Evaluation of residual aberration in fifth-order geometrical aberration correctors

Evaluation of residual aberration in fifth-order geometrical aberration correctors Abstract Higher order geometrical aberration correctors for transmission electron microscopes are essential for atomic-resolution imaging, especially at low-accelerating voltages. We quantitatively calculated the residual aberrations of fifth-order aberration correctors to determine the dominant aberrations. The calculations showed that the sixth-order three-lobe aberration was dominant when fifth-order aberrations were corrected by using the double-hexapole or delta types of aberration correctors. It was also deduced that the sixth-order three-lobe aberration was generally smaller in the delta corrector than in the double-hexapole corrector. The sixth-order three-lobe aberration was counterbalanced with a finite amount of the fourth-order three-lobe aberration and 3-fold astigmatism. In the experiments, we used a low-voltage microscope equipped with delta correctors for probe- and image-forming systems. Residual aberrations in each system were evaluated using Ronchigrams and diffractogram tableaux, respectively. The counterbalanced aberration correction was applied to obtain high-resolution transmission electron microscopy images of graphene and WS2 samples at 60 and 15 kV, respectively. transmission electron microscope, aberration corrector, high-resolution image, low accelerating voltage Introduction The resolution of transmission electron microscopes (TEM) has drastically improved since aberration correctors were developed in the 1990s [1]; resolutions better than 50 pm have been achieved at 300 kV [2–4]. In recent years, correctors have been used at various accelerating voltages ranging from 15 kV [5, 6] to 1.2 MV [7] for high-resolution imaging. Aberration correctors have become indispensable in the atomic level analysis of materials [8]. A corrector of spherical aberration was developed by Haider et al. [1] for TEM and scanning TEM (STEM) with double hexapole fields. A corrector with quadrupole–octapole fields was developed by Krivanek et al. [9] for STEM. There have also been reports of a higher order aberration corrector [10] and chromatic aberration corrector [11, 12]. The most widely used correctors consist of magnetic fields with 3-fold symmetry [1]. The 3-fold symmetric field is generated by a hexapole or a dodecapole. A corrector consisting of double hexapole-fields can correct geometrical aberrations up to the fourth-order, including third-order spherical aberration. The uniform phase angle realized by these correctors, which can be observed in a Ronchigram, is limited up to 30–40 mrad [13] because the fifth-order aberration of a 6-fold astigmatism is intrinsically introduced in this type of corrector and dominantly limits the uniform phase angle. To correct the 6-fold astigmatism in the hexapole field corrector, two types of correctors were developed. One is a double hexapole-field corrector with optimized multipole thickness [14] and the other is a triple hexapole-field corrector [10]. Under the correction of the 6-fold astigmatism by these fifth-order aberration correctors, the flat-phase angle was extended to 70 mrad. Towards the larger flat-phase angle beyond that achieved by the advanced correctors, we need to determine which aberration is dominant after the correction of the 6-fold astigmatism and identify the aberrations quantitatively. Moreover, after this identification, we need to explore ways to reduce or correct these aberrations. In this paper, we present the residual aberrations after the correction of the 6-fold astigmatism and determine methods to reduce the aberration through calculations and experiments. Calculation of residual aberration in fifth-order aberration corrector In this section, we present the calculated residual aberrations on a double hexapole-field corrector (advanced double-hexapole corrector [14]) and triple hexapole-field corrector (delta corrector [10]). To calculate the electron trajectory in a hexapole field that is required to estimate the aberrations, an iterative method with complex geometrical aberration coefficients [15] was used. Fringe fields near the entrance and exit plane of the hexapole fields were neglected by assuming a sharp cut-off field approximation in the calculation. When an electron beam passes into a hexapole field parallel to the optic axis, the electron trajectory at the exit plane of the hexapole field contains aberrations which are 3-fold astigmatism, third-order spherical, fourth-order three-lobe, fifth-order spherical, 6-fold astigmatism and sixth-order three-lobe aberration (see Table 1 and Supplementary data Appendix 1). The symmetry of these aberrations is circular or 3N-fold (N: integer) because these aberrations are generated as combination aberrations due to the 3-fold astigmatism. This can be predicted by the combination rule [16]. The electron trajectories at the exit of the double-hexapole and the delta correctors show same aberrations with different coefficients, since these aberrations are always generated after a hexapole field. Table 1. List of aberrations. Those in bold can be generated in correctors using hexapole fields. Wave aberrations are shown by using a complex angle ω and its complex conjugate ω̅ Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] Table 1. List of aberrations. Those in bold can be generated in correctors using hexapole fields. Wave aberrations are shown by using a complex angle ω and its complex conjugate