TY - JOUR
AU1 - Borisevich, Albina Y.
AU2 - Lupini, Andrew R.
AU3 - Travaglini, Samuel
AU4 - Pennycook, Stephen J.
AB - Abstract The implementation of aberration correction for the scanning transmission electron microscope (STEM) enables the use of larger probe-forming apertures, improving the transverse resolution significantly and also bringing depth resolution at the nanometer scale. This opens up the possibility of three-dimensional imaging by optical sectioning, and nanometer-scale depth resolution has been demonstrated for amorphous and off-axis samples. For crystalline materials it is usual to image in a zone axis orientation to achieve atomic resolution. In this case, the tendency for the beam to channel along the columns complicates the simple optical sectioning technique. Here we conduct a series of simulations which demonstrate that higher beam convergence angles available in next generation aberration correctors can overcome this limitation. Detailed simulations with realistic values for residual aberrations predict nanometer-scale depth resolution for Bi dopant atoms in Si (110) for an instrument corrected up to fifth order. Use of a monochromator appears to significantly improve the depth resolution. depth sectioning, electron channeling, aberration correction, scanning transmission electron microscopy Introduction The recent successful correction of magnetic lens aberrations to third order [1,2] has led to revolutionary improvements in point resolution and signal-to-noise ratio. The range of accessible lattice spacings has expanded into the sub-angstrom regime [3], and simultaneous detection of light and heavy elements became possible [4]. Single-atom imaging is almost routine [5–7], unlike before aberration correction (e.g. [8–11]), and spectroscopic identification of a single atom in its bulk environment has recently been reported [12]. One unanticipated advantage of aberration correction is the wider probe-forming aperture, which gives a reduced depth of focus. This now makes it possible to optically section through a sample, in a way similar to confocal optical microscopy [13], but with nanometer-scale depth resolution and single-atom sensitivity. For amorphous and nanoparticulate samples, the resulting three-dimensional (3D) datasets can be approximated as a convolution between a 3D point spread function of the probe and the object function. The anticipated nanometer-scale resolution and single-atom sensitivity have recently been demonstrated for amorphous layers in semiconductors [14] and catalytic nanopowders [15]. In the case of a heavy dopant atom in a crystal aligned along a high-symmetry crystallographic axis electron channeling effects are prominent and no longer allow the approximate treatment of the image as a simple 3D convolution. Fertig and Rose [16] investigated probe propagation in thin and thick crystals and likened channeling effects to the atomic columns acting as ‘waveguides’, trapping the probe and altering the intensity distribution within the crystal. This effect greatly complicates quantitative interpretation and often leads to counterintuitive results. For example, it was demonstrated using both classical and modified multislice simulations (for 200 kV and C3 = 1 mm, where C3 is the coefficient of third-order spherical aberration, often referred to as Cs) that the intensity of the image of a single Au atom on the exit surface of Si is greatly enhanced compared with the image of the same atom on the entrance surface for thicknesses of Si up to ∼30 nm [17,18]. Voyles et al. [19,20] simulated the images of single Sb atoms inside Si (110) (for 200 kV and C3 = 1 mm) and pointed out that for thicknesses in excess of 60–80 nm the image of the dopant atom shifts from the column where the dopant is located to the neighboring column. This reflects the well-known phenomenon of the coupling of two closely spaced s-states [21]. (However thicknesses of 60–80 nm are outside the experimental range where single atoms have been detected so far in a 200 kV uncorrected microscope, making this concern largely academic.) At the same time, Bloch wave calculations of the scanning transmission electron microscope (STEM) images of GaAs for sub-angstrom probe sizes conducted by Peng et al. [22] showed that the contribution of channeling decreases for larger illumination angles, suggesting that advanced aberration-corrected microscopes, such as those with full fifth-order aberration correctors, can demonstrate depth sensitivity even inside crystals. It is thus very interesting to evaluate the possibility of 3D imaging in crystals and estimate the effective focal depth of substitutional atoms for different probe conditions. Methods The first series of calculations was conducted using ‘frozen phonon‘ multislice codes [23], for a 300 kV aberration-free probe with varying illumination angles (35 and 70 mrad). One practical difficulty is that larger aperture angles require larger detector angles, which in turn require larger unit cells for the calculations, while the sub-angstrom resolution of the resulting data also requires fine sampling in real space. Thus our simulations used unit cells of ∼30 × 30 Å, and both specimen transmission functions and probe wavefunctions of 2048 × 2048 pixels for the line traces; 1024 × 1024 pixels were used for the full images to reduce calculation time. A total of 10 phonon configurations generated assuming the Einstein model dispersion were used; a root mean square displacement of 0.078 Å was used for Si; for CaTiO3 the values for Ca, Ti and O were computed from the room temperature crystallographic data [24], and the values for Bi and La dopants were derived from those for Si and Ca, respectively, assuming inverse square root mass proportionality. The simulations for realistic probe parameters required changes to the multislice code, starting with the modification of the probe-forming procedure to include the higher-order geometric aberrations. Chromatic effects were incorporated into the frozen phonon routine: for each phonon configuration the defocus was set at a slightly different value (in the range of ±1.15 Cc × ΔE/E0 from the center (where Cc is the coefficient of chromatic aberration, E0 is the electron energy and ΔE is the gun energy spread)). The final image was computed from the phonon contributions taken with Gaussian weights for a Gaussian with FWHM equal to Cc × ΔE/E0 as opposed to equal weights in the original code. (While this procedure somewhat reduces the statistical value of the phonon averaging, we found that the crystal in our simulation is thick enough so that the averaging within just one configuration is sufficient. Comparison of the results utilizing randomly displaced atoms to those where all configurations were identical and perfect did not reveal any artifacts and only demonstrated some ‘flattening’ of the intensity maxima normally expected when phonons are introduced.) To avoid artifacts it was found to be critical to have a sufficient number of steps for the Cc integration: the defocus step should be small compared with the focus depth of the Cc-free probe. A total of 13 defocus steps were thus used for the case of the non-monochromated microscope and 9 steps for the case of the monochromated. Results and discussion Aberration-corrected probes and the depth of field One of the commonly used definitions of the depth of field, which was formulated by Born and Wolf for incoherent light optics [25], gives the distance within which the intensity on the optic axis stays within 20% of the maximum as Δz = λ/θ2 (full width at 80% maximum of intensity, FW80%M). We therefore use this as a convenient measure of the depth of field, although in a real, noisy image the actual depth resolution may be rather worse. Thus with increasing orders of aberration correction, and the associated increase in aperture size, the depth of field will decrease very quickly. This change is visualized in Fig. 1, which shows reconstructions of the probe shape in both transverse and vertical directions for three microscopes, starting with the ORNL's 100 kV VG Microscopes HB501UX at Oak Ridge National Laboratory (ORNL) before correction (Fig. 1a), then ORNL's 300 kV C3-corrected VG Microscopes HB603U (Fig. 1b) and finally a hypothetical future microscope with C5 corrector (Fig. 1c) (C3 and C5 denote the coefficients of third- and fifth-order spherical aberrations, respectively). For the uncorrected 100 kV instrument (Fig. 1a) with cold field emission gun (CFEG) (C3 = 1.3 mm, Cc = 1.3 mm, ΔE = 0.3eV, θ = 10 mrad) the depth of field is ∼45 nm; indeed, over the range of −10 to 10 nm from the maximum intensity the probe changes very little. For the C3-corrected 300 kV microscope (C3 = −40 µm, C5 = 10 cm, Cc = 1.6 mm, ΔE = 0.3eV, θ = 23 mrad) the projected depth of field is 3.7 nm, suggesting nanometer-scale depth sensitivity. For the hypothetical C3/C5-corrected STEM with a monochromator (C3 = 2.3 µm, C5 = −2.1 mm, C7 = 50 cm, Cc = 1.6 mm, ΔE = 0.1eV, θ = 50 mrad) the estimated depth of field is 1.0 nm. Changes are also evident laterally: with aberration correction (Figs 1b and 1c), the probe rapidly becomes sub-angstrom. Also, since the optimal probe is achieved via using terms of different order to balance higher order, uncorrected aberrations, the probe acquires some asymmetry in the vertical direction. In Fig. 1b the intensity on the optic axis decays monotonically in the overfocus region, while in the underfocus region it decays sharply and forms subsidiary maxima. Fig. 1 View largeDownload slide Probe intensity profiles in free space as a function of defocus for microscopes of three generations: (a) uncorrected 100 kV microscope, (b) 300 kV microscope with third-order corrector and (c) a hypothetical 200 kV instrument with C3/C5 corrector (see detailed probe parameters in text). Note different scales vertically and laterally. Intensity maxima are normalized to zero defocus. Fig. 1 View largeDownload slide Probe intensity profiles in free space as a function of defocus for microscopes of three generations: (a) uncorrected 100 kV microscope, (b) 300 kV microscope with third-order corrector and (c) a hypothetical 200 kV instrument with C3/C5 corrector (see detailed probe parameters in text). Note different scales vertically and laterally. Intensity maxima are normalized to zero defocus. Another consequence of the aberration compensation is that the focal depth estimates from the more realistic probe profiles (Fig. 1) are somewhat larger than calculated for the aberration-free case: for the vertical profile of Fig. 1b the FW80%M is ∼3.9 nm (vs 3.7 nm), and for the profile of Fig. 1c it is ∼1.05 nm (vs 1.00 nm). These probe profiles incorporate chromatic aberration Cc and gun energy spread ΔE, which also contribute to the probe widening in the vertical direction compared with the aberration-free case. Interestingly, as the defocus spread caused by geometric aberrations Δzgeom ≈ λ/θ2 decreases, it can become comparable with defocus spread introduced by the gun energy spread and chromatic aberration Δzchrom ≈ CcΔE/(1.76E0) (the factor of 1.76 is needed for the conversion of FWHM to FW80%M). If we consider ORNL's 300 kV microscope, Δzgeom is ∼3.7 nm while Δzchrom for a CFEG (Cc = 1.6 mm, ΔE = 0.3 eV) is just 0.91 nm; therefore geometric effects dominate the depth of field. So even for a hypothetical microscope with the same aberration corrector but Shottky-type gun (ΔE = 0.6 eV), the ‘chromatic’ defocus spread of 1.8 nm would still be below the ‘geometric’ one and the aggregate focal depth would not change substantially. This is illustrated by the comparison of the two probe intensity profiles vs defocus in Fig. 2a, where the two guns would give very similar depth of field. The situation, however, changes for the 200 kV C3/C5-corrected microscope (Δzgeom = 1 nm), where the CFEG (Cc = 1.6 mm, ΔE = 0.3 eV) would correspond to Δzchrom = 1.4 nm, making the two contributions comparable. In this case reducing the gun energy spread to 0.1 eV by the addition of a monochromator should have a beneficial effect on the vertical resolution of the microscope. Indeed, the comparison of the probe profiles for this hypothetical instrument with CFEG, monochromated CFEG, as well as with a Shottky gun (Fig. 2b), reveals clear differences. Thus the effects introduced by chromatic aberration and gun energy spread become very important for the later generations of aberration correctors. Fig. 2 View largeDownload slide Vertical electron probe profiles in free space for different values of gun energy spread: (a) for ORNL's 300 kV instrument with its existing gun or hypothetical Shottky gun, (b) for a 200 kV C3/C5-corrected microscope; 0.6 eV corresponds to a Shottky-type gun, 0.3 eV to a cold field emission gun (CFEG), and 0.1 eV to a monochromated gun. Intensity maxima are normalized to zero defocus. Fig. 2 View largeDownload slide Vertical electron probe profiles in free space for different values of gun energy spread: (a) for ORNL's 300 kV instrument with its existing gun or hypothetical Shottky gun, (b) for a 200 kV C3/C5-corrected microscope; 0.6 eV corresponds to a Shottky-type gun, 0.3 eV to a cold field emission gun (CFEG), and 0.1 eV to a monochromated gun. Intensity maxima are normalized to zero defocus. Test simulations: aberration-free probe The first series of simulations was intended to determine whether increasing illumination angle enables depth sensitivity in crystals and to address dependence of the channeling contribution on atomic number. To that effect, two different doped crystals were considered: (110) Si doped with Bi and (100) CaTiO3 doped with La. Figure 3a shows the model of the Si crystal used in the simulation. The dopant atoms are distributed over four neighboring atomic columns, while separated vertically by ∼4 nm. The entrance surface of the 16 nm crystal is pure Si, and the last of the dopant atoms is located at the exit surface. Figure 3b shows line profiles of the simulated HAADF images for the illumination semiangle of 0.035 rad taken with the probe focused, respectively, in the entrance plane and then in the plane of each subsequent dopant atom. In the profiles taken with the probe focused in the defect planes, the Bi-containing columns are immediately identifiable. Moreover, the intensities of both doped and undoped columns stay quite consistent—and distinct—throughout the focal series, including the profile taken at the entrance plane. The simulations for intermediate focus values, omitted for clarity, illustrated a gradual transition between the presented profiles. In the last two profiles, where the probe is focused deeper within the crystal, it undergoes significant