# Sub-microradian angular detection limits for field mapping by Lorentz 4D scanning transmission electron microscopy on a Si p-n junction

Alexis Wartelle, Bruno C da Silva, David Cooper, Martien I den Hertog

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Alexis Wartelle, Bruno C da Silva, David Cooper, Martien I den Hertog. Sub-microradian angular detection limits for field mapping by Lorentz 4D scanning transmission electron microscopy on a Si p-n junction. Journal of Applied Physics, 2025, 138 (10), pp.105703. 10.1063/5.0273467. hal-05252002

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# Sub-microradian angular detection limits for field mapping by Lorentz 4D scanning transmission electron microscopy on a Si p-n junction

Alexis Wartelle ✉ ; Bruno C. da Silva; David Cooper ✉ ; Martien I. den Hertog ✉

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J. Appl. Phys. 138, 105703 (2025)

https://doi.org/10.1063/5.0273467

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# Sub-microradian angular detection limits for field mapping by Lorentz 4D scanning transmission electron microscopy on a Si p-n junction

Cite as: J. Appl. Phys. **138**, 105703 (2025); doi: 10.1063/5.0273467

Submitted: 31 March 2025 · Accepted: 12 August 2025 ·

Published Online: 8 September 2025

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Alexis Wartelle,<fnref n="1" />,ª) ID Bruno C. da Silva,<fnref n="1" /> David Cooper,<fnref n="2" /> ID and Martien I. den Hertog<fnref n="1" />,ª) ID

## AFFILIATIONS

## ABSTRACT

We have examined a focused ion beam prepared silicon p-n junction lamella with different thicknesses by Lorentz 4D scanning transmission electron microscopy using center-of-mass (CoM) analysis. Using convergence semi-angles of 992, 207, and 83 µrad, we demonstrate diffraction limited spatial resolution combined with deflection detection limits on the sample as good as 0.49 µrad. We assess electron beam deflection detection limits away from the p-n junction, as well as the variation along the junction, and compare these with the variation in a vacuum region. With the larger semi-convergence angle of 992 µrad, a rigid shift of the transmitted disk was observed in the junction, while using smaller angles, a redistribution of intensity in the transmitted beam occurs, leading to an underestimation of the electric field related to the larger electron probe size. It is shown that electric field detection limits as good as 0.033, 0.0072, and 0.0051 MV cm⁻<fnref n="1" /> could be obtained for decreasing semi-convergence angles (at 376 nm sample thickness), equivalent to angular detection limits of 3.2, 0.68, and 0.49 µrad, respectively. For an optimal detection limit, we show that about 50 counts per pixel within the diffracted disk are needed for all settings, allowing rapid acquisition of diffraction maps, mitigating sample drift and beam damage issues. Moreover, we demonstrate that the CoM shift increases linearly with electrically active sample thickness, as must be the case for a quantitative signal.

© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0273467

## INTRODUCTION

The ongoing trend of miniaturization of devices demands tools capable of not only looking at the chemistry and crystal structure at nanometer and atomic length scales, but also at the electric fields present in the material. As most devices obtain their functionality from the internal electric field distribution, we need to be able to quantitatively measure these fields and their spatial distribution at small length scales. Transmission electron microscopy (TEM) combines high spatial resolution with the advantage that as the electron is a charged particle it is inherently sensitive to electric fields. One of the proposed techniques for field mapping<fnref n="1" />⁻⁶ is 4D Scanning TEM (4D-STEM), where a convergent electron beam is scanned over the sample point by point, and a diffraction pattern is recorded at each scan position. The diffraction pattern can be recorded on a segmented detector,⁴,⁷ typically referred to as

differential phase contrast (DPC). The diffraction pattern can be better sampled using a pixelated detector. Typical examples in the literature have 128 or more pixels. In this work, a 1R medipix detector was used with 256 × 256 pixels, mounted on a JEOL NEOARM TEM. With respect to other TEM-based field mapping techniques, the advantage of 4D-STEM is the in principle unlimited field of view, and the absence of the need for a vacuum reference region, that is required for off-axis electron holography.⁸ However, this comes at the cost of large data sets.