ω̅ Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] We first evaluated the residual aberrations in the advanced double-hexapole corrector [14]. Figure 1a shows a schematic of the double-hexapole corrector. We implemented an aberration correction system in which the focal lengths of the transfer lens doublet between the hexapole fields were the same for simplicity; however, these can be different in an asymmetric-type corrector [17]. We ignored parasitic aberrations, such as coma and star aberrations as these can be compensated by the beam alignments using deflectors in the correctors. By using the calculated combination aberrations shown in Supplementary data Appendix 1, we calculated aberrations generated by the double-hexapole corrector. All equations shown below were obtained from our calculation. Three-fold astigmatism (A3) and three-lobe aberration (R5) become A3=A3,HF1L1M3−A3,HF2L2M3 (1) R5=3A3,HF1̅A3,HF12L15+A3,HF2̅L23(−5A3,HF12L12+5A3,HF1A3,HF2L1L2−3A3,HF22L22)12M5fOL4 (2) where A3,HF1 and A3,HF2 denote the coefficients of the 3-fold astigmatism, L1 and L2 denote the lengths of the hexapole fields, M is the demagnification ratio between the second hexapole field and the objective lens and fOL is the focal length of the objective lens. A3 and R5 become zero when the corrector satisfied the condition A3,HF1 = A3,HF2 and L1 = L2. The negative third-order spherical aberration generated in the hexapole fields was used for compensation of the positive third-order spherical aberration of the objective lens. Consequently, all aberrations up to fourth-order were eliminated. A residual aberration higher than fifth-order, which cannot be compensated, is a 6-fold astigmatism as shown in equation below: A6=−9M2O42L56fOL2+9CsTLfOL2O4M2fTL2L (3) where O4 is the absolute coefficient of the negative spherical aberration generated by the hexapole fields, and fTL and CsTL are the focal length and the spherical aberration coefficient of the transfer lenses between the hexapole fields, respectively. Arg(A3,HF1), which is the direction of the 3-fold astigmatism generated by hexapole field 1 (HF1), was set to 0°. The first term in Eq. (3) is the 6-fold astigmatism generated by the two hexapole fields. The second term is the 6-fold astigmatism due to the combination aberration of the first hexapole field and the spherical aberration of the transfer lenses. These two terms with opposite signs can cancel each other out for a particular condition. We can find that particular length of the hexapole fields where A6 = 0 as in [14] LA6=0=214CsTLfOL2M2fTLO4. (4) Fig. 1. View largeDownload slide Schematic of (a) the double-hexapole corrector and (b) the delta corrector used in the aberration calculation. Fig. 1. View largeDownload slide Schematic of (a) the double-hexapole corrector and (b) the delta corrector used in the aberration calculation. Other than the 6-fold astigmatism, the fifth-order spherical aberration appears in the double-hexapole corrector. This aberration can be compensated by adjusting the transfer length between the principal plane of the hexapole field 2 (HF2) and the front focal plane of the objective lens, by using the combination aberration [18]. As described above, the aberrations up to fifth-order can be compensated in the double-hexapole corrector. From the calculation, the next influential aberration is the sixth-order three-lobe aberration (R7). R7 is a dominant residual aberration in the double-hexapole corrector after A6 is corrected, which is expressed as: R7=−7M3L3/2O45/286fOL3+732CsTLfOL(L2−12fTL2)O43/24MLfTL4. (5) The first term is generated by the two hexapole fields. The second term is introduced as the combination aberration of the hexapole fields and the spherical aberration of the transfer lenses. Assigning typical values to the parameters in Eq. (5); fOL = 1.0 mm, O4 = 0.5 mm, M = 1.5, fTL = 30 mm, CsTL = 500 mm and L = 15 mm, the three-lobe aberration coefficient becomes R7 = 13.2 mm. For an acceleration voltage of 60 kV, the phase difference from the axis exceeds π/4 at 44 mrad. We also calculated the residual aberrations in the delta corrector. Figure 1b shows a schematic of the delta corrector. Under the condition L1 = L2 = L3, A3 and R5 become zero when |A3,HF1 |= |A3,HF3 |, |A3,HF2 |=22/3 |A3,HF1 |≈1.63 |A3,HF1 | and Arg(A3,HF3)−Arg(A3,HF1)=±arctan(22)≈±70.5°. Since this relationship between vectors of the three hexapole fields resembles the Greek symbol Δ [10], this condition is called the delta condition. In the delta condition, A6 is small, but not zero. A3 and A6 become 0 when |A3,HF1 |= |A3,HF3 |, |A3,HF2 |=(9+281)/10 |A3,HF1 |≈1.61 |A3,HF1 | and Arg(A3,HF3)−Arg(A3,HF1)=±arctan(−2+22281−11+281)≈±73.3°, under the condition: L1 = L2 = L3 and CsTL = 0. Therefore, the two conditions, when A3 = R5 = 0 and A3 = A6 = 0, are slightly different in the angle and excitation ratio between the three hexapole fields for the delta corrector. When CsTL is not zero, the above solved conditions change slightly depending on CsTL. In practice, however, R5 can be controlled by changing the excitation of the transfer lenses between the hexapole fields as with the double-hexapole corrector. This adjustment changes R5 and keeps the A6 ≈ 0 condition. Therefore, all aberrations up to fifth-order can be corrected in the delta corrector. As for the advanced double-hexapole corrector, the next influential aberration is the sixth-order three-lobe aberration. In the delta condition, the sixth-order three-lobe coefficient (R7) becomes R7=(−21002ⅈ+32532)M3L3/2O45/2352807fOL3−CsTLfOL((−24ⅈ+2)L2+12(22ⅈ+2)fTL2)O43/247MLfTL4 (6) when L1 = L2 = L3, and Arg(A3,HF1) = 0°. R7 is a dominant residual aberration in the delta corrector as well. Assigning the typical values used above, R7 = 8.8 mm. The phase difference from the axis caused by this R7 exceeds π/4 at 47 mrad for 60 kV. Residual amount of sixth-order three-lobe aberration The sixth-order three-lobe aberration becomes the dominant residual aberration in both, the advanced double-hexapole and delta correctors, as a result of the calculations. Collecting coefficients from Eqs. (5) and (6), R7 can be expressed as R7=k1M3L3/2O45/2fOL3+k2CsTLfOLO43/2MLfTL2+k3L3/2CsTLfOLO43/2MfTL4=k1M3L3/2O45/2fOL3+CsTLfOLL3/2O43/2MfTL2(k21L2+k31fTL2) (7) with different complex coefficients k1, k2 and k3. The coefficients of the double-hexapole corrector are k1 ≈ −0.35, k2 ≈ −25.7, and k3 ≈ 2.14, which are real because the direction of each term corresponds to the direction of the 3-fold astigmatism generated by HF1. The coefficients of the delta corrector are k1=(−21002ⅈ+32532)352807≈0.23e−0.43πi, k2=(264ⅈ+122)47≈24.9e−0.52πi, and k3=(−24ⅈ+2)47≈2.27e0.51πi. The absolute values of k1 and k2 in case of the delta corrector are smaller than those in case of the double-hexapole corrector, which means that R7 shown by the first and second terms is smaller in the delta corrector than the double-hexapole corrector. The absolute value of the third term, which shows almost the opposite direction of the first and second terms, is larger in the delta corrector than in the double-hexapole corrector. In addition, the absolute value of the second term is larger than that of the third term when L < fTL, which is a standard condition for both the correctors. Thus, R7 of the delta corrector becomes smaller than that of the double-hexapole corrector when the variables, such as fOL, M, L, O4, fTL and CsTL are the same. For a reduction of R7, the delta corrector is preferable to the double-hexapole corrector. Experimental evaluation of the residual aberration We experimentally corroborated the residual aberrations by using a TEM dedicated to low acceleration voltages equipped with delta correctors for probe- and image-forming systems. First, we evaluated the residual aberrations for a probe-forming system at an acceleration voltage of 60 kV. Figure 2a–c shows experimental Ronchigrams at the underfocus, in-focus, and overfocus conditions, respectively. By assigning the experimental parameters in Eq. (6), R7 was calculated to be 17 mm, resulting in π/4 limit of 43 mrad. In the experiment, R7 was counterbalanced by R5 and A3, because they have a 3-fold symmetry [6]. By counterbalancing R7 = 17 mm with R5 = 92 μm and A3 = 98 nm, the π/4 limit is improved from 43 mrad to 72 mrad. The simulated Ronchigrams after counterbalancing are shown in Fig. 2d–f. The figures are consistent with the experimental results. Fig. 2. View largeDownload slide (a–c) Experimental Ronchigrams with different defoci using the delta corrector at 60 kV. (d–f) Simulated Ronchigrams with R7 = (17 mm, 70°), R5 = (92 μm, 10°) and A3 = (98 nm, 70°). Fig. 2. View largeDownload slide (a–c) Experimental Ronchigrams with different defoci using the delta corrector at 60 kV. (d–f) Simulated Ronchigrams with R7 = (17 mm, 70°), R5 = (92 μm, 10°) and A3 = (98 nm, 70°). We also evaluated the residual aberrations for an image-forming system with those experimentally measured using a diffractogram tableau method at 60 and 15 kV. Using Eq. (6), calculated value of R7 for the image-forming system was found to be 14 and 34 mm, at 60 and 15 kV, respectively. In the experiments, these were measured as 9 and 15 mm by the diffractogram tableau method (Supplementary data Appendix 2), and had good agreement with the simulated results. The discrepancy between the experiment and calculation may be due to measurement errors, or could have been caused by slightly different values of the parameters. For example, the demagnification ratio M was changed to correct the third- and fifth-order spherical aberrations; this affects the value of R7 with the third power of M. Both the values of R7 lead to a π/4 limit of <50 mrad, which can be expanded to more than 70 mrad by the counterbalance of R7, R5 and A3. Figure 3 shows the TEM images of monolayer graphene at 60 kV. To reduce the effects of chromatic aberration, the energy spread of the electron source was decreased to 0.05 eV by using a monochromator [19, 20]. The power spectrum of the graphene image shows the spots corresponding to 0.071 nm, which indicates that the information limit is 15 times the wavelength of a primary electron (4.87 pm at 60 kV). The experimental image shows each carbon atom clearly by resolving the C-C dumbbells with a separation of 0.142 nm. The simulated image with R7 of 9 mm also shows carbon atoms clearly and is consistent with the experimental image. The residual R7 of the image-forming system with the delta corrector is sufficiently small to observe single carbon atoms in graphene. Fig. 3. View largeDownload slide (a) Experimental TEM image of monolayer graphene at 60 kV. The measured sixth-order three-lobe R7 was 9 mm. The third- and fifth-order spherical aberrations were corrected to ~0. The energy width was reduced to 0.05 eV using a monochromator. The underfocused image was taken by Gatan OneView with an exposure time of 8 s with drift compensation. No filtering process was applied to the image. (b) Modulus of the Fourier transform of (a). Spots of 14.1 nm−1 = (0.071 nm)−1 correspond to (15 λ)−1. (c) Averaged image of small areas in (a). (d) Simulated image