The continuing development of aberration correctors for the scanning transmission electron microscope (STEM) offers the possibility of locating single atoms in crystals in 3D via optical depth sectioning. The main factors that determine the feasibility of such an approach are visibility and dose requirements. Here, we show how Poisson's statistics can be quantitatively incorporated into STEM image simulations and demonstrate that the 3D location of single cerium atoms in wurtzite-type aluminum nitride is indeed feasible under large-angle illumination conditions with a relatively low dose. We also show that chromatic aberration does not presently represent a limitation provided a cold field emission source is used. These results suggest efforts into improved aberration corrector designs for larger illumination angles that offer significant potential for 3D structure determination of materials.

Aberration correction in electron optics has dramatically improved the spatial resolution of electron microscopes and observation with sub-Ångstrom resolution is now routinely available. Such electron microscopes have become indispensable tools in condensed matter physics, materials science, and engineering fields.1 The current record for spatial resolution in the scanning transmission electron microscope (STEM) is 45 pm.2 Since this is significantly finer than typical projected interatomic distances, it has been argued that the resolution of current microscopes suffices for the most physical or chemical problems and that therefore there is little need to seek to further improve spatial resolution.3 This argument, however, applies only in the lateral two dimensions: the attainable resolution along the remaining dimension, depth, is still limited to several nanometers, significantly larger than typical interatomic spacing. Because material properties are governed by 3D atom positions and chemical bonds, and for the full understanding and control of materials properties, boosting depth-resolution to atomic dimensions is essential for the development of 3D imaging and spectroscopy.4–6 Approaches to high-resolution 3D imaging include electron tomography,7 hollow-cone illumination STEM8 and scanning confocal electron microscopy.9,10 Optical depth sectioning with large-angle illumination STEM11 is a very promising approach for both imaging and spectroscopy.12–14 The critical obstacle for large-angle illumination STEM is the ability to correct higher-order geometric aberrations, which has recently been demonstrated up to including sixfold-astigmatism by Sawada et al., who were able to increase the illumination angle (α) up to 70 mrad at 60 kV.15,16 This result is greatly encouraging for large-angle illumination STEM optical sectioning, and we can significantly improve the depth resolution if we can develop the aberration corrector to the same angles at higher acceleration voltages.

However, to perform atomic-scale optical depth sectioning via large-angle illumination STEM, some other difficulties remain. One is an electron dose problem: there is a trade-off between atom visibility and specimen damage due to beam irradiation. In large-angle illumination, the electron probe is very finely focused within a narrow 3D space—ultimately a single atom size—and multiple scattering along the atomic column (so-called electron channeling) is significantly suppressed. Therefore, with increasing illumination angle, a higher electron dose is required to maintain a good signal-to-noise ratio (SNR). However, no specimen can survive for long under a finely focused probe with high electron dose and high accelerating voltage. The other central problem is that, because of chromatic aberration, the energy spread of the electrons causes a blurring of the probe along the depth direction, which becomes more severe for larger illumination angles. Recent remarkable progress in monochromator design has demonstrated extremely good energy resolution of 10–30 meV at 60–200 kV (Refs. 17–19) and may ultimately assist large-angle illumination STEM. Unfortunately, owing to the use of energy-selecting slits in such monochromators, the coherent current is reduced, meaning that simultaneously maintaining monochromation, spatial resolution and probe current becomes increasingly difficult. Thus, the key question remains unanswered: Will noise related to the low electron dose (or shot noise) and/or chromatic aberration prevent single atom location in large-angle illumination STEM under realistic dose and energy spread limitations? Here, we demonstrate the direct determination of single dopant depth location in a bulk material with atomic precision via large-angle illumination STEM optical sectioning through image simulations including appropriate shot noise and chromatic aberration. Our result indicates that it should be possible to identify the depth location of a single dopant with tolerable electron dose and a current typical cold field emission source.

