et al., 2009; Peralta et al., 2013).

The energy barrier is also influenced by the field sensitivity, S<sub>T</sub>, of the evaporation rate close to the ZBEF. As defined by Brandon (Brandon, 1966), this sensitivity can be defined at the first order mathematically by:

$$S_T = \frac{d\ln(K_{EV})}{d\ln(F)} = -\frac{F}{k_B T} \left(\frac{dQ}{dF}\right)_T \quad (6)$$

Since AP works close to F<sub>EV</sub>, we can define the quantity C = dQ/d(F/F<sub>EV</sub>)<sub>T</sub>, so that C ~ k<sub>B</sub>TS<sub>T</sub> which is an energetic parameter (written in eV) expressing the sensitivity to the field of the field evaporation for a given temperature. Assuming the sensitivity almost constant for a sufficient range of field, a simple assumption takes the activation energy barrier for the evaporation of the considered atom to be given adequately by the linear approximation:

$$Q(F) = C \left(1 - \frac{F}{F_{EV}}\right) \quad (7)$$

Values of C are experimentally in the range [0.5-3] eV with operating fields close to F<sub>EV</sub> (Ernst, 1979; Kellogg, 1984; Ashton et al., 2020a; Rousseau et al., 2023; Vurpillot et al., 2024). The higher the C value, the more abrupt is the variation of the evaporation rate with the field. C and F<sub>EV</sub> are generally higher for materials of high sublimation energy, such as refractory metals (W, Ir, Pt, etc...). The stronger these values, the more deterministic the evaporation process looks like. Non-linearities of Q linked to the complexity of the field evaporation mechanism obviously exist (Waugh et al., 1976; Forbes et al., 1982; Wada, 1984; Miller and Forbes, 2014a; Ashton et al., 2020a; Vurpillot et al., 2024) (Waugh et al., 1976; Forbes et al., 1982; Wada, 1984; Miller & Forbes, 2014; Ashton et al., 2020; Vurpillot et al., 2024). However, when they can be considered a small effect in the regime used experimentally (typically F<sub>op</sub> in the range 0.8 to 0.95 F<sub>EV</sub>).
b. Voltage vs Laser Pulsing

In APT a high voltage V<sub>DC</sub> (3 ÷ 14 kV) is applied between the specimen and the detection system in order to generate a high electric field. The latter is settled just under the evaporation field F<sub>ev</sub>. During the laser-assisted field evaporation of a tip, positive charged ions A<sup>n+</sup> are generated and accelerated towards a detector. The initial energy of an ion emitted from a tip surface is equal to the electric potential energy neV<sub>DC</sub>, where ne is the ion electric charge (e = 1.6×10<sup>-19</sup> C, n is an integer) and V<sub>DC</sub> is the DC voltage applied to the tip. As an ion reaches the detector, its final energy corresponding to the kinetic energy 1/2mv<sup>2</sup>, where m is the ion mass and v = L/t its velocity. L represents the flight path, which as a first approximation corresponds to the distance between the tip and the PSD and t is the time of flight (TOF) interval of the ion (Müller et al., 1968). The mass/charge ratio m/n of ions is derived from the conservation energy 1/2mv<sup>2</sup> = ne V<sub>DC</sub>:

$$
\frac{m}{n} = 2eV_{DC} \left(\frac{t}{L}\right)^2 . \qquad (8)
$$

where *t* is the TOF of ions (*t* = *L*/*v*).

Therefore eq. (8) can be used to transform a TOF spectrum into a mass spectrum, where the different ionic species can be recognized. For each ion detected both the impact position on the PSD and the TOF are recorded. In order to measure a TOF, the evaporation should be triggered in time. This occurs by electric or laser pulses, depending on the material which composes the specimen and the atom probe design. Typically, sub-nanosecond pulses are used for conductive materials (i.e. metals). Their effect is lowering the activation barrier *Q* through the modulation of the surface field *F* (eq. (Fev4)). Voltage pulsing has been applied to 1D-AP or APT of semiconductors before the introduction of femtosecond lasers, but these analyses are rare, and mostly applied to sufficiently conductive systems (Krishnaswamy et al., 1981). In fact, nanosecond voltage pulses cannot be properly transmitted along a poorly conductive tip. Semiconductors can be analyzed by voltage pulsing, which can be achieved through a conductive coating of the specimen as a final step of preparation (Adineh et al., 2017).

For non-conductive materials, such as semiconductors, oxides and polymers, current practice is using femtosecond laser pulses, which rapidly increase the temperature T of the specimen (eq. (Fev3)) (Gault et al., 2006; Bunton et al., 2007; Cerezo et al., 2007). The mechanisms of laser absorption and energy relaxation are described in several works (Vella, 2013; Kelly et al., 2014; Vella and Houard, 2016; Vella et al., 2018). Significantly, the presence of a strong static field can enhance the optical absorption in dielectrics through the modification of the band structure and the accumulation of positive charges (holes) at the surface (Silaeva et al., 2014). Typical temperature increases are of the order of several tens to one to two hundred K, and have been estimated by exploiting the modifications induced by laser pulses to photoluminescence signals from oxide specimens (Gautam et al., 2025). One of the main drawbacks of laser pulses is that heat relaxation occurs on the timescale of several tens of ns, which translates into the appearance of transients of evaporation probability after the pulse, i.e. into an asymmetric shape of mass/charge peaks (thermal tails on the high mass/charge side of a peak). This phenomenon leads to peak superposition, which generally complicates the analysis of mass/charge spectra and therefore composition assessment (see section 4.b). In order to improve field/temperature for minimal background noise while maintaining evaporation rate, a hybrid pulsing (voltage and laser) has been recently proposed (Larson et al., 2023a).

c. Detector in Atom Probe Tomography (APT): Principles, Metrological Limitations, and Performance

The detector used in Atom Probe Tomography (APT) is arguably the most critical component of the instrument, as many of the technique’s metrological limitations are inherently tied to its operation. The
final 3D point atom cloud - comprising the reconstructed 3D positions and elemental identities of atoms - is generated by identifying the (X, Y) position and TOF of ion impacts on a time- and position-sensitive detector placed in front of the specimen. Field evaporation of ions, triggered by laser or voltage pulses at repetition rates ranging from 25 to 1000 kHz, produces ions that are accelerated by an applied voltage of 1–20 kV toward the detector (see Fig. [2]). To ensure a detection rate dominated by single-hit events, the evaporation rate is carefully controlled to less than one atom per pulse (typically one evaporation event per 100 pulses, or a detection rate of ~1%). This rate can be practically adjusted from 0.01% to ~10%, with a typical detection throughput of 10²–10⁴ hits per second.

While most events are single hits, a significant fraction involves multi-hit events, defined as the detection of two or more impacts within the detection window following a trigger pulse. Multi-hits can be sequential (with a measurable delay between impacts) or simultaneous (same arrival time but different spatial positions). An ideal detector would capture all events to prevent data loss. However, the physical mechanisms of ion emission often result in correlated, non-random ion emission, leading to frequent species-dependent multi-hits that are closely spaced in time and position. If the detector fails to resolve these events accurately, they may introduce biases into the measurement. A large recent literature on these processes may be found (Meisenkothen et al., 2015; Schiester et al., 2024; Jakob and Thuvander, 2024; E. Di Russo et al., 2020; De Geuser et al., 2007; Thuvander et al., 2019, 2013; Costa et al., 2012; Peng et al., 2018; Morris et al., 2022; Russo et al., 2018; Cuduvally et al., 2020; Kitaguchi et al., 2014; Larson et al., 2018).

### i. Detection Principle: Ion-to-Electron Conversion and Amplification

Detection in APT relies on ion-to-electron conversion using a stack of two microchannel plates (MCPs). Each MCP consists of an array of glass capillaries (or "channels") with high secondary electron emission coefficients, arranged in a hexagonal lattice. A high DC voltage applied across the MCP generates an electron avalanche upon ion impact, providing high gain (10⁴–10⁶ electrons per hit for ions in the keV range with masses between 1 and 1000 Da for a single MCP). MCPs exhibit low intrinsic background noise, typically ~1 count/cm²/s across the detector (Guest, A.J., 1971; Woodhead and Ward, 1977).

The channels, with diameters of 10–20 µm and lengths under 1 mm, contribute to an intrinsic sub-nanometer positioning resolution due to the instrument’s magnification (10⁵–10⁶). Using two MCPs in a chevron configuration (channels tilted by ~20° to ensure uniform gain) achieves a total gain of 10⁷–10⁸ electrons per hit. The open area ratio (OAR) of conventional MCPs is ~60%, though recent improvements (e.g., funneled channel inputs) have increased this to 80% (Fehre et al., 2018). Detection efficiency decreases significantly for high-mass elements at low energies (<2–3 keV) (Fraser, 2002). For example, at 1 keV, the detection efficiency for argon (Ar⁺) and xenon (Xe⁺) drops by over 30% and 60%, respectively, compared to 5 keV. Special attention is required for large, slow-moving molecules (mass > 1000 Da), as their detection efficiency is severely reduced.

The transit time of the electron burst is less than 1 ns, with a transit time spread of a fraction of a nanosecond. The resulting electronic signals are brief (~ns duration) and detectable (1–100 mV) with standard electronics, exhibiting low sub-nanosecond jitter. MCPs are thus well-suited for ion-to-electron conversion in APT, where ToF typically ranges from 100 to 1000 ns, ensuring precise timing measurements.

### ii. Event Encoding: Spatio-Temporal Localization

To achieve optimal metrological performance, the electron burst must be localized in time and space with high precision. This is accomplished by the event-encoding readout system positioned behind the MCPs. Early APT systems experimented with various readout methods, such as Charge-division
readouts (e.g., wedge-and-strip detectors in the PoSAP atom probe (Cerezo et al., 1989), 10×10 multianodes in the TAP (Bostel et al., 1989)) or Optical detectors (electron-to-photon conversion via a phosphor screen, as in the oPoSAP (Cerezo et al., 1996) and OTAP (Deconihout et al., 1998)).

Nevertheless, for the past two decades, crossed delay lines (DLDs) have been the standard in modern APT instruments (see Fig. [2]-(c) (Jagutzki et al., 2002a; Kelly et al., 2003; Da Costa et al., 2005; Monajem et al., 2025). In a DLD, the electron burst is intercepted by two perpendicular propagation lines (one for each axis, X and Y). The generated electronic signals propagate to both ends of each line, and their arrival times (TU, TD, TR, and TL) are used to reconstruct the impact's position and timing. Only four preamplifiers and time-to-digital converters are required, simplifying operation. The electronic pulse from the electron burst has a full-width-at-half-maximum of 3–4 ns. The sum of the propagation times measured at both ends of a delay line (relative to the start signal) equals the line's total propagation time (TL), which ranges from 25 to 100 ns depending on detector size and geometry (e.g., ~0.5 ns/mm for serpentine delay lines on an 80 mm MCP stage, ~1 ns/mm for helical bifilar delay lines). The difference in arrival times determines the (X, Y) position, while the arrival times (corrected for TL) provides sub-100 ps timing accuracy. Sequential multi-hits separated by more than TL are detected unambiguously. Adding a third, tilted delay line or measuring the MCP arrival signal can resolve nearly simultaneous particles if they are sufficiently separated. The sequential multihit decomposition is then limited by the signal separation in time. An intrinsic dead time of ~4 ns (the physical width of the electronic signal) creates a minimum spatiotemporal dead zone of ~8×8 mm² × 4 ns. Advanced systems using high-frequency digital sampling (>1 GS/s) on all four DLD ends can reduce this dead time to 1.5–2 ns, shrinking the dead zone to ~2×2 mm² × 2 ns. Nonetheless, multi-hits detected inside this zone are registered as a single event with incorrect position and timing (see section 4.a.ii).

### iii. Event Encoding: Performance Trade-offs: MCP-DLD Combination

To conclude, the MCP-DLD system strikes an optimal balance for APT applications, offering:

* High throughput (up to ~50 kHits/s without loss),

* Low intrinsic noise (<0.1% in the best cases),

* High detection efficiency (up to 80%),

* High spatial precision (<50 µm),

* High timing accuracy (<100 ps).

Despite these advantages, limitations persist, particularly for correlated multi-hits and dead zones, which may affect the accuracy of 3D reconstructions, especially for complex materials producing a high amount of correlated evaporation. For this reason, research on detector technology is ongoing and the future developments will be shortly discussed in the perspective section 6.a.i.
<figure><img src="image_3.png" /></figure>

**Figure [2]**. (a) Schematic illustration of the APT detection process. The 3D reconstruction of atomic positions and elemental identities relies on measuring both the (X, Y) impact position and the time-of-flight (ToF) of ions using a time- and position-sensitive detector (b) The detector's ion-to-electron conversion system consists of a stack of two microchannel plates (MCPs) arranged in a chevron configuration, with diameters ranging from 40 to 120 mm depending on the atom probe's geometry. Each MCP is biased at ~1 kV, generating an electron avalanche within its tubular channels upon ion impact. This avalanche produces an output charge of 10³–10⁶ electrons, which is directed toward the readout system. The resulting electron burst (~1 ns duration) encodes the event's spatial and temporal information (Source: Hamamatsu) (c) In modern APT instruments, a crossed delay-line detector (DLD) decodes the (X, Y, ToF) coordinates by analyzing electronic pulse signals (VR, VL, VU, VD) collected at the ends of serpentine or helical propagation lines along the X and Y axes. (Image courtesy of C. Bacchi).

d. Mass spectrometry

Since the development of 1D APs, time-of-flight mass spectrometry based on eq. (8) has been the basis of element-specific detection (Müller et al., 1968). The simple assumptions and basic equations established at that time are still routinely used in modern APs to determine the chemical nature of field evaporated atoms. Both composition measurements and three-dimensional reconstructions rely on mass spectrum quality and processing. The quality of the mass spectrum depends not only on the physics of field evaporation but also on AP design (flight length) and hardware devices eventually used to reduce energy deficits (reflectrons). It also depends on the operator and analysis conditions including voltage, detection rate, sample temperature, pulse fraction, or laser energy. The final result is also related to post-processing carried out by the operator, including calibration, mass resolution optimization, background noise subtraction, peak identification, mass range definition, and peak deconvolutions. These issues have been reported or reviewed in detail in several works (Baptiste Gault et al., 2012; Pareige et al., 2016).

In APT, the TOF contribution to composition accuracy is governed by any effect that disrupts the ideal eq. (8) or distorts peak shapes and areas, since composition is derived from integrated peak intensities. Poor calibration or calibration drift can shift peaks and alter ranging of overlapping species, while dispersion in evaporation timing, especially in laser-assisted APT, broadens peaks and increases overlap ambiguity. Variations in flight path and geometry, as well as energy spread arising from field variations,
molecular dissociation, or post-ionization, further broaden or skew peaks, complicating integration and masking minor species. Detector timing resolution, jitter, dead-time, saturation, and multi-hit limitations reduce mass resolving power and introduce systematic undercounting, often in a composition-dependent manner during burst or correlated evaporation events. Finally, non-ideal peak shapes, background contributions, imperfect deconvolution, and limits in mass range and dynamic range can bias peak integration, leading directly to errors in quantified composition, particularly for low-abundance elements or closely spaced mass/charge ratios (Pareige et al., 2016). Some of these aspects will be further developed in section 4.b.

### e. 3D reconstruction

On modern APT instruments, different geometrical configurations exist. Straight flight path instruments integrate a detector of diameter D at a distance LOF (length of flight distance). D is limited in current technology by MCP diameters (~40 mm, 80 mm or 120 mm, with 80 mm being the most common tradeoff). LOF between the specimen and the detector is in the range 80 mm–400 mm for straight flight path instruments. Some instruments integrate an electrostatic device that enable to collect a larger field of view (FOV) on the specimen. This is the case for reflectron-fitted instrument (Panayi et al., 2006; Tegg et al., 2023; Heller et al., 2024), or more recently with instrument incorporating electrostatic lens. Here, since the projection features, specific to the geometrical and electrostatic parameters of the instrument are taken into account by manufacturers in the reconstruction software, we will focus on the reconstruction process for the former and simple case of the straight flight path atom probe.

The 3D point cloud is built point-by-point by assuming a simple reverse-projection from the detector impact position onto a 'virtual' emitting surface located at the end of an APT specimen. With a specimen radius below 100 nm the projected image of the surface on the detector has a magnification that is in the range of $10^6$. This is the consequence of the highly curved nature of the specimen surface. Ions initially have a near-radially fly path. The trajectory progressively curves towards the detector forming a highly compressed imaged on the surface on the detector. The trajectories of ions are fully determined by the distribution of the electrostatic field, making them independent of the ion's charge, mass, and applied voltage. Final trajectories of ions are very close of a point emitting source of constant position.

The seminal paper of Bas et al. (Bas et al., 1995) based on the recipes from a previous work by Hyde et al. (Hyde et al., 1994) proposed a simple method to 'de-magnify' the detector coordinates $(X_D, Y_D)$ into the real-space location of each ion on the specimen surface $(x,y,z)$. We may note that the projection relationship used to de-magnify ignore completely the real complexities of the exact trajectories to establish a bijective relationship between a point on the emitting surface and a point on the detector. Some other projections models have been proposed, with the most commonly implemented is a quasi-stereographic point-projection (Wilkes et al., 1974; Smith and Walls, 1978; Cerezo et al., 1999; De Geuser and Gault, 2017). To make simple, the projection is close to a simple angular projection with a compression factor typically between 1.4 and 2 which depends on the specimen geometry, i.e. radius of curvature and shank angle (Hyde et al., 1994; Geiser et al., 2009; B. Gault et al., 2011; Loi et al., 2013). Hence, it varies over the duration of the analysis (Baptiste Gault et al., 2011; Hatzoglou et al., 2023).

