# Determination of electric field in the vicinity of polarized atom probe tomography specimens using scanning transmission electron microscopy

Mohammed Ilhami, Juan Macchi, Celia Castro, Gerald da Costa, Francois

Vurpillot, Williams Lefebvre

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Mohammed Ilhami, Juan Macchi, Celia Castro, Gerald da Costa, Francois Vurpillot, et al. Determination of electric field in the vicinity of polarized atom probe tomography specimens using scanning transmission electron microscopy. 2025. hal-05388042

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Determination of electric field in the vicinity of polarized atom
probe tomography specimens using scanning transmission
electron microscopy

Mohammed ILHAMI¹, Juan MACCHI¹, Celia CASTRO¹, Gerald Da COSTA¹, Francois VURPILLOT¹,
Williams LEFEBVRE¹

¹Univ Rouen Normandie, INSA Rouen Normandie, CNRS, Normandie Univ, GPM UMR

6634, F-76000 Rouen, France

Corresponding author: [mohammed.ilhami1@univ-rouen.fr](mailto:mohammed.ilhami1@univ-rouen.fr)

**Abstract**

Determination of electrostatic field distribution at the nanometer scale at the vicinity of a polarized specimen is of great interest for solving the reconstruction issue in atom probe tomography. In this work, we present a method based on the Rutherford scattering theory to quantify the electric field maps obtained by scanning an electron beam across on a needle like specimen submitted to a high voltage in a transmission electron microscope. After a validation of the geometrical model on simulated datasets, electron beam deflection maps were acquired in scanning mode using a holder developed to perform atom probe tomography experiments in transmission electron microscopes. The experimental results allow pointing
out the link between the crystal orientation and the electrostatic field distribution at the
vicinity of a pure tungsten specimen. The methodology presented, which does not depend on
the thickness of the sample, is foreseen to serve as an additional input for present and future
atom probe reconstruction methods.

**Keywords:** Atom probe tomography, Transmission electron microscopy, 4D-STEM, Electric field mapping, correlative microscopy.

# 1. Introduction

The ongoing quest for atomic-scale materials characterization has led to the development of complementary analytical techniques, each offering unique insights into the fundamental properties of materials. Atom probe tomography (APT) and transmission electron microscopy (TEM) are the pillars of nanoscale characterization: APT provides three-dimensional chemical mapping with near-atomic resolution and part-per-million sensitivity(Gault et al., 2021; Kelly and Miller, 2007; Williams Lefebvre Ulrikson, François Vurpillot, Xavier Sauvage, 2016), while TEM provides structural and compositional information at atomic resolution through two-dimensional projections(Hawkes and Spence, 2019).

APT involves evaporating atoms by applying a high voltage to a needle-shaped sample of a tip's radius of few tens of nanometres. Under the effect of the intense electric field generated at the apex of the tip, atoms are gradually expelled and simultaneously ionized according to
the field evaporation mechanism(Miller and Forbes, 2014; Vurpillot, 2016a). The projection
of the evaporated ions is then recorded by a position-sensitive detector(Gault et al., 2021;
Larson et al., 2013; Miller and Forbes, 2014; Williams Lefebvre Ulrikson, François Vurpillot,
Xavier Sauvage, 2016). Insofar as the trajectories of ions are affected by the shape of the tip
surface, its surrounding electrostatic field, and the presence of areas with varying evaporation
fields(Beinke et al., 2016), the original positions of atoms in the evaporated volume are
reconstructed assuming simple projection models(Gault et al., 2012; Larson et al., 2013;
Williams Lefebvre Ulrikson, François Vurpillot, Xavier Sauvage, 2016). These models rely
on the prior knowledge or on some hypotheses about the physical and geometrical
parameters(Arslan et al., 2008; Devaraj et al., 2014; Herbig et al., 2015; Lefebvre-Ulrikson,
2016) governing the specimen shape.