broadening before scattering from the dopant atom, and the image features are no longer round in shape. Fig. 3 View largeDownload slide (a) The model doped Si crystal with four Bi atoms at different depths (Bi, black; Si, gray); (b) simulated HAADF image intensity scans for an aberration-free probe with 35 mrad semiangle, detector angle from 70 to 300 mrad. Defocus varies from 0 (top) to 38.4, 76.8, 115.2 and 153.6 Å, consecutively. Fig. 3 View largeDownload slide (a) The model doped Si crystal with four Bi atoms at different depths (Bi, black; Si, gray); (b) simulated HAADF image intensity scans for an aberration-free probe with 35 mrad semiangle, detector angle from 70 to 300 mrad. Defocus varies from 0 (top) to 38.4, 76.8, 115.2 and 153.6 Å, consecutively. The images in Fig. 3 clearly show that in this case the vertical positions of dopants can be determined with nanometer precision without simulations, i.e. the data can be interpreted directly as a convolution of the effective probe size with a 3D object function. Thus for Bi-doped Si the illumination semiangle of 0.035 rad (somewhat above the currently available value of 0.023 rad) is sufficient to operate qualitatively in the depth-sensing regime. However, channeling still affects the effective probe size in the crystal. Figure 4a shows the model of the CaTiO3 crystal with similarly distributed dopant atoms; the distorted orthorombic perovskite structure of CaTiO3 is approximated by the ideal cubic structure with the same unit cell volume. The results of the simulation for the illumination semiangle of 0.035 rad are given by the line profiles in Fig. 4b. It is evident that for the heavier CaTiO3 these conditions are not enough to switch to a predominantly depth-sensing mode. Even though within each image taken in the defect plane the brightest column is the one with the La atom in that plane, only the shallowest La (∼38 A) has its maximum intensity as a function of focus close to that plane. For the deeper La, the highest intensity is still obtained when the beam is focused on the entrance surface. Clearly, this dataset deviates from the simple convolution picture. Thus, the contribution of channeling is qualitatively significant, and although by comparison with image simulations the La depth could be extracted from the experimental data with nanometer resolution, the data do not allow for quick intuitive interpretation. These probe parameters are thus of limited use for 3D-imaging of strongly channeling samples such as CaTiO3 that have not been otherwise extensively characterized. Fig. 4 View largeDownload slide (a) The model doped CaTiO3 crystal with four La atoms at different depths (La, black; Ca, dark gray; Ti, gray w/horizontal stripes; O, white). (b and c) simulated HAADF image intensity scans at different defocus values using an aberration-free probe with (b) 35 mrad semiangle, detector angle from 70 to 300 mrad and (c) 70 mrad semiangle, detector angle from 150 to 300 mrad. Defocus varies from 0 (top) to 38.2, 76.4, 114.6 and 152.9 Å, consecutively. Fig. 4 View largeDownload slide (a) The model doped CaTiO3 crystal with four La atoms at different depths (La, black; Ca, dark gray; Ti, gray w/horizontal stripes; O, white). (b and c) simulated HAADF image intensity scans at different defocus values using an aberration-free probe with (b) 35 mrad semiangle, detector angle from 70 to 300 mrad and (c) 70 mrad semiangle, detector angle from 150 to 300 mrad. Defocus varies from 0 (top) to 38.2, 76.4, 114.6 and 152.9 Å, consecutively. The situation changes once the illumination semiangle is doubled to 0.07 rad (Fig. 4c). This set of simulated image profiles displays clear dominance of depth sensitivity over channeling. The intensity of undoped columns is consistent throughout; the intensity of the La-doped columns varies but stays far above that for the undoped columns, thus enabling direct interpretation of the results. Between Figs 4b and 4c there is a clear switch with the increase of the illumination angle from the regime where channeling predominates to the depth-sensing regime where channeling is suppressed, in agreement with earlier results by Peng et al. [22]. Therefore our results suggest that further advances in aberration correction will overcome many of the difficulties related to channeling and enable 3D characterization of crystalline samples without the need to tilt away from the zone axis. Realistic probe parameters: focus depth inside the crystal Having obtained the qualitative validation, it is now very interesting to generate a more quantitative estimate of the capabilities of the more realistic instruments for the imaging of point defects in crystals. The simulations discussed in the previous section suggest that the C3/C5-corrected 200 kV microscope with the illumination semiangle of 0.05 rad should be capable of generating depth-sensitive data for Bi-doped Si. We can thus generate a sequence of the simulated images at different defocus values to determine the focal depth of the defect. A 16 nm thick crystal of Si in (110) orientation was used in the simulation, with one Bi atom in the middle. Simulated image profiles were obtained for defocus values from the exit to the entrance plane of the crystal, with defocus sampled every two unit cells (∼8 Å) and finer in the vicinity of the dopant atom. The intensities of the Bi-doped column and the neighboring pure Si columns were plotted as a function of defocus. In Fig. 5, the solid lines correspond to the C3/C5-corrected microscope with CFEG and the dashed lines correspond to the monochromated instrument. Apparently, in both cases the Bi atom is visible to some extent over a wide range of defocus values. However, the Bi column intensity increases more sharply for the monochromated microscope (FW80%M ≈ 2.5 nm), than for the instrument with CFEG (FW80%M ≈ 3.7 nm). The effective focus depth is thus decreased by about a third through the use of a monochromator; the residual defocus spread is likely dominated by the combination of the geometric aberrations and channeling effects. Residual channeling is also manifested by the shift of the maximum of the incremental intensity 3–4 Å deeper inside the crystal than the impurity position in the model. Even with this limitation, however, the impurity position can be determined directly with better than 1 nm accuracy, while with the aid of simulation it can be localized with angstrom precision. Fig. 5 View largeDownload slide Simulated intensity of the two neighboring columns in a dumbbell vs defocus in Si (110) ∼16 nm thick doped with one Bi atom. Squares denote the column containing Bi atom, stars denote a pure Si column; solid lines are simulations for a 200 kV C3/C5-corrected microscope with CFEG and dashed lines are simulations for the instrument with a monochromated gun. Probe semiangle 50 mrad, detector angle 100–300 mrad. Fig. 5 View largeDownload slide Simulated intensity of the two neighboring columns in a dumbbell vs defocus in Si (110) ∼16 nm thick doped with one Bi atom. Squares denote the column containing Bi atom, stars denote a pure Si column; solid lines are simulations for a 200 kV C3/C5-corrected microscope with CFEG and dashed lines are simulations for the instrument with a monochromated gun. Probe semiangle 50 mrad, detector angle 100–300 mrad. These results indicate that microscopes with aberration correction to fifth order, which give decreased vertical probe sizes and thus improved capability for 3D imaging in non-periodic systems, will also allow evaluation of the 3D structure of periodic crystals with nanometer-scale resolution. Concluding remarks Aberration correction improves the resolution of STEM by enabling higher convergence angles for the STEM probes. Probe simulations show that the increased convergence angles result in a depth sensitivity of currently available aberration-corrected STEM instruments on the nanometer scale; this sensitivity is expected to improve when the next generation of instrumentation is available. As the influence from geometric aberration decreases, other factors such as chromatic aberration and gun energy spread become more prominent and are expected to play a decisive role in determining the depth resolution of the C5-corrected instruments. Unlike amorphous or nanoparticulate materials, oriented crystals are affected by electron channeling—preferential propagation of the beam along the atomic columns. Frozen phonon multislice simulations for aberration-free probes demonstrate that for a reasonably high probe angle the contribution of channeling is diminished, and depth sensitivity within an aligned crystal can be achieved; the onset of depth sensitivity occurs at higher convergence angles for higher atomic number crystals, which channel more strongly (CaTiO3vs Si). A simulation with a modified frozen phonon multislice code, performed using realistic probe parameters (C3, C5, C7, Cc and ΔE) for a C3/C5-corrected column, suggests that such an instrument will be capable of localizing a Bi dopant atom inside a Si crystal with nanometer resolution. We also show that reducing the energy spread of electrons in such systems through monochromation would further improve the depth resolution. References 1 Urban K,  Kabius B,  Haider M,  Rose H.  A way to higher resolution: spherical-aberration correction in a 200 kV transmission electron microscope,  J. 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TI - Depth sectioning of aligned crystals with the aberration-corrected scanning transmission electron microscope
JF - Journal of Electron Microscopy
DO - 10.1093/jmicro/dfi075
DA - 2006-01-01
UR - https://www.deepdyve.com/lp/oxford-university-press/depth-sectioning-of-aligned-crystals-with-the-aberration-corrected-JAUwz3aAdF
SP - 7
EP - 12
VL - 55
IS - 1
DP - DeepDyve
ER -