Crystallographic Dependence of Field Evaporation Energy Barrier in Metals Using Field Evaporation Energy Loss Spectroscopy Mapping

Abstract

Atom probe tomography data are composed of a list of coordinates of the reconstructed atoms in the probed volume. The elemental identity of each atom is derived from time-of-flight mass spectrometry, with no local chemical information readily available. In this study, we use a data processing technique referred to as field evaporation energy loss spectroscopy (FEELS), which analyzes the tails of mass peaks. FEELS was used to extract critical energetic parameters that are related to the activation energy for atoms to escape from the surface under intense electrostatic field and dependent of the path followed by the departing atoms. We focused our study on pure face-centered cubic metals. We demonstrate that the energetic parameters can be mapped in two-dimensional with nanometric resolution. A dependence on the considered crystallographic planes is observed, with sets of planes of low Miller indices showing a lower sensitivity to the field. The temperature is also an important parameter in particular for aluminum, which we attribute to an energetic transition between two paths of field evaporation between 25 and 60 K close to (002) pole. This paper shows that the information that can be retrieved from the measured energy loss of surface atoms is important both experimentally and theoretically.

Introduction

Atom probe tomography (APT) provides three-dimensional (3D) material microanalysis at the nanometer or subnanometer scale (Larson et al., 2013; Miller & Forbes, 2014; Lefebvre-Ulrikson et al., 2016; Gault et al., 2021). At the core of APT is the process of field evaporation, wherein atoms located at the surface are removed from the solid in the form of ions under the influence of an intense surface electrostatic field (Waugh et al., 1976). Typically, the critical field required for field evaporation, devoid of any thermal activation, falls within the range of 10–60 V/nm. Achieving such intense electric fields is easily accomplished through the classical tip effect, with the tip relatively close to a counter electrode. For APT, the material is fashioned into a sharply pointed needle with an end radius (R) ranging from 10 to 100 nm and subjected to a high voltage (V) of a few kV. With careful control of field evaporation through adjustment of V, atoms are liberated individually from the surface, ionized, and accelerated by the electrostatic field away from the surface. The specimen itself acts as an electrostatic lens, projecting ions in a deterministic manner from the surface onto a position-sensitive detector located typically 10 cm away from the tip. Elemental identification of the projected ion is achieved through time-of-flight mass spectrometry. The atoms are field evaporated by the superimposition of ultrashort voltage pulses (ns) or laser pulses (ps or fs pulse duration) onto the DC voltage, providing a start time for ions. The impact of the ions on the detector determines the stop time. Postprocessing algorithms are employed to associate to each mass-to-charge an elemental nature, and by recalculating the original positions of the detected ions assuming a known projection law (De Geuser & Gault, 2017), reconstruct a 3D point cloud with near-atomic resolution (De Geuser & Gault, 2020).

However, the resulting 3D volume provides limited information about the chemical nature of the bonds, i.e., metallic, covalent, or ionic, between atoms prior to their departure from the surface. The nature of chemical bonding remains usually imperceptible from studies of the mass spectra, which is a histogram of the counts at a given mass over charge ratio. While extensive research has explored the potential of field evaporation to directly and locally measure the binding energy of single atoms, practical challenges persist in accurately measuring local surface fields due to surface self-reorganization under field evaporation. Additionally, intricate electrostatic devices are required to measure precise binding energies linked to local atomic environments, local fields, and crystallographic locations on the specimen surface (Ernst, 1979). More recent work has also postulated that traces of the bonding mechanism can be retrieved from the analysis of the rate of multiple-ion field evaporation in APT (Zhu et al., 2018). These studies have proposed assessing the occurrence of multiple field evaporation events to infer bonding mechanisms in thermoelectric and phase-change materials. However, retrieving local physical and chemical properties at the nanoscale in 3D remains challenging, and the details of fine-scale energetics within each family are for the moment out of reach. It is worth noting that access to this information is also of fundamental importance for the practical operation of APT. Variations in both the nature of a surface atom and its local neighborhood significantly impact the field evaporation behavior of the material (Tsong, 1978; Ge et al., 1999). Analyzing alloys with atoms of diverse chemical nature can prove arduous, necessitating precise tuning of the analysis conditions to avoid species-specific losses, for instance (Miller, 1981; Yamaguchi et al., 2009; Takahashi et al., 2011; Chang et al., 2019; Jin et al., 2022). This underlines the intricate interplay between the physical and chemical nature of the local zone under investigation and the field evaporation process.