In experiments, the number of detected electrons deviates from that predicted by a noise-free simulation, especially under low-dose conditions. Such fluctuations can be modeled by Poisson's statistics,20,21 where the probability of detecting k electrons is determined by the number of expected electrons ν as: Pr(k,ν)=νkeν/k!. Each probe position (or pixel) is an independent event, and we simulate the intensity fluctuation by the following process: (1) multiply the number of incident electrons per pixel (Dp: pixel dose) by the simulated fractional STEM image (normal to the incident beam) and (2) read out each pixel value as the expected value ν and create a noisy image using Poisson's random values generated by a Monte Carlo method. Figures 1(a)–1(f) show the frame-averaged experimental images of wurtzite aluminum nitride (w-AlN) along the [112¯0] direction,22,23 where 30 frame images were sequentially acquired at the same area. These experimental images are fully quantified by a single electron counting method: the intensity is normalized by the incident probe (0%–2%)24 and consequently, by comparison with simulation, it can be concluded that the sample is 23 ± 1 atoms or 7.2 ± 0.3 nm thick. As shown on the top-right section of Figs. 1(a)–1(f), the noise level of the simulated images qualitatively appears to reproduce the experiment fairly well. To more quantitatively compare to the experiment, we introduce the two parameters of “column-contrast” (CcolA) and “column-SNR” (ScolA) as an extension of conventional measures25 

CcolA=Icol,iAIbk,jAIcol,iA+Ibk,jA,ScolA=Icol,iAIbk,jAσbk,
(1)

where Icol,iA is the mean value within the area A at a particular i-column (A is the white circle in (f) with the radius of 2σg), where σg is the full-width-at-half-maximum of the fitted 2D Gaussian, Ibk,jA is the mean value within the background area A′ (where we use the location between atomic columns as shown in (f)), σbk is the standard deviation of the background intensity within the area A′. Figs. 1(g) and 1(h) show the experimental and shot-noise-simulated CAl2σg and SAl2σg of Al atomic columns as a function of pixel dose, which gives quantitatively excellent agreement. We note that column-contrast is a robust parameter and that it is possible to measure the values of CAl2σg even at low-dose conditions, although the image is very noisy. An atomic column intensity typically has approximately a 2D Gaussian distribution, and therefore we can theoretically calculate the column-contrast and column-SNR based on the Gaussian approximation (see supplementary material): the calculated results are shown in Figs. 1(g) and 1(h) as solid lines, showing excellent quantitative agreement with experiments. This agreement allows us to quantitatively simulate shot noise and moreover it can be possible to derive a minimum electron dose to observe an object of Ccolnσg at a prerequisite Scolnσg. Using the column-contrast and column-SNR based on the Gaussian approximation, the minimum electron dose is given by

Dp(Scolnσg)2×(1Ccolnσg)24IbkA.
(2)

We note that if we know the target structure, it is possible to estimate Ccolnσg and Ibk,jA of the target object by using image simulation in advance. We therefore, using Eq. (2), can examine different electron dose conditions to determine the optimum dose for seeing the object before actually performing the experiment.

FIG. 1.

(a)–(f) Atomic-resolution experimental ADF-STEM images obtained from w-AlN along the [112¯0] direction, where the number of frames averaged are N = 1, 2, 4, 8, 16, and 30 (200 kV, 30 mrad). The white circles in (f) indicate the used area size of A and A′ in Eq. (1). The overlaid top-right images are simulated images including shot noise. (g) Scol2σg and (h) Ccol2σg measured from the experiment (blue) and simulated (red) shot noise images are plotted as a function of pixel dose (×103). The green solid lines are calculated from the Gaussian approximation model. The scale bar in (a) is 3 Å.

FIG. 1.