We use the center-of-mass (CoM) approach for data analysis of the 4D-STEM diffraction maps, where the center-of-mass is individually calculated on each diffraction pattern.⁹⁻¹⁴ As explained in Ref. 15, and other literature, the pixel (px) shift of the electron beam measured by CoM can be related to the electric field in the plane of the sample, perpendicular to the electron beam path. In the presence of a uniform electric field along a thickness *t*, an

<footnote n="1">¹Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, Grenoble 38000, France</footnote>

<footnote n="2">²Univ. Grenoble Alpes, CEA-LETI, Grenoble F-38000, France</footnote>

<footnote>ª) Authors to whom correspondence should be addressed: alexis.wartelle@neel.cnrs.fr and martien.den-hertog@neel.cnrs.fr</footnote>
incoming electron beam is deviated from its initial direction by an angle $\gamma$ due to the Coulomb force. The electron beam deflection can be related to the projected electric field along the sample thickness $t$ by$^{6,16}$

$$E_{\perp}^{\text{proj}} = -\frac{p_0^2}{m_e^* q t} \gamma, \quad (1)$$

where $m_e^*$ is the relativistic mass of the electron, $q$ the elementary charge, and $p_0$ the initial electron momentum. If the electron probe is of similar size or larger than the electrical object, the measured electric field is a convolution of the probe size with the real electric field profile. Conveniently, CoM analysis has sub-pixel accuracy and is to a large degree insensitive to the reciprocal pixel size.$^{15,17}$ More details can be found in SI 1 in the supplementary material.

For successful semiconductor device design, tools are required that can reliably measure the internal fields. As an illustration, for a device of thickness 30 nm (for example, the fin width of a fin field-effect transistor, a.k.a.), an electric field of $E_{max} = 0.12$ MV cm$^{-1}$ and 200 keV electrons, a beam deflection of around 1 µrad is expected.$^{15}$ Thus, information about the sensitivity relative to the spatial resolution is required for accurate device characterization. In the literature, a theoretical analysis for DPC estimated that a detection limit better than 1 µrad is possible$^{18}$ and experimentally 3 sigma angular detection limits ranging from 0.92, 1.2,$^{19}$ or 1 µrad$^{15}$ can be found, which account for precision limits due to the experimental setup,$^{19}$ or that are related to the sample.$^{15,19}$

The aim of this paper is to assess detection limits on a TEM (JEM-ARM200F “NEOARM” by JEOL) equipped with a cold field emission gun, optimized STEM Lorentz alignments and direct electron detector as a function of semi-convergence angle and electron dose, and compare it with detection limits from the literature.$^{15,19}$ Let us first emphasize the distinction between “detection limit,” which we consider as the smallest signal value clearly above the noise, and the related terms “precision,” “sensitivity,” and “accuracy.” In some papers,$^{6,16,20}$ the standard deviation of the field values in a field-free reference region, i.e., 1 sigma considering a Gaussian distribution of the measurement values, defines the error bars on an electric field measurement. In these Refs. 6, 16, and 20, this quantity is referred to as the sensitivity of the measurement. We note, however, that some ambiguity remains concerning the term “sensitivity,” as it can also refer to the ratio between the extracted signal and the physical effect of interest.$^{3,4,11,20}$

Precision and accuracy also typically refer to the error bars on a measurement that may be defined with different methods (standard deviation, 3 sigma, etc.). Assuming a Gaussian distribution, a choice is possible to use either the 1 sigma variation (that will describe about 68% of measurements) or 3 sigma variation (describing over 99% of measurements) as precision or error on a measurement. In several publications,$^{3,15,18}$ the three sigma interval was used as the error or the detection limit. Indeed, if the detection limit is defined as the three sigma variation, a signal larger than this value will stand out from the background with a good confidence level. On the other hand, a detection limit defined as one sigma may have an overlap with the measurement values (for about 34% of measurements). In this contribution, we have

evaluated the detection limit as the 3 sigma variation in a field-free region of the sample similar to Refs. 3, 15, and 18. We used a fixed evaluation area (here $60 \times 70$ nm$^2$). Comparing our detection limits with measurement errors from the literature, the amount of lateral averaging needs to be taken into account.