In order to improve APT reconstructions, correlative microscopy approaches have proven
their relevance(Arslan et al., 2008; Cojocaru-Mirédin and Devaraj, 2022; Devaraj et al., 2014;
Herbig et al., 2015; Lefebvre-Ulrikson, 2016; McCarroll et al., 2022) insofar as they may
provide useful additional information about the morphologies to reconstruct. Meanwhile, to
account for the instantaneous evolution of the main reconstruction parameters (field factor
and image compression factor mainly) an enhanced dynamic reconstruction algorithms has
been developed(Hatzoglou et al., 2019) based on the idea that geometrical parameters
governing the ion trajectories vary along the evaporation sequence(Gault et al., 2011). But
still, APT reconstructions assume a uniform curvature at the tip apex whereas, even for the
field evaporation of a single crystal, many evidences of a non-uniform tip surface during field
evaporation have been given(Müller and Tsong, 1969; Southon et al., 1975). This is for
instance illustrated in Figure 1. Furthermore the tip surface can evolve in complex ways when
field evaporated materials are composed of phases with different evaporation fields .This can
lead to unresolved distortions in the reconstructed volume(Vurpillot, 2016b) which may
require strong help of field evaporation simulations, use of specific reconstruction
algorithms(De Geuser et al., 2007; Lawitzki et al., 2021) or of a correlative microscopy
approaches.

<figure><img src="image_2.png" /><figcaption>Figure 1: Field evaporation sequence of a Fe-51,3at%Cr alloy performed at room temperature and at constant voltage of 5,0 kV. Three images of the APT needle at various</figcaption></figure>
times of field evaporation are superimposed. A red circle of 130 nm radius indicates the validity of the hemispherical cap hypothesis at the onset of field evaporation (time $t_0$). At times $t_1$ and $t_2$, the curvature has become asymmetric, with either a large curvature (blue arrows) or a small one (red arrows).

To solve the reconstruction issue in APT, the knowledge of the electric field distribution around the tip apex(Beleggia et al., 2014) can form the basis of an APT aberration correction scheme, which could help improving reconstruction methods and spatial precision. Such an approach requires the quantitative measurement of the electrostatic field around a polarized tip during its field evaporation. Moreover, tracking the shape of the sample apex during field evaporation may provide a basis for correcting trajectory aberrations in APT. To this end, several ideas have been suggested for combining atom probe tomography with electron microscopy techniques(Kelly et al., 2022, 2013; Kirchhofer et al., 2018). Recently, Da Costa et al.(Da Costa et al., 2024) have succeeded in integrating an atom probe into a commercial TEM. This instrumental development made it possible to track the change in tip geometry before, during and after field evaporation. In addition, 2D information on the electric field distribution at the tip apex can also be used to correct ion trajectory aberrations(Zheng et al., 2020). Simulations have shown that it is possible to determine ion trajectories from the modelled electric field around the tip apex(Larson et al., 2012; Smith and Walls, 1978).
The ability to map electric fields has emerged with progress in electron holography (EH)(Brodovoi et al., 2022; Cumings et al., 2002; Matteucci et al., 1991; Midgley, 2001; Zheng et al., 2020) and pixelated scanning transmission electron microscopy (commonly referred to as 4D-STEM)(Ophus, 2019; Wu et al., 2019). Winkler et al.(Winkler et al., 2020), Boureau et al.(Boureau et al., 2021), Cooper et al.(Cooper et al., 2024) and more recently Jo et al.(Jo et al., 2025) have compared these two techniques, indicating that EH provides greater sensitivity to electrostatic potentials, while 4D-STEM is more sensitive to potential gradients, which are directly linked to electric fields(Boureau et al., 2021). It should be noted that long range electric fields could affect the reference wave(De Knoop et al., 2014), which makes EH limited in terms of field of view(Migunov et al., 2015).

A significant challenge in electric field mapping using EH and 4D-STEM is the quantification of results. While qualitative field maps can be obtained either from the interference pattern between an object and a reference wave in EH or by measuring the shift of the centre of mass (CoM)(Müller et al., 2014; Krajnak et al., 2016; Müller-Caspary et al., 2017; Toyama et al., 2020; Heimes et al., 2023) of each diffraction pattern in 4D-STEM, quantitative analysis are more complex. In 4D-STEM, electric field maps are often derived from the deflection angle of the transmitted electrons(Cooper et al., 2024; Wu et al., 2019). However, the methods used for this calculation generally depend on the thickness of the sample(Heimes et al., 2023; Haas et al., 2019; Taplin et al., 2016; Pöllath et al., 2021), which is not suitable in the case of a needle-shaped sample where there is an effect of the 3D aspect.
In this work, we present a new method for quantifying the electric field maps obtained by 4D-
STEM on a needle like specimen submitted to a DC voltage. The experiments involve
electrically polarizing a metallic tip in situ within a TEM using a specially designed sample
holder, which is routinely used to perform APT experiments in a commercial TEM(Da Costa
et al., 2024). The interpretation of the quantitative measurement is based on the Rutherford
scattering theory. By considering the polarized tip as a positively charged object and the
incident electrons as accelerated charged particles, the electron trajectories are deviated by the
effect of the electric field generated by the tip. This straightforward method, which does not
rely on assumptions such as small beam deviation or specimen symmetry, is suitable for
quantifying and mapping large electric fields (>2 V/nm) in the vacuum around the sample.
Experimental results obtained from a test tungsten specimen are compared with corresponding
modelling. Interestingly, the method enables visualization and quantification of the influence
of an emerging grain boundary at the tip's apex on the electric field distribution. The potential
of this approach for modelling field evaporation and improving APT reconstruction is also
discussed.
## 2. Materials and Methods

### 2.1. Sample's preparation

A needle-shaped metal sample was prepared from a GGB Industries Picoprobe® T-4 series tungsten tip using a Plasma Focused Ion Beam Scanning Electron Microscope (PFIB-SEM). Annular milling was used to thin the tip, starting with an inner diameter of 4 µm at an accelerating voltage of 30 kV and a beam current of 4 nA. The diameter and current were progressively reduced until reaching an inner diameter of 0.8 µm at 1 nA. Finally, the sample was cleaned using a voltage of 5 kV and a current of 0.1 nA.

### 2.2. Experimental setup

Electric field maps derived from beam deviation maps were acquired at room temperature by performing 4D-STEM acquisitions in a JEOL-F200 microscope operating at an accelerating voltage of 200 kV. A specially designed TEM sample holder (Da Costa et al., 2024) was used to apply the high voltage up to 8 kV. Using a pixelated Cheetah Timepix3 ASI detector with a matrix of 512x512 pixels and a pixel size of 55 µm, a 2D diffraction pattern was recorded at each position of the scanning probe. A total map of 150x200 positions was acquired with a scan step of 2.5 nm, corresponding to a scan area of 375nm x 500nm.

<caption>Table 1: Experimental parameters for the 4D-STEM acquisitions</caption>
<table><thead><tr><td>Acceleration voltage (kV)</td><td>Convergence semi-angle (mrad)</td><td>Camera length (mm)</td><td>Dwell time (ms)</td><td>Precession angle (°)</td></tr></thead><tbody><tr><td>200</td><td>2</td><td>1500</td><td>5</td><td>0.25</td></tr></tbody></table>

During the experiments, the microscope was operated in diffraction mode, using a condenser lens aperture of 10 µm, a convergence semi-angle of 2 mrad, and a probe current of 50 pA. A camera length CL of 1500 mm was selected to ensure that only the central spot of the transmitted beam remained within the detector area. To minimize dynamic diffraction effects, a precession angle of 0.25° degrees was applied at a frequency of 200 Hz. Two maps were acquired to determine the deviation of the electron beam induced by the electric field: an object signal on the polarized tip subjected to 1kV, and a reference signal acquired in the vacuum without the biased tip. Table 1 resumes the experimental parameters used to perform pixelated STEM experiments.

<figure><img src="image_3.png" /></figure>
Figure 2: Schematic of 4D-STEM field map measurements. (a) 2D diffraction patterns and (b) centre of mass CoM measurements at each scanning position, and (c) map of deflection angle.

## 2.3. Theoretical approach

The electric field can be estimated by measuring the electron beam shift induced by electromagnetic fields in 4D-STEM datacube. Specifically, at each scanning position, the shift of the centre of mass (CoM) from the reference CoM is measured, yielding a displacement amplitude map in pixels. This pixel map is then converted to distance unit by multiplying by the pixel size (55 µm). Subsequently, the deflection angle $γ$ is obtained from the CoM shift $S_{COM}$ and the camera length $CL$ via the equation

$$ \gamma = \tan \left( \frac{S_{COM}}{CL} \right) \qquad (1) $$

Datacube processing was performed using LiberTEM Python library(Clausen et al., 2020). Figure 2 schematically illustrates the principle and the steps of 4D-STEM data processing used to obtain qualitative electric field maps, (a) record a 2D dataset of diffraction patterns; (b) quantify the shift of the CoM of each diffraction pattern at each scanning position; and (c) measure the deflection angle at each scanning position to produce a qualitative electric-field map.
Quantitative electric field measurements can be obtained from the deflection angle of the beam due to the connection between the electric field E and the shift of the CoM in the diffraction patterns in 4D-STEM datasets by the widely used equation (Heimes et al., 2023)

$$E = -\frac{hv \sin(\gamma)}{\lambda t e} \qquad (2)$$

with Planck's constant $h$, the relativistic speed $v$, the shift angle $\gamma$, the electron wavelength $\lambda$, the specimen thickness $t$ and the electron charge $e$. The major drawback of using this equation is that it depends on the thickness of the specimen.

To get quantitative electric field measurements from the deflection angle map, we propose in this study, an approach based on considering the electron beam as a single accelerated electron passing through an electric field generated by a positive charge (the biased tip) at a distance $b$ from the centre of the charges.

In the simplest approach, we may consider the end apex as a spherical geometry with constant surface charge at its surface. Considering Gauss law, this approximation is similar to consider the tip as a point charge $Q$ depending on the surface field. The charge is situated at the centre of the spherical apex. According to Rutherford scattering model (Rutherford, 1911), the deflection angle $\gamma$ can be expressed in terms of the impact parameter $b$ as
$$ \gamma(x, y) = 2 \arctan \left( \frac{eQ}{4\pi\epsilon_0 m v^2 b(x, y)} \right) \qquad (3) $$

Where e is the elementary charge, $\epsilon_0$ is the vacuum permittivity, v is the electron velocity and m its mass. Here Q is the total charge of the diffusing object ($Q = 4\pi R^2 \times \sigma$) with $\sigma = \epsilon_0 E_0$ the surface charge. In the spherical approach, $E_0(x, y) = V/b(x, y)$, so that we can write

$$ E_0(x, y) = \frac{mv^2}{eb(x, y)} \tan \left( \frac{\gamma(x, y)}{2} \right) \qquad (4) $$

For a non-spherical object, because of the shank of the specimen, the surface electric field is lower than the electric field for a sphere. At the tip surface ($b = R$), it is generally admitted that $E = V/kR$ (Gomer, 1961), with k a semi-constant factor that varies slowly with R (or b). We can deduce that $E \sim E_0/k$ close to the tip surface where most of the deflection builds up due to the presence of the field. As a result, an approximate expression of the field is given by

$$ E(x, y) = \frac{mv^2}{ekb(x, y)} \tan \left( \frac{\gamma(x, y)}{2} \right) \qquad (5) $$

In this case, we consider the tip as an ellipse. The axis of the tip is first determined from the virtual bright field image, and the focal parameter c, which defines the separation of the foci of the associated family of confocal ellipses(Hilbert and Cohn-Vossen, 1999), is positioned at the apex along this axis. For each point $(x, y)$ in the scanning area, we then associate the
confocal ellipse that passes through the local surface position with respect to this axis and the parameter c, which defines at that location the two principal radii of curvature $R_1(x, y)$ and $R_2(x, y)$ in the meridional and parallel directions, respectively. The corresponding principal curvatures $\kappa_1(x, y)$ and $\kappa_2(x, y)$ are evaluated using the standard expressions for axisymmetric spheroidal surfaces written in terms of the tip axis and c. The mean curvature is defined as $H(x, y) = \frac{1}{2}(\kappa_1(x, y) + \kappa_2(x, y))$, and its inverse $R_m(x, y) = \frac{1}{|H(x, y)|}$ is used as an effective mean radius of curvature. In our reconstruction, this mean radius $R_m(x, y)$ is taken as the local impact parameter $b(x, y)$ in equations (3)-(5), thereby consistently incorporating the elliptic tip geometry into the electric field determination.

In order to validate the quantitative method proposed in this study, we applied it to electrostatic simulations. Finite element (FE) method simulation using LORENTZ-2E™ software, and an atomistic simulation method.

## 2.4. FE simulations

LORENTZ-2E™ is a finite element simulation-based software that relies on cylindrical symmetry. Therefore, to simulate the electric field around a virtual tip, only half of the tip geometry needs to be modelled; the software then reconstructs the complete field by symmetry around the axis of the tip. The field factor k can be estimated from modelling a virtual tip that replicates the real tip's cone angle and radius of curvature. However, it is
limited in terms of modelling asymmetric forms which makes this software limited in this manner.

## 2.5. Atomistic simulations

In order to bypass the limitations of the cylindrical symmetry in LORENTZ-2E™, we developed an atomistic simulation method based on Rutherford scattering theory. This simulation method mimics the principal functioning of 4D-STEM experiments. A virtual tip is constructed using the Atomic Simulation Environment (ASE)(Hjorth Larsen et al., 2017), and its non-zero surface-atom charges are calculated (for an applied voltage) using the Robin-Rolland method(Klaes et al., 2021; Rolland et al., 2015). An electron grid is then initialized above this virtual tip, defining the scanning region and step size in the x and y directions. Each electron in the grid is assigned an initial velocity corresponding to the chosen acceleration voltage. When the electrons interact with the electric field generated by the charged atoms, the electrons undergo a deflection due to the Lorentz force

$$ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \qquad (6) $$

At each simulation time step, the positions, velocities, and accelerations of all the electrons are updated using the relativistic Leapfrog integration algorithm(Van Gunsteren and Berendsen, 1988), which ensures more accurate handling of high-speed particles. Specifically, the electron momenta are updated by half a time step, positions are advanced, and then
momenta are updated again, incorporating relativistic corrections to the velocity ($v = \frac{pc}{\sqrt{m^2c^2+p^2}}$). The total electrostatic force $F_T$ acting on each electron due to the charged atoms on the tip surface is calculated from Coulomb's law, according to the following equation

$$ \vec{F}_T = \sum_i \frac{K_e q_e q_i}{r_i^2} \vec{r}_i \qquad (7) $$

where $K_e$ is Coulomb's constant, $q_e$ is the electron charge, $q_i$ is the charge of atom $i$, $r_i$ is the distance between the electron and atom $i$, and $\vec{r}_i$ is the unit vector pointing from the electron toward atom $i$.

Throughout the simulations, the instantaneous positions and velocities of the electrons are saved at each time step. By analyzing the final trajectories of the electrons and using the proposed quantitative method, the electric field generated can be calculated. To simplify the problem, we simulate a spherical geometry instead of a tip. Specifically, a sphere of radius 10nm is constructed and the non-zero charges of surface atoms are calculated. Theoretically, applying a constant voltage of 1 kV to this sphere generates an electric field of around 100 V/nm. The deflection angle and the corresponding electric field maps components for each electron are calculated and represented in Figure 3. The simulation estimates an electric field of 93 V/nm, instead of the theoretical value of 100 V/nm. This difference indicates that the method presented in this study estimates the electric field with an error of around 7%. Despite
this imprecision, which may be due to inaccuracy in the charge calculation method or to the
effect of volume on the accelerated electrons, this method remains reliable for quantifying the
electric field and can be used to estimate the electric field from experimental 4D-STEM
deflection angle maps.

<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Projection of the simulated sphere into the scanning area represented by a rectangle, in x-y coordinates. The colors of atoms in the sphere represent the charges calculated in Coulomb units. (b - d) The deflection angle magnitude, y and x components, respectively. (e – g) The corresponding electric field magnitude, y and x components.</figcaption></figure>
### 3. Results

In order to retrieve the geometrical descriptors to be used for both simulations and 4D STEM based quantification of electric field, the sample was first imaged in TEM. Figure 4 shows a bright field (BF) image of the tip, which displays the morphology of a typical field emitter analysed in APT. The cone's half angle and the tip's apex radius were measured, and found to be 8.0 ± 0.2 degrees and 32 ± 2 nanometres, respectively. These measurements were used as input to model a tip of similar geometry in LORENTZ-2E and to estimate the corresponding electric field generated by applying the same voltage as the one applied for the 4D STEM acquisition. The simulations modelled the tip as a conical shape with a sharp apex, designed to replicate the real tip's cone angle and radius of curvature, and biased at 1 kV to match the experimental conditions. From the geometrical description of the sample in BF TEM and the results of LORENTZ-2E simulations, the field factor k to be implemented in the quantification of the electric field based on 4D STEM acquisitions can be determined.
<figure><img src="image_5.png" /><figcaption>Figure 4: Bright Field (BF) TEM image (bottom) of the tungsten tip with a measured half cone angle of 8.0 ± 0.2 ° and a tip radius at the apex of 32 ± 2 nm. The rectangle outlines the entire scanning area from which four color-coded examples of electron diffraction patterns (top) were selected, indicating the electron beam shift (CoM when 1kV in dark-green dots, and in reference in red).</figcaption></figure>

The electric field distribution around the modelled tip is shown in Figure 5. From these simulations of the electric field maps, the field factor k could be determined with a value of 4.23 ± 0.05 for a tip radius R = 32 ± 2 nm. This value of k is close to the typical value of 3 for
a LEAP geometry, which is consistent with the usual morphology of metallic tips analysed in APT(Pogorelov et al., 2009).

<figure><img src="image_6.png" /><figcaption>Figure 5: Electric field E measurements around a virtual biased tip simulated with an applied voltage of 1 kV using LORENTZ-2E™. The magnitude of the electric field E is shown in (a), while (b) and (c) display its y- and x-components, respectively.</figcaption></figure>

CoM shifts in the diffraction patterns were measured at each scanning position (between dark-green and red dots as indicated in Figure 4), enabling the calculation of deflection angle for both magnitude and directional components (x and y). The 4D-STEM deflection angle maps displayed in Figure 6(a-c) provide essential information about the spatial distribution of the electric field around the polarized APT tip. The method described in the theoretical approach section was further used to quantify the electric field from the deflection angle maps. Equation (5) was used to construct quantitative maps of the electric field using k value determined earlier by Lorentz simulation. The resulting electric field maps magnitude and directional components (x and y) are presented in Figure 6(d-f).
<figure><img src="image_7.png" /><figcaption>Figure 6: (Top) Deflection angle maps in milliradians, representing the magnitude of deflection (a), the angular deflection along y-axis (a) and x-axis (c) respectively. (Bottom) Calculated electric field maps representing the magnitude of electric field in (d) and its components along y-axis (e) and x-axis (f) respectively. Maps in (d–f) are superimposed a superimposed with BF TEM image of the tip.</figcaption></figure>

A rapid glance at the electric field maps calculated from 4D-STEM acquisition indicate values in the typical range encountered for APT field emitters(Miller and Forbes, 2014; Vurpillot, 2016a). Here, the specimen was observed at room temperature and the voltage was fixed at 1.0 kV, below the onset of field evaporation (no variation of the specimen geometry was visible in BF TEM). The calculated values are also consistently below the reported evaporation fields for tungsten (W), which however strongly ranges from 47 V/nm to 98 V/nm depending on the theoretical model used for its estimation(Vurpillot, 2016a).
As expected, the electric field maps display maximum values in the vicinity of the tip apex.

This is the location at which the atom subjected to the highest electric field is expected to evaporate once the threshold field is exceeded. However, more attention needs to be paid to the asymmetry of the electric field maps, which is only observed in the experimental case.

Indeed, for a similar specimen geometry (tip radius and shank angle), the simulated maps in Figure 5 show a perfect symmetry with respect to the long axis of the tip. On the opposite, and though of a small amplitude with respect to the magnitude of the electric field, a clear asymmetry is visible along the y-direction (Figure 6(e)).

In order to reveal the origin of this asymmetry in the electric field distribution, additional acquisitions were performed to identify the crystal orientation in each location of the specimen using the ACOM-TEM methodology (Rauch and Veron, 2014). Results are displayed in Figure 7. This additional investigation allows the identification of two distinct tungsten crystals, separated by a grain boundary which intersect the specimen's apex. The asymmetry in the electric field map can hence be put into relation with the presence of a grain boundary at the tip's surface, where the magnitude of the electric field is the largest. This is what is going to be further discussed in the next section.
<figure><img src="image_8.png" /></figure>

Figure 7: Combined image of deflection angle map, crystal orientation map and a BF TEM image of the W tip polarized at 1 kV. The crystal orientation map indicates the presence of a grain boundary which intersects the tip's apex. Diffraction patterns recorded in the upper and lower crystals are shown on the left and right respectively, together with the crystals' direction perpendicular to the image plane and the corresponding orientation of the body centered cubic structure of W in each grain.

# 4. Discussion

## 4.1. Quantification of the electric field at the APT specimen's apex

The quantified distribution of the electric field obtained in this study is compared with the results of Lorentz-2E simulations, as shown in Figure 8. A good agreement is observed between the maximum electric field values estimated at the tip apex in both the simulated and experimental data.
For polarized APT specimens, 4D-STEM remains a comparatively straightforward methodology relative to off-axis electron holography. In the detailed comparison between electric field mapping by 4D-STEM and off-axis electron holography proposed by Cooper et al. (Cooper et al., 2024), the weaker robustness of 4D-STEM was attributed to a diffraction-related artefact. The authors noted that the convergence of the beam can induce variations of diffraction contrast which may lead to a loss of accuracy of the field measurement but mentioned that this artefact can be partially reduced by the application of precession. In the present work, a precession angle of 0.25 was employed to minimize dynamical diffraction artefacts. It is important to note that only the field distribution in vacuum is considered here. Although a more rigorous quantification of the electric field in 4D-STEM would ideally account for beam convergence effects, the present results already demonstrate good agreement between experiment and simulation (Figure 8).

The proposed methodology can thus be applied to APT specimens to provide an accurate estimation of the actual electric field at the specimen surface during the onset of field evaporation. In contrast to previously reported 4D-STEM or DPC based quantification approaches, the present method circumvents common assumptions typically imposed in electric field quantification, such as the consideration of a constant specimen thickness (Haas et al., 2019) or the assumption of a cylindrical symmetry of the tip (Wu et al., 2019). Our methodology hence proposes a straightforward way to map quantitatively the anisotropic distribution of the electric field in the vicinity of atom probe specimen.
<figure><img src="image_9.png" /><figcaption>Figure 8: Variation of the magnitude of the electric field along the x-axis (parallel to the long axis) in the APT specimen observed in Figure 6 compared to the simulated values obtained using the Lorentz-2E software. These plots have been drawn from left to right in the datasets of images Figure 5(a) and Figure 6(a).</figcaption></figure>

## 4.2. Variations of electric field related to crystal orientation

Being able to link the dynamic variations of specimen geometry with the sequence of detection in APT is a key objective of the coupling between APT and TEM in a single instrument(Ceguerra et al., 2021; Da Costa et al., 2024; Kelly et al., 2020, 2013; Lefebvre, 2016). This is all the more crucial so as strong variations of the specimen geometry (Figure 1) hinder APT reconstruction algorithms based on the hypothesis of a hemispherical cap(Gault et al., 2012; Larson et al., 2013; Miller and Forbes, 2014). However, even more relevant would be to map the trajectories of ions thanks to the prior knowledge of electric field maps in 2D or 3D, which are accessible thanks to off-axis holography(Beleggia et al.,
2014; Zheng et al., 2023) or 4D-STEM(Ophus, 2019; Wu et al., 2019). Meanwhile, getting an estimate of the distribution of electric field magnitude along the specimen's surface would be useful to identify the physical conditions experienced by field-evaporated ions. This would allow to feed physical models for field evaporation(Oberdorfer et al., 2013; Qi et al., 2022; Rolland et al., 2015; F. Vurpillot et al., 1999), field-induced molecular dissociation(Peng et al., 2019) and to link the variations observed in the mass spectra with the electric field conditions at the tip's apex. These various goals are within reach with the available combination of APT and STEM(Da Costa et al., 2024).

The present work proposes to combine, on polarized APT specimens, crystal orientation and quantitative electric field mapping in two dimensions. The asymmetric field distribution revealed in the y-component of the electric field showed in Figure 6(e) could be explained by the crystallographic orientations of the two grains present at the apex of the tip. Automated crystal orientation mapping(Rauch and Veron, 2014) analysis was performed and the results show that there are two W crystals with different orientations at the apex of the tip. In a single-phase metal, it is hence demonstrated that the presence of a grain boundary results in an asymmetry in the electric field distribution (Figure 7). Such a result is very useful to understand how ion trajectories can be affected in APT reconstruction by the presence of a crystal defect. But here, the effect of the defect is more likely related to the fact that the presence of two distinct orientations of the same phase (body-centered cubic tungsten) result in atomic scale variations of the equilibrium surface geometry of the specimen. Fletcher et
al. (Fletcher et al., 2019) showed by the mean of finite and boundary element method FEM-BEM simulations that grain misorientation can be encoded as a rotation of the crystallographic anisotropy function, producing orientation-dependent faceting and asymmetric field evolution at a vertical grain boundary.

Before the advent of combined APT-(S)TEM techniques, or observations of field-evaporated specimens in TEM, field ion microscopy (FIM) played a crucial role in revealing that, under high voltage polarization, single crystals develop facets corresponding to crystallographic planes emerging at the tip apex(Mikhailovskii et al., 1997). Such observations were made in the early days of FIM across various metals. For instance, FIM analysis of pure aluminum revealed dark {111} planes(Southon et al., 1975; Vurpillot et al., 2025), forming relatively flat facets, in contrast to the bright appearance of 〈110〉 orientations(Lefebvre et al., 2007). This indicates that atomic terraces associated with {111} planes tend to field evaporate prior to being imaged in FIM(Southon et al., 1975). FIM has also shown that single-phase grain boundaries can induce contrast variations(Rendulic and Müller, 1967). While brightness contrast in FIM provides a qualitative indicator of electrostatic field variations near the tip surface, particularly in single crystals, this technique is now rarely applied. With APT alone, however, an indirect estimation of the surface electrostatic field is sometimes possible through the analysis of charge-state ratios, based on the phenomenon of post-ionization(Kingham, 1982; Tegg et al., 2024). This approach has been applied to materials such as titanium aluminides(Lefebvre et al., 2002), Al-based alloys(Lefebvre et al., 2007; Tegg et al., 2024)
and semiconductors(Mancini et al., 2014) and allowed to account for composition biases (e.g. preferential loss of an element(Mancini et al., 2014), chemical reactions at the specimen surface at DC voltage(Lefebvre et al., 2002)) or to identify the occurrence of local magnification artifact due to low field or high field precipitates(Lefebvre et al., 2007).

The theoretical understanding of this crystallographic effect remains incomplete. It is generally assumed that the equilibrium tip surface under field evaporation adopts a polyhedral morphology, either due to crystallographic variations in the evaporation field or due to local field enhancements from the atomic terrace structure(Fletcher et al., 2020, 2019; François Vurpillot et al., 1999). One possible explanation lies in the intrinsic anisotropy of crystals, which leads to distinct work function values associated with specific crystal planes. Within the framework of the Müller–Schottky field evaporation model, this anisotropy can justify the formation of facets. In its simplest form, the model describes the evaporation field F<sub>EV</sub> as

$$F_{EV} = \frac{4\pi\epsilon_0}{n^3e^3}(\Lambda + \sum I_n + n\phi)^2 \quad (8)$$

Where Λ is the heat of sublimation of a neutral atom, I<sub>n</sub> the n<sup>th</sup> ionization energy of the atom and φ<sub>e</sub> is the work function of the emitting surface. For tungsten, the dependence of electron work function on crystal orientation has been quantified(Swanson and Crouser, 1967) and has led to the use of specific orientations of W monocrystals in cold field emission guns and Schottky field emission guns(Crewe, 1971; Swanson and Schwind, 2008). The physics used
to describe the field evaporation mechanism can be found in some recent reviews(Miller and Forbes, 2014; Vurpillot, 2016a). Early models emphasized the role of the work function, thus providing a rationale for the anisotropy of the evaporation field in single crystals. However, this classical model, developed in the 1950s–60s, is now known to be inaccurate. More recent models involving mechanical bound rupture induced by the Maxwell electrostatic stress involve mechanical constants that are also crystallographic direction dependent. It has been found, for instance, that in this approach

$$F_{EV} \propto \left( \frac{\Lambda \times C_{11}}{r_{WS}^3} \right)^{1/4} \qquad (9)$$
$r_{WS}$ being the Wigner Seitz radius, and $C_{11}$ being the elastic constant in the considered direction. This last value is known to vary with the crystallographic direction (McMullen et al., 1982; McMullen and Perdew, 1987; Vurpillot, 2016a).

Ab initio simulations now show that field evaporation in metals inherently probes the local bond strength along the pre-evaporation trajectory of atoms near the tip surface (Ashton et al., 2020; Ohnuma, 2022; Qi et al., 2022). Since this trajectory depends on crystal orientation, the electric field required to induce evaporation also varies accordingly. Experimental confirmation of such crystallographic dependence in field evaporation has recently been reported (Rousseau et al., 2023; Vurpillot et al., 2025).

Nevertheless, a comprehensive and predictive link between tip geometry and crystal orientation remains elusive. Such a connection may eventually be established through in-situ APT-TEM studies (Da Costa et al., 2024; Mayer et al., 2023). The present work contributes additional evidence that the crystal orientation governs the surface field distribution and thereby influences the trajectories of field-evaporated ions toward the detector.

# 5. Conclusion

By means of a theoretical model applied on 4D STEM acquisitions, this study provides a method to quantify and map the electric field distribution in the vicinity of an atom probe specimen polarized to a high constant voltage. Experimental quantification is compared to
modelling using a commercial software which imposes a rotational symmetry of the needle
shape specimen. Good agreement is found, especially for the largest value of the electric field,
which is reached at the vicinity of the tip's apex. It is hence foreseen that this method is used
to provide estimates of evaporation fields at the onset of field evaporation.

Interestingly, the method allowed to clearly show the effect of an emerging grain boundary at
the tip's apex on the electric field distribution. Such a result highlights the relevance of in situ
coupling of APT and TEM(Da Costa et al., 2024; Kelly et al., 2013; Mayer et al., 2023) for a
better understanding of the physics of field evaporation of all classes of materials and for the
emergence of TEM informed APT reconstruction protocols.

Acknowledgements

The Normandy Region is acknowledged for its financial support through the RIN Tremplin
FusionSATMET project. ANR is acknowledged for its support through the project
DynaMETSAT (ANR-25-CE42-2131-01).

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