Using the hypothesis of full ionization of surface atom, average and constant detection efficiency, and assuming a conservation of the evaporated volume, an estimation of the depth of analysis is easily produced. The result is a list of 3D coordinates for each ion detected and labelled as an elemental species. This is the basic reconstructed dataset. The accuracy of positioning is linked to simple geometric hypothesis that are used by the reconstruction process. The tip surface is assumed to be mostly spherical,
with an estimation of the radius of curvature R (radius of the sphere) that can come from external measurements (e.g. from an electron microscopy image, SEM or TEM), or deduced from the applied voltage V. Indeed, each material is known to have its own evaporation field F<sub>EV</sub> and electrostatic impose F=V/k<sub>F</sub>R, with k a field factor (slowly varying with R, see equation (10)). Assuming a specimen evaporating at F<sub>EV</sub>, R is simply defined by R=V/kF<sub>EV</sub>. (e.g. from an electron microscopy image (Geiser et al., 2009)). Despite its simplicity, this method leads to impressive results, allowing for the identification of nanometric or even atomic scale details in the analysis of many materials. Some distortions of the image exist, but do not affect the compositional accuracy of the instrument. Further information on reconstruction methods can be found in dedicated book chapters and references therein (Larson et al., 2013; Miller and Forbes, 2014b; Vurpillot, 2016).

f. Correlation tables and (Saxey) diagrams

Additional insights into evaporation processes can be obtained by analyzing frequency correlations in multiple detection events. Two data analysis methods are introduced to visualize and interpret these correlations, both introduced in the APT community mainly by D. Saxey (Saxey, 2011). Both of them can be applied for an advanced interpretation of mass spectra and may provide useful information for the assessment of possible compositional errors.

The first, known as the correlation table, helps determine whether the simultaneous detection of two ions in an event occurs more frequently than would be expected under independent evaporation. This involves evaluating all possible ion pairs within each event. For instance, if ions A, B, and C are detected, the resulting pairs are AB, BC, and AC. The strength of the correlation between ions i and j is measured using the significance value d<sub>ij</sub>, given by:

$$d_{ij} = \frac{p_{ij} - e_{ij}}{\sqrt{e_{ij}}} \qquad (9)$$

Here, p<sub>ij</sub> is the observed number of ij pairs in the dataset, and e<sub>ij</sub> is the expected number of such pairs if no correlation exists, based on the frequency of each ion type. It's important to note that d<sub>ij</sub> doesn't directly quantify correlation strength - it also reflects the population sizes of the ions involved. Thus, even weak correlations can yield high d<sub>ij</sub> values if the ions are abundant. For meaningful comparisons between ion pairs or datasets, the ratio (p<sub>ij</sub> - e<sub>ij</sub>) / e<sub>ij</sub> should be used, assuming the d<sub>ij</sub> value confirms a significant correlation. The d<sub>ij</sub> values are displayed in a chart where each ion is represented by a row and column as in Fig. [3].
<figure><img src="image_4.png" /></figure>

**Figure [3].** Correlation table illustrating the statistical analysis of ion pair combinations detected within the same multiple event in the analysis of GaSb under high field conditions (19.75 V/nm). Strong colors highlight significant correlations (in red) and anti-correlations (in green). The color intensity corresponds to the *d*<sub>ij</sub> values described in eq. (9), while grey entries correspond to a statistically insignificant set of pairs. As the order of ions in each pair is not considered, these tables are symmetric. Additionally, the vertical axes display the total number of ion pairs that each ion species is involved in. This tables indicates a strong correlation of Sb<sub>2</sub><sup>2+</sup>-Sb<sup>2+</sup> pairs and a strong anticorrelation of Sb<sub>2</sub><sup>2+</sup>-Sb<sup>+</sup> pairs (Reproduced with permission from (Müller et al., 2011), Elsevier).

The second method for analyzing correlations in multiple events is the correlation histogram (Saxey, 2011), originally introduced in molecular physics for the covariance analysis of multiphoton ionization and dissociation experiments of gas jets (Frasinski et al., 1989). This plot resembles the correlation table but instead of identifying ions by detector range, it displays their mass/charge ratios on a two-dimensional histogram. Each entry represents one ion pair, with the x and y axes corresponding to the mass/charge ratios of the two ions. Since the pair AB is equivalent to BA, each is counted twice, making the histogram symmetric along the diagonal. Figure [4]-(a) shows a typical correlation histogram for SiC, where the redundant part below the diagonal has been partly masked. The interpretation of the main features visible in the histogram is assisted by the scheme of Fig. [4]-(b).

Peaks in the histogram correspond to ion pairs and are often followed by tails extending in various directions. The most prominent tails have a positive slope and align with the diagonal. When the same data is plotted using corrected time-of-flight (adjusted for voltage variation and flight path differences), these tails appear as straight lines with unit slope. This corresponds to correlated evaporation shortly after the laser pulse, during the cooling phase, with identical delays. Other tails in the correlation histogram appear horizontal or vertical. These represent cases where one ion evaporates at the time of the laser pulse, while the second ion evaporates later without any time correlation.

Additionally, tracks with a negative slope can be seen. These tracks usually indicate the presence of frequent molecular dissociation events during flight (Ivan Blum et al., 2016; Saxey, 2011). Dissociations involve metastable ions. Some of these ions are known to dissociate in specific directions with respect to the electric field (Tsong and Cole, 1987; David Zanuttini et al., 2017), and it has been assessed that evaporated ions should be in an excited state in order to dissociate (Zanuttini et al., 2018). Such dissociation affects the time of flight of the resulting ions due to their change in mass/charge ratio as they traverse the accelerating field (Tsong and Liou, 1985). More details about molecular dissociation will be given in section 4.c.iv. These features are found in straight flight-path instruments, while do not apply to instruments with a reflectron, in which trajectories are strongly modified and focused in time (Di Russo and et al., 2020).
<figure><img src="image_5.png" /><figcaption>**Figure [4]**. Use of correlation histograms for advanced interpretation of mass spectra and insight in field evaporation mechanisms. (a) The correlation histogram extracted from an analysis of SiC. The red letters and arrows indicate several dissociation tracks. The inset is a zoom in the neighborhood of m/z=12 and shows the detail of the peak related to the detection of C⁺-C⁺ pairs; here, the diagonal track marked with a red "b" is due to the homolytic dissociation C₂⁺ → C⁺ + C⁺. (b) Schematics indicating the different possible interpretation of features (dots refer to correlation peaks, lines to tracks) appearing in the correlation histogram (Adapted with permission from (Ndiaye et al., 2023a), American Chemical Society). (c) Schematic representation of how the different processed highlighted in the correlation diagram would appear within the mass spectrum containing multiple events only.</figcaption></figure>

The interest of correlation histograms in APT compositional metrology is twofold. On one hand, they allow for the identification of ions that would be typically out of the ranges defined for standard mass spectrometry. Despite most of the reported studies assessed that the impact of dissociation products on the composition measurement is moderate in dielectrics (Santhanagopalan et al., 2015; Russo et al., 2018; Cuduvally et al., 2020; Ndiaye et al., 2023a), the situation could be different in the expanding domain of soft materials and organic compounds. On the other hand, correlation histograms (as well as correlation tables) allow for the direct visualization of dissociations and ion correlations. They constitute thus a very important tool for the investigation of the physico-chemical processes occurring at the specimen surface or during flight. This can in turn be useful for the interpretation of compositional biases (see sections 4.c.iv and 5.a.iv).
### 3. The role of surface field in APT

The driving force of almost all physical mechanisms involved in APT is the strong electrostatic field F existing from the specimen surface to the micron scale around it. This field is generated by the applied voltage V to the pointed needle of interest with reference to surrounding electrodes existing generally held to ground voltage. These electrodes can be placed a few centimeters ahead (in simple atom probe designs such as the LaWATAP), or more locally, if an extraction microelectrode is used (case of the local electrode atom probe). Different fields can be defined and used in APT, as reported in Tab. [II], corresponding to different mechanisms or phenomena. The electrostatic field distribution is at the source of atomic effects such as field polarization, induced field driven diffusion, field ionization effects, field evaporation effect, field dissociation effect, and field projection. The surface field and the local field in the micron scale distance to the surface is of interest to understand all these processes. In this review, we will address only electrostatic effect, which is valid generally in Laser Assisted Atom Probe Tomography (The intrinsic electromagnetic oscillation is neglected in LA-APT) and in Voltage Pulsed Atom Probe Tomography (VP-APT) for metals or good electronic conductors. However, we may note that more advanced treatment should be necessary to understand Terahertz Pulse driven Atom Probe (TP-APT) that was recently developed, where electric field oscillation with sub picosecond timescale is used to trigger field evaporation. TP-APT is therefore out of the scope of this review.

<table><caption>Table [II]. A set of differently defined electric fields used in APT and field ion emission theory and experiments. Ref. a = (Müller, 1941, 1956; Tsong, 1978b; Miller and Forbes, 2014a); Ref. b = (Ernst, 1979; Kingham, 1982; Mancini et al., 2014a; Tegg et al., 2024); Ref c = (Müller, 1960; Tsong, 1978b, 2005); Ref. d = (Gomer, 1994; Rigutti et al., 2017); Ref. e = (Suchorski et al., 1995);</caption><thead><tr><td>Type of field</td><td>Typical Notation</td><td>Definition / description</td><td>Refs.</td></tr></thead><tbody><tr><td>Evaporation field, escape field</td><td>F<sub>e</sub>, F<sub>ev</sub>, F<sub>EV</sub>, ZBEF</td><td>Field value for which the energy barrier for the transition of an atom from a bound state to an ionized state vanishes</td><td>a</td></tr><tr><td>Equivalent field, effective field</td><td>F<sub>eq</sub>, F<sub>eff</sub></td><td>Electric field calculated through charge sate ratio (CSR) statistics. Within the limits of the post-ionization models, it yields an approximation of the microscopic field F<sub>μ</sub>.</td><td>b</td></tr><tr><td>Best Image field (FIM)</td><td>BIF</td><td>Field at which the images formed in FIM have the best atomistic contrast and spatial resolution. The value of the BIF is primarily related to the ionization energy of the image gas used and therefore has a given value for each image gas.</td><td>c</td></tr><tr><td>Average, nanoscale field</td><td>F<sub>a</sub>, F<sub>s</sub></td><td>Average value of the electric field on the surface of a field emitter. Determines the stress state at few layers from the surface and in the sample bulk.</td><td>d</td></tr><tr><td>Microscopic field</td><td>F<sub>μ</sub></td><td>Field at a given point of the surface of a field emitter. Varies following the atomic roughness and the crystal symmetry. In FIM images recorded at the BIF, the BIF yields an approximate measure of the microscopic field in correspondence of the imaged atoms.</td><td>e</td></tr></tbody></table>
a. Field distribution from analytical to atomistic distribution

The field vector **F**(**r**) can be derived from the electric potential distribution **F**(**r**)=−∇**V**(**r**). **r**(x,y,z) is here the position vector describing the space. **F** can also be derived from the distribution of charge existing on the surfaces of the electrodes surrounding the ion considering that any surface *dS* with a surface charge σ. The APT specimen, a sharply pointed needle, in vacuum, is polarized to a DC voltage V, with V in the kV regime. The surface electric field generated on APT specimen is about 10-50 V/nm, which means surface charge of about 1 electron charge per nm². The end apex of the sample is roughly described by a hemispherical cap of radius R. (R in the range 10-100 nm). The specimen is situated at a close distance of grounded electrodes. LEAP instrument uses conical extraction electrodes with aperture of about 50 microns placed at 50 microns ahead. The 3D distribution of the electric field in this geometry is a classical problem that can be answered by solving the Laplace equation (electrostatic approximation) using a known geometry. Note that in DC voltage, the approximation of the specimen as a sufficiently good electrically conductive medium is fairly sufficient. Indeed, as we will show, the amount of total charge existing at the polarized specimen apex is about 10⁴ electrons for the application of the voltage V. It means any variation of surface electric field in the sub-second range is equivalent to current in the sub-fA regime, that can be easily drained in the micron scale apex, with limited voltage drop and internal field. Anyway, over the surface and in the free space in front of the specimen the field intensities and gradients are strong.

Conditions of constant voltage are used on the specimen surface. For certain smooth, symmetrical emitter shapes, the problem is easy to solve analytically. A number of mathematical models of emitter shape have been used and some have an exact solution to Laplace’s equation for the electrostatic potential. The simplest approach imposes a specimen shape very close to a surface of constant curvature radius R, at least close to the specimen axis. For a polarized sphere F is simple V/R. So it is tempting to adapt this expression for a real APT geometry. To a first approximation, the electric field magnitude F was found to follow then classical expression

$$F = \frac{V}{k_F R} \qquad (10)$$

where k<sub>F</sub> a dimensionless semi-constant, i.e. that is slowly varying as the specimen is field evaporated. k<sub>F</sub> is in the range of 2–10, and depends primarily on the specimen’s shank angle and end radius (Vurpillot and Oberdorfer, 2015). For a smooth emitter, the hemispherical approximation ±45° about the specimen’s main axis is a good approximation of the morphology of the specimen’s tip. However, when approaching the outer edge of the end apex, the electric field decreases, the steady state radius of curvature tends to decrease, which makes the specimen surface more elliptical. Nevertheless, in atom probe, ±45° is approximately the maximal field of view of most of instrument. The spherical approximation is used for field evaluation, and standard process of reconstruction of the dataset. For a homogeneous material, analyzed in atom probe under a limited variation of voltage, k<sub>F</sub> is considered as a constant, and the variation of field (lateral to the surface and in the first nanometers above the surface) are neglected. It is also considered that the voltage penetration in the material is extremely abrupt. A first order evaluation of the local electric field at the tip surface is to consider locally the surface curvature (1/R<sub>C</sub>), which is defined as the average of the principal curvatures 1/R<sub>1</sub> and 1/R<sub>2</sub> for an arbitrary shape surface. Field is thus inversely proportional to R<sub>C</sub> (R=R<sub>C</sub> for a sphere) in a first approach. An evaluation of the surface geometry and its local curvature is a first indicator of the Field distribution existing on the surface. This approach was used for instance to study morphology variation of heterogeneous materials under field evaporation and related distortion in image projection of ions (Figure [5]) (N. Rolland et al., 2015; Vurpillot et al., 2016; Beinke et al., 2016).
a

<figure><img src="image_6.png" /></figure>

b

<figure><img src="image_7.png" /></figure>

Figure [5]. (a) Geometric modeling of a specimen geometry composed of two phases having evaporation field different by 20%. (b) Modeling of the tip shape at the evaporation depth $z_e$ in the case of a high field material B (in red) seated on a low field material A (in blue). (Adapted with permission from (N. Rolland et al., 2015), Elsevier).

Anyway, when differences in specimen surface becomes too strong (about 20% is curvature), the validity of a constant $k_F$ is not sufficient, and non-linear effects are observed. In addition, the notion of curvature is an ill-posed problem. Classical electrostatic textbook formulates that the relationship between the local curvature and the amplitude of the field is not straightforward, since the buildup of charge on the surface giving rise to the electric field is a long-distance process that cannot be solved only by local inspection of the morphology. This over-simplification is the main cause of misinterpretation and errors in the final dataset analysis. In addition, curvature can be defined at different scale, for macroscopic, micro or mesoscopic, but also atomic. A single atom on a flat surface can be considered to produce a sub-nanometer roughness. Ab-initio calculation shows geometric curvature of material surfaces at this scale are not smeared out and induces local enhancement of field close to the geometric electrostatic description (Silaeva et al., 2013). To resume, the control of the electric field from the atomic scale to the mesoscale is at the roots of metrological performances and experimental, empirical or computational means are necessary to evaluate dynamically the field distribution over the specimen surface.

Assuming physical and geometrical properties of the specimen in the configuration of an atom probe enables a direct modelling of electrostatic field distribution over the specimen. In older models applied to an arbitrary emitter shape, determining this field was difficult because of the nine or ten orders of magnitude variation in length scale going from the specimen (defined at the nanometer scale) to the surrounding electrodes (defined on the microscale) to the detector (defined at nearly the meter scale). Currently, the electrostatic field is often calculated using finite element methods (FEM) when the space between electrodes is meshed so that electrostatic voltage can be deduced everywhere or boundary
element methods (BEM), where the distribution of surface charge on electrodes can be calculated in 3D without the need to mesh the 3D space entirely (Figure 6)-(a-d) (Loi et al., 2013; Vurpillot et al., 2013; Oberdorfer et al., 2013; Vurpillot and Oberdorfer, 2015; Nicolas Rolland et al., 2015a; Rousseau et al., 2020; Fletcher et al., 2022; Hatzoglou et al., 2023b; Lüken et al., 2024). Using this approach, it is possible to use a fine mesh to define the specimen (sub-nm) and a much coarser mesh for electrode of macroscopic dimensions. Both models (Laplace computation or BEM computation) are equivalent, since they both assume that the density of charges are located on the surface with a penetration depth smaller than the size of a single atom (Figure 6). Models can describe mesoscopic details in continuum approaches (Fletcher et al., 2022; Hatzoglou et al., 2023a), or atomistic details (Vurpillot and Oberdorfer, 2015). This last approach requires more computational resources due to the large number of points to be defined in an atomically rough model (Figure 6)-(f). The field distribution determination on realistic modelled specimen enable to explore surface effects (Katnagallu et al., 2018; Vurpillot et al., 2018), of dynamic effects during the flight of ions from the surface to the detector (Figure 6)-(d)) (Ivan Blum et al., 2016; E. Di Russo et al., 2020; Gault et al., 2016; Rousseau et al., 2020).