In a recent study (Rousseau et al., 2023), we demonstrated that additional information relative to the local field evaporation activation energy related in part from binding energy of the field evaporating atoms could be extracted from the shapes of the mass peaks in atom probes operated in voltage pulsing mode and not equipped with energy compensation devices. This method, termed “field evaporation energy loss spectrometry” (FEELS), not only aids in identifying and quantifying elements in the analysis but also encodes information regarding an atom’s binding with its neighborhood under high electrostatic field and energetic aspects related to the field evaporation process (Rousseau et al., 2023). We showcased on various examples how information related to the chemical and structural nature of evaporated atoms could be relatively easily extracted from APT datasets with nanometer spatial resolution.

FEELS relies on the measurement, in logarithmic scale, of the slope of the mass peak tails to extract an energetic parameter, C, consistent with a relative electric field sensitivity. It was found, in good agreement with density functional theory (DFT), that this term could be significantly different at grain boundaries, for instance, in pure aluminum (Al). The measurement of this term for the same element but in different phases highlighted the possibility to map the local environment of atoms in terms of the chemical composition or structural properties. However, the deeper physical meaning of C is still unclear, indeed this energetic parameter is linked not only to the binding energies and the original situation of the atom under the electric field, but also to the local path that is followed by these atoms in the first stages of the field evaporation process from the surface (Moore & Spink, 1974; Forbes, 1976; Waugh et al., 1976; Schmidt et al., 1993; Schmidt & Ernst, 1994; Forbes, 1995; Ashton et al., 2020; Ohnuma, 2021). From the DFT calculation, it was found that the energy barrier height variation was different when considering an evaporation path including the roll-up motion on surrounding neighbors before being expelled from the surface compared with a direct lift-out from the surface. As a result, predicting the field evaporation behavior is extremely complicated since the local path is strongly dependent on the local atomic configuration, which is also related to the crystallography, and possibly the field and temperature at which the field evaporation proceeds.

Here, we focused our attention on pure face-centered cubic (FCC) metals to evaluate experimentally the dependency of C with physical and instrumental parameters. We have measured C as a function of the crystallographic location of atoms on nickel (Ni), rhodium (Rh), and Al varying base temperature and detection rate (DR). The influence of the rolling-up on C has also been explored.

Materials and Methods

Experiments on the LEAP 5000XS

Ni and Al APT specimens were prepared using electrochemical polishing, starting from pure metal wire specimens and using the solutions and conditions reported in Larson et al. (2013), Lefebvre-Ulrikson et al. (2016), Miller & Forbes (2014), and Gault et al. (2021). APT specimens from Rh were prepared by using a HELIOS 5 dual-beam Xe plasma scanning electron microscope/focused ion beam via classical lift-out specimen preparation process. At the start of the analysis, all specimens possessed a curvature radius of <50 nm. The needle-shaped specimens were analyzed using the voltage pulsing mode on a local electrode atom probe with a direct flight path (LEAP 5000XS). The analysis temperatures varied between 25 and 60 K for Al and Ni specimens. Rh specimens exhibited greater brittleness; thus, the analyses were successfully conducted only at 80 K in this study. The pulse fraction (PF) was consistently set to 20% across all analyses. During APT specimen evaporation, the DR was maintained constant, with no restrictions on the gradually increasing DC voltage (V_DC) to offset the progressive blunting of the specimen radius.

C-Value Measurement

In APT, the field evaporation rate-constant (on a single atom) for thermally activated field evaporation follows an Arrhenius law K_EV (equation (1)) that is dependent on the energy barrier Q(F) and the temperature T of analysis.

where $kB$ is the Boltzmann constant, T is the temperature, and $Q(F)$ is the activation energy barrier that depends on an external electric field (F), which can be approximated using the following linear approximation:

where $FEV$ is the evaporation field of the considered atom, i.e., the field at which the energy barrier is reduced to zero, and C is the abovementioned energetic parameter. Note that F_EV is also called zero-barrier evaporation field, and F is the operating field. Q empirically monotically decreases with F at low temperature and high field. A linear Q(F) approximation is generally used even if nonlinearities of Q(F) linked to the path of the adatom at the surface during the escaping process exist (Waugh et al., 1976; Ashton et al., 2020), but the linearity of Q with F can empirically be considered as a good approximation at the conditions used in the voltage pulsing mode (Wada, 1984).