To put these detection limits into more perspective, we also show the variation of the beam deflection signal along the p-n junction as well as on sample regions of different thicknesses and compare it with the variation scanning the same region in vacuum. Moreover, assessing the beam deflection as a function of the sample thickness is instructive to assess the quantitative nature of the CoM beam shift. Quantification of the electric field in the junction was presented elsewhere$^{21}$ combining *in situ* biasing and finite element modeling and is beyond the scope of this paper.

## METHODS

The specimen we studied was a focused ion beam (FIB) prepared silicon p-n junction, made from a well-studied wafer.$^{15,16,20-22}$ It comprised a $1 \times 10^{19}$ cm$^{-3}$ phosphorus doped n-type layer grown on a $1 \times 10^{19}$ cm$^{-3}$ boron doped p-type layer.$^{23}$ This FIB specimen was investigated in the work by Cooper *et al.*$^{20}$ The sample contains six ion beam polished regions with different thicknesses, ranging from t1 (104 ± 5 nm) to t6 (459 ± 5 nm). Data were acquired in the Lorentz mode$^{24}$ using three different convergence angles, 83, 207, and 992 µrad. The Lorentz mode refers to the use of a Lorentz lens as image or probe forming lens with the objective lens turned off. The advantage of this mode is that almost parallel probes and long camera lengths are available, in (near) magnetic field-free conditions at the sample. An acceleration voltage of 200 kV (incidentally 80 kV, see Figs. S8 and S9 in the supplementary material) was used. The transmitted beam impinges on a direct electron detector with $256 \times 256$ pixels,$^{25,26}$ the MerlinEM camera by Quantum Detectors. See SI 2 in the supplementary material for more details. All experimental settings are summarized in Table 1. The most important experimental setting when measuring the electric fields by 4D-STEM is to reduce the dynamical diffraction present in the transmitted beam. The effect of the dynamical diffraction leads to very large inaccuracies in the measurements of the beam shift. In Si, this step is rather straightforward for the low convergence angles used and was achieved by tilting the specimen by 4° around the (001) growth direction and 0.5° around the (110) in plane direction.

## RESULTS AND ANALYSIS

Figure 1(a) shows a schematic of the three different modes we used, with the beam incident on the modulus of the electric field maps, as well as examples of experimental diffraction patterns. For 207 and 83 µrad convergence angles, a redistribution of intensity was observed in the transmitted beam as it passed from a region not containing a field and into the p-n junction (SI 3 in the supplementary material). For the 992 µrad convergence angle, a rigid shift was observed (SI 4 in the supplementary material). These observations are consistent with previous work$^{11,15}$ where a redistribution of intensity occurs when the electron probe is larger than the changes in the electric field and as such different parts of the probe interact with different field strengths. In the case of a
<table><caption>**TABLE I.** Comparison of performed 4D-STEM experiments realized in a Si p-n junction carried out at 200 kV, corresponding to the data shown in Fig. 1 at t5 (376 ± 5 nm). The signal-to-noise ratio is calculated by dividing the total signal by the noise, $3\sigma_E$. The standard deviation was calculated from a region with no field (area of 70 × 60 nm²). For comparison, note that the precision in Ref. 20 is given only as the standard deviation $\sigma$.</caption><thead><tr><th>Convergence semi-angle (μrad)</th><th>83</th><th>207</th><th>992</th></tr></thead><tbody><tr><td>Settings</td><td></td><td></td><td></td></tr><tr><td>Dwell time (ms)</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Beam current (pA)</td><td>214</td><td>145</td><td>392</td></tr><tr><td>Camera length (m)</td><td>40</td><td>15</td><td>2.5</td></tr><tr><td>Pixel size real space (nm)</td><td>5.4</td><td>5.4</td><td>5.4</td></tr><tr><td>Pixel size reciprocal space (μrad)</td><td>0.97</td><td>2.26</td><td>12.88</td></tr><tr><td>N<sub>per pixel</sub><sup>CBED</sup> (counts)</td><td>34.2</td><td>49.4</td><td>264.7</td></tr><tr><td>Diffraction limited beam diameter (nm)</td><td>37</td><td>15</td><td>3</td></tr><tr><td>Measurements—beam deflection</td><td></td><td></td><td></td></tr><tr><td>Beam deflection δ<sub>max</sub><sup>measured</sup> (pixels)</td><td>11.15</td><td>5.09</td><td>1.02</td></tr><tr><td>Noise level 3σ<sub>δ</sub> (pixels)</td><td>0.50</td><td>0.3</td><td>0.25</td></tr><tr><td>Noise level 3σ<sub>δ</sub> in vacuum (pixels)</td><td>0.24</td><td>0.12</td><td>0.05</td></tr><tr><td>Measurements—electric field</td><td></td><td></td><td></td></tr><tr><td>E<sub>max</sub><sup>measured</sup> (MVcm<sup>-1</sup>) modulus</td><td>0.115</td><td>0.140</td><td>0.150</td></tr><tr><td>Noise level 3σ<sub>E</sub> (MVcm<sup>-1</sup>)</td><td>0.0051</td><td>0.0072</td><td>0.033</td></tr><tr><td>Signal-to-noise ratio (SNR) E<sub>max</sub><sup>measured</sup>/(3σ<sub>E</sub>)</td><td>22.4</td><td>19.5</td><td>4.5</td></tr><tr><td>Noise level 3σ<sub>E</sub> (μrad)</td><td>0.49</td><td>0.68</td><td>3.2</td></tr><tr><td>Noise level 3σ in vacuum (μrad)</td><td>0.23</td><td>0.27</td><td>0.64</td></tr></tbody></table>