<figure><img src="image_8.png" /></figure>

Figure 6. (a) TEM image of a real specimen composed of heterogeneous materials with strong difference in evaporation fields (SiO₂ and Si) evaporated in an APT. Note the complexity of local curvatures of the surface (negative curvature which not exist considering equation (10)). Simulation of the field evaporation of this specimen, and representation of (b) surface charge and electric field (c) local temperature under laser pulsing showing the interplay between geometry, temperature and local field for this structure (Adapted with permission from (Hatzoglou et al., 2023b), IOP). (d) Simulation of the field evaporation of a heterogeneous material with representation of the electric field in a realistic Atom Probe geometry. This field distribution can be used to study the trajectories of ions or cationic molecule from the surface to the detector (Adapted with permission from (Vurpillot et al., 2013), Elsevier). (e-f) Modelling field evaporation of a heterogeneous material composed of 2 crystallographic structures. The internal and surface structure can be observed. Note the strong atomic variation of the local electric field induced by local roughness at the atomic scale.
b. Field polarization

In the presence of the strong electric field, any atom or molecule reaching the apex region are submitted to polarization effect. Under this electric field, a neutral particle is transformed into a dipole interacting with the strong variation of field existing in the first 100 nm to the surface. Due to polarization forces, these particles migrate towards the highest field region at the apex and accommodate to the temperature of the specimen. The end apex field distribution acts as a potential well with an energetic depth depending on the intrinsic polarizability of the particle. The same forces may act on any neutral particle emitted from the surface. The volume isotropic polarizability of most of atoms and molecules may be found in literature (in Å³) and are generally around ~1 Å³ for single atom or diatomic molecule. Under the surface electric field intensity (10-50 V/nm), final binding energy is in the range 0.5-10 eV ($\frac{\alpha F^2}{2}/4\pi\epsilon0$).

The effect of the electric field on gas atoms or molecules near a field emitter has been studied in the context of field-ion microscopy (FIM) (Miller and Smith, 1981). FIM is a predecessor of APT, utilizing similar specimen geometries and electric field intensities. In FIM, mono-atomic gases (e.g. He, Ne) are typically used to produce a highly-magnified image of the surface of an emitter via field ionization near the specimen surface. Eventually, the atoms cross a region where the electric field is sufficient to cause their ionization and projection towards a detector screen. The polarization effects act as a local well, increasing the local concentration of atoms. Maximal enhancement of this concentration to $10^4$ was evaluated In a first approach the local pressure at the apex may be described using Maxwell Boltzmann distribution of velocities to

$$P = P_0 e^{\frac{\alpha F^2}{2kT}} / 4\pi\epsilon0 \quad (11)$$

In FIM, this effect enables a strong enhancement of the ionization probability (described in the next section), close to high field region as it participates to the increase of the image gas supply function.

The same effect was recently used to elucidate local oxidation of Fe in an Environmental Atom Probe. The specimen (pristine Fe single crystal) is simultaneously exposed to low pressures of pure oxygen gas, on the order of $10^{-7}$ mbar, while applying intense electric fields on their surface of several tens of volts per nanometer. The local composition of the different surface structures is probed directly and in real time and successfully compared with first principles-based models. Rough Fe{244} and Fe{112} facets were found are more reactive toward oxygen than compact Fe{024} and Fe{011} facets. Results demonstrate that the influence of electric field on the oxidation kinetics depends on the timescales that are involved as the system evolves toward equilibrium. The initial oxidation kinetics show that electric fields facilitate the formation of an oxide through an increase in pressure due to polarization effects. However, as one approaches equilibrium, high field values mitigate this formation. Ultimately, externally applied electric fields can be used to dynamically exploit reaction dynamics at the nanoscale towards desired products in a catalytic reaction (Lambeets et al., 2025).

Field polarization is also an important phenomenon limiting the probability of emitting desorbed atoms of molecules. Indeed, if atoms are emitted from the specimen surface, under the application of laser pulses, the kinetic energy of these atoms may not exceed the induced surface thermal energy (<1000K). As a result, thermally desorbed energy would never exceed 100 meV. This energy is much smaller than the binding energy induced by the polarization (>1 eV in practice). We may note that the spatial extension of the high electric field region may achieved several nanometers in thickness, so that a displacement of the emitted particle is however possible inside the potential well induced by the gradient of field (Gault et al., 2016).
### c. Field ionization

Field ionization is a quantum mechanical mechanism that occurs when atoms are subjected to sufficiently strong external electric fields. In the absence of a field, an electron in a bound atomic state is confined by the Coulomb potential of the nucleus. The application of an intense electric field, however, modifies this potential by introducing a linear term that effectively lowers and narrows the barrier on the vacuum side. As a result, the electron wavefunction acquires a finite probability to tunnel through the barrier, a process formally described by quantum tunneling theory and closely related to the Fowler-Nordheim formalism originally developed for field electron emission (equation (12)). (Fowler and Nordheim, 1928). As a result, the theoretical treatment is very similar to the calculation of electron tunneling on a field emitter surface.

When the local field strength reaches several tens of volts per nm, as is the case at the apex of sharply pointed needle, the tunneling probability becomes appreciable and field ionization takes place. This process underpins the operation of field ion microscopy, where surface atoms at the tip apex are ionized and the resulting ions are projected onto a detector, enabling imaging with near-atomic resolution (Müller and Bahadur, 1956; Forbes, 1995; Gomer, 1959).

<figure><img src="image_9.png" /><figcaption>Figure [7]. Schematic potential energy diagram for field ionization of an atom of ionization energy $I_1$ close to a clean metallic surface defined by $\Phi$ the work function of electron (fermi level) situated at the optimum tunneling conditions. A clear relationship between the maximum tunneling rate and the field $F$ is found, with $x_c$ being the critical distance for tunneling.</figcaption></figure>

Potential energy diagrams (V(x)) for field ionization, in close proximity to a material surface, are simplified in Figure [7]. The barrier tunneling rate D(V,E) for an electron confined in the atomic wavefunction close to the surface is generally calculated using the Jefferies-Wentzel-Kramer-Brillouin (JWKB) approximation, following
$$D(E, V) \propto \exp \left\{ -\sqrt{\frac{32\pi^2 m_e}{h^2}} \int_{\text{barrier}} \sqrt{V(x) - E} dx \right\} \quad (12)$$

With $m_e$ the electron mass, and E the energetic level of the electronic wavefunction. V(x) is the sum of electrostatic potentials induced by the material surface, the electric field, and the coulomb potential of the atom nucleus.

For a given surface, the transparency can be calculated as a function of physical constant of the selected atom (essentially the ionization energy I), the work function of electron in the material (Φ), and a refined expression of the shape of potential energy close to the surface (V(r) or V(x), in a one-dimensional form). In the simplest approach a triangular shape of the barrier may be assumed giving a transparency with following

$$D(E, V) \propto \exp \left\{ -\frac{b}{F} \right\} \quad (13)$$

With b, a constant related to I and Φ. It reveals that there is a strong dependency of the ionization rate over the surface field. We may note that the field is generally higher at close distance (more precisely, around a given critical distance $x_c$) from surface roughness, so that ionization can be used as a local probe of the surface field. The atom of interest acts like the tip scanning the surface in Scanning Tunneling Microscopy (STM).

From $\tau=1/D$, the probability of ionization P(t) for a single atom in a time interval t may be evaluated given by

$$P(t) = 1 - \exp \left\{ -\frac{t}{\tau} \right\} \quad (14)$$

if $\tau$ remains constant. Generally, there is a change in position during t so that it varies. Equation must be replaced by

$$P(t) = 1 - \exp \left\{ -\int_{x} \frac{dx}{v(x)\tau(x)} \right\} \quad (15)$$

Where $v(x)=dx/dt$ the velocity of the particle

The probability of ionization is therefore not only dependent of the quantum ionization rate, but also from the time of residence of the particle in the high field region. We may note that it was recently proven using ab-initio computation, that the probability was also dependent of the electronic density of state of the surface. Using the Tersoff-Hamann approximation known from scanning tunneling microscopy it is possible to study the tunneling of gas atoms hovering above the surface in the presence of a very strong electrostatic field. It was demonstrated that the tunneling rate could vary significantly at the atomic scale depending on the nature of the atom of the surface. This different being link not only to the work function of electrons but to the electronic density of states of the surface (Bhatt et al., 2023).

### d. Post-ionization

The process called post-ionization can be understood as the sequential ionization of an ionized atom, as described in the previous section, beyond the escape charge state $n=n_i$ (Fig. [Field3]). A field-evaporated or field-ionized atom can lose one or more extra electrons provided the tunneling probabilities of these electrons are sufficiently high. It finally acquires the arrival charge state $n=n_f$. This process occurs therefore after an ion has already been created and is being accelerated away from the surface by the intense field. Post-ionization processes occur nevertheless extremely close to the surface. Microscopically, post-ionization has the same features of ionization: it is field-assisted electron
tunneling from the ion to the sample surface. The role of the external field is lowering and thinning down the potential barrier, so that electrons can tunnel out with appreciable probability. The theory of post-ionization has been developed by Ernst and Jentsch for $Rh^+$ (Ernst and Jentsch, 1981) and later generalized by Kingham (Kingham, 1982) based on an analytical approximation of a WKB Hamiltonian. This probability is largest very close to the surface, where the local field is of the order of tens of V/nm, at a specific critical distance $x_c^n$. Most importantly in APT metrology, it states that sequential charge states $M^+$, $M^{2+}$, $M^{3+}$ etc. have increasing field threshold for tunneling probability, as well as increasing values of critical ionization distances $x_c^n$ depending on the ionization state n. This mirrors the fact that ionization energies increase with ionization state. The relative abundance of different ion species becomes therefore a function of the electric field and may be used to determine or, at least, estimate it. This finding lies at the foundation of the charge state abundance or ratio metrics.

### e. Charge state ratio metrics (Kingham curves)

As mentioned above, the phenomenon of post-ionization translates into a monotonous increase of the abundances of higher charge states from a field emitter with increasing field intensity. This behavior is illustrated in Fig. [8]-(a): the abundance of the $Ga^{2+}$ ion increases by almost one order of magnitude when the $V_{DC}$ bias is increased from 4.5 kV to 6.5 kV during a series of parametric analyses of a GaN specimen. As the Kingham post-ionization theory clearly identifies a univocal relationship between the abundance of charge states and the field, it becomes possible to exploit the former in order to estimate the field at the surface of a field emitter. An example of this is reported in Fig. [8]-(b), where the charge state abundances of W ions are reported as a function of the field intensity (Kingham, 1982; Tegg et al., 2024). For practical purposes it is common to introduce the quantity known as Charge State Ratio

$$CSR(A^{(n+1)}/A^{n+}) = \frac{N(A^{(n+1)})}{N(A^{n+})}, \qquad (16)$$

Where $N(A^{i+})$ is the number of ions of charge state $i+$ counted within a given region of the dataset. If only two charge states are present in the dataset, the shorter expression $CSR(A)$ can be adopted without ambiguity. The expressions relating CSR and field $F$ in the theory by Kingham are the result of a full quantum mechanical calculations, and their application to common situations in APT is not straightforward. In order to circumvent this problem, Tegg and co-authors have recently provided a set of simplified expressions of this form:

$$F_{eq} = a \left( 1 - \frac{b}{\left[ CSR(A^{(n+1)}/A^{n+}) \right]^{0.3} + b + 0.256} \right), \qquad (17)$$

Where a and b are parameters listed in the reference (Tegg et al., 2024). The $F_{eq}(CSR)$ function (also commonly called the equivalent field or the effective field) is given for three combinations of charge states of W in Fig. [8]-(c), which also visualizes the excellent approximation of the analytic expression (17) to the relations directly issued from the more complex expressions given by Kingham. The CSRs provide thus a commonly used metrics for the estimation of the microscopic field $F_μ$ at the surface of a field emitter, as $F_{eq} \sim F_μ$. The main advantages of it are the following:

(i) Instrument and sample independence. The CSR is issued by a quantum model in which the only environmental parameter is the surface field $F$. In other words, it is independent of temperature and also of the used instrument. Notice that other environmental parameters, such as the laser energy per pulse or the DC voltage, are instrument and/or sample dependent, i.e. e given set of their values does not yield the same effect in different instruments and for different samples.

(ii) Material independence. As the phenomenon of post-ionization occurs in vacuum, the CSR is in principle independent of the material from which a given element is evaporated. However, this statement
is only approximately valid. A given element may indeed be found in different oxidation/reduction states within different materials, which could translate into a significant abundance of higher charge states at fields for which post-ionization is not predicted to produce them. As an example, Al<sup>2+</sup> is not supposed to be post-ionized to Al<sup>3+</sup> at the typical fields at which AlGaN is evaporated, yet Al<sup>3+</sup> is found in the mass spectrum (Rigutti et al., 2016a).

(iii) *A global or microscopic quantity.* The possibility of representing the CSR not only as a *global quantity*, but to define it within an arbitrary region of a given APT dataset is perhaps the main advantage of this metrics. An example of the detector space distribution of the CSR(Ga) from the analysis of GaN is reported in Fig. [8]-(d), while the CSR(Ga) distribution within a thin slice extracted from the 3D reconstructed volume of a GaN/AlGaN heterostructure is reported in Fig. [8]-(e). In the first example the CSR distribution provides information about the inhomogeneity of the surface field and its correlation with the specimen crystallography (Russo et al., 2018); in the second example, the alternation of layers with higher and lower fields correlates with the presence of layers of different composition and, consequently, different binding energy; in this same example, the gradual decrease of the field from the beginning to the end of the analysis is linked to the decrease of the evaporation rate necessary to keep the detection rate constant while the imaged surface increases (Di Russo et al., 2018). These two examples illustrate how the CSR can be displayed as a *microscopic quantity*, which can yield important information about the details of the APT analysis.

The CSR has become in the last two decades an extremely important parameter for the assessment of
the surface field intensity, and its use is nowadays current in most APT works studying the fundamental
aspects of field evaporation of new systems and/or their compositional metrology. The application of
the CSR to specific problems will be illustrated in the case studies of part 5.

<figure><img src="image_10.png" /></figure>

**Figure [8].** Relevance of charge state ratio (CSR) metrics in APT. (a) Example of experimental dependence of the fractions of Ga²⁺ and Ga⁺ on the applied voltage during several analyses of a GaN field emission specimen. (b) Field dependence of the probability of finding a field-evaporated atom in a given charge state according to the Kingham post-ionization model applied to tungsten. (c) Relationship between the CSR(W(n+1)⁺/Wⁿ⁺) and the surface electric field as calculated by Kingham (Kingham, 1982) and by (Tegg et al., 2024) (Adapted from (Tegg et al., 2024), Oxford Academic). (d) Two-dimensional detector space map of the CSR(Ga²⁺/Ga⁺) issued from the analysis of a GaN field emitter. (e) Two-dimensional map of the CSR(Ga²⁺/Ga⁺) issued from the analysis of a GaN/AlGaN heterostructure realized within a thin slice extracted from the 3D reconstructed space.
f. Field-induced stress

The average surface electric field $F_s$ introduces a tensile stress state in the specimen. Due to the accumulation of a high density of free holes on the apex surface, the field does not penetrate deep into matter but “pulls” on the system, generating a Maxwell stress on the emitter surface (Gomer, 1994):

$$ \sigma_{apex} = \frac{1}{2} \varepsilon_0 F_s^2 \qquad (18) $$

Which is independent of the analyzed material. Significant stress intensities can be achieved during APT analyses, as reported in Fig. [9]. Approximating the apex as a hemisphere and integrating the forces $dT$ that develop on a surface element $dS$, as illustrated in the inset of Fig. [9], the tension $T$ along the field emitter axis is

$$ T = \pi R_{apex}^2 \sigma_{apex} \hat{n}_{<axis>} \qquad (19) $$

where $\hat{n}_{<axis>}$ is a unit vector pointing along the field emitter axis. The stress at the apex may be assumed as hydrostatic, as the Maxwell forces are directed approximately along the needle radius. At a distance from the apex approximately equal to several $R_{apex}$, the stress transforms into uniaxial, with the only non-negligible component

$$ \sigma_{zz}(z) = |T|/S(z), \qquad (20) $$

where

$$ S(z) = \pi \cdot \left( R_{apex} + \frac{(R_{base} - R_{apex}) \cdot z}{L_{tip}} \right)^2 \qquad (21) $$

is the area of the axial cross section at the axial coordinate $z$, $R_{base}$ is the specimen radius at its base and $L_{tip}$ is the specimen length (inset of Fig. [9]) (Rigutti et al., 2017). An upper limit for the intensity of stress may be estimated for an ongoing experiment though the CSR metrics. So far, a measurement of stress intensity cannot be achieved within typical APT setups. Photonic Atom Probe measurements, exploiting the known dependence of photoluminescence lines as a function of stress could assess stress levels up to 9 GPa in diamond (Rigutti et al., 2017) and around 1.25 GPa in ZnO (Dalapati et al., 2021). Mechanical stress may produce plastic relaxation, which may turn into specimen fracture (some specific values of yield strength are reported in Fig. [9] for reference). This mechanism is particularly critical during the analysis of multilayers with strongly differing evaporation fields (Nicolas Rolland et al., 2015b). Due to the complexity of the stress measurement, the role of mechanical stress on composition measurements in APT has not been studied yet, but it is known that stress gradients may promote formation, diffusion and migration of point defects (Aziz, 2001; Connétable and Maugis, 2020) as well as dislocation motion (Chakravarthy and Curtin, 2011) in bulk materials.
<figure><img src="image_11.png" /></figure>

Figure [9]. The hydrostatic tensile stress generated at the apex of a field emitter as a function of the electric field. Reference values of stress and field are reported from the literature in the diagram: ref. (a) = (Ono et al., 2005); ref. (b) = (Rigutti et al., 2017), ref.(c) = (Chen et al., 2020), ref.(d) = (Pavlina and Van Tyne, 2008), ref. (e) = (Dalapati et al., 2021). In the inset, schematics illustrating the spatial evolution of the stress field within the specimen.