The atom is assumed to leave the surface at the top of the pulse and the formed ion acquires almost instantaneously all the kinetic energy induced by the potential energy. The experimental mass-to-charge ratio (M = m/n) is determined using the following equation for the conservation of energy:

where t_f is the ion's time of flight, L is the flight distance, e is the elementary charge, and V_T is the maximum voltage from the DC and pulsed high voltages. Due to the probabilistic nature of the field evaporation process (equation (1)), some ions can leave the specimen slightly before or after the top of the pulse; hence, some ions have an actual energy deficit δ (V = V_T(1 − δ)), leading to an increase in their time of flight, which creates a tail in the mass spectrum (Regan et al., 1976). For a mass-to-charge ratio of the evaporated element M₀, and to a first-order approximation, energy deficit δ can be derived from

allowing the field at which the ion was evaporated to be extracted from the measured mass-to-charge ratio. Indeed, for a certain deficit, the number of evaporated atoms is proportional to equation (1), with Q(F) derived from equation (2). In the mass spectrum, F being proportional to V, the number of ions at a mass-to-charge M, I(M) (being proportional to K_EV in a first approximation), can be seen as linearly dependent on δ, and C can be written as

In practice, a peak is chosen from an APT mass spectrum plotted in log scale, and the slope of the mass peak tail is −C/k_BT. An illustration of this process is provided in Figure 1a, using an Al specimen evaporated in voltage pulse mode on a LEAP 5000XS at T = 25 K. Approximately, 5 × 10⁶ atoms were collected with a DC voltage variation of <10%. To accurately measure the C value, a statistically significant number of atoms must be assessed within the mass bins of the mass spectrum. Note that a more correct expression of the shape of the mass peak as a function of δ was found both analytically and numerically. It was found that the linear approximation was sufficient for M > 10, above which dynamic effects could be neglected, and for range of mass, M > 1.005 M₀. More details can be found in Rousseau et al. (2023).

Provided that a sufficient number of ions were collected, mass spectra can be calculated from selected locations across the field of view and local measurements of C can be performed. Typically, a minimal 1,000 to 10,000 ions in the considered selected peak are deemed adequate to perform the linear regression, falling between 1.005 and 1.015 M₀ in the mass spectrum. This process has been automated under MATLAB™ by discretizing the hitmap into a regular mesh, with a spatial binning selected to accommodate the necessary local number of ions. As depicted in Figure 1b, the local C value can be calculated as C(x, y) = −k_BT × a(x, y), where a(x, y) denotes the slope obtained from the linear regression, and T represents the specimen temperature. Note that the detector hitmap shown inset displays the characteristic pattern of poles and zone lines that form in the analysis of many crystalline materials, and is due to crystallography-induced trajectory aberrations and faceting of the specimen around sets of crystallographic planes (Gault et al., 2012; Vurpillot & Oberdorfer, 2015). This approach facilitates the generation of a C-map, as detailed in the subsequent section.

Results

Aluminum Experiments

A pure-Al specimen was analyzed at 60 K at a DR of 0.5 ions/100 pulses, and a PF of 20%. A subset of 10 million ions was extracted across a moderate DC voltage increase (<5%). Figure 2a plots the map of the density of impacts on the detector or hitmap, which highlights the crystallographic nature of the FCC Al specimen. This image was compared with a modeled atomic surface of a sphere intercepting an FCC structure in Figure 2b, with indexed main poles. Using the mass spectrum (similar to Fig. 1), the local fraction of Al⁺⁺ (Fig. 2c) to the total number of hits and the measured local fraction of hydrogen (Fig. 2d) are displayed. These maps are compared with the C values measured locally on the detector (Fig. 2e, C-map). C ranges from 0.6 to 1.2 eV, with apparent local contrasts. The histogram of the C values across the entire detector, using an adjusted Jet colormap, with C ranging from 0.6 to 1.2 eV is given in Figure 2f. Notably, this histogram displays a bimodal behavior, featuring main peaks at ∼0.65 and 0.95 eV, with an average value near 0.85 eV. The corresponding C-map, with the same colormap, is shown inset. It is noteworthy that regions with the lowest hydrogen fraction coincide with regions of lowest Al⁺⁺ fraction, specifically the (002) and (111) regions, indicating a lower electric field (Ernst, 1979; Kingham, 1982).