rigid shift, we expect to be able to obtain quantitative results directly from Eq. (1) and S3 in the **supplementary material**. The inset in Fig. 1(a) shows a high angle annular dark field STEM image of the sample under study indicating the regions with different thicknesses.

We first consider the modulus of the electric field maps obtained with an acquisition time of 10 ms for each acquired diffraction pattern and compare profiles with a width of 11 and 92 nm,¹⁵ respectively, acquired on region t5 (376 ± 5 nm).²⁰ The p-n junction can be seen as the vertical line in the STEM images (we scan perpendicular to the junction). Looking at the field maps, Figs. 1(b)-1(d), we notice three things: (i) the junction seems to become narrower at higher convergence angles, (ii) the maximum electric field at the junction increases at higher convergence angles, and (iii) the variation of the field, mainly visible outside of the junction also increases at these higher angles. These observations are qualitatively similar to the ones reported elsewhere.¹⁵ To obtain quantitative information, electric field profiles with widths from 11 to 92 nm were extracted from the maps, Figs. 1(e) and 1(f). As expected, the precision of the measurements is improved for larger averaging. For decreasing convergence angle, we can appreciate lower noise, both in the maps as well as in the profiles.

To better assess the precision in the maps, the standard deviation $\sigma_E$ was extracted from a region containing no field of an area of 70 × 60 nm².¹⁵ The signal-to-noise ratio (SNR) is calculated by dividing the total signal by the noise, defined as $3\sigma$. The results of these analyses on the different data sets are summarized in Table I.

The measured maximum field is 0.15 and 0.14 MV cm⁻¹ for convergence angles of 992 and 207 μrad, respectively. The maximum field decreases to 0.115 MV cm⁻¹ at 83 μrad convergence angle. This decrease is expected as the probe size is larger for low convergence angles and the theoretical diffraction limited beam sizes are 3, 15, and 37 nm according to the Rayleigh criterium²⁷ for the convergence angles of 992, 207, and 83 μrad, respectively. The observed decrease in the measured field strength is well explained considering a convolution with the expected probe sizes¹⁵ (see Eq. S2 in the **supplementary material**).

From the field maps presented in Figs. 1(b)-1(d), we calculate field detection limits of 0.033, 0.0072, and 0.0051 MV cm⁻¹, angular detection limits of 3.2, 0.68, and 0.49 μrad, and signal-to-noise ratios of 4.5, 19.5, and 22.4 for decreasing convergence angles. The conversions of detection limits to electric field or angular values proceed from the acceleration voltage and the sample thickness, prior semi-convergence angle calibration defining the angular pixel size, and Eq. (1). While we clearly see the familiar trade-off between spatial resolution and momentum transfer (precision) due to the uncertainty principle,²⁸ we show that improved alignments and stability demonstrate detection limits beyond the state of the art,¹⁵,¹⁹ at or very close to the diffraction limited spatial resolution, and present a clear improvement with respect to the results presented previously.¹⁵,¹⁹