### g. Field-induced migration

Some elements are quite prone to movements on the nanoscale under the effect of electric or strain fields (Tsong and Kellogg, 1975). Electromigration – a thermally activated transport mechanism where an applied electric field reduces the activation energy barrier, enables directional ion movement driven by thermal energy. This effect allows ions with lower migration energy barriers to move efficiently, even against concentration gradients, and typically occurs faster than diffusion. The migration, also known as directional walk, has usually specific directions, towards one or more crystal poles, and primarily concerns solute species in both metals and semiconductors. In metals this process has been studied for solute species such as P, Si, Mn, B, C and N, and (B. Gault et al., 2012). The proneness to directional walk is generally increasing with increasing elemental evaporation field and decreasing atomic radius (B. Gault et al., 2012). In semiconductors, an example of directional walk is illustrated in Fig. [10], showing that for low (actually: less homogeneous) field conditions the distribution of B in Si is apparently concentrated close to the axis of the specimen (Guerguis et al., 2024). These solute species are accumulated in one subregion of the surface. On one hand, this induces a degradation in the associated spatial precision and a bias in the calculation of concentrations. On the other hand, migrating species may evaporate with a strong degree of correlation (section 4.c.iii), which can lead to specific losses by pile-up effects (section 4.b.ii). In battery materials, for instance, the in situ delithiation process can be understood as an electromigration process (Greiwe et al., 2014; Kim et al., 2022). Notably, at the low temperatures used in atom probe tomography (APT) analyses (typically 30–80 K), the electrostatic field still facilitates ion migration. To mitigate unwanted effects, effective shielding—such as coating the specimen with a conductive material—can be employed (Kim et al., 2022).
<figure><img src="image_12.png" /><figcaption>Figure [10]. Normalized detector density maps of Si and B from the APT analysis of Si:B at two different field conditions according to the CSR(Si) metrics. The red arrow indicates the laser direction. The analysis conducted at low field clearly point out that B is subjected to field-driven migration towards the axis of the specimen.
(Adapted with permission from (Guerguis et al., 2024), Elsevier)</figcaption></figure>
## 4. Mechanisms of loss

In APT there are different mechanisms leading to losses. These can be classified into three categories, which are often interdependent, i.e. detector losses, losses related to difficulties in the interpretation of mass spectrum, and physical mechanisms, related to the evaporation process itself. It is crucial that the operator applies empirical methods allowing for the assessment of a loss. In some cases, the assessment and sufficient complementary information may allow for the correction of the error in the composition measurement. Table [III] reports the main mechanisms of loss and lists some of the systems impacted by the different mechanisms.

<table><caption>Table [III]. Synoptic view of the main channel of loss or error in composition measurement in APT, the conditions at which they typically occur, and examples of the systems impacted, which are detailed in part 5.</caption><thead><tr><th>Category</th><th>Mechanism</th><th>Conditions of occurrence</th><th>Examples of systems affected</th></tr></thead><tbody><tr><td rowspan="2">Detector Performance</td><td>Intrinsic detection efficiency – Open Area Ratio</td><td>All; instrument-dependent</td><td>All</td></tr><tr><td>Pile-up – dead or dazzled zone</td><td>Large fraction of multiple detection events, high evaporation rates</td><td>See entries for correlated evaporation and/or molecular dissociation</td></tr><tr><td rowspan="4">Data Analysis</td><td>Peak Overlap</td><td>Composition-dependent</td><td><sup>64</sup>Zn<sup>2+</sup>,<sup>0</sup>O<sub>2</sub><sup>+</sup> in ZnO; <sup>14</sup>N, <sup>28</sup>Si in Si, N containing systems (e.g. TiSiN, GaN:Si);</td></tr><tr><td>Peak Ambiguity</td><td>Composition-dependent; Molecular ions at low field.</td><td>As, P clusters in III-V semiconductors; SiGeC;</td></tr><tr><td>Thermal tails</td><td>Poor thermal management due to low conductivity or conductance; High Laser Intensity</td><td>GaAs; SrTiO<sub>3</sub></td></tr><tr><td>Background noise</td><td>Instrument-dependent (DUV, EUV); High-Field</td><td>SrTiO<sub>3</sub></td></tr><tr><td rowspan="4">Physical Channel</td><td>Neutral Evaporation</td><td>Systems ejecting neutral species directly or upon molecular dissociation; Low-Field</td><td>As in III-As, N in nitrides, O in oxides</td></tr><tr><td>Preferential Evaporation</td><td>Systems with chemical species with different ZBEF; High-Field, Low Field-Fraction, Low Laser Intensity</td><td>Ga in GaAs, GaN, AlGaN, InGaN; In in InN; Zn in (Mg,Zn)O;</td></tr><tr><td>Correlated Evaporation</td><td>Chemical bond-dependent; Possibly related to retention and migration; Feeding Pile-up</td><td>GaSb; GaAs; GaN; TiN; Fe<sub>n</sub>O<sub>m</sub>; ErMnO<sub>3</sub>; SiC; B in Si:B and SiGe:B;</td></tr><tr><td>Molecular Dissociation</td><td>Sample-dependent (Low-field)</td><td>InP, GaN, Al(Ga)N; Ga<sub>2</sub>O<sub>3</sub>; SiC;</td></tr></tbody></table>
a. Detector losses

The average system detection efficiency DE is a key parameter to obtain accurate reconstruction of atom coordinates, but also to get reliable measurement of composition, in particular if the efficiency is not homogeneous over the mass range or the energy range of impinging ions. It is therefore important to develop ways to measure efficiently this parameter as a function of the system of interest and conditions of analysis. When focusing on the detection system, we need to evaluate the contribution of the amplification system (the MCPs), and the contribution of the Time and position encoding system (the delay-line anode) (Sijbrandij et al., 1996; Deconihout et al., 2002; Prosa et al., 2014).

i. Multi-channel plate performance

Microchannel plates (MCPs) are electron multiplier devices that can detect charged particles and fast neutral particles, among other high-energy particles. The MCP consists of micropores that form a honeycomb structure, as shown in Figure [2]. In the event that an incoming particle generates secondary electrons at the input surface of the MCP and these electrons enter the pores under the electric field, then avalanche multiplication of the electron bunch provides a pulsed current. The first step toward generating secondary electrons in the pore is important. In APT, the ratio of the number of detected ions to the number of effective signals that reach the delay-line anode, that is, the MCP detection efficiency ($DE_{MCP}$), is therefore critical.

$DE_{MCP}$ varies with the incident energy, mass, incident angle of detection, and charge state of an ion, and several experimental studies of detection efficiency in an MCP operated in the counting mode have been reported (Fraser, 2002). Recent MCPs have generally a large angle of channels with respect to the face entrance of MCPs (~20°), to avoid variation of gain with angle of incidence of striking ion. $DE_{MCP}$ is therefore considered almost homogeneous over MCPs surface. The main factor affecting $DE_{MCP}$ is the ion of interest velocity. To summarize these previous investigations, the detection efficiency generally increases with the ion impact energy above the threshold for secondary electron emission, and the detection efficiency varies with the ion species. Above 1-2 keV for low mass element (high velocity) and above 3-6 keV for high mass element (slow velocity), the detection efficiency reaches a plateau roughly proportional to the open-area ratio (OAR) of the MCP. The fact that the OAR limits the detection efficiency indicates that particles are poorly detected if they do not enter an MCP pore. The OARs of conventional MCPs (hereafter MCPs) are typically 50-65%. Note that this value was confirmed experimentally in APT measurement using the lattice of tungsten atoms as a reference (Deconihout et al., 2002).

In the early 2000s, several attempts to exceed the OAR limit in APT have been made by applying a repelling electric field to reflect secondary electrons back to the MCP surface (Sijbrandij et al., 1996; Deconihout et al., 2002) however this solution generally increase the occurrence of noise signal due to emission of secondary electrons by the mesh used to generate the field region. A solution was to widen the open area. An efficient way to accomplish this goal was to narrow the walls of the pores. However, a certain wall thickness is needed to maintain mechanical durability. That is, the OAR has a practical upper limit, usually 80%, and the OAR never approaches 100%. An alternative solution is to expand the entrance of the pores, leaving the inner walls untouched. Detection efficiency up to 90% were reported with some prototypes (Matoba et al., 2011). Commercial MCPs are certified to 80% detection efficiency. Using a protocol of encapsulating a defined volume of a known material, an overall detection efficiency measurement was performed. In Nickel/Chromium multilayer samples (NIST SRM 2135c), DE was found to reach 77±2% close to the manufacturer value (Prosa et al., 2014).
## ii. Dead or dazzled zone

As soon as ions are detected one by one, the detection efficiency DE is equal to $DE_{MCP}$. However, difficulties arise when more than one ion is evaporated and detected from the probed area on a single evaporation pulse. Incident ion hits on the walls of MCP channels generate a short electronic (~ns) pulse from the output. We may note that at high gain, the recovery time of a MCP channel is relatively long (up to 100 µs (Beavis et al., 1989; Westman et al., 1997)), so that the amplitude of electronic pulse is lower for successive impacts at the same position, but this effect generally affects MCP gain at a high flux, much higher than usually used detection rate in APT. The main cause of DE loss, is failure of the position-time encoding system to detect some impacts. Indeed, when two ions reach the detector closely separated in time and in space, one of these ions can be lost due to a masking effect. This phenomenon is named ion pile-up (Figure [11]-(a)), (Tsong et al., 1978; Cerezo et al., 1984; Rolander and Andrén, 1989) and was known to affect atom probe measurement very early in the development of the instrument. It affects in particular correlated field evaporation events and dissociation events, which are the main cause of multi-hit events (or multiple events). We may recall, that even working at low detection rate (<<1 ion/pulse in average), the probability of multiple event (PME), due to intrinsic physical mechanism of emission (De Geuser et al., 2007; Cojocaru-Mirédin et al., 2024), can be large. In some non-metallic alloy the PME can reach 100% of the detection events. In addition, the probability that these multiple events arise from close location is generally high.

Ion pile-up and relative DE loss is linked to capability of the position-time encoding system (namely the DLD) to detect close impacts, in latest atom probe designs. A typical DLD system has an electronic dead-time that is between 1.5 and 20 ns depending on the acquisition electronic system (Da Costa et al., 2005; Gribb et al., 2002; Jagutzki et al., 2002a; Costa et al., 2012; Prosa and Oltman, 2022). During this dead-time, multiple ion impacts on the detector cannot be differentiated from one another. However, if the ions in the multi-hit event arrive sufficiently far on the detector, individual ions positions may be resolved even if the ions have the same time of arrival. In DLD this is possible through the time-to-position conversion, 1 ns corresponding to about ~1 mm on large DLD. This defines a 3D dead region in (X,Y,time) space, The physical dead-zone is thus a few millimeters (5 to 20 mm) wide around each ion impact site and span over a few nanosecond after the initial hit. When plotting the time and space distribution for every second ion related to each first ion hit in multiple events, a depleted region appears; where TOF differences and relative positions do not exceed, respectively, the dimensions of the dead zone around the first point of impact. It was highlighted by Peng et al. (Peng et al., 2018) and Meisenkothen et al. (Meisenkothen et al., 2015) with the analysis of materials producing high PME Figure [11]-(b,c). We may note that the exact shape of the dead region is neither homogeneous nor isotropic, because of the complexity of the encoding process as it was demonstrated by Bacchi et al. (Bacchi et al., 2019).

Depending on the PME and on the complexity of the mass spectra, the selective losses and the exact estimation of the DE can be complex to determine. Some recent corrections procedure, assuming a model of multiple events, were proposed in some cases, but it is difficult to generalize (Ndiaye et al., 2024; Prosa and Oltman, 2022). However, we may note that some additional information can be used to reduce or compensate the detection losses induced by multiple events. In 1978, Tsong et al. noted that the amplitudes of detection signal vary in height as a function of the number of hits (Tsong et al., 1978). The MCPs are indeed used in a saturation regime, which means that every single-ion impact induces almost the same signal amplitude. However, during a multiple impact when the electrical signals of the ions are superimposed on the MCP, their amplitude is added together. In 1987, using an imaging atom probe, Kellogg demonstrated the good linearity between the signal amplitude and the number of spots induced by field evaporation (Kellogg, 1987). As a result, on modern DLD atom probes, the dead region is indeed not completely “dead”, if amplitude of the signal is measured. This region should be rather referred to as the “dazzled region” because the metaphor indicates a reversible state and the region is not completely “blind” with respect to the generation of a signal issued by a second impact. The
comparison of the amplitude histograms of single events in Figure [11]-(d,e) shows that the histograms relative to carbon in a SiC APT analysis which appear to be composed of successive convolutions of a histogram like the one produced by silicon. The multiple peaks appearing in the carbon amplitude histogram correspond thus to the trace of double (triple, etc.) impacts. This phenomenon has been consistently interpreted due to the specific tendency of C to correlate evaporation in carbides, yielding a higher fraction of multiple evaporation and detection events for this element, which is consequently more exposed to measurement errors due to the limitations in the detector performances. The observed trend of increase of C loss with increasing depth of analysis can then be explained by the decrease in geometric magnification, making it more and more likely for correlated evaporated ions to fall within the so-called “dazzled region” of the detector (Figure [11]-(f), (Ndiaye et al., 2023a)).

<figure><img src="image_13.png" /></figure>

**Figure [11]**. Detector losses in APT. (a) Schematic representation of the so-called “dead” or “dazzled” region of a detector, approximately corresponding to a volume within a radius $\Delta r$ and a time $\Delta t$ from the first impinging ion issued from a multiple evaporation event; (b, left) Dead zone characterization for a LEAP 5000 XS: amount of ion pairs detected in multiple events as a function of the TOF difference between the two constituent ions and (b, right) 3D time and spatial distribution of the second ion in the ion pair with respect to the first ion; selected iso-density surfaces are marked in red and green; (c) serial top views of density maps over thin slices at different positions of (b, right) (Reproduced with permission from, Elsevier). (d) Sketch of the dependence of MCP amplitudes on the number of impact at neighboring distance (i.e. within the dazzled region); (e) Histograms of the amplitudes of $Si^{2+}$ and $C^{2+}$ detection events interpreted as single impacts in the analysis of SiC: the histogram for $C^{2+}$ shows a higher presence of single detection events which large amplitudes, which implies the presence of a significant number of multiple impacts erroneously interpreted as singles. (f) Dynamical effects influencing detector losses. At increasing time points of the evolution of the specimen, correlated ions of the same mass/charge evaporating from sites at the same distance on the specimen surface impinge with decreasing distance on the detector. This may translate into an increased probability of detector loss via the pile-up effect.
### iii. Detection efficiencies

An evaporated atom has a certain probability of detection, which is related on one hand to the intrinsic efficiency of the detector, and on the other hand to possible physical mechanisms limiting its probability of being detected. For this reason it is useful to introduce some classification in what is referred to as “detection efficiency” (Rigutti et al., 2016a; Di Russo et al., 2018).

*Efficiency of the detector system.* (section 4.a.i.) The intrinsic efficiency of the detector $DE_{MCP}$ (sometimes indicated as $η_D$) corresponds approximately to the OAR. A further reduction in the intrinsic efficiency must be taken into account for the instruments including a reflectron. It can be considered as independent of the incident ionic species (Da Costa, 2016). Typically $DE_{MCP}$ = 60-80% depending on the detector type and generation.

*Specific detection efficiency.* It is convenient to define a specific detection efficiency $DE_i$ ($η_i$), i.e. relative to a chemical species i (atomic ion, molecular ion, element, isotope ...) that can be identified by the analysis of the spectrum of mass:

$$DE_i = \alpha_i DE_{MCP} \qquad (22)$$

where $α_i$ corresponds to the fraction of detectable ions, i.e. those which do not evaporate in a loss channel such as preferential evaporation, neutral evaporation or affected by molecular dissociation mechanisms after evaporation (which will be discussed in the following sections). This detection efficiency is not only a characteristic of each ionic species, but can also be considered as a function of the electric field, and of the starting position of the ion on the tip. The specific efficiency is therefore also a local, microscopic efficiency.