In Figure 2c, we report the local radii measured using the Drechsler method on the (002), (113), and (022) poles based on the rings imaged (Drechsler & Wolf, 1958; Gault et al., 2008), superimposed over an overlay of the C map and the hit impact density.

In Figures 2b and 2c, the maps of the locally measured composition of H (maximum of 2% in red) and the local composition of Al⁺⁺ (maximum of 8%) are displayed, respectively.

A similar analysis was performed for APT data obtained from the very same specimen but at a base temperature of 25 K under identical analysis conditions, as reported in Figure 3a–e. In Figure 3a, the hitmap is reported, Figure 3b reports the Al⁺⁺ composition map, the lowest Al⁺⁺ composition is around the (111) pole, just outside the field of view. This demonstrates the change in the faceting of the specimen observed previously at different temperatures (Nakamura & Kuroda, 1969; Gault et al., 2010) likely that had been attributed to changes in the field evaporation behavior of different sets of atomic planes (Chen & Seidman, 1971). Comparatively, the radius of curvature at the (002) pole is slightly lower than that observed in the 60 K experiment, decreasing from 103 to 91 nm, whereas the curvature of the (113) poles remains similar. The H composition, Figure 3c, increases slightly at lower temperature, with the lowest values near the low-index poles (002) and (111). The C-map (Fig. 3d) is now less contrasted with values varying from 0.4 to 0.8 eV, but with fine-scale contrasts corresponding to the crystallography of the surface. Note the lowest values at the exact locations of pole centers. The histogram of the C value (Fig. 3e) now exhibits a globally monomodal distribution with an average value of 0.6 eV. The spatial distribution illustrates crystallographic variations ranging from 0.75 eV around the (022) pole to 0.5 eV near the center of the (002) pole. The H fraction, Figure 2b, increases slightly at lower temperature, with the lowest values near low-index poles (002) and (111) in Figure 2c.

Rhodium

In Figure 4, the same method is applied to the analysis of Rh. A selection of 3 million extracted from the analysis of a pure-Rh specimen conducted at 80 K, a DR of 5 ions/1,000 pulses, and a PF of 20% is presented. Figure 4a is the Rh mass spectrum centered on the main peak used for measurement (Rh⁺⁺). Note that Rh is principally detected in the 2+ charge state (local field measurement using postionization is not possible). Figure 4b illustrates the hitmap density, while Figure 4c presents the overlay of the C-map and the hitmap. A notable correlation between the C value and the crystallographic structure of the specimen surface is observed, with higher values consistently associated with zone lines and areas having low density of impacts. Exact centers of (111) and (002) poles are found to have low C values. Figure 4d depicts the histogram of C values obtained from Figure 4c. The average C value is 1.55 eV, with a range of 1.2–1.9 eV. Note that the correlation of the C-map with local radii of curvature is here difficult since the measurement is quite imprecise especially on the (022) pole, which appears asymmetric. The average values are here similar between poles [i.e., 65 ± 5 nm on the (111) pole, 72 ± 10 nm on the (022) pole, and 70 ± 5 nm on the (002) pole].

Nickel

A pure-Ni specimen was analyzed at 25 and 60 K, and Figure 5 summarizes the mapping of the C values in both cases. Specifically, Figure 5a (and Fig. 5c) depicts the C-map at 60 K (and 25 K), yielding average values of 1 eV (and 0.65 eV), respectively. The figures are overlaid with the hitmap to highlight crystallographic structures. Additionally, Figure 2b (and Fig. 2d) represents the hitmap histogram for the 60 K analysis (respectively, 25 K), showing the relatively low dispersion of C values, with distributions at 60 and 25 K having the same global shape. The C-maps obtained for two DR, namely, 1 ion/100 pulses and 5 ions/100 pulses, are plotted in Figures 5a and 5f, respectively, showing no significant differences. At both temperatures, and both DR, the highest C values are consistently observed on high-index poles.