(a)–(f) Atomic-resolution experimental ADF-STEM images obtained from w-AlN along the [112¯0] direction, where the number of frames averaged are N = 1, 2, 4, 8, 16, and 30 (200 kV, 30 mrad). The white circles in (f) indicate the used area size of A and A′ in Eq. (1). The overlaid top-right images are simulated images including shot noise. (g) Scol2σg and (h) Ccol2σg measured from the experiment (blue) and simulated (red) shot noise images are plotted as a function of pixel dose (×103). The green solid lines are calculated from the Gaussian approximation model. The scale bar in (a) is 3 Å.

Close modal

The depth resolution, given by (2λ/α2)2+(CcΔE/E0)2, is shown in Figure 2(a) as a function of illumination angle for the selected values of the energy spread ΔE (Cc = 1 mm at 300 kV). (Note: the above expression is only valid at atomic-resolution observations, the 3D contrast transfer function26 being such that the depth resolution is significantly worse for lower spatial frequency objects.) For increasing illumination angle, the depth resolution is strongly limited by chromatic aberration, as is evident in the case of 100 mrad with ΔE ≥ 0.5 eV. However, even with a current cold field emission gun (FEG) (ΔE ∼ 0.3 eV), it is possible to achieve ∼1 nm depth resolution, which is roughly five times better than a current depth resolution. We note that the depth resolution with a current cold FEG is saturated around 60–80 mrad, and therefore it would be appropriate to select the smaller illumination angle of 60 mrad to minimize irradiation damage.

FIG. 2.

(a) Depth resolution as a function of illumination angle at different energy spreads (ΔE = 0, 0.1, 0.2, 0.3, 0.5, 1.0 eV), where Cc = 1 mm at 300 kV. (b) Experimental zero-loss profile (crosses) in a vacuum and fitted-skew-Gaussian profile (solid line) as a function electron energy-loss or defocus spread. (c) Probe intensity along the optical axis as a function of depth for illumination angles (mrad) as given in the legend. Dotted and solid lines are obtained without and with chromatic aberration. The 2D inset images are xy-plane (right, Δf = 0 nm), xz-plane of the probe with or without (left, top/bottom) chromatic aberration.

FIG. 2.

(a) Depth resolution as a function of illumination angle at different energy spreads (ΔE = 0, 0.1, 0.2, 0.3, 0.5, 1.0 eV), where Cc = 1 mm at 300 kV. (b) Experimental zero-loss profile (crosses) in a vacuum and fitted-skew-Gaussian profile (solid line) as a function electron energy-loss or defocus spread. (c) Probe intensity along the optical axis as a function of depth for illumination angles (mrad) as given in the legend. Dotted and solid lines are obtained without and with chromatic aberration. The 2D inset images are xy-plane (right, Δf = 0 nm), xz-plane of the probe with or without (left, top/bottom) chromatic aberration.

Close modal

To implement chromatic aberration into image simulation, we adopt a zero-loss EELS (electron energy-loss spectroscopy) profile as a defocus-spread-function. Figure 2(b) shows the experimental zero-loss profile in a vacuum (crosses) obtained from the JEOL ARM300CF (300 kV) installed at the University of Tokyo, where the horizontal axis of energy scale (eV) is converted into defocus (nm) via Δf=Cc(ΔE/E0). The image intensity with chromatic aberration can be approximated by the discrete convolution of the defocus spread function D(τ) (τD(τ)=1) and aberration-free defocused image intensity I(Δf)

ICc(Δf)=τD(τ)×I(Δfτ).
(3)

In the following calculation, we refer to the fitted-skew-Gaussian profile (solid line) as the defocus-spread-function. We first check the effect of chromatic aberration on the probe. Figure 2(c) shows the 2D image intensity and the intensity profile (on the optical axis) of the probe as a function of defocus. Solid and dotted lines are obtained with and without chromatic aberration, respectively, with plots shown for illumination by circular apertures of semi-angle 30, 60, and 80 mrad and annular apertures spanning 30–60 and 45–60 mrad. Evidently, chromatic aberration produces probe spreading along the depth direction, particularly at larger illumination angles. As noted above, with the cold FEG there is no significant improvement in depth resolution between 60 and 80 mrad, and therefore 60 mrad is likely to be the best practical choice. It is also noteworthy that although annular illumination is expected to reduce the channeling effect, at large illumination angles, the full circular illumination is more suitable than annular illumination for optical sectioning. The residual channeling effects present are discussed in the supplementary material.

To further examine the performance of depth sectioning by large-angle illumination STEM, we implement shot noise and chromatic aberration in image simulations. For the determination of a single dopant depth, we selected Ce-doped w-AlN as a model system, where the single Ce atom is in the form of a substitutional defect at the Al site. The used structure model and image simulation conditions are given in the supplementary material (the Ce atom is located at the 7th site from the entrance surface in a 74.6 Å (24 atom) specimen). Figures 3(a) and 3(b) show the simulated aberration-free focus-series ADF-STEM images of the defect structure assuming a probe angle of (a) 30 mrad at 200 kV and (b) 60 mrad at 300 kV. The pure Al columns and a single Ce-containing Al atomic column are shown for a focal step of 3.11 Å, corresponding to the Al-interatomic distance along the [112¯0] direction. With α = 30 mrad, the strong Z-contrast intensity related to the Ce dopant is delocalized over a wide depth range of ±7 unit cells (or ±2 nm) about the true depth location, which means that determining the dopant depth requires accurate intensity analysis combined with extensive image simulations.23,24 However, for the large illumination angles, the strong Z-contrast intensity is well localized within the narrower depth range of ±2 unit cells (or ±0.6 nm) about the true depth location, and, moreover, the Z-contrast intensity is clearly maximized at the depth of the Ce atom. Therefore, it is in principle possible to identify the dopant depth with atomic precision by simply finding the maximum Z-contrast. Figs. 3(c) and 3(d) show the focus-series ADF-STEM images (60 mrad at 300 kV) (c) without and (d) with chromatic aberration as a function of dose of 25 × 2m electrons per probe position (m = 0, …, 7, infinite). When, as is usually the case, the scan step (Δs) is much smaller than the probe width, the pixel dose can be converted into an electron dose (e2) through multiplication by Δs–2. Assuming a scan step of Δs = 0.1 Å, the electron dose used in Fig. 3 is given by 25 × 2m× 102 e2. The measured CCe2σg and the mean value within the area A (within the radius of 2σg) of the Ce-containing column with chromatic aberration are plotted in Figs. 3(e) and 3(f), as a function of defocus, with different electron doses per probe position. The pink-colored solid lines in the background are the infinite dose profiles with chromatic aberration and the cyan-colored solid lines in the bottom are the infinite dose profiles without chromatic aberration. For increasing pixel dose, the noise level of the column-contrast becomes smaller and converges onto the infinite dose profile.

FIG. 3.

Simulated focus-series (Δf = 3.11 Å) ADF STEM images of the defect structure of Fig. S1, where a single Ce dopant is located at the depth marked by the white rectangle (Δf = 21 Å). The calculated conditions are (a) 200 kV with 30 mrad and (b) 300 kV with 60 mrad, and the ADF detector angle is 70–200 mrad. The left and right atomic columns are pure Al and single Ce containing Al columns. (c), (d) Tableaux of the calculated focus-series images under pixel dose of 25 × 2m electrons (m = 0,…, 7, infinite), where we use 300 kV with 60 mrad (c) without and (d) with chromatic aberration (cold FEG). (e) The CCe2σg profiles are plotted as a function of defocus, where the plot range is 0.6–1.0. The colors of the plot correspond to the different pixel doses, and the background pink- and cyan (bottom) -colored solid lines are the infinite dose profiles with and without chromatic aberration. (f) The mean value profiles (within the Ce-containing atomic column) obtained from (d) in the same manner (0%–1.5%). The contrast ranges are: (a) 0%–5% and the others 0%–3%.

FIG. 3.

Simulated focus-series (Δf = 3.11 Å) ADF STEM images of the defect structure of Fig. S1, where a single Ce dopant is located at the depth marked by the white rectangle (Δf = 21 Å). The calculated conditions are (a) 200 kV with 30 mrad and (b) 300 kV with 60 mrad, and the ADF detector angle is 70–200 mrad. The left and right atomic columns are pure Al and single Ce containing Al columns. (c), (d) Tableaux of the calculated focus-series images under pixel dose of 25 × 2m electrons (m = 0,…, 7, infinite), where we use 300 kV with 60 mrad (c) without and (d) with chromatic aberration (cold FEG). (e) The CCe2σg profiles are plotted as a function of defocus, where the plot range is 0.6–1.0. The colors of the plot correspond to the different pixel doses, and the background pink- and cyan (bottom) -colored solid lines are the infinite dose profiles with and without chromatic aberration. (f) The mean value profiles (within the Ce-containing atomic column) obtained from (d) in the same manner (0%–1.5%). The contrast ranges are: (a) 0%–5% and the others 0%–3%.

Close modal

Using Eq. (3), an optimum electron dose can be estimated for a target column-SNR. The electron doses for the Ce-containing Al column at SCe2σg = 2, 4, 6 are estimated to be Dp,Ce = 14, 58, 130 e/pix (m = 0, 1, 2), respectively (Δf = 21 Å). These estimated electron doses explain well the image quality and the profile of Figs. 3(d) and 3(e). The Ce dopant at Δf = 21 ± 6 Å can be clearly seen even in the very low-dose range of 25–100 e/pix (2 < SCe2σg < 5), though the noise level of CCe2σg is relatively large. It is noteworthy that the pure Al column requires 10 times higher electron doses than the Ce containing column, and therefore, in experiment, if the Al atomic column is visible (200 e/pix), then we are able to determine the dopant depth with high confidence. With an electron dose higher than 400 e/pix, the finite-dose images almost perfectly reproduce the infinite-dose images and profiles CCe2σg. We find that the minimum pixel dose for just visualizing the Ce dopant is 14 e/pix, and to see the pure Al atomic column, we require 458 e/pix (SAl2σg = 4), corresponding to a beam current of 10 pA and scanning rate of 7.3 μs/pix (similar pixel dose to the single frame of Fig. 1(a)). We note that a typical acquisition condition is 20 pA with 30 μs/pix or ∼3700 e/pix, and hence the required electron dose for a single image acquisition is one order of magnitude lower dose. For the identification of dopant depth, we need to sequentially acquire a focus-series of about the number of unit cells along the depth, i.e., 20 or 30 images for 10 nm thickness. The required minimum electron dose is thus reasonably similar to the experimental conditions of Fig. 1.

The chromatic aberration with the cold FEG does not vary column-contrast noticeably, but does give a slight reduction of the depth sensitivity: the dopant contrast is delocalized in the under-defocus (shallow depth) direction, owing to the asymmetric zero-loss profile. We note that the mean value profile of Fig. 3(f) has a much sharper peak on the dopant depth location and smaller noise level than that of column-contrast, which could be useful for the depth identification. The mean value profiles with 100 or 200 e/pix are well converged to the infinite-dose profile, which is consistent with our optimum pixel dose. Therefore, even under low-dose conditions, if we have a focal-series images with sufficiently fine step (unit-cell-step), we can determine the dopant depth by finding the defocus value at which the mean value profile maximum.

Although chromatic aberration does not introduce a serious difficulty for the depth identification in the single dopant case, the further question is whether this will still be the case for multiple dopants in a single atomic column. Figs. 4(a) and 4(b) show simulations of an xz-plane-view of Figs. 3(a) and 3(b) (single Ce dopant) along the X-X′ direction in the top image with the different electron dose images arranged in the vertical direction. In the same manner, three Ce dopants (zCe = 21.0, 42.8, and 64.5 Å) without and with chromatic aberration are given in Figs. 4(c) and 4(d). The left and right vertical bright lines in each image correspond to the pure and Ce-containing Al columns, respectively. As already discussed for the single dopant case, it can be clearly seen that the bright Z-contrast is well localized at the dopant depth, even with chromatic aberration. In the three dopants case with chromatic aberration, the bright Z-contrast is delocalized along the whole depth (Fig. 4(d)) and it would be difficult not only to identify the dopant depths and but even to reliably count the number of dopant atoms. However, if we can sufficiently reduce chromatic aberration and/or source of energy spread (<0.1 mm eV), it becomes possible to identify the depths of the three dopant atoms, as shown in Fig. 4(c), and is more evident in the mean value profile of Fig. 4(e). With chromatic aberration, the individual dopant atoms are more difficult to resolve; however, three peaks at the proper depths are detectable at a higher pixel dose of 200 e/pix (∼2 × 104 e2), suggesting that the development of a monochromator or chromatic aberration corrector should be useful for future investigations. Finally, we note that we have not employed any noise reduction algorithms which can be highly successful in reconstructing images from noisy data,27–29 indicating significant future potential for the method.

FIG. 4.

(a)–(d) Cross-sectional-defocus-view across X-X′ direction as a function of pixel dose of 25 × 2m electrons (m = 0, …, 7, infinite), with 300 kV and 60 mrad, where all the images are simulated. A single Ce dopant is located at zCe = 21.0 Å in (a), (b), whereas three Ce dopants are located at zCe = 21.0, 42.8, and 64.5 Å in (c), (d). Chromatic aberration is included in (b) and (d). (e) The mean value profiles of (c) as a function of defocus and pixel dose, where the background cyan-colored and pink-colored (bottom) solid-lines are obtained from the infinite dose image with or without chromatic aberration, respectively.

FIG. 4.

(a)–(d) Cross-sectional-defocus-view across X-X′ direction as a function of pixel dose of 25 × 2m electrons (m = 0, …, 7, infinite), with 300 kV and 60 mrad, where all the images are simulated. A single Ce dopant is located at zCe = 21.0 Å in (a), (b), whereas three Ce dopants are located at zCe = 21.0, 42.8, and 64.5 Å in (c), (d). Chromatic aberration is included in (b) and (d). (e) The mean value profiles of (c) as a function of defocus and pixel dose, where the background cyan-colored and pink-colored (bottom) solid-lines are obtained from the infinite dose image with or without chromatic aberration, respectively.

Close modal

In summary, we have quantitatively demonstrated the feasibility of locating a single Ce dopant in depth via large-angle illumination STEM optical sectioning through image simulations including finite electron dose effects. Our image contrast analysis reveals that large-angle illumination STEM imaging can perform well under low-dose conditions, thereby avoiding severe specimen damage. We also considered the effect of chromatic aberration, which we found does not limit the depth sectioning even with a current cold FEG for the case of single isolated dopants. However, chromatic aberration correction, monochromation, or a better electron source will be required for a depth resolution better than 1 nm or for multiple dopants in a single column. Therefore, there is a compelling case to continue to develop geometric and chromatic aberration correctors to achieve atomic-resolution electron microscopy in all three dimensions.

See supplementary material for the derivation of contrast and SNR with Gaussian approximation, the conditions of image simulation,30 and a discussion of channeling effects.

A part of this work was supported by the Research and Development Initiative for Scientific Innovation of New Generation Batteries (RISING II) project of the New Energy and Industrial Technology Development Organization (NEDO), Japan. R.I. used resources of the National Energy Research Scientific Computing Center, which was supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. A.R.L. was supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, U.S. Department of Energy. This research was supported under the Australian Research Council's Discovery Projects funding scheme (Project No. DP110102228).

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