Since the detection limits presented above are sample-dependent, we also evaluate them in the same way but on 4D-STEM maps acquired in vacuum (these are shown in Figs. SI 5-SI 7 in the **supplementary material**). We point out that no descan correction was applied, and thus these results are upper bounds for the detection limits. If the instrument was perfect, no CoM shift should be measured and all $3\sigma$ values should be exactly zero. As can be seen in Table I, this is not the case. The $3\sigma$ variation of beam deflection in vacuum is about two to three times smaller than the deflection on the sample reference region. The ratio $\sigma_{vacuum}/\sigma_{sample}$ decreases when the semi-convergence angle increases with values going from 47% through 40% to 20% for 83, 207, and 992 μrad, respectively. It must be noted that this happens despite the increase in $\sigma_{vacuum}$. Our hypothesis is that the sample-related contributions to the detection limit are linked to the decreasing beam size. A certain roughness scale on the sample will only contribute to the uncertainty if the beam is small enough to “see” this variation. While isolating the different sample-related contributions is beyond the scope of our work, we have gathered in SI 8 in the **supplementary material** detection limit measurements as a function of sample thickness. Our findings are compatible with an electric field detection limit decreasing with thickness [consistent with the 1/t prefactor in Eq. (1)], and an angular detection limit varying less with the sample thickness, although more measurements would be needed to conclude.

When looking at the field maps in Figs. 1(b)-1(d), we can see that small variations are visible along the p-n junction. They are most clearly observed in the map acquired in Fig. 1(d) with the largest convergence angle. To assess the inhomogeneities in Fig. 1(d) in more detail, Fig. 2 shows the electric field as well as beam deflection profiles in pixel acquired along the junction, compared to an equivalent profile along a reference line next to the junction. We can compare this field/beam deflection variation with
<figure><img src="image_13.png" /><figcaption>FIG. 1. 4D-STEM maps at different convergence angles: (a) schematics of the three Lorentz STEM modes with different convergence angles that were used. An HAADF STEM image of the FIB lamella is shown in the inset with the dotted line indicating the approximate p–n junction location. (b)–(d) Modulus electric field maps acquired at region t5 (376 ± 5 nm) of the Si p–n junction specimen with 10 ms pixel dwell time using these three modes, increasing the semi-convergence angle going from left to right. (e) and (f) Profiles made in the maps shown in (b)–(d) as indicated, averaged along 11 and 92 nm, respectively.</figcaption></figure>

the beam deflection measured in a similar sized vacuum region at
the same experimental conditions, (Fig. SI 5 in the [supplementary
material], see also Figs. SI 6 and SI 7 in the [supplementary material
for equivalent maps at semi-convergence angles of 207 and

83 µrad), where about four times smaller beam deflections are
observed. This comparison further demonstrates that the variations
along the beam deflection profile come at least partly from the
sample itself or from electrostatic/dynamic charge effects, as
<figure><img src="image_14.png" /><figcaption>FIG. 2. (a) Electric field map with 992 µrad convergence angle (alpha 3). (b) Profiles made at the indicated locations in (a). Red dotted profile: horizontal pixel deflection/electric field profile along a reference line away from the junction. Blue solid profile: horizontal px deflection/electric field profile along the junction. The yellow bands indicate the 3σ variation for this data set calculated from a region with no field (area of 70 × 60 nm²).</figcaption></figure>

indicated by the above-mentioned evaluation of detection limits in vacuum. The yellow bands in Fig. 2 indicate the 3σ variation for this data set. The field/beam deflection variation values are within the 3σ<sub>E</sub> band in the reference region (red dotted profile) but are larger along the junction (blue profile). One may first consider sample thickness variations: while the error on the measured thickness is estimated around ±5 nm,<sup>20</sup> such thickness fluctuations would be clearly visible by other imaging techniques such as High-Angle Annular Dark Field (HAADF) STEM or holography, and can, therefore, be ruled out.<sup>20</sup> Another possible source of increased noise can be small beam positioning errors considering the pyramidal shape of the field profile. Yet, a position error of about 1 nm would only lead to an additional error of about 10% of the 3σ<sub>E</sub> value; thus, unreasonably large positioning errors would be required to account for the larger noise level on the junction. We speculate that an increased disorder at the junction after specimen preparation could alter the beam propagation, or that the surface charging at the p-n junction is different with respect to the homogeneously doped regions, leading to larger fluctuations in beam deflection.

In terms of the dose vs detection limit, it has been observed
that increasing the dose, the detection limit of the experiment
(defined as 3σ<sub>E</sub>) first rapidly improves, until reaching a plateau
where more dose will not increase the detection limit further. We
attribute this plateau to combined effects of some remaining influ-
ence of diffraction contrast, sample thickness variations, dynamic
charging/discharging effects, and the combined effect of micro-
scope instabilities (current ripple, high tension jitter, sample drift,
etc.). In Fig. 3, we show this trade-off, comparing our new data
with literature values.<sup>15</sup> We can easily see that the same trends
apply for all three considered convergence angles; however, the
detection limits are about two to five times better than in the litera-
ture at similar settings. We demonstrate an improved spatial resolu-
tion: only 7% underestimation of the electric field with respect to
30% underestimation<sup>15</sup> at similar convergence angle and camera

lengths. We attribute our improved spatial resolution and approxi-
mately diffraction limited probe size to an improved alignment,
this may be the result from a simpler column in our case (probe
corrector only), without a monochromator and image corrector,¹⁵
or could indicate a better quality Lorentz lens in our setup. On the
other hand, this will not explain the reduced noise levels in the

<figure><img src="image_15.png" /><figcaption>FIG. 3. Dependence of the noise level of the measurement of the electric field, 3σE, on the number of average counts per pixel within the transmitted disk and the measurement conditions at t5 (376 ± 5 nm). The standard deviation σE was calculated from a region with no field (area of 70 × 60 nm²). Round symbols connected by guide to the eye lines indicate the results from Ref. 15, star symbols indicate the new results obtained on the JEOL NEOARM. See SI 3 in the supplementary material for images of the transmitted beam acquired at different px times.</figcaption></figure>

12 September 2025 12:01:26
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>FIG. 4. (a) Pixel beam shift measured by CoM as a function of the sample thickness for different experimental conditions. (b) Measured electric field with 3σ error bars for different experimental conditions at 80 kV (disks) and 200 kV (crosses), for a common dwell time of 10 ms. The vertical dashed lines indicate the nominal thicknesses t1–t5. Our measurements on the t6 region have not been included in the above analysis due to their systemic outlier behavior; however, they are presented in SI9 in the supplementary material.</figcaption></figure>

sample reference region since they are not expected to be influ-
enced by probe size. In fact, a smaller 3σ variation is expected for a
larger probe size since sample thickness inhomogeneities will be
more averaged by the larger probe. Also sample drift is not

expected to influence the 3σ variation in a reference region. On the
other hand, the 3σ variation will be expected to be influenced by
the electrical stability (jitter) in any of the optical components.
Therefore, the more likely hypothesis is that the simpler column of
the JEOL NEOARM TEM (probe corrector) with respect to the TEM used in prior work (monochromator, probe corrector, and image corrector),¹⁵ potentially combined with improved electrical stability, explains the improved detection limits we show.

In terms of quantification, it is important to validate if the CoM beam deflection increases linearly with the increasing sample thickness. Previous work that used both a four segment detector as well as a CMOS pixelated detector on the exact same sample we use in this work showed that when using the normalized DPC or CoM signal no linearity was observed with increasing thickness.²⁰ In that work, a convergence angle between 270 and 410 µrad was used in low magnification conditions. A linear relation was observed using a fast pixelated PnCCD detector working at a 20 µrad semi-convergence angle.¹⁴

Figure 4(a) shows the measured CoM beam shift in pixel as a function of the sample thickness, including 3σ error bars. We present fits for different acquisition conditions. It can be seen that no beam deflection will be measured for sample thicknesses below about 25–100 nm. For increasing thickness beyond these values, the data show that the beam deflection increases linearly with the sample thickness. The absence of beam deflection for small thicknesses is due to the so-called dead layer,¹ a crystalline but electrically inactive surface layer, not contributing toward the measurement of the field across the sample. This can arise from both mechanical specimen damage and surface charging. In this case, the inactive surface layer has been measured to be 47 nm in this sample, by measuring the step in phase at each region by off-axis electron holography, and plotting it as a function of the crystal thickness calculated from convergent beam electron diffraction (CBED) measurements.²⁰ Our measurements by CoM 4D-STEM indicate a dead layer of about 35 ± 12 nm at 83 µrad semi-convergence angle, taking into account the 3σ error bars: the green band in Fig. 4(a), in agreement with the previous study.²⁰ These data show that the CoM signal measured here is potentially quantitative if recorded on a pixelated direct electron detector, in the sense that the beam deflection increases linearly with the electrically active sample thickness, whereas such a behavior has not been derived from measurements on the same wafer with a CMOS camera or a 4 quadrant DPC detector.²⁰

The data sets made using 83 µrad semi-convergence angle at 200 kV are represented by green crosses and fit (dotted green line). Data measured using the largest convergence semi-angle, 992 µrad alpha3@ 200 kV [red crosses in Fig. 4(a)] demonstrate small beam deflections and relatively large error bars. Figure 4(b) shows the calculated electric field values obtained with a 10 ms dwell time at the different experimental conditions from Fig. 4(a) (crosses) as well as results from 203 µrad, 80 kV data sets (blue disks), all together with their 3σ error bars. Considering all of our data with decreasing convergence angle, increasingly underestimating the electric field with increasing beam size, we find field values between 0.16 and 0.105 MV cm⁻¹ which is in accordance with other observations.¹⁵,²⁰,²¹ We have performed complementary measurements at 203 µrad semi-convergence angle and 80 kV, which illustrate the change in field sensitivity with acceleration voltage for a constant convergence angle, presented in SI 9(a) in the supplementary material.

Figures 4(a) and 4(b) also demonstrate that with the larger convergence angle, it becomes increasingly difficult to measure

beam deflections well below one pixel on the thinnest regions with good precision even with a pixelated direct electron detector. To illustrate this problem better, we present a more detailed analysis of a beam deflection map and profile on a thin region (t2: 150 ± 5 nm) in SI 10 in the supplementary material, showing that even with these excellent detection limits, extracting quantitative information remains challenging, especially if the field containing region is small and the sample is thin.

CONCLUSION

Therefore, we believe to have shown detection limits beyond literature reports¹⁵,¹⁹ combined with diffraction limited spatial resolution. The CoM signal may be quantitative, increasing with the sample thickness. At the highest convergence angle (992 µrad), the diffraction limited beam diameter is about 3 nm, which is needed to provide the required spatial resolution for devices employing increasingly small features. For example, 30 or 12 nm are reported as the width of the field containing region in a FinFET,¹⁵,²⁹ where a deflection detection limit below 1 µrad is needed.¹⁵ In this work, we have reached this threshold on our p-n junction sample for two out of the three used semi-convergence angles, albeit unfortunately not for 992 µrad. Nevertheless, our detection limit measurements in vacuum indicate that for all three settings, values below 1 µrad can be reached. Therefore, the FinFET-related requirement may be met if the sample contributions to the detection limit can be reduced. On the other hand, the error on the field value can still be significant, especially if thinner samples are considered.

Moreover, many samples can be more challenging for 4D-STEM CoM analysis than such a perfect Si p-n junction grown by epitaxy. For example, in nanowires or other nano materials, often small rotations of the crystal lattice are present, that will also generate a shift in the CoM signal that is not related to any electrical property change within the sample, typically referred to as diffraction contrast. Analyzing fields at hetero-interfaces is very challenging as the material change and its gradient also affect the beam deflection and strain can give rise to diffraction contrast.¹² In such examples, diffraction contrast results in Kikuchi deficiency or excess (dark or bright) lines in the diffraction pattern.²⁷ These features have no relation with the electric properties, but they will move or broaden when the crystal orientation or strain changes while scanning the electron beam along the sample. This movement of dark and bright features will strongly influence the CoM result, complicating even qualitative analysis. We expect that the main improvements of the 4D-STEM technique for field mapping for most samples, where diffraction contrast will be the major problem, will be in how to mitigate this problem best. Indeed, studying hetero-interfaces encouraging results were obtained using beam precession and energy filtering.³⁰

More experimental development or an additional degree of freedom to influence the electrical properties *in situ* is still needed to obtain more precise quantitative values. For example, *in situ* biasing will allow only looking at the part of the CoM signal that changes with bias, thereby excluding diffraction contrast.⁶,¹⁶,²¹,³¹ However, the cleanroom or FIB technology can be challenging to develop. Another option to minimize diffraction contrast is
precession.¹⁶ We, therefore, aim to combine the demonstrated Lorentz STEM modes with precession in the future.

In summary, an FIB prepared silicon p-n junction lamella has been examined by Lorentz 4D-STEM on a JEOL NEOARM equipped with a medipix detector from Quantum Detectors using CoM analysis and different convergence angles ranging from 83 to 992 µrad and px times ranging from 1 to 50 ms. With 992 µrad semi-convergence angle, a hard shift of the transmitted disk was observed, while a redistribution of intensity in the CBED pattern occurs at smaller convergence angles. With 83 and 207 µrad convergence angle, an underestimation of the electric field maximum compared to 992 µrad was observed, related to the diffraction limited larger electron probe size. However, an improved SNR is achieved using the smaller semi-convergence angles (combined with longer camera lengths). Field detection limits as good as 0.033, 0.0072, and 0.0051 MV cm⁻<fnref n="1" />, angular detection limits of 3.2, 0.68, and 0.49 µrad, and signal-to-noise ratios of 4.5, 19.5, and 22.4 are observed for decreasing convergence angle at the 376 nm sample thickness. In vacuum, angular detection limits of 0.64, 0.27, and 0.23 µrad are observed for decreasing convergence angle.

For an optimal detection limit, we show that about 50 counts per pixel within the diffracted disk are needed at any setting, allowing fast diffraction maps and mitigating sample drift and beam damage issues. Finally, we demonstrate that the CoM shift may be a quantitative signal increasing linearly with the electrically active sample thickness. On the other hand, at the largest convergence angle we assessed, error bars are often significant in thinner sample regions. We demonstrate that the larger error bars in thinner regions seem to be mainly related to the sample.

We demonstrate excellent field detection limits and SNR on this equipment with improved stability, perfectly suited to characterize weak, built-in electric fields (with angular deflections below 1 µrad) in junctions in today and tomorrow's semiconductor devices while maintaining nm range spatial resolution.

## SUPPLEMENTARY MATERIAL

See the supplementary material for literature background for CoM analysis (SI.1); methods (SI.2); experimental CBED patterns with and without field (SI.3); profiles along CBED patterns at 992 µrad convergence angle (alpha 3) (SI.4); variation of beam deflection in the vacuum at 992 µrad convergence angle (alpha 3) (SI.5); variation of beam deflection in the vacuum at 207 µrad convergence angle (alpha 2) (SI.6); variation of beam deflection in the vacuum at 83 µrad convergence angle (alpha 1) (SI.7); detection limits in terms of electric field and angle, as determined at different sample thicknesses (SI.8); analogue of Fig. 4 from the main text, including our measurements on the t6 sample region as well as measurements at 203 µrad semi-convergence angle and 80 kV made on t1-t3 in Fig. SI9(a) (SI.9); and beam deflection map and profiles at t2 (150 ± 5 nm, around 100 nm electrically active thickness) (SI.10).

## ACKNOWLEDGMENTS

This project received funding from the European Research Council under the European Union's H2020 Research and Innovation programme via the e-See project (Grant No. 758385) as

well as from the French National Research Agency (ANR) via the PEPR "DIADEME" project METSA (No. ANR-22-PEXD-0013). A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the author accepted manuscript arising from this submission, in accordance with the grant's open access conditions. The TEM facility JEOL NEOARM was co-financed by the European Union under the European Regional Development Fund (ERDF) (Contract No. RA0023813).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Alexis Wartelle:** Formal analysis (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Bruno C. da Silva:** Conceptualization (equal); Investigation (equal); Methodology (equal). **David Cooper:** Methodology (equal); Writing – review & editing (equal).

**Martien I. den Hertog:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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