with R7 = 9 mm, 30°, and a defocus of −1.5 nm calculated using the software elbis [21]. Fig. 3. View largeDownload slide (a) Experimental TEM image of monolayer graphene at 60 kV. The measured sixth-order three-lobe R7 was 9 mm. The third- and fifth-order spherical aberrations were corrected to ~0. The energy width was reduced to 0.05 eV using a monochromator. The underfocused image was taken by Gatan OneView with an exposure time of 8 s with drift compensation. No filtering process was applied to the image. (b) Modulus of the Fourier transform of (a). Spots of 14.1 nm−1 = (0.071 nm)−1 correspond to (15 λ)−1. (c) Averaged image of small areas in (a). (d) Simulated image with R7 = 9 mm, 30°, and a defocus of −1.5 nm calculated using the software elbis [21]. Figure 4 shows the TEM image of monolayer tungsten disulfide WS2 at 15 kV. The power spectrum showed spots corresponding to 0.091 nm which is 9.2 times the electron wavelength (λ = 9.94 pm at 15 kV). These spots indicate that the information limit is sufficiently small to obtain atomic-resolution images at 15 kV. In the experimental image of WS2, an atomic distance of 0.182 nm was resolved. The simulated image, with a value of 16 mm for R7, did not show atomic-resolution image because the calculated phase shift exceeded the π/4 limit at 48 mrad which corresponds to 0.21 nm at 15 kV. R7 can be counterbalanced with R5 of 108 μm and A3 of 143 nm. For this condition, the π/4 limit expanded to 80 mrad, which corresponds to 0.12 nm. The simulated image using the counterbalance shows each atom clearly, as the one from the experiment. This result suggests that the atomic-resolution imaging at the ultra-low accelerating voltage of 15 kV essentially requires the wide flat-phase angle realized by a small value of R7 and its counterbalance with R5 and A3. Fig. 4. View largeDownload slide (a) Experimental TEM image of monolayer WS2 at 15 kV. The measured R7 was 16 mm. O4 and O6 were corrected to ~0. An energy width of 0.05 eV and an exposure time of 5 s were used. No filtering process was applied to the image. Various contrasts of the atoms appear in the image due to the specimen bend. (b) Modulus of the Fourier transform of (a). Spots of 11.0 nm−1 = (0.091 nm)−1 correspond to (9.2 λ)−1. White vertical stripes were caused by a characteristic of the camera. (c) Magnified image of indicated area in (a). (d) Simulated image with R7 = (16 mm, −8°), R5 = (108 μm, 52°), A3 = (143 nm, −8°) and (e) R7 = (16 mm, −8°). A defocus of −1.5 nm was used in both (d) and (e). Fig. 4. View largeDownload slide (a) Experimental TEM image of monolayer WS2 at 15 kV. The measured R7 was 16 mm. O4 and O6 were corrected to ~0. An energy width of 0.05 eV and an exposure time of 5 s were used. No filtering process was applied to the image. Various contrasts of the atoms appear in the image due to the specimen bend. (b) Modulus of the Fourier transform of (a). Spots of 11.0 nm−1 = (0.091 nm)−1 correspond to (9.2 λ)−1. White vertical stripes were caused by a characteristic of the camera. (c) Magnified image of indicated area in (a). (d) Simulated image with R7 = (16 mm, −8°), R5 = (108 μm, 52°), A3 = (143 nm, −8°) and (e) R7 = (16 mm, −8°). A defocus of −1.5 nm was used in both (d) and (e). Discussion of the compensation of the sixth-order three-lobe aberration In the case of the advanced double-hexapole corrector, A6 is corrected by the combination aberration using the spherical aberration CsTL of transfer lenses. We first discuss below, the possibility to compensate R7 in the advanced double-hexapole corrector. The first term of Eq. (5) represents a portion of R7 generated only by the hexapole fields. The second term represents another portion of R7 generated as the combination aberration between the hexapole fields and CsTL. Since the focal length of the transfer lens is larger than half the thickness of the hexapole field (fTL > L/2), (L2−12fTL2) in the second term becomes negative. Therefore, both the first and second terms in Eq. (5) are negative and consequently both terms of R7 produce an aberration in the same direction. Therefore, R7 in the configuration of an advanced double-hexapole corrector is inevitable. For the case of the delta corrector, the direction of R7 generated by only the hexapole fields [the first term in Eq. (6)] is similar to that generated by the combination of the hexapole fields and CsTL [the second and third terms in Eq. (6)], and therefore cannot be canceled out either. In conclusion, R7 cannot be compensated in either the double-hexapole or delta correctors. To compensate R7 for wider uniform phase areas, different types of correctors might be required. Summary The aberration calculation showed that the sixth-order three-lobe aberration (R7) is the dominant residual aberration after the correction of the fifth-order aberration in the double-hexapole and the delta correctors. R7 is generally smaller in the delta corrector than in the advanced double-hexapole corrector. Counterbalancing R7 with the fourth-order three-lobe aberration (R5) and the 3-fold astigmatism (A3), the flat-phase angle was extended to more than 70 mrad, which was demonstrated experimentally in the probe- and image-forming systems. The small residual R7 and the effect of the counterbalance were experimentally identified in low-kV TEM imaging using samples of monolayer sheets of graphene and WS2. Supplementary data Supplementary data are available at Microscopy online. Acknowledgments The authors thank M. Mukai and T. Sasaki of JEOL for their cooperation in the development and installation of the microscope. The authors also thank Y.