*Reconstruction, or global detection efficiency.* It is also possible to define a detection efficiency $DE_{rec}$ ($η_{rec}$), related to the reconstruction (i.e. to the algorithm used in order to retrieve volume information from the evaporation sequence of a sufficiently large specimen). This parameter is a constant that the user determines when reconstructing a certain volume. The chosen value is the one that makes it possible to obtain the best spatial fidelity of the 3D reconstruction (section 2.e). Its physical meaningfulness is conditioned by the knowledge of the specific atomic volumes within the analyzed compound. For a structure with known thickness layers the global detection efficiency is that which makes it possible to recover these thicknesses in the 3D reconstruction. It is possible, however, that for particularly deep volumes a constant value of $DE_{rec}$ is not sufficient to ensure the spatial fidelity of the entire reconstruction. In this case, in the absence of a method for determining the detection efficiency as a function of the analysis depth, it is appropriate to carry out the reconstruction of separated layers, each layer adopting a different value of $DE_{rec}$. An estimation of measurement of this detection efficiency was performed in various alloys and compounds by Diercks and Gorman (Diercks and Gorman, 2018), by using correlated electron microscopy measurement of APT specimen before and after APT measurements (Fig. [12]).
<figure><img src="image_14.png" /></figure>

Figure [12]. Curves of the detection efficiency and image compression factor values for GaN, CdTe, Ceria, and some metallic specimen (Al, NiNbZr alloy) that produce a self-consistent reconstruction. Assuming common values of ICF in the range 1.4-1.6, the detection efficiency in the analysis of GaN, CdTe and Ceria is found much lower than metallic samples, indicating a strong channel of undetected emission (reproduced with permission from (Diercks and Gorman, 2018), Elsevier).

b. **Mass peak interpretation**

After detection and data collection, attention must be paid to the interpretation of mass spectra, which should be as accurate as possible. This is not always straightforward in mass spectrometry and in ToF spectrometry in particular, as ambiguities may affect mass spectra, depending on the system under study.

i. *Mass peak superposition*

Mass peak superposition is a current problem in APT. Some examples of this are given in Fig. [13]-(a,c,d,e). In part (a) the typical mass spectrum of ZnO is affected by the superposition of the O₂⁺ dication and of the ⁶⁴Zn²⁺ atomic ion at 32 m/z. In this case it is quite straightforward to separate the amount of O and the amount of Zn within the peak, under the hypothesis that the isotopic abundances of Zn²⁺ correspond with the natural abundances, and that no other O-containing species. A slightly more complex case is presented in part (c), showing a close up of a mass spectrum issued from the analysis of Cu-duped ZnS. The family of Zn⁺ isotopes can be quite well resolved. However, these peaks are superimposed to ZnH⁺ peaks, which count for about 1/20 of the total amount of Zn and which are superimposed to the ⁶⁷Zn⁺ and ⁶⁸Zn⁺. Furthermore, the detection of Cu is complicated by two factors: first, the ⁶³Cu⁺ peak is quite clearly resolvable, but it is affected by a high background originating from the low-mass tail of ⁶⁴Zn⁺. A case in which hydride molecules are in even larger amount is presented in part (d) relative to the analysis of a ZrH phase. In this close up, the ZrH²⁺ peaks are superimposed to the ZrHn²⁺ molecular ions. The peak decomposition is shown in the inset, and has been obtained under the hypothesis that ZrH²⁺ ions are present only (Diagne et al., 2025). Peak decomposition or, with a deprecated term, deconvolution methods are not within the scope of this review (Johnson et al., 2013; Coakley and Sanford, 2022; Li et al., 2026). They can provide quite accurate composition measurements, but it must be kept in mind that the identity of individual detection events cannot in general be provided (for exceptions and for the statistical interpretation of individual detection events,
see for instance (London et al., 2017) and (London, 2019)). This, in turn, complicates and questions the
possibility of performing further statistical analyses on the events contained in superimposed
mass/charge peaks. Strategies for overcoming these problems are quite demanding and imply either the
synthesis of materials using isotopic substitution (David L.J. Engberg et al., 2018) or the implementation
of the instrumentation. The close-up of the spectrum in Fig.[13]-(e) shows the possibility of resolving
$^{28}$Si$^+$ and $^{14}$N$_2$$^+$ at 20 m/z in the analysis of SiN. This spectrum has been obtained by exploiting the
capabilities of an instrument featuring a dual Einzel lens configuration and an elongated flight path of
about 1m. Furthermore, the data have been selected from a reduced zone of the detector, which increases
the mass separating power up to $\Delta m/m \sim 1400$ but also reduces the field of view (Chae et al., 2025).

ii. *Mass peak ambiguities*

*Mass peak ambiguities* occur when the identity of a peak presents some uncertainty between two or
more ion species. This situation may involve a mass peak superposition or not, and is often related to
the evaporation of molecular ions or clusters. An example of this is provided in Fig. [13](a), where the
peak at 16 m/z can be attributed to O+, O₂²⁺ or to a mixture of both. A similar situation applies to the
peak at 14 m/z in N-containing systems. Currently, arguments based on the high ionization energy for
the process O₂⁺ → O₂²⁺+e⁻ tend to suggest that the peak is actually composed of atomic O⁺ (or N⁺,
respectively) only (Jaroń-Becker et al., 2004). This sort of ambiguities may be present also in spectra
with a large number of relatively complex molecular ions. III-V materials such as arsenides (Fig. [13]
(b)) or phosphides produce a large number of molecular clusters Asn k⁺ or Pn k⁺. In these cases, ambiguities
may occur concerning mass peaks potentially formed by ions with the same mass/charge ratio but with
different masses. In all these cases, the multiplicity of the atom within the molecular ion is a factor
directly influencing the calculation of the composition (Russo et al., 2017; E. Di Russo et al., 2020).

The occurrence of molecular ions, usually enhanced in low-field conditions, is a common phenomenon
in field evaporation of dielectrics and often complicates the interpretation of mass/charge spectra. If they
are not producing mass peak superposition or other ambiguities, and if they are not associated to more
complex phenomena such as molecular dissociation and correlated evaporation (see sections 4.c.iv and
4.c.iii, respectively), their presence does not prevent the user from the determination of a correct
composition, provided the multiplicity of each atomic species is correctly accounted for. However,
molecular ions are generally a hint about the complexity of chemical processes at the surface of field
emitters (Cojocaru-Mirédin et al., 2024), and are in general associated with a degradation of the spatial
precision (Gault et al., 2026). Therefore, the user should always be cautious in the application of
advanced data mining protocols to datasets containing large amounts of molecular ions.

iii. *Thermal tails and background noise*

*Thermal tails.* Thermal tails are due to delayed evaporation during the phase of heat relaxation after a laser pulse. They induce broadening and skewness in mass/charge peaks, and can therefore lead to the increase of background for neighboring peaks or even to significant difficulties in identifying them and quantifying the associated signal. The impact of thermal tails can be reduced by appropriate choice of experimental parameters such as laser intensity and wavelength (Vurpillot et al., 2009; Vella, 2013; Kirchhofer et al., 2013; Caplins et al., 2023), as well as sample shape (Arnoldi et al., 2012). On the other hand, various methods are available for the optimization of the mass spectrum analysis in case of peak overlaps with thermal tails (London, 2019).

*Background noise* is generally referred to the random detection of ionized gaseous species coming from
the analysis chamber or species adsorbed at the surface of the specimen. These atoms/molecules being
ionized at the DC voltage, are detected at any time during the opening of the detection window and
constitute a continuous signal in the time-of-flight spectra. Another small contribution may be originated from the detector itself (Pareige et al., 2016). In instrument pulsed with deep or extreme UV light, pulsing may contribute to the increase of background noise too (Miaja-Avila et al., 2021). Solute species are obviously the most exposed to loss related to background noise, as this increases the LOD.

<figure><img src="image_15.png" /></figure>

**Fig. [13].** Examples of mass/charge spectra potentially exhibiting problems for the attribution of ion species. (a) The shown mass/charge spectrum of ZnO is affected by (i) the significant presence of atomic and molecular hydrogen (ii) the potential ambiguity in attributing the peak at 16 m/z to O<sub>+</sub> or O<sub>2</sub><sup>2+</sup>, (iii) the superposition of the peaks of O<sub>2</sub><sup>+</sup> and <sup>64</sup>Zn<sup>2+</sup> at 32 m/z and (iv) the presence of a large amount of molecular ions (highlighted with the red text). (b) The spectrum of AlGaAs exhibits a large amount of As<sub>n</sub><sup>k+</sup> molecular clusters (red text). (c) The close-up of the mass/charge spectrum of ZnS:Cu shows the superposition of Zn<sup>+</sup> peaks (indicated by blue squares) with the peaks of ZnH<sup>+</sup> (green triangles). The <sup>63</sup>Cu<sup>+</sup> peak can be recognized but is largely superimposed to the low-mass tail of majoritarian <sup>64</sup>Zn<sup>+</sup>. Due to the presence of <sup>64</sup>ZnH<sup>+</sup> (and, secondarily, to the high-mass tail of <sup>64</sup>Zn<sup>+</sup>), the signal of <sup>65</sup>Cu<sup>+</sup> is not recoverable. (d) Close up of the mass/charge spectrum of ZrH in the region containing the superimposed signals of Zr<sup>2+</sup> and ZrH<sup>2+</sup> ions. The decomposition of these peaks is shown in the histogram in the inset. (e) High-resolution mass/charge spectrum of SiN showing the possibility of resolution of <sup>28</sup>Si<sup>+</sup> and <sup>14</sup>N<sub>2</sub><sup>+</sup> peaks (Adapted with permission from (Chae et al., 2025), Springer Nature).
### c. Physical channels

Physical channels of loss reflect the complexity of field ion evaporation, and the departure of its reality from the picture of ionization of atomic species. Neutral evaporation, preferential evaporation, corelated evaporation, molecular evaporation with or without dissociation can significantly complicate not only the appearance and the interpretation of mass spectra, but also the detection process itself. These phenomena can often be assessed and sometimes quantified; however, they should be managed as carefully as possible because they are not avoidable in most non-metallic materials.

#### i. Neutral evaporation

Neutral evaporation is the most “devious” mechanism of loss in APT. Because neutral evaporation may affect differently chemical species of the sample of interest, it can lead to a direct systematic bias in the measurement of composition. We recall that MCPs detection gain is strongly dependent of the particle kinetic energy (or more rigorously particle velocity) below 1-2 keV. Neutral particles, directly emitted from the specimen surface, are not accelerated by the distribution of field close to the emitter, and are therefore completely undetectable in an APT configuration. Furthermore, the change in vacuum pressure that could be linked to an emission, in form an atomic gas, of neutral particles, is extremely limited due to the limited size of the probed volume in APT. A simple calculation of the increase of the gas pressure from the completed sublimation of the top apex will be significantly lower than that of the residual pressure in the analysis chamber (~$10^{-9}$ Pa in the best case). In this review we will also differentiate the case of neutral emission induced by dissociation of cationic molecules to the direct emission of atoms, or some neutral molecules from the surface. This emission will be described in the section related to molecular dissociation and Saxey’s plot representation.

<figure><img src="image_16.png" /></figure>

**Figure [14].** Sketch illustrating the proposed field evaporation behavior for the Ba:STO layers (a) the sample structure (assuming a perovskite structure); (b) Field evaporation of the $TiO_{2}^{+}$ leaving behind a Ba/Sr rich surface; (c) the Ba/Sr-O bonds become stretched due to ionic polarization caused by the field; (d) $Ba^{2+}/Sr^{2+}$ are field evaporated; (e) The oxygen left behind is then thermally desorbed as neutrals. (Scheme adapted from (Morris et al., 2024))

Because there is no direct proof of the direct emission of neutrals in the high electric field conditions of an APT, the literature of the effect is scarce. Several studies have proposed that the discrepancy between the APT measured composition of compound semiconductors and oxides and their actual compositions (which are normally stoichiometric and well established) was mainly due to thermal desorption of the
nonmetallic element present in the material. This was for instance proposed for oxides (Devaraj et al., 2013) (Figure [14]) or nitrides. The literature often proposed the emission of gaseous N₂ and O₂ in nitrides and oxides directly from the surface. The formation of a significant amount of these molecules which are at the end desorbed or sublimated without ionization is evoked. This hypothesis is linked to the variation of the measured composition with the surface electric field. When increasing the electric field, the non-metallic element proportion is always seen to increase progressively, indicating a strong relationship between the probability of ionization and the electric field. On a zinc oxide (ZnO) surface, Xia et al. simulated the surface stability under a positive electric field ranging from 25 to 30 V/nm (Karahka et al., 2015; Xia et al., 2015). Above 15 V/nm, the surface undergoes reorganization due to the electrostatic field effect, and beyond 25 V/nm, the geometry optimization no longer converges. The surface rapidly emits metallic species (in the form of Zn²⁺), while oxygen atoms rearrange quickly to form O₂ molecules.

The hypothesis of neutral emission is also reinforced by the absence of evidence of strong biases that may be due to multi-hits events and detector pile-up. This last hypothesis was for instance ruled out using of a specially designed meshed electrode placed in front of the specimen to manipulate ion pile-up from co-evaporated ions (Morris et al., 2022). Pile-up is therefore not the major cause of composition inaccuracy in GaN and AlN, or TiN (Schiester et al., 2024). The preferential field evaporation (Hatzoglou et al., 2020) (described hereafter) was also ruled out, since the amount of unwindowed detection events in mass spectra is not found to drastically increase at high laser pulse energy / low field. In addition, an estimation of measurement of the detection efficiency was performed in various alloys and compounds by Diercks and Gorman (Diercks and Gorman, 2018), by using correlated electron microscopy measurement of APT specimen before and after APT measurements. Using this approach, an empirical relationship between the image compression factor and detection efficiency is found for all of the analyzed specimens. Since ICF is generally in the range [1.4-1.6] in APT, the detection efficiency is found to be lower than expected reinforcing the hypothesis of undetected neutral emission (figure [12]). The same observed low estimated value was also measured in ternary semiconductor alloys by Di Russo et al. (Di Russo et al., 2018) A more robust similar approach, reinforced by numerical computation of the ICF in the geometry of the specimen, was also proposed by Veret et al in the case of monazite CePO₄ indicating a systematic degradation of the detection efficiency, and loss of the non-metallic element probably by neutral emission (Veret et al., 2025).

The neutral emission mechanism appears to correspond to experimental observations in various oxides and nitrides. However, this mechanism relies on the fundamental hypothesis that a neutral atom or small neutral molecule may first exit the extreme field region, and second may survive the extreme electric field that exists in front of the specimen. While many authors seem to identify the direct emission of neutral molecules from the surface, one may question the likelihood of such an event occurring on our sample, which is subjected to an electric field exceeding 20 V/nm in most of experimental cases.

Indeed, two physical phenomena oppose this mechanism. First, under these field strengths, if neutral particle emission were to occur, the particle would be strongly polarized by the electric field. Gault indicated that under the influence of the electric field, and due to the atomic polarizability of neutral molecules, the binding energy associated with surface polarization is on the order of 0.2 to 0.45 eV under a field between 20 and 30 V/nm (see field polarization, section 3.b). A molecule desorbed due to thermal excitation (a few hundred K, 10 to 100 meV) have a relatively low velocity induced normally by the thermal kinetic energy due to the surface specimen temperature. We may note that the potential well induced by polarization effect is linked to the gradient of field in front of the surface (so extending on several tens of nanometer). A neutral molecule at the temperature induced by the laser pulse would normally shift only by a few nm from its initial position before being recapture by the specimen. During the travel, density functional theory (DFT) models predicts the ionization of this neutral particles (as found by Karakha for instance). Nevertheless, this is fundamentally linked to the limitations of DFT,
which neglects reaction kinetics and assumes the electronic ground state at each time step during
relaxation. In reality, it should be necessary to evaluate the probability of ionization throughout the
entire process of molecule or atom expulsion. This kind of calculation was done in the process of field
ionization or post field ionization, in the context of a single atom or molecule escaping the surface
(Haydock and Kingham, 1981; Kingham, 1982).

To evaluate this probability, we make use of the ionization theory as described in section 3.c. This
approach calculates the integrated probability of ionization of a particle traveling from the specimen
surface to the infinite using Schrödinger equation in simple model potential of the atom under the
presence of a surface electrostatic field. For a single particle of charge leaving a planar surface submitted
to a field F, the probability of ionization or post ionization depends essentially on the transparency of
the electronic tunnel barrier existing between the center of the atom and the specimen surface. The
transparency follows an equation mostly exponentially dependent of the inverse of the local field similar
to equation (13) (D(E) ∝ exp{-b/F} with b a constant essentially related to I the ionization energy of the
neutral molecule). The lifetime τ of an atom or a small molecule in the neutral state at position x in front
of the specimen is therefore equal to τ(x)=1/D(x). A neutral molecule escaping the surface will have a
final probability to be ionized following P = 1 - exp {-∫x dx/v(x)τ(x)}, integrating the local probability
over the flight path, with v(x)=dx/dt the velocity of the particle. The probability of ionization for various
atom can be evaluated theoretically to the experimental value of best image field in FIM observed
experimentally. Muller and Tsong (Müller, 1969) gave the values for Xe (I=12.1; BIF=12 V/nm), N(
I=15.6eV; BIF=17 V/nm), Kr( I=14.0eV; BIF=15 V/nm), Ar( I=15.8eV; BIF=22 V/nm), Ne( I=21.6eV;
BIF=35 V/nm), and He( I=24.6eV; BIF=44 V/nm). For O₂, or N₂ no BIF value is given but the estimated
BIF should fit the same trend, with the main parameter being the ionization energy (I O₂=12.07 eV, and
I N₂=15.58 eV, so BIF respectively ~12V/nm and 17V/nm).

This simple estimation shows that under cryogenic temperature, and under field evaporation condition (F>20V/nm), these molecules have a very high probability of be directly ionized. A more refined and accurate calculation was proposed by Haydock and Kingham in 1981 (Haydock and Kingham, 1981). They estimated a probability of ionization of 0.5 at the BIF (at 45 V/nm for He) with an initial thermal velocity of the cryogenic temperature applied to the specimen (T~1 K). We may note that the ionization occurs very close to the specimen surface, with an ionization zone of width thinner than a nm.