Discussion

The three FCC materials studied in this paper have common characteristic features in terms of the C value. Lowest value of C is always found at the center of the crystallographic poles, and generally the highest values of C are found in the vicinity of the high-index poles. The average values of C are consistent with previous experimental and computational reports. In Al, an average C value of 0.5 ± 0.1 eV was measured by using the Ernst–Kellogg procedure, i.e., through measurements of the evaporation rate at different base temperatures and applied fields (Rousseau et al., 2023). The experiments of Ernst on Rh enable to derive a C value of 1.5 (±0.5) eV (Ernst, 1979), and recent experiment on Ni alloy by Hatzoglou et al. reported a value below 1.1 eV, depending on the composition of the alloy [10]. Here, we observe a large crystallographic dependence, with variations of about 30% between poles in the best case.

We have observed that there is no obvious correlation between the C value and the surface field, as assessed based on the fraction of Al⁺⁺, which is strongly correlated with the magnitude of the surface electric field according to the postionization theory (Kingham, 1982). Overall, the C value tends to increase with the electric field strength (F). Interestingly, regions with the highest electric field, e.g., near the (024) poles in Al, exhibit relatively lower C values compared with regions with a slightly lower electric field strength, e.g., near the (022) poles. We have also attempted to correlate the presence of hydrogen with the C-map, but the relationship appears to be complex and not easily interpretable (Figs 2d and 3c).

For all three metals, we have observed similar trends in the variation of the C value with temperature. Lower specimen temperatures correspond to lower C values. For example, the C_Al value varies from 1 to 0.5 eV as the temperature decreases from 60 to 25 K, while the C_Ni changes from 1 to 0.65 eV. Although complete data for Rh at 30 K were not available, partial measurements (not presented here) suggest a similar decrease. This behavior could be attributed to the relative nonlinearity of Q as a function of F at higher temperature, previously reported for a range of metals (Forbes et al., 1982; Wada, 1984). In our approach, we assume a first-order linear expression of Q(F), which is a simplification as confirmed by experimental observations (Wada, 1984). This behavior is also evident in DFT simulations (Ashton et al., 2020; Ohnuma, 2021). In previous works by Ernst (1979) and Kellogg (1984), the Q(F) function consistently exhibited an increased slope at low-field values.

The variation of C with temperature can be explained based on the simplified model illustrated in Figure 6a. The variation of Q with F is represented by the solid black curve. At low temperatures (25 K), the maximum pulse reaches a field F(25 K), resulting in a lowered energy barrier to Q(F)_25K—for Al, this is estimated to be 25 meV. Note that this value obtained from an evaporation rate variation with temperature was taken from Rousseau et al. (2023). The C value is extrapolated from the slope of Q(F) with respect to F, where energy deficits provide access to ions emitted at lower electric field strengths. The mass peak tail arises from a field variation of 0.5–2% below the maximum field. Conversely, at 60 K, to maintain the same evaporation rate, the energy barrier must be higher, since Q/kT is constant. Consequently, the electric field is slightly reduced, resulting in an estimated energy barrier of 60 meV in Al at F(60 K). At this position, the derivative of Q(F) is more pronounced, leading to a higher C value.

It is interesting to note that this behavior is not systematic over the whole surface in Al. Looking at the histogram of the C values in Figures 2f and 3e, the lowest peak at about 0.4–0.7 eV is present in both distributions. However, the average value of this peak does not vary strongly with T (from 0.65 to 0.5 eV). This peak comes from the region around the (002) pole. In this case, the variation of Q(F) can be understood through the case illustrated in Figure 6b. In this case, the slope is nearly constant with F, resulting in reduced variations of C with T, C₀₀₂ (25 K) ≈ C₀₀₂(60 K).

The observed behavior is particularly intriguing in the case of the Al specimen, specifically within the vicinity of the (002) pole region. We could argue that this phenomenon is related to the relatively high amount of hydrogen detected in APT analysis for Al, which is different at 25 and 60 K. H is known to lower the field evaporation of metals and H is indeed only detected as trace for Ni and Rh (<1%). However, the H composition maps presented in Figures 2d and 3c are relatively similar, whereas the C-maps evolve strongly between the two cases [especially looking around the (002) pole region]. Alternatively, we have correlated this phenomenon with the hitmaps presented in Figures 2a and 3a, focusing on the (002) pole region. While the characteristics of other regions remained consistent, the (002) region exhibited notable changes with decreasing temperature, as detailed in Figure 7. Figure 7a is a section of the detector hitmap obtained at 60 K, along with, in Figure 7b, a section of the ball model showing the edge of a set of (002) plane. Figures 7c and 7d are similar, but at 25 K. At 60 K, the zone line between the (002) and (113) poles appears dense, contrasting with its depleted appearance at 25 K. These observations are attributed to trajectory artifacts occurring during the initial flight steps (Waugh et al., 1976; Vurpillot & Oberdorfer, 2015). A dense line indicates ions moving inward toward the zone line, as indicated by the red arrows in Figures 7a and 7b, while a depleted line implies ions moving away from the zone line in Figures 7c and 7d. The Al desorption hitmap is effectively interpreted through electrodynamic simulations of the field evaporation process. The presence of depleted zone lines can be attributed to the slight repulsion of ions during the initial flight steps, induced by the local electric charges of neighboring atoms at the terrace edges (as observed in Figs 7b and 7d). Above 60 K, a transition in the contrast of some zone lines is observed, indicative of a rolling-up motion of the departing atoms. This suggests that Al atoms move over their neighbors before lifting off from the surface above a critical temperature, which is not predicted at 0 K using DFT models (Sánchez et al., 2004). Consequently, the energetic landscape experienced by these atoms differs significantly. While other surface atoms experience an increase in the energy barrier for field evaporation, atoms from the (002) pole follow a smoother path, resulting in a reduced increase in the energy barrier for field evaporation and, hence, a lower C value.

It is also worth noting that these observations are crucial for APT applications, as significant differences between crystallographic locations may alter local analysis conditions. Strong variations in the C value could affect the curvature across the field of view and lead to spatial distortions in the reconstructed data. Recent works have shown the potential impact to introduce local variations in curvature of the specimen to improve reconstruction algorithms (Hatzoglou et al., 2019; Fletcher, 2021; Dialameh et al., 2024). However, field evaporation models involved in these works generally neglect the variation of the C values as a function of the material or crystallography directions. Our work draws a more complex picture where the physical parameters (essentially C and F_EV) have to be carefully measured. Additionally, it is important for controlling the field evaporation conditions in alloys, as a variation in the C value could change the preferential field evaporation of one element over another from one crystallographic location to another, leading to compositional inaccuracies (Hatzoglou et al., 2020). It is worth noting that in the general case, the nonlinearity of the energy barrier with the applied field should be considered. The C value, which also measures the field sensitivity of the energy barrier, changes slightly with the base temperature.

Conclusions

In this paper, we present the application of FEELS to three FCC pure metals, namely, Al, Ni, and Rh. In the linear approximation formulated in equation (2), the field sensitivity of activation energy is quantified by the energetic parameter C, referred to as the C value deduced from the tails of mass peaks in the mass spectrum obtained in the voltage pulsed atom probe without energy compensation devices. We were able to reveal an influence of the crystallography of the specimen surface on this C value, likely associated to the local atomic arrangements at the surface, with low-index poles, i.e., more closed-packed planes, exhibiting a lower C value. However, at low temperatures, the slope of the energy barrier with field remains relatively similar (within 30–40%) as a function of crystallography.

The local variation of the C value as a function of specimen temperature was measured to be stronger in Al than for Ni and Rh. This variation could be attributed to the rolling-up effect (Waugh et al., 1976), which modifies the sensitivity of the energy barrier to the applied field. The impact of rolling-up is highlighted through a correlation of the C-map with zone line features in hitmaps. The region close to the (002) pole, where rolling-up is suspected to be significant, exhibits a different behavior in the variation of the energy barrier as a function of the field.

Finally, the FEELS method provides new insights into the physical process of field evaporation that are spatially resolved. However, it remains qualitative for the moment. Extracting local chemical information would require a systematic comparison of the measured C value with DFT field evaporation models, which are still in their infancy.

Availability of Data and Materials

The authors have declared that no datasets apply for this piece.

Acknowledgments

F.V. acknowledges the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-21-CE42-0024 (project HiKEAP).

Financial Support

This work was funded by French Agence Nationale de la Recherche (ANR), under grant ANR-21-CE42-0024 (project HiKEAP).

References

Author notes