-C. Lin of AIST for the sample preparation and donation for the TEM observations. The authors acknowledge Y. Kondo for his careful reading of this manuscript. Funding This work was supported by the JST (Japan Science and Technology Agency), Research Acceleration Program (2012–2016). References 1 Haider M , Uhlemann S , Schwan E , Rose H , Kabius B , and Urban K ( 1998 ) Electron microscopy image enhanced . Nature 392 : 768 – 769 . Google Scholar CrossRef Search ADS 2 Erni R , Rossell M D , Kisielowski C , and Dahmen U ( 2009 ) Atomic-resolution imaging with a sub-50-pm electron probe . Phys. Rev. Lett. 102 : 096101 . Google Scholar CrossRef Search ADS PubMed 3 Sawada H , Tanishiro Y , Ohashi N , Tomita T , Hosokawa F , Kaneyama T , Kondo Y , and Takayanagi K ( 2009 ) STEM imaging of 47-pm-separated atomic columns by a spherical aberration-corrected electron microscope with a 300-kV cold field emission gun . J. Electron Microsc. 58 : 357 – 361 . Google Scholar CrossRef Search ADS 4 Sawada H , Shimura N , Hosokawa F , Shibata N , and Ikuhara Y ( 2015 ) Resolving 45-pm-separated Si-Si atomic columns with an aberration-corrected STEM . Microscopy 64 : 213 – 217 . Google Scholar CrossRef Search ADS PubMed 5 Sasaki T , Sawada H , Hosokawa F , Sato Y , and Suenaga K ( 2014 ) Aberration-corrected STEM/TEM imaging at 15 kV . Ultramicroscopy 145 : 50 – 55 . Google Scholar CrossRef Search ADS PubMed 6 Morishita S , Mukai M , Suenaga K , and Sawada H ( 2016 ) Atomic resolution imaging at an ultralow accelerating voltage by a monochromatic transmission electron microscope . Phys. Rev. Lett. 117 : 153004 . Google Scholar CrossRef Search ADS PubMed 7 Akashi T , Takahashi Y , Tanigaki T , Shimakura T , Kawasaki T , Furutsu T , Shinada H , Müller H , Haider M , Osakabe N , and Tonomura A ( 2015 ) Aberration corrected 1.2-MV cold field-emission transmission electron microscope with a sub-50-pm resolution . Appl. Phys. Lett. 106 : 074101 . Google Scholar CrossRef Search ADS 8 Suenaga K , Sato Y , Liu Z , Kataura H , Okazaki T , Kimoto K , Sawada H , Sasaki T , Omoto K , Tomita T , Kaneyama T , and Kondo Y ( 2009 ) Visualizing and identifying single atoms using electron energy-loss spectroscopy with low accelerating voltage . Nat. Chem. 1 : 415 – 418 . Google Scholar CrossRef Search ADS PubMed 9 Krivanek O , Dellby N , and Lupini A ( 1999 ) Towards sub-Å electron beams . Ultramicroscopy 78 : 1 – 11 . Google Scholar CrossRef Search ADS 10 Sawada H , Sasaki T , Hosokawa F , Yuasa S , Terao M , Kawazoe M , Nakamichi T , Kaneyama T , Kondo Y , Kimoto K , and Suenaga K ( 2009 ) Correction of higher order geometrical aberration by triple 3-fold astigmatism field . J. Electron Microsc. 58 : 341 – 347 . Google Scholar CrossRef Search ADS 11 Kabius B , Hartel P , Haider M , Müller H , Uhlemann S , Loebau U , Zach J , and Rose H ( 2009 ) First application of Cc-corrected imaging for high-resolution and energy-filtered TEM . J. Electron Microsc. 58 : 147 – 155 . Google Scholar CrossRef Search ADS 12 Linck M , Hartel P , Uhlemann S , Kahl F , Muller H , Zach J , Haider M , Niestadt M , Bischoff M , Biskupek J , Lee Z , Lehnert T , Borrnert F , Rose H , and Kaiser U ( 2016 ) Chromatic aberration correction for atomic resolution TEM imaging from 20 to 80 kV . Phys. Rev. Lett. 117 : 076101 . Google Scholar CrossRef Search ADS PubMed 13 Sawada H , Tomita T , Naruse M , Honda T , Hambridge P , Hartel P , Haider M , Hetherington C , Doole R , and Kirkland A ( 2005 ) Experimental evaluation of a spherical aberration-corrected TEM and STEM . J. Electron Microsc. 54 : 119 – 121 . 14 Müller H , Uhlemann S , Hartel P , and Haider M ( 2006 ) Advancing the Hexapole Cs-corrector for the scanning transmission electron microscope . Microsc. Microanal. 12 : 442 – 455 . Google Scholar CrossRef Search ADS PubMed 15 Sawada H , Hosokawa F , Sasaki T , Kaneyama T , Kondo Y , and Suenaga K ( 2011 ) Chapter 6 Aberration correctors developed under the triple C project. In: Hawkes P (ed.) Advances in Imaging and Electron Physics Vol 168 : pp. 297 – 336 ( Academic Press , San Diego). 16 Krivanek O , Dellby N , Keyse R , Murfitt M , Own C , and Szilagyi Z ( 2008 ) Chapter 3 Advances in aberration-corrected scanning transmission electron microscopy and electron energy-loss spectroscopy. In: Hawkes P (ed.) Advances in Imaging and Electron Physics Vol 153 : pp. 121 – 160 ( Academic Press , New York ). 17 Hosokawa F , Sawada H , Kondo Y , Takayanagi K , and Suenaga K ( 2013 ) Development of Cs and Cc correctors for transmission electron microscopy . Microscopy 62 : 23 – 41 . Google Scholar CrossRef Search ADS PubMed 18 Beck V ( 1979 ) A hexapole spherical aberration corrector . Optik 4 : 241 – 255 . 19 Mukai M , Kim J S , Omoto K , Sawada H , Kimura A , Ikeda A , Zhou J , Kaneyama T , Young N P , Warner J H , Nellist P D , and Kirkland A I ( 2014 ) The development of a 200 kV monochromated field emission electron source . Ultramicroscopy 140 : 37 – 43 . Google Scholar CrossRef Search ADS PubMed 20 Morishita S , Mukai M , Suenaga K , and Sawada H ( 2016 ) Resolution enhancement in transmission electron microscopy with 60-kV monochromated electron source . Appl. Phys. Lett. 108 : 013107 . Google Scholar CrossRef Search ADS 21 elbis, BioNet Laboratory Inc., http://www.bio-net.co.jp/SP-elbis_e/index.html. © The Author(s) 2018. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Microscopy Oxford University Press

Evaluation of residual aberration in fifth-order geometrical aberration correctors