Assuming this last model, the only parameter that could explain the desorption of neutral species is the
presence of a significant initial velocity of the emitted particle. Ionization probability is limited in
quantum model by the time of residence of the particle in the ionization zone. This time is inversely
proportional to the velocity of the atom or the molecule. Haydock and Kingham calculated the
probability of ionization for He, with initial thermal velocity, and found a reduction by one range of
order from 4K to 200K at the BIF (45 V/nm) or 80% of the BIF (35 V/nm). A significant increase in
velocity is possible if we assume that the molecule leaves the surface in an excited state, with kinetic
energy of more than 1 eV for instance. In addition, this process is compatible with the required energy
to be liberated from the polarization effect described previously (the kinetic energy must be higher than
0.2-0.4 eV). We may note that molecules in excited states in APT were observed concurrently in some
other studies during the process of field evaporation (Vurpillot et al., 2018; David Zanuttini et al., 2017).
Indeed, some dissociation channels observed experimentally were associated to the presence of strongly
excited internal states in SiO₂ or GaN, normally not existing at moderate temperature (<<1000K). Shifts
in the energy curves of electrons could reach several eV, indicating that the process of field evaporation
significantly populates these states. We postulate that the atomic surface reconstruction following the
emission of metallic species, as modelled by Kreuzer et al. and as proposed in Figure [14], could produce
excited molecules. It could be likely for Oxygen atoms of Nitrogen atoms, that are not linked together
in the original structures.
## ii. Preferential evaporation

Historically, preferential field evaporation was identified as the most obvious physical phenomenon that induced biases in the composition measurement. The term “preferential” indicates that one or more species are easier to evaporate and may also do it in really not convenient moments, i.e. out of laser or voltage pulse, which makes them either undetectable or unidentifiable in mass spectra. Anomalous behavior and subsequent biases have been observed in the quantitative analysis of solute alloying elements, depending on the analysis conditions. The most sticking example is the case of FeCu and FeSi, where the apparent concentration of solute copper (Cu) in ferritic iron (Fe) obtained by APT tends to be much lower than the actual concentration as the specimen temperature increases, whereas the apparent concentration of solute silicon (Si) in Fe tends to be higher than the actual concentration (Miller and Smith, 1981; Worrall and Smith, 1986; Danoix et al., 2001; Miller et al., 1996). Such phenomena have been qualitatively interpreted by the mechanism referred to as preferential field evaporation. A qualitative interpretation is given in both laser or voltage pulsing mode, using theoretical evaporation rate equations expressed in eq. (5). The basic theory of field evaporation can be used to predict how the evaporation field (more precisely the field at which the evaporation rate reaches the needed value for analysis, and not the theoretical ZBEF). Experimentally, this corresponds to changing both the field strength and temperature simultaneously, so as to maintain a constant rate of field evaporation. For a material A of ZBEF $F_A$ and field sensitivity $C_A$, equation (5) may be rearranged to

$$ \frac{F}{F_A} = 1 - \frac{k_B T}{c_A} \ln \left( \frac{K_0}{K} \right) \quad (23) $$

K is here the evaporation rate at the tip surface, proportional to the detection rate. $K_0$ is the evaporation rate for a zero barrier. This critical field strength is proportional to the applied voltage V, and following this simple approach, linearly dependent of applied temperature T. We may note that the slope of this linear expression is dependent of the sensitivity to the field as defined in equation (6). $C_A$ is generally linked to $F_A$, ZBEF is normally stronger for element with higher C, since C and ZBEV are both dependent of the binding energy of atoms to the surface. Field evaporation of the surface is therefore easily tuned by either increasing the field (through the applied voltage in VPAP) or the specimen temperature (through laser pulsing in La-APT). In practical, the DC field for a standing temperature $T_0$ is set at a field F ensuring a very low evaporation rate $K_{DC}$ between pulses, and evaporation rate on pulse is increased to a rate $K_{pulse}$, so that the DC field evaporation is significantly lower than the detection rate on pulses generating mass peak on the mass spectrum. We may note that this difference must integrate the duty cycle of the acquisition sequence (~$10^4$). Assuming classical values $F_A=3.5V/Å$, and $C_A=1eV$, considering than less than 1% of atoms are evaporated between pulses (DC noise), and following (23), the required evaporation field pulse (through voltage pulse $V_P$) can be estimated. $V_P$ must be larger than 4% of the DC voltage $V_{50}$ at 50K and 8% of the DC voltage $V_{100}$ at 100K in VP mode. At the same $V_{50}$ and $V_{100}$, temperature pulse of respectively ~150K and ~300K are necessary to generate the same evaporation rate in laser assisted mode (Figure [15]). The critical parameter controlling the required pulse fraction (i.e. the ratio $V_P/V_{DC}$) to ensure low DC field evaporation for a given temperature is here the amplitude of the sensitivity to field $C_A$. An example for a case B with $C_B=0.5eV$ and $F_B=2.5V/Å$ is shown in (Figure [15]). A pulse fraction of 16% is necessary in this case to ensure the good conditions of analysis.
<figure><img src="image_17.png" /><figcaption>Figure [15]. Comparison between the constant evaporation probability of two species A (left) and B (right) having different ZBEF ($F_A$, $F_B$) and activation energy barrier ($C_A$, $C_B$) – a case potentially leading to the loss of B atoms by preferential evaporation.</figcaption></figure>

When dealing with a compound, composed of these two elements A and B, preferential field evaporation is often interpreted by superimposing the two graphs of the two elements on the same scale of F and T. With an evaporation field difference of 40% between A and B, only with a large pulse fraction (larger than this value), good conditions of analysis may be found, by setting the DC field below the lowest curves of both species. Nevertheless, this interpretation is over simplistic, since the actual microscopic field $F_μ$ over each atom may vary significantly due to the local depletion or retention of the neighbor atoms under field evaporation. When analyzing an alloy, the low evaporation field atoms are first removed from the surface, increasing the protrusion degree of high evaporation field neighbor atoms ($F*=γF$, with $γ$ a field enhancement factor). Classical electrostatics showed that the local microscopic field (see Table [II]) can be theoretically enhanced by a factor $γ=2$ for an atom situated on top of a planar surface for instance (Suchorski et al., 1995), with actual measured value that can reach $γ=1.5$. We may assume that due to this process, a quasi-equilibrium shape of the surface builds up so that the microscopic field over each atom will be close to the evaporation field of each atom. This interplay between geometry and field was demonstrated in atomistic modelling approach for instance (Vurpillot et al., 2000). In this case, a direct use of these diagram is clearly difficult in the context of predicting composition biases.

Recently more advanced phenomenological models were developed to give a proper interpretation. These models consider the coupled evaporation rate of elements on pulse and between pulse in the analysis of binary or ternary alloy. A first analytical model was developed by Takahashi et al. to predict such phenomena in VP mode (Takahashi and Kawakami, 2014), and was improved by Hatzoglou et al. some years after (Hatzoglou et al., 2020). This model provides the more favorable analysis condition while maintaining high measurement accuracy. This model explained the temperature dependence on the apparent concentration of solute element whereas the dependence of the pulse fraction and pulse frequency is not completely explained.

An example of result is presented in Figure [16] for an FeCu alloy with 1.2% of Cu dispersed in the Fe matrix. However, the application of these models to laser assisted field evaporation is more tedious since
there is no variation of the local field (V is constant) and therefore no variation of the evaporation barrier on the evaporation pulse, or between pulses. Experimentally, the base temperature has also a second order effect on the composition bias, indicating that it is not possible to control the DC field evaporation of the low evaporation field species.

<figure><img src="image_18.png" /><figcaption>Figure [16]. Cu compositions (at%) as a function of the analysis temperature (K) for different pulse fractions $fp=V_p/V_{DC}$ (15, 20, and 25%). Experimental data (dots) have been extracted from Takahashi and Kawakami (2014). Full lines are obtained from the analytical model of this study using variables energetic constants of Fe and Cu ($C_{Fe} \sim 0.5$ eV; $C_{Cu} \sim 0.3$ eV). Reproduced with permission from (Hatzoglou et al., 2020), Oxford University Press)</figcaption></figure>

Although preferentially field evaporation is suspected in the analysis of binary or ternary semiconductor compound, a quantitative model for evaluating the difference between the real and apparent concentrations of alloying elements has not yet been proposed in laser assisted mode. Quantitative discussion concerned with measurement condition has not carried out sufficiently (Di Russo et al., 2018).

If the quantitative estimation of the apparent concentration can be accomplished, preferable measurement conditions can be easily suggested for different alloys. It is generally accepted that quantitative analyses are preferably realized under low specimen temperature, high-pulse fraction, and high-pulse frequency. A low temperature reduces the difference in the evaporation field between the alloying element and a high-pulse frequency reduces the period of d.c. evaporation. However, it is often not convenient to apply these preferable measurement conditions because of an increased possibility of the fracture of needle-shaped specimens under such strict conditions. Therefore, it is necessary to choose measurement conditions that realize a better balance between a reduced possibility of specimen rupture and high quantitative performance.

### iii. Correlated evaporation and multiple detection events

Correlated evaporation occurs when two or more ions are emitted from the tip within the time window of the evaporation trigger pulse, regardless of whether the pulse is electrical or laser-based. If significant correlated evaporation occurs, the frequency of multiple detection events as a function of the number of detected ions is significantly different from the Poissonian statistics that would be observed if evaporation proceeded randomly in space and time, as illustrated in Fig. [17]-(a) (De Geuser et al., 2007). Evaporated ions may be emitted from adjacent regions or from points on the tip that are
sufficiently separated so that the departure of one ion does not alter the electric field experienced by the second atom on the surface. In the first case, the departure of an ion can immediately trigger the evaporation of its nearest neighbor atoms (Müller et al., 2011). This typically occurs because these atoms are located at the top of a terrace, or at neighboring sites along terrace edges, as reproduced in the simulation data of Fig. [17]-(b). These sites correspond to regions of high local electric field due to the reduced curvature radius at the atomic scale. The departure of an ion can significantly alter the local field distribution, thereby inducing the evaporation of one or more additional atoms. Because of their close proximity on the tip, neighbouring atoms are projected onto nearby regions of the detector, leading to spatially correlated detections (Fig. [17]-(c)). In the second case, the departure of two or more ions is simply due to the temporal coincidence of these events (i.e., occurring during the same trigger pulse). However, their projection points on the detector are generally well separated, and the multi-ion detection arises solely from a temporally correlated event (Fig. [17]-(c)). Spatially correlated detection events may or may not be resolved by the detector, depending on its multi-hit capabilities, which should be sufficient to prevent the occurrence of so-called pile-up phenomena (see sections 2.c and 4.a.ii).

The field evaporation of B in Si provides a clear example of compositional bias induced by correlated evaporation effects. In 2012, Da Costa et al. have analyzed [100]-oriented Si wafers implanted with a very high dose of ¹¹B atoms (10¹⁷ at./cm²) (Costa et al., 2012). Because the evaporation field of B (64 V/nm) is higher than that of Si (33 V/nm), B is prone to significant retention effects. The use of a conventional delay-line detector (DLD) led to a selective loss of B atoms due to pronounced pile-up effects. In contrast, the use of an advanced delay-line detector (aDLD) allowed the detection of up to 30 ions from a single evaporation pulse. The capability to record multi-ion events is therefore critical for accurate compositional analysis and has been associated with an increase in the measured B implant peak concentration from 5 to 12 at.% (section 2.c.iii).

Spatial correlation is not exclusively acting over the short time scale of a single pulse, but may persist over several periods. In this case spatial correlations do not yield multiple detection events, but may be assessed through the analysis of detection events occurring after one or several pulses, also referred to as pseudo-multiples (Müller et al., 2011; Ndiaye et al., 2024) (section 4.d.v).

<figure><img src="image_19.png" /><figcaption>**Figure [17].** Correlated evaporation in APT. (a) Comparison of statistics of multiple detection events for purely random (uncorrelated) evaporation (green squares), experimental data (blue diamonds) and simulation accounting for spatial correlations in evaporation. (b) Surface of a simulated sample (blue dots) at a given time. The red circles are the next 100 atoms in the evaporation sequence and their positions clearly indicate spatial correlation in the sequence (Reproduced with permission from (De Geuser et al., 2007), Elsevier). (c) Statistics of distances of ion impacts on the detector resulting from the analysis of SiC.</figcaption></figure>
iv. *Molecular dissociation*

In APT, molecular dissociation occurs when field-evaporated molecular ions – such as metal-oxygen or nitride complexes—fragment during flight to the detector due to intrinsic metastability or bond softening in the high electric field. If this process is fast enough, i.e. if it produces neutral atoms or fragments within the first few angstroms from the surface, the neutral fragments evade detection, leading to systematic undercounting of specific elements. For example, in ZnO, dicationic species like ZnO²⁺ dissociate into charged Zn ions and neutral O atoms, resulting in apparent oxygen loss and metal enrichment. Simulations indicate that stronger fields (~10 V/nm) favor neutral O emission over fully ionized products, directly biasing bulk compositions, particularly in oxides and non-metals where molecular ions dominate (see section 5.a.iii on ZnO and related reference (David Zanuttini et al., 2017)). These problems also concern dissociation reactions in which neutral fragments can still be detected, as in SiO₂, (D. Zanuttini et al., 2017). Undetected neutral fragments may also leave a trace of their presence by influencing the position of two charged fragments in correlation histograms, as in the case of the reaction GaN₃²⁺ → Ga⁺ + N⁺ + N₂ occurring during field evaporation of GaN (Gault et al., 2016).

Three types of dissociations can be distinguished based on the analysis of the correlation histograms (Section 2.f):

(i) *Heterolytic dissociations with production of two charged fragments.* If the dissociation occurs near the tip surface, both daughter ions travel through the entire potential drop and are detected with mass/charge ratios reflecting their own identities. If the dissociation happens further from the tip, the measured values are closer to the mass/charge ratio of the parent ion. When dissociation occurs mid-flight, the measured ratios are intermediate, depending on how much of the potential drop was traversed before and after dissociation. Assuming velocity conservation, the measured mass/charge ratio m₁' of one fragment is given by (here and in the following, we assume that the parent ion is doubly charged and the daughter ions are simply charged):

$$m'_{1} = m_{1} \left[ 1 - \frac{V_{d}}{V_{DC}} \left( 1 - \frac{m_{1}}{m_{p}} \right) \right]^{-1} \qquad (24)$$

Here, m₁ is the actual mass/charge of the daughter ion, mₚ is the mass of the parent ion, V₉ is the potential at the dissociation point, Vₐ₉ is the tip voltage (Saxey, 2011). The second ion's measured value, m₂', is then:

$$m'_{2} = m_{2} \left[ 1 - \frac{m_{p}-m_{2}}{m_{p}-m_{1}} \left( 1 - \frac{m_{1}}{m'_{1}} \right) \right]^{-1} \qquad (25)$$

In the histogram (Fig. [4]-(a), see also the green features in Fig. [4]-(b,c)), such dissociation tracks start at positions corresponding to the m/z values of the fragments and converge towards the point of the diagonal corresponding to the m/z value of the parent molecule. A large number of such tracks are visible in SiC. Similar tracks have been found in other materials, including GaN (Saxey, 2011; Russo et al., 2018), GaAs (Cuduvally et al., 2020), SiO₂ (D. Zanuttini et al., 2017) and LiFePO₄ (Santhanagopalan et al., 2015). Phosphorus molecular clusters as produced from InP are also prone to molecular dissociation (E. Di Russo et al., 2020).

(ii) *Heterolytic dissociations with production of a neutral fragment.* Even neutral atoms formed by dissociation can sometimes be detected. Though they can't be accelerated in the electric field, if a neutral fragment is produced during flight, it retains the parent ion's velocity and may reach the detector. In such cases, for a dissociation producing two fragments the following relation holds:

$$m'_{2} = \frac{m_{2}m'_{1}}{m'^{1}-m_{p}+m_{2}} \qquad (26)$$
Since dissociation strongly affects the apparent mass/charge ratios, accounting for it is crucial for identifying the ions involved. The track produced by neutral-producing dissociation reactions has an end on the diagonal, at the $m_p/z_p$ value when the dissociation happens close to the detector, and an asymptotic behavior approximating the mass/charge of the charged fragment when the dissociation happens at nm distance from the specimen surface (see the blue features in Fig. [4]-(b,c)). Such dissociation tracks have been clearly observed in several studies on $SiO_2$ (David Zanuttini et al., 2017) as well as in pure water specimens (Schwarz et al., 2020). In the case of reactions producing three or more fragments, the interpretation of the track is less straightforward but it is sometimes possible, as for the reaction $GaN_3^{2+} \rightarrow Ga^+ + N^+ + N_2$ in field evaporation of GaN (Gault et al., 2016; Russo et al., 2018). Neutral-producing dissociations also appear to be quite frequent in the analysis of organic molecules, such as tetradecane (Meng et al., 2022a).

(iii) *Homolytic dissociations with Coulomb repulsion.* Homolytic dissociation corresponds to the reaction $A_2^{2+} \rightarrow A^+ + A^+$. If the reaction involves the same isotope, the m/z values of the parent and of the daughter ions coincide, so all reaction products are grouped in the correlation histogram close to the diagonal. However, due to the spatial proximity of the produced fragments undergoing similar acceleration, these reactions also produce a diagonal track with negative slope which is due to the Coulomb repulsion between the fragments (Inset of Fig. [4]-(a) and red feature in Fig. [4]-(b,c)). If the fragments are ejected with a component of their motion along the trajectory of the parent ion, they will tend to repel each other: the forward ion will be accelerated, the backward ion decelerated. This does not happen when the fragments are ejected in a direction perpendicular to the trajectory of the parent ion. In this case, Coulomb repulsion will rather tend to increase the distance of the events on the detector. This information is not retrievable from correlation histograms in m/z, but can be accessed if the correlation between time of flight difference and distance of events on the detector is considered (Ivan Blum et al., 2016).

Correlation histograms may thus reveal systematic errors in the composition measurement, which may be more or less significant depending on the amount and on the identity dissociation events (Saxey, 2011; Santhanagopalan et al., 2015; Ivan Blum et al., 2016; Zanuttini et al., 2018). Dissociation fragments also project back to incorrect spatial origins, blurring 3D reconstructions. A careful analysis of correlation histograms may help quantifying the number of ions involved in dissociation reactions. In SiC, for instance, they sum up to about 0.5% of the total amount of ions, and are not sufficient to account for the observed errors in composition measurements (Ndiaye et al., 2023a).

### d. Methods for assessment and (possibly) correction of losses

An old proverb of unknown origin cited in chapter 9 of “Atom Probe Tomography – Put theory into practice” reminds to the APT user that “If you don’t promptly identify the bias, you may feel much more uncomfortable later” (Lefebvre-Ulrikson et al., 2016). Though the authenticity of the proverb is questionable, its message is not. The need of assessing whether a compositional bias affects an APT dataset is so common that applying a set of methods for this assessment has become just a good practice for any APT user. In most cases, an inaccuracy in compositional measurements in APT depends on some environmental parameter or is the result of detector limitations. Methods i. and ii. Refer to environmental variables or surface dynamics, methods iii. to v. to detector performances. Furthermore, the user should take care about the way environmental variables influence the occurrence of multiple evaporation events (correlated evaporation, dissociation, etc.) and thus the specific detection efficiencies. The applicability of these methods clearly depends also on the material under study, and certain systems could be impossible to test due to their fragility or configuration. In these cases, the principle should be followed
that a biased analysis is better than a too high risk of fracture, leading to potentially no analysis at all
(Kelly and Larson, 2000).

i. Variation of environmental parameters

The simplest test for ascertaining a compositional bias is varying the environmental parameters of the experiment. The most common move is varying the laser energy at constant detection rate and constant base temperature (Agrawal et al., 2011a; Mancini et al., 2014a; Morris et al., 2018). An example of this approach is illustrated in Fig. [18]-(a). In this case, the atomic fraction of Ga and N in GaN is clearly deviating from stoichiometry and is correlated with the laser energy. However, varying the laser energy at constant detection rate also implies that the DC voltage applied to the specimen adjusts accordingly. So, the same variation of measured composition can be displayed as a function of the DC voltage, as shown in Fig. [18]-(b). Thus, the variation of the laser pulse energy at constant detection rate often reveals field-driven mechanisms of loss. On the other hand, varying the laser energy allowing the detection rate to change can be a means to identify mechanisms of loss directly related to the absorption of the light, as well as other non-ideal phenomena such as the laser-induced asymmetry of the apex curvature. This effect arises because the laser light is often absorbed unevenly on the tip surface, resulting in spatially nonuniform heating (Fleischmann et al., 2018; Müller et al., 2011). Another important environmental parameter is the laser frequency. The upper limit for laser frequency is usually the inverse of the time of flight of the heaviest ions. Typically, frequencies above 500 kHz are not used in most APT systems. Low frequencies may enhance phenomena related to preferential evaporation or to the supply of foreign species (most commonly, oxygen or hydrogen (Meier et al., 2022)).

<figure><img src="image_20.png" /></figure>

Figure [18]. Examples of methods for the assessment of physical-chemical mechanisms leading to compositional biases. (a,b) The variation of environmental parameters. In the reported example, (a) the variation of the laser pulse energy at constant base temperature and detection rate clearly indicates a variation of the measured composition in GaN. (b) As the measurement is performed at constant detection rate, a second environmental variable varies, i.e. the DC voltage (the blue arrow indicates the dependence of VDC on the laser pulse energy). The composition may be thus shown to correlate also to this environmental variable. Measurements performed without the constraint of constant detection rate may lift the ambiguity on which parameter is primarily influencing the measurement. (c,d) Investigation of distributions within a given dataset. In the two reported examples, detector space histograms indicate a correlation between (c) the CSR(Ga²⁺/Ga⁺) and (d) the measured composition in GaN. While the diagrams on the left hand-side indicate just a sort of axial symmetry which could be attributed to the specimen shape, those on the right hand-side also contain clear information about the crystallographic symmetry of the system (Data reproduced from (Mancini et al., 2014a; Russo et al., 2018)).
## ii. Variations within a given dataset

APT datasets collected with a fixed set of experimental parameters may also offer the opportunity of assessing compositional inaccuracies. These become visible as spatial gradients of the measured composition having no relationship with the nanostructure of the sample. These spatial gradients may be assessed both in the detector space or in the reconstructed space. Such gradients are generally related to surface field non-uniformities, visualized via the CSR, which may have different origins: they may be related to aforementioned the laser-induced apex curvature asymmetry, to a non-constant curvature of the apex curvature at equilibrium related to specimen preparation or to specimen crystallography (as visible in Fig. [18]-(c,d)). In specific cases, comparison between the composition pattern and the CSR may indicate different roles of field and crystallography in the determination of composition (Diagne et al., 2025). Long APT analyses at constant detection rate on specimens with a significant evolution of the apex radius may also reveal field and composition gradients when investigated in the reconstructed space. If the radius increases indeed, the imaged surface increases and the surface field must thus decrease in order to reduce the evaporation rate and keep the detection rate constant. The gradual decrease of the surface field with the analyzed depth may thus translate into a gradient of measured composition if a field-dependent bias is present (Rigutti et al., 2016b).

## iii. Analysis of multiple events and detector pile-up

The analysis of multiple detection events is crucial in APT composition metrology as the physical channels leading to losses in terms of molecular dissociation and correlated evaporation leave their trace in this subset of data. This part of data mining constitutes a sort of bridge between correlated evaporation and detector performance. In the framework of quantitative analyses, multiple data may also offer the possibility to quantify the loss and therefore correcting the errors in composition measurements.

Beyond providing the key to quantifying the effects of molecular dissociation (section 4.c.iv), multiple events are most useful for assessing and quantifying the possibility of pile-up and its effect on the measured composition. In most compounds, a first effect of correlated evaporation on the measured composition can be traced by looking at the statistics of multiple detection events resolved for the different chemical species present in the sample. An example of this is reported in Fig. [19]-(a) relative to the analysis of SiC. The statistics shows clearly that, while Si is mostly detected as a single event, C has the opposite behavior, i.e. high PME. This is a hint about a higher degree of correlation for C evaporation, leading to higher potential losses by detector pile-up, which is indeed the case for SiC and likely for many other carbides (Ndiaye et al., 2023a). The effect of pile-up can be estimated or quantified especially when an element has more detectable isotopes (see next section) and can be visualized as a dip in the statistics of distances for homoisotopic pairs close to zero, as reported in Fig. [19]-(b) for the frequencies of $ ^{71}\text{Ga}^{+} - ^{71}\text{Ga}^{+} $ and $ ^{69}\text{Ga}^{+} - ^{69}\text{Ga}^{+} $. This occurs because homoisotopic pairs corelated in space are also extremely close in time when they impinge on the detector. This is not the case for heteroisotopic pairs correlated in space, which have a sufficiently large difference in TOF to allow the heavier isotope to escape from the dead/dazzled zone (Russo et al., 2017).

An alternative approach was proposed in order to neutralize the impact of multiple detection events on the composition measurement, consisting in strongly reducing the detection efficiency by placing a grid on the opposite side of the local electrode aperture, thereby significantly decreasing the number of multi-hit events. This method was first proposed by Thuvander et al. in 2013 and later (Thuvander et al., 2013, 2019) and more recently applied by Morris et al. (Morris et al., 2022) to study compositional biases of both GaN and AlN. Even with a substantial reduction in multi-hits, particularly at high fields, the overall composition of GaN and AlN remained essentially unchanged, indicating that correlated evaporation phenomena have little effect on the measured composition. It should be reminded that spatial filters do
not filter multiple evaporation events closely correlated in space, which would be still subjected to pile-
up and be detected as single events.

<figure><img src="image_21.png" /><figcaption>**Figure [19].** Analysis of multiple events in APT. (a) Statistics of multiple events as a function of multiplicity in SiC resolved by species. C is much more frequently found in multiple events than Si. (b) Histogram of distances on the detector of Ga⁺ pairs from multiple detection events. The homoisotopic pairs present a dip at short distances due to detector pile-up (Reproduced with permission from (Russo et al., 2018), Elsevier). (c) Calculated variation of detected isotope abundances of Ga in double evaporation events as a function of the detection efficiency for same-isotope pairs. (d) Isotope abundances detected in multiple hits for Si⁺ during the evaporation sequence of a B-doped Si sample. The data indicate a depletion of the majority isotope ²⁸Si⁺ and an enhancement of those of the minority isotopes ²⁹,³⁰Si⁺ (Reproduced with permission from (Guerguis et al., 2024), Elsevier).</figcaption></figure>

iv. Investigation of isotopic ratios

A robust approach for the assessment of biases induced by correlated evaporation involves analyzing
the occurrence of homo- and heteroisotopic multiple events for elements composed of two or more
isotopes (Thuvander and Andrén, 2011; Gopon et al., 2022; Russo et al., 2018; Ndiaye et al., 2024). An
example is Ga, which consists of two isotopes, ⁶⁹Ga and ⁷¹Ga. Hetero-isotopic multi-hits typically occur
with a time delay in spatially correlated events, due to the different m/n ratios of the isotopes, which is
generally sufficient to avoid pile-up effects. In contrast, homo-isotopic hits can occur within the
detector's so-called “dead zone,” resulting in the detection of only a single ion (Fig. [19]-(b)). In terms
of effect on abundances, the process can be understood as follows. Natural gallium consists of two stable
isotopes, ⁶⁹Ga and ⁷¹Ga, with (approximated) natural abundances p₁=60% and p₂=40%, respectively.
Assuming random pair evaporation, the probability of evaporating an isotope pair is given by p₁²=36%
for ⁶⁹Ga–⁶⁹Ga, p₂²=16% for ⁷¹Ga–⁷¹Ga, and 2p₁p₂=48% for mixed pairs. Due to detector pile-up effects,
same-isotope pairs are detected with a reduced efficiency factor ηpile-up, while mixed-isotope pairs are
detected without suppression. This leads to a modified detection frequency per pair, where same-isotope
contributions are scaled by ηpile-up. As a consequence, the measured isotopic abundance deviates from
the natural abundance, as shown by the plot in Fig. [19]-(c): with decreasing detection efficiency for
same-isotope pairs, the apparent abundance of ⁶⁹Ga (majority) decreases while that of ⁷¹Ga (minority)
increases. In the extreme (though hypothetical) case in which all homoisotopic pairs were undetected, only heteroisotopic pairs would be responsible for detected isotope abundances leading to 50% abundance for both isotopes. Similar considerations, though slightly more complex, apply to elements with more isotopes. A further complication to this picture is that detected isotope pairs may also results from events with multiplicity higher than 2. Applied to GaAs, this approach indicated that only ~2% of the total $Ga^+$ ions were missed due to pile-up, an amount insufficient to fully explain the observed compositional biases (Russo et al., 2018, 2017).

An experimental example of biased isotopic abundances is reported in the analysis of B-doped Si (Fig. [19]-(d)). The abundance of the majority isotope $^{28}$Si$^+$ decreases during the analysis, while that of minority isotopes $^{29,30}$Si$^+$ increases. This phenomenon is related to the increase of the electric field during the evaporation sequence and to the observed increase of multiple events. Along with the analysis of $Si^{2+}$ isotopic abundance, the observation allowed the authors for an estimation of lost Si atoms and for a correction of the B doping concentration (see section 5.b.i for details, (Guerguis et al., 2024)).

## v. Pseudo-singles and Pseudo-multiples

The terms pseudo-singles and pseudo-multiples refer to two different categories of detection events (Müller et al., 2011; Ndiaye et al., 2024).

Pseudo-singles are single detection events resulting from a double event in which the second ion was undetected (detector dead zone) or was misinterpreted/unresolved as a double event (detector dazzled zone). Pseudo-singles can bias isotopic abundances of single detection events, and are undistinguishable from real single events unless the instrument allows for recording the amplitude of the MCP signal (Fig. [11]-(d,e)). In this case, it becomes possible to estimate the amount of pseudo-singles $N_{pseudo singles}$, which may help in the estimation of total losses by pile-up (Ndiaye et al., 2023a, 2024).

Pseudo-multiples are defined as successive detection events separated by exactly $\Delta(N_{pulse})=1$ (consecutive laser pulses), mimicking the spatial correlations of true multiples (same-pulse events) but without dead-time losses, serving as a lossless reference (Müller et al., 2011; Ndiaye et al., 2024).

By comparing the spatial distributions of isotopic pairs in true multiple events and pseudo-multiples, it becomes possible to identify and quantify the depletion of same-isotope pairs in true multiples, directly attributable to pile-up losses and to estimate the number of expected pairs $N_{expected pairs}$. Mixed-isotope pairs, which are less affected by pile-up due to their larger time-of-flight differences. Using the measured number of mixed-isotope pairs and their natural probabilities, it is possible to calculate the expected total number of pairs in the absence of loss, which is given by

$$N_{corrected} = N_{expected pairs} - \frac{N_{pseudo singles}}{2}. \qquad (27)$$

In the analysis of $LaB_6$, the deficit between expected and measured pairs reveals that about 58% of B pairs are lost in multiple events, primarily involving singly charged ions (Ndiaye et al., 2024). This approach restores the material's stoichiometry to near its nominal value, demonstrating that isotopic and pseudo-multiple event analysis can effectively quantify and correct detection biases in APT. The method provides a physically grounded framework for improving compositional accuracy in materials analysis.

## vi. Isotopic substitution

Isotopic substitution consists in purposely altering the isotopic abundances of a given element, most often by replacing a majority isotope with a minority one. This operation is meant to shift mass peaks without changing the chemistry of the compound, and to avoid mass peak superpositions or ambiguities.
It can also be used in order to study the accumulation of specific isotopes at specific locations in the sample, resulting from transport, precipitation or other microscopic mechanisms. The most common application of this approach is probably deuterium charging in order to distinguish hydrogen from the sample from environmental hydrogen in APT of metals (Mouton et al., 2019; Chen et al., 2023). Isotopic substitution has been reported in studies on oxides, with the investigation of ¹⁸O-enriched hematite, revealing that the only singly charged O⁺ monomers are detected (and not O₂²⁺) (Bachhav et al., 2013). Annealing in ¹⁸O was also applied in order to study oxygen transport along metal-perovskite heterostructures (Taylor et al., 2023). Another noteworthy case concerns the resolution of Si and N signatures via the substitution of ¹⁴N by ¹⁵N, yielding a composition in good agreement with complementary analyses for Ti-Si-N (David L.J. Engberg et al., 2018), as well as more reliable quantification of N in Fe₄N phases in steels (Takahashi et al., 2022).

vii. Statistical correction of biased composition from random alloys

Once the error has been assessed and globally corrected, a question remains whether the correction can be extended to other properties than the average composition. A correction method has been proposed for assessing the correct statistical properties of the random alloys which would be affected by compositional inaccuracies conserving their randomness. This is a quite restrictive scenario, but it is instructive on how low detection efficiencies affect the information on APT datasets. The datasets displayed in Fig. [20] are simulated, but the protocol has profitably been applied on real APT datasets (Rigutti et al., 2016a). The correction protocol aims to recover the statistical properties of a biased distribution while conserving its spatial characteristics, i.e. the positions of local minima and maxima. In principle, the statistical properties of a random alloy are known. It is thus possible to define a reference binomial function corresponding to the statistical distribution of the alloy, sampled by voxels of constant volume and number of atoms (Fig. [20]-(c)). In case depicted in Fig. [20]-(a), the bias is described by a different detection efficiency for Al ($η_{Al}=0.3$) and for Ga ($η_{Al}=0.15$) in the AlGaN alloy (the composition is determined in terms of the III-site fraction, see section 5.a.iii). The main effects of this bias are a shift of the average III-site fraction from the pristine $\langle y^{reference} \rangle=0.25$ to $\langle y^{bias} \rangle=0.4$, and an increase of the standard deviation of the distribution, i.e. of the amplitude of the compositional fluctuations. Based on the average site fraction $\langle y^{bias} \rangle$ and the average number of atoms per voxel $\langle N_{bin}^{bias} \rangle$, an associated binomial distribution and its partition function are defined.

The latter is used in the correction protocol schematized in Fig. [20]-(b). Similarly, the reference binomial distribution is used to obtain an interpolated reference partition function. The partition functions of the biased and reference datasets, interpolated to obtain a strictly monotonically increasing behavior, are shown in the diagram of Fig. [20]-(b). The interpolated partition functions are thus used to define the correction of the site fraction measured locally in any voxel (for mathematical details, see ref. (Rigutti et al., 2016a)). The application of this correction protocol makes it possible to make the biased distributions similar to the reference one, as can be seen in the diagrams in Fig. [20]-(c). However, when the efficiency becomes very low, the correction protocol cannot completely bring averages and standard deviations back to the reference values.
<figure><img src="image_22.png" /></figure>

**Figure [20].** Statistical correction of random alloy distributions affected by compositional biases conserving the alloy randomness. (a) The frequency distribution analysis (left) of a simulated Al<sub>0.25</sub>Ga<sub>0.75</sub>N dataset in which the specific detection efficiency is for η<sub>Al</sub>=0.3 for Al and η<sub>Ga</sub>=0.15 for Ga and the corresponding 2D III-site fraction map (right). The statistical distribution can be approximated by a binomial function. Not only the average III-site fraction is affected by an error, but also alloy fluctuations exhibit enhanced amplitudes. (b) The correction scheme is based on the knowledge of the biased partition function derived from the binomial function in (a) and on the knowledge of the reference partition function that would describe the statistical distribution of an ideally unbiased alloy. The corrected composition y<sup>Corrected</sup> can be evaluated for any voxel or subvolume starting from the y<sup>Biased</sup> composition measured. (c) The resulting corrected distribution reasonably fits the reference binomial function, has a correct average (frequency distribution analysis, left) and exhibits weaker fluctuations (right) (Adapted with permission from (Rigutti et al., 2016a), AIP).
**5. Review of literature data**

In this part we review the main studies on compositional metrology of APT present in the literature. This part is meant to depict the state of the art, identify common problems and trends in different material systems, and provide an overview of the methods adopted to identify, describe and understand the inaccuracies in composition measurements. This part is divided into two main sections, one dealing with the measurement of the composition of the matrix, the other with the assessment of the concentration and distribution of impurities.

**a. Composition of the matrix**

The measurement of the composition of the matrix of a compound material is obviously the first step in the understanding of its functional and chemical properties. While a relatively large number of techniques can determine global compositions and even global statistical properties with excellent precision and accuracy, APT is needed when the materials properties are determined by the distribution of chemical species at the nanoscale or at the atomic scale. For this reason, it is critical to understand whether the elements present in a compound material are detected with specific detection efficiencies, and how these may depend on the experimental parameters chosen. A representative number of studies has appeared in the recent past focusing on this problem.

**i. Arsenides, Antimonides, Phosphides**

Classical III-V semiconductors (i.e. III-V except III-Nitrides) were among the first compound semiconductor to be analyzed by atom probe, namely with 1D APT (either voltage- or laser-assisted) during the 1980s (Sakurai et al., 1984; Hashizume et al., 1986; Cerezo et al., 1986). These works also provide the first examples of APT compositional metrology in semiconductors. A renewed interest in the analysis of III-V compounds occurred after the introduction of the 3D APT assisted by fs laser pulses.

Field evaporation of III-Vs is generally complicated by the occurrence of large molecular clusters of element V atoms, which can be found in the mass spectra of GaAs (B. P. Gorman et al., 2011; Russo et al., 2017), InAs (Perea et al., 2006), GaSb (Müller et al., 2011), and InP (E. Di Russo et al., 2020). The presence of such clusters introduces an ambiguity in the interpretation of mass spectra, which could account for up to 10% of the element V atomic fraction (Cuduvally et al., 2020). This is not the only source of inaccuracy, as the measured composition is also field-dependent in these compounds (section 4.b).

*Binary compounds.* Figure [21]-(a) and (b) report two consistent sets of measurements performed with very different instruments in the case of GaAs, a stoichiometric compound. The measurements performed by Hashizume et al. (Figure [21]-(a)) show a significant excess of Ga at low field and a slight Ga deficit at high field (Hashizume et al., 1986). These authors do not adopt the CSR metrics, but plot their results as a function of the effective DC field fraction, i.e. the quantity $V_{DC}/(V_{DC}+V_E)$, where $V_E$ is the evaporation field as determined for a given detection rate. The abscissa is therefore a growing function of the surface field. The more recent results obtained by Di Russo et al. (Figure [21]-(b)) are reported as a function of the CSR($As^{2+}/As^{+}$) and reveal a consistent common trend for a set of measurements performed at different temperatures, laser wavelengths and pulse laser energy, with the atomic fraction of Ga varying from 60% at low field to 30% at high field. These results can be interpreted as following: the preferential evaporation of Ga atoms at high field can explain the Ga loss, while at low field, a possible mechanism accounting for the loss of As atoms is the formation of As clusters on the tip surface and their direct dissociation in smaller subunits with the formation of neutral As
molecules/atoms. Furthermore, the clustering of V-element atoms exhibits only a weak dependence on the laser pulse energy at fixed DC voltage (Russo et al., 2017).

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

**Figure [21].** Field-dependent compositional biases in classical III-V semiconductors. (a) Atomic fraction of Ga measured in 1D-LAAPT of GaAs as a function of the effective DC field fraction (see main text, a quantity approximately proportional to the surface field) (Data adapted from (Hashizume et al., 1986), American Institute of Physics). An excess of Ga is measured at low field, while a deficit of Ga is measured at high field. A similar situation is reproduced in (b), in which measurements performed at different temperatures, laser wavelengths and intensities align on a common trend (Adapted with permission from (Russo et al., 2017), Oxford Academic,). (c) Field-dependence of the measurement of the In III-site fraction in the In<sub>0.53</sub>Ga<sub>0.47</sub>As pseudobinary alloy. (Adapted with permission from (Cuduvally et al., 2020), Elsevier)

Further works on GaSb have shown a similar complexity (Müller et al., 2011), but also that compositional gradients may occur due to the different effect of the laser light on the illuminated and
on the dark side of the specimen. This effect is particularly critical for low-bandgap materials but it may
be reconducted to a field effect because the steady state curvature of the field emitter apex becomes
larger on the illuminated side and smaller on the dark side (Müller et al., 2012). Finally, an excess
composition in As has also been reported for the magnetic compound Ga<sub>0.963</sub>Mn<sub>0.037</sub>As, but
compositional biases in this systems have not been investigated systematically yet (Kodzuka et al.,
2009).

*Pseudobinary alloys.* More critically than in stoichiometric compounds, field-dependent compositional biases have also been assessed in pseudobinary III-V alloys. Figure [21](b) displays the dependence on the CSR(As²⁺/As⁺) of the measured III-site fraction of In. Differently from III-N materials (see section 5.a.ii-iii) there is no clear trend in the compositional bias. This is also due to the interplay of the possible loss mechanisms, which have been classified by relevance. Among the various mechanisms, cluster peak overlaps are identified as the most significant contributor to As loss. This effect is more pronounced at low field, where larger clusters ionize more easily, causing overlaps that obscure As detection. Multi-hit events are the second most significant loss mechanism, also more prevalent at low field—70% of multi-hits contain As versus 57% at high field. This suggests additional As loss may occur via dissociation into indistinguishable fragments. Dissociation is the third channel and is slightly more significant at high field, contributing about 0.05 at.% at low field and 0.1 at.% at high field. Neutral production through dissociation is less important and occurs mainly at high field, but experimental evidence and DFT calculations indicate such neutrals likely re-ionize, minimizing loss. DC evaporation is the least significant mechanism but increases at high field due to the higher evaporation field required for As compared to In and Ga. While this is a qualitative argument, evaporation fields defined by Müller theory are not exactly applicable to semiconductor compounds and should be used cautiously (see section 2.a). At low field, background accounts for ~2 at.%, rising to ~30 at.% at high field. Significant DC evaporation of In and Ga likely increases the measured As fraction at high field, partially offsetting losses but reducing the overall detection efficiency. (Cuduvally et al., 2020).

## ii. III-Nitrides : GaN

Compositional biases have been reported in several III-nitride materials, with GaN being by far the most extensively investigated. This prominence can be attributed to three main factors. First, GaN dominated semiconductor materials research during the first two decades of the 2000s because of its exceptionally broad technological relevance, spanning power electronics and optoelectronics coupled to potential for optimization due to the persistence of defects. Second, the material is readily accessible, as it is routinely grown by epitaxial techniques in many laboratories, both as thin films and as nanowires. Finally, GaN is particularly well suited for atom probe tomography (APT): it can be analyzed with high reliability while still exhibiting pronounced compositional biases under specific experimental conditions. This behavior is especially striking given that the intrinsic composition of GaN deviates by less than 0.1 at.% from ideal 1:1 stoichiometry.

*First reports and thermal effects.* The first report of compositional biases in GaN was provided by Agrawal et al. in 2011, based on a study of nanowires grown along the <0001> direction using LEAP 3000X Si and LEAP 3000X HR instruments operated with a green laser (λ=532 nm) (Agrawal et al., 2011a). Because GaN has a wide bandgap of approximately 3.6 eV, laser absorption at 532 nm is intrinsically limited (photon energy ≈ 2.3 eV), which can hinder optimal atom probe analysis. Measurements were conducted at a fixed detection rate φ, with the laser pulse energy *E*<sub>las</sub> varied between 3 and 20 pJ and the DC bias applied to the tip V<sub>DC</sub> was adjusted to keep φ constant during tip evaporation. Under these conditions, the charge-state ratio CSR(Ga⁺/Ga²⁺) decreased from 1 to 0.03, accompanied by a corresponding reduction in the measured Ga fraction from 0.8 to 0.5 (Fig. [22]). The reported mass
spectra reveal the presence of N⁺, N₂⁺, Ga²⁺, and Ga⁺ ions, together with less abundant species such as N²⁺, GaN⁺, N₃⁺, GaN₃⁺. However, no information is provided on how the mass spectra evolve as a function of the analysis parameters. Varying the laser repetition rate between 200 and 500 kHz did not measurably affect the composition, while a modest reduction in signal-to-noise ratio led to improved spectral quality through a decrease in uncorrelated inter-pulse events. Accordingly, a repetition rate of 200–250 kHz was selected. Analysis of two-dimensional density and composition maps extracted from the cross-section revealed the presence of a c-axis pole and a sixfold symmetry, consistent with the stereographic projection along the <0001> direction of the wurtzite hexagonal crystal structure of GaN. At the location of this pole, a reduction in both Ga and N densities was observed. This behavior was attributed to the divergence of ion trajectories caused by a local decrease in the tip curvature radius. Such a reduction also results in a local enhancement of the electric field, which in turn leads to a Ga-deficient composition at the pole. Thermal modelling indicates that the low thermal conductivity of GaN nanowires leads to non-uniform heating at high *E*<sub>las</sub>, which in turn promotes preferential nitrogen loss and, consequently, a spatially non-uniform compositional distribution. This inhomogeneous heating is likely driven by surface absorption, potentially associated with defects or surface states. Similar conclusions were drawn by Dawahre et al. in 2013 (Dawahre et al., 2013) on bulk GaN grown along the c-axis, analyzed using a LEAP 3000 XS operated in laser mode (λ = 532 nm) at a repetition frequency of 250 kHz.

*The role of crystal symmetry and polarity.* In 2012, Riley et al. analysed GaN nanowires oriented along the non-polar a-axis using a LEAP 4000X Si (Riley et al., 2012). In this case, no artifacts related to non-uniform heating were observed when employing a UV laser (λ=355 nm). Analysis of hit maps showed that Ga⁺ and N₂⁺ follow the same trend, exhibiting low hit densities along a band connecting the (011̄0), (11̄20) and (10̄10) poles, as well as at the Ga-polar surface. This reduced hit density is attributed to the locally enhanced electric field at the edges of these low-index planes, where the increased field promotes the preferential evaporation of doubly charged ions such as Ga²⁺ and N₂²⁺. It should be noted that the hexagonal wurtzite GaN crystal structure contains no N–N bonds. Consequently, the observed low abundance of N⁺ ions, in contrast to the predominance of N₂⁺ ions, can be explained by the relatively low diffusion barrier of N atoms (0.66 eV for diffusion on the Ga-polar surface (Fernández-Garrido et al., 2008)) compared with the desorption barrier of N₂. This difference facilitates the diffusion of N atoms across the surface and the subsequent formation of N₂ molecules. After optimizing the analysis conditions to achieve overall stoichiometry, Ga deficiency was observed on N-polar facets, whereas N deficiency occurred on non-polar and Ga-polar facets. Stoichiometric composition was detected only near the N-polar facet. These findings suggest that facet polarity critically modulates Ga and N evaporation, resulting in local compositional deviations even when the material is globally stoichiometric. This behaviour is attributed to the preferential loss of N atoms: during multiple-hit events, more N₂ than Ga is evaporated, leaving a higher fraction of N ions undetected. N₂ loss is particularly pronounced at non-polar and Ga-polar surfaces, likely due to variations in atomic bonding coordination at these facets. After optimizing the analysis conditions to achieve overall stoichiometry, Ga deficiency was observed on N-polar facets, whereas N deficiency occurred on non-polar and Ga-polar facets. Stoichiometric composition was detected only near the N-polar facet. These observations indicate that facet polarity strongly governs the evaporation rates of Ga and N, leading to local deviations from stoichiometry despite an overall balanced composition. The observed compositional biases were attributed to the preferential loss of N atoms, either between laser pulses or during multiple-hit detection events. In particular, during multiple-hit events, N₂ evaporates more readily than Ga, resulting in a larger fraction of undetected N ions. This N₂ loss mechanism is especially pronounced at non-polar and Ga-polar surfaces, likely due to variations in atomic bonding coordination at these facets. Despite local compositional variations arising from crystallography, the primary factor influencing the detection rate is the DC voltage applied to the tip. In particular, increasing V<sub>DC</sub> appeared to bring the Ga/N ratio closer to stoichiometry; however, this effect is largely apparent, resulting from a decrease in Ga detection. This behavior was explained by the lower evaporation field of Ga (≈15 V/nm) compared with N. This is
consistent with previous observations of Ga depletion at high applied DC voltages in GaAs, but the
reader should consider that the application of the values of evaporation fields (see section 2.a) does not
reflect the nature of the Ga-N bonding in GaN, and even less the crystal polarity and the possible state
of the surface. Both As and N, as group-V elements, display similar evaporation behavior, with
thermally assisted surface migration and emission as higher-order molecular ions. (Liddle et al., 1988;
Gorman et al., 2007; Brian P. Gorman et al., 2011).

Further insights into the atom probe tomography analysis of GaN nanowires grown along the <0001>
direction were provided by Diercks et al. in 2013 (Diercks, 2013). In this study, atom probe analyses
were performed using both a straight-flight-path LEAP 4000X Si and a reflectron-equipped LEAP
4000X HR, the latter providing enhanced mass resolution. Both instruments operated with a UV laser
(λ=355 nm) laser. Hit maps revealed that Ga and N ions exhibit sixfold symmetry around the (0001)
pole. Although Riley et al. reported that nanowire polarity influences evaporation behavior and,
consequently, the detected composition, Diercks et al. observed no such variations when nanowires were
mounted in the opposite orientation. The specimens exhibited the same sixfold pattern, indicating that
the observed evaporation behavior depends primarily on crystallography rather than bond direction.
Constant detection rate measurements (0.007 ions/pulse) were then carried out while increasing Elas
from 0.002 to 10 pJ. At the lowest laser energies, concentration maps revealed N₂ depletion and Ga
enrichment at the central pole and along the six zone lines. At 0.1 pJ, this trend reverses, with N₂
becoming enriched at these sites and in a region opposite the incident laser direction. At 10 pJ, this effect
dominates, and any correlation between local composition and crystallographic orientation disappears.
A similar behavior is observed for the charge-state ratio CSR(Ga²⁺/Ga⁺), which serves as an indicator of
local electric field strength. At low laser energies, a high-field region appears at the (0001) pole and
along the zone lines. This feature vanishes at 0.1 pJ, and at 10 pJ only a high-field region is observed on
the side opposite the laser. The high-field region at the (0001) pole coincides with the area of highest
multiple-hit detection events.

*Influence of laser and temperature.* In a subsequent study published in 2015 (Diercks and Gorman, 2015), Diercks and Gorman. further investigated the influence of tip base temperature on the measured composition by varying Elas from 0.006 to 0.05 pJ, a range in which ion emission is not expected to be affected by the laser incidence direction. Using a LEAP 4000X Si instrument, it was observed that increasing the tip base temperature from 20 to 120 K produced no measurable change in the detected Ga fraction. The variation of the N fraction measured as a function of Elas was then examined. As Elas increased from approximately 0.001 to 10 pJ, the detected N fraction decreased from about 0.6 to 0.1, with stoichiometric composition observed at Elas ≈ 0.01 pJ (Fig.[22]). The same behavior was observed using both UV (λ=355 nm) and green (λ=532 nm) lasers, indicating comparable laser–matter absorption mechanisms despite the different excitation wavelengths. In contrast to earlier reports, the study shows that at the lowest laser energies the apparent nitrogen content exceeds 50%. These observations indicate that the apparent deviation from the 1:1 Ga:N ratio is likely due to under-detection of one or both species. To test this hypothesis, TEM images of four nanowires analyzed at Elas ranging from 0.005 to 0.1 pJ were acquired before and after atom probe analysis and used to determine the reconstruction efficiency, ensuring consistency with the known evaporation depth. The resulting detection efficiencies were 0.20 ± 0.08 for nanowires analyzed with the LEAP 4000X HR and 0.37 ± 0.08 for those analyzed with the LEAP 4000X Si. Both values are lower than the nominal instrument detection efficiency, which is approximately 0.5. Despite the reduced detection efficiency, nominal GaN stoichiometry is recovered at intermediate laser energies (0.005–0.01 pJ), indicating that both Ga and N ions are partially lost outside the assigned mass ranges. At the lowest laser energies, where a N-rich composition is measured, the electric field required to maintain the target detection rate is sufficiently high to promote ion evaporation between laser pulses, predominantly affecting Ga and potentially N. As Elas increases, Ga losses due to between-pulse evaporation are reduced owing to the lower DC field at the specimen surface. In contrast, higher laser energies increase the probability of nitrogen loss through sublimation of neutral N₂ species rather than through uncorrelated evaporation of ionic nitrogen species (see section