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Abstract

Abstract Higher order geometrical aberration correctors for transmission electron microscopes are essential for atomic-resolution imaging, especially at low-accelerating voltages. We quantitatively calculated the residual aberrations of fifth-order aberration correctors to determine the dominant aberrations. The calculations showed that the sixth-order three-lobe aberration was dominant when fifth-order aberrations were corrected by using the double-hexapole or delta types of aberration correctors. It was also deduced that the sixth-order three-lobe aberration was generally smaller in the delta corrector than in the double-hexapole corrector. The sixth-order three-lobe aberration was counterbalanced with a finite amount of the fourth-order three-lobe aberration and 3-fold astigmatism. In the experiments, we used a low-voltage microscope equipped with delta correctors for probe- and image-forming systems. Residual aberrations in each system were evaluated using Ronchigrams and diffractogram tableaux, respectively. The counterbalanced aberration correction was applied to obtain high-resolution transmission electron microscopy images of graphene and WS2 samples at 60 and 15 kV, respectively. transmission electron microscope, aberration corrector, high-resolution image, low accelerating voltage Introduction The resolution of transmission electron microscopes (TEM) has drastically improved since aberration correctors were developed in the 1990s [1]; resolutions better than 50 pm have been achieved at 300 kV [2–4]. In recent years, correctors have been used at various accelerating voltages ranging from 15 kV [5, 6] to 1.2 MV [7] for high-resolution imaging. Aberration correctors have become indispensable in the atomic level analysis of materials [8]. A corrector of spherical aberration was developed by Haider et al. [1] for TEM and scanning TEM (STEM) with double hexapole fields. A corrector with quadrupole–octapole fields was developed by Krivanek et al. [9] for STEM. There have also been reports of a higher order aberration corrector [10] and chromatic aberration corrector [11, 12]. The most widely used correctors consist of magnetic fields with 3-fold symmetry [1]. The 3-fold symmetric field is generated by a hexapole or a dodecapole. A corrector consisting of double hexapole-fields can correct geometrical aberrations up to the fourth-order, including third-order spherical aberration. The uniform phase angle realized by these correctors, which can be observed in a Ronchigram, is limited up to 30–40 mrad [13] because the fifth-order aberration of a 6-fold astigmatism is intrinsically introduced in this type of corrector and dominantly limits the uniform phase angle. To correct the 6-fold astigmatism in the hexapole field corrector, two types of correctors were developed. One is a double hexapole-field corrector with optimized multipole thickness [14] and the other is a triple hexapole-field corrector [10]. Under the correction of the 6-fold astigmatism by these fifth-order aberration correctors, the flat-phase angle was extended to 70 mrad. Towards the larger flat-phase angle beyond that achieved by the advanced correctors, we need to determine which aberration is dominant after the correction of the 6-fold astigmatism and identify the aberrations quantitatively. Moreover, after this identification, we need to explore ways to reduce or correct these aberrations. In this paper, we present the residual aberrations after the correction of the 6-fold astigmatism and determine methods to reduce the aberration through calculations and experiments. Calculation of residual aberration in fifth-order aberration corrector In this section, we present the calculated residual aberrations on a double hexapole-field corrector (advanced double-hexapole corrector [14]) and triple hexapole-field corrector (delta corrector [10]). To calculate the electron trajectory in a hexapole field that is required to estimate the aberrations, an iterative method with complex geometrical aberration coefficients [15] was used. Fringe fields near the entrance and exit plane of the hexapole fields were neglected by assuming a sharp cut-off field approximation in the calculation. When an electron beam passes into a hexapole field parallel to the optic axis, the electron trajectory at the exit plane of the hexapole field contains aberrations which are 3-fold astigmatism, third-order spherical, fourth-order three-lobe, fifth-order spherical, 6-fold astigmatism and sixth-order three-lobe aberration (see Table 1 and Supplementary data Appendix 1). The symmetry of these aberrations is circular or 3N-fold (N: integer) because these aberrations are generated as combination aberrations due to the 3-fold astigmatism. This can be predicted by the combination rule [16]. The electron trajectories at the exit of the double-hexapole and the delta correctors show same aberrations with different coefficients, since these aberrations are always generated after a hexapole field. Table 1. List of aberrations. Those in bold can be generated in correctors using hexapole fields. Wave aberrations are shown by using a complex angle ω and its complex conjugate ω̅ Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] Table 1. List of aberrations. Those in bold can be generated in correctors using hexapole fields. Wave aberrations are shown by using a complex angle ω and its complex conjugate ω̅ Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] Symbol Order (wave) Order (geometrical) Symmetry Aberration Wave aberration P1 1 0 1 Shift Re[P1ω̅] O2 2 1 Round Defocus Re [12O2ω̅ω ] A2 2 1 2 2-Fold astigmatism Re [12A2ω̅2 ] P3 3 2 1 Second-order axial coma Re [13P3ω̅2ω ] A3 3 2 3 3-Fold astigmatism Re [13A3ω̅3 ] O4 4 3 Round Third-order spherical aberration Re [14O4ω̅2ω2 ] Q4 4 3 2 Third-order star aberration Re [14Q4ω̅3ω ] A4 4 3 4 4-Fold astigmatism Re [14A4ω̅4 ] P5 5 4 1 Fourth-order axial coma Re [15P5ω̅3ω2 ] R5 5 4 3 Fourth-order three-lobe aberration Re [15R5ω̅4ω ] A5 5 4 5 5-Fold astigmatism Re [15A5ω̅5 ] O6 6 5 Round Fifth-order spherical aberration Re [16O6ω̅3ω3 ] Q6 6 5 2 Fifth-order star aberration Re [16Q6ω̅4ω2 ] S6 6 5 4 Fifth-order rosette aberration Re [16S6ω̅5ω ] A6 6 5 6 6-Fold astigmatism Re [16A6ω̅6 ] P7 7 6 1 Sixth-order axial coma Re [17P7ω̅4ω3 ] R7 7 6 3 Sixth-order three-lobe aberration Re [17R7ω̅5ω2 ] T7 7 6 5 Sixth-order pentacle aberration Re [17T7ω̅6ω ] A7 7 6 7 7-Fold astigmatism Re [17A7ω̅7 ] We first evaluated the residual aberrations in the advanced double-hexapole corrector [14]. Figure 1a shows a schematic of the double-hexapole corrector. We implemented an aberration correction system in which the focal lengths of the transfer lens doublet between the hexapole fields were the same for simplicity; however, these can be different in an asymmetric-type corrector [17]. We ignored parasitic aberrations, such as coma and star aberrations as these can be compensated by the beam alignments using deflectors in the correctors. By using the calculated combination aberrations shown in Supplementary data Appendix 1, we calculated aberrations generated by the double-hexapole corrector. All equations shown below were obtained from our calculation. Three-fold astigmatism (A3) and three-lobe aberration (R5) become A3=A3,HF1L1M3−A3,HF2L2M3 (1) R5=3A3,HF1̅A3,HF12L15+A3,HF2̅L23(−5A3,HF12L12+5A3,HF1A3,HF2L1L2−3A3,HF22L22)12M5fOL4 (2) where A3,HF1 and A3,HF2 denote the coefficients of the 3-fold astigmatism, L1 and L2 denote the lengths of the hexapole fields, M is the demagnification ratio between the second hexapole field and the objective lens and fOL is the focal length of the objective lens. A3 and R5 become zero when the corrector satisfied the condition A3,HF1 = A3,HF2 and L1 = L2. The negative third-order spherical aberration generated in the hexapole fields was used for compensation of the positive third-order spherical aberration of the objective lens. Consequently, all aberrations up to fourth-order were eliminated. A residual aberration higher than fifth-order, which cannot be compensated, is a 6-fold astigmatism as shown in equation below: A6=−9M2O42L56fOL2+9CsTLfOL2O4M2fTL2L (3) where O4 is the absolute coefficient of the negative spherical aberration generated by the hexapole fields, and fTL and CsTL are the focal length and the spherical aberration coefficient of the transfer lenses between the hexapole fields, respectively. Arg(A3,HF1), which is the direction of the 3-fold astigmatism generated by hexapole field 1 (HF1), was set to 0°. The first term in Eq. (3) is the 6-fold astigmatism generated by the two hexapole fields. The second term is the 6-fold astigmatism due to the combination aberration of the first hexapole field and the spherical aberration of the transfer lenses. These two terms with opposite signs can cancel each other out for a particular condition. We can find that particular length of the hexapole fields where A6 = 0 as in [14] LA6=0=214CsTLfOL2M2fTLO4. (4) Fig. 1. View largeDownload slide Schematic of (a) the double-hexapole corrector and (b) the delta corrector used in the aberration calculation. Fig. 1. View largeDownload slide Schematic of (a) the double-hexapole corrector and (b) the delta corrector used in the aberration calculation. Other than the 6-fold astigmatism, the fifth-order spherical aberration appears in the double-hexapole corrector. This aberration can be compensated by adjusting the transfer length between the principal plane of the hexapole field 2 (HF2) and the front focal plane of the objective lens, by using the combination aberration [18]. As described above, the aberrations up to fifth-order can be compensated in the double-hexapole corrector. From the calculation, the next influential aberration is the sixth-order three-lobe aberration (R7). R7 is a dominant residual aberration in the double-hexapole corrector after A6 is corrected, which is expressed as: R7=−7M3L3/2O45/286fOL3+732CsTLfOL(L2−12fTL2)O43/24MLfTL4. (5) The first term is generated by the two hexapole fields. The second term is introduced as the combination aberration of the hexapole fields and the spherical aberration of the transfer lenses. Assigning typical values to the parameters in Eq. (5); fOL = 1.0 mm, O4 = 0.5 mm, M = 1.5, fTL = 30 mm, CsTL = 500 mm and L = 15 mm, the three-lobe aberration coefficient becomes R7 = 13.2 mm. For an acceleration voltage of 60 kV, the phase difference from the axis exceeds π/4 at 44 mrad. We also calculated the residual aberrations in the delta corrector. Figure 1b shows a schematic of the delta corrector. Under the condition L1 = L2 = L3, A3 and R5 become zero when |A3,HF1 |= |A3,HF3 |, |A3,HF2 |=22/3 |A3,HF1 |≈1.63 |A3,HF1 | and Arg(A3,HF3)−Arg(A3,HF1)=±arctan(22)≈±70.5°. Since this relationship between vectors of the three hexapole fields resembles the Greek symbol Δ [10], this condition is called the delta condition. In the delta condition, A6 is small, but not zero. A3 and A6 become 0 when |A3,HF1 |= |A3,HF3 |, |A3,HF2 |=(9+281)/10 |A3,HF1 |≈1.61 |A3,HF1 | and Arg(A3,HF3)−Arg(A3,HF1)=±arctan(−2+22281−11+281)≈±73.3°, under the condition: L1 = L2 = L3 and CsTL = 0. Therefore, the two conditions, when A3 = R5 = 0 and A3 = A6 = 0, are slightly different in the angle and excitation ratio between the three hexapole fields for the delta corrector. When CsTL is not zero, the above solved conditions change slightly depending on CsTL. In practice, however, R5 can be controlled by changing the excitation of the transfer lenses between the hexapole fields as with the double-hexapole corrector. This adjustment changes R5 and keeps the A6 ≈ 0 condition. Therefore, all aberrations up to fifth-order can be corrected in the delta corrector. As for the advanced double-hexapole corrector, the next influential aberration is the sixth-order three-lobe aberration. In the delta condition, the sixth-order three-lobe coefficient (R7) becomes R7=(−21002ⅈ+32532)M3L3/2O45/2352807fOL3−CsTLfOL((−24ⅈ+2)L2+12(22ⅈ+2)fTL2)O43/247MLfTL4 (6) when L1 = L2 = L3, and Arg(A3,HF1) = 0°. R7 is a dominant residual aberration in the delta corrector as well. Assigning the typical values used above, R7 = 8.8 mm. The phase difference from the axis caused by this R7 exceeds π/4 at 47 mrad for 60 kV. Residual amount of sixth-order three-lobe aberration The sixth-order three-lobe aberration becomes the dominant residual aberration in both, the advanced double-hexapole and delta correctors, as a result of the calculations. Collecting coefficients from Eqs. (5) and (6), R7 can be expressed as R7=k1M3L3/2O45/2fOL3+k2CsTLfOLO43/2MLfTL2+k3L3/2CsTLfOLO43/2MfTL4=k1M3L3/2O45/2fOL3+CsTLfOLL3/2O43/2MfTL2(k21L2+k31fTL2) (7) with different complex coefficients k1, k2 and k3. The coefficients of the double-hexapole corrector are k1 ≈ −0.35, k2 ≈ −25.7, and k3 ≈ 2.14, which are real because the direction of each term corresponds to the direction of the 3-fold astigmatism generated by HF1. The coefficients of the delta corrector are k1=(−21002ⅈ+32532)352807≈0.23e−0.43πi, k2=(264ⅈ+122)47≈24.9e−0.52πi, and k3=(−24ⅈ+2)47≈2.27e0.51πi. The absolute values of k1 and k2 in case of the delta corrector are smaller than those in case of the double-hexapole corrector, which means that R7 shown by the first and second terms is smaller in the delta corrector than the double-hexapole corrector. The absolute value of the third term, which shows almost the opposite direction of the first and second terms, is larger in the delta corrector than in the double-hexapole corrector. In addition, the absolute value of the second term is larger than that of the third term when L < fTL, which is a standard condition for both the correctors. Thus, R7 of the delta corrector becomes smaller than that of the double-hexapole corrector when the variables, such as fOL, M, L, O4, fTL and CsTL are the same. For a reduction of R7, the delta corrector is preferable to the double-hexapole corrector. Experimental evaluation of the residual aberration We experimentally corroborated the residual aberrations by using a TEM dedicated to low acceleration voltages equipped with delta correctors for probe- and image-forming systems. First, we evaluated the residual aberrations for a probe-forming system at an acceleration voltage of 60 kV. Figure 2a–c shows experimental Ronchigrams at the underfocus, in-focus, and overfocus conditions, respectively. By assigning the experimental parameters in Eq. (6), R7 was calculated to be 17 mm, resulting in π/4 limit of 43 mrad. In the experiment, R7 was counterbalanced by R5 and A3, because they have a 3-fold symmetry [6]. By counterbalancing R7 = 17 mm with R5 = 92 μm and A3 = 98 nm, the π/4 limit is improved from 43 mrad to 72 mrad. The simulated Ronchigrams after counterbalancing are shown in Fig. 2d–f. The figures are consistent with the experimental results. Fig. 2. View largeDownload slide (a–c) Experimental Ronchigrams with different defoci using the delta corrector at 60 kV. (d–f) Simulated Ronchigrams with R7 = (17 mm, 70°), R5 = (92 μm, 10°) and A3 = (98 nm, 70°). Fig. 2. View largeDownload slide (a–c) Experimental Ronchigrams with different defoci using the delta corrector at 60 kV. (d–f) Simulated Ronchigrams with R7 = (17 mm, 70°), R5 = (92 μm, 10°) and A3 = (98 nm, 70°). We also evaluated the residual aberrations for an image-forming system with those experimentally measured using a diffractogram tableau method at 60 and 15 kV. Using Eq. (6), calculated value of R7 for the image-forming system was found to be 14 and 34 mm, at 60 and 15 kV, respectively. In the experiments, these were measured as 9 and 15 mm by the diffractogram tableau method (Supplementary data Appendix 2), and had good agreement with the simulated results. The discrepancy between the experiment and calculation may be due to measurement errors, or could have been caused by slightly different values of the parameters. For example, the demagnification ratio M was changed to correct the third- and fifth-order spherical aberrations; this affects the value of R7 with the third power of M. Both the values of R7 lead to a π/4 limit of <50 mrad, which can be expanded to more than 70 mrad by the counterbalance of R7, R5 and A3. Figure 3 shows the TEM images of monolayer graphene at 60 kV. To reduce the effects of chromatic aberration, the energy spread of the electron source was decreased to 0.05 eV by using a monochromator [19, 20]. The power spectrum of the graphene image shows the spots corresponding to 0.071 nm, which indicates that the information limit is 15 times the wavelength of a primary electron (4.87 pm at 60 kV). The experimental image shows each carbon atom clearly by resolving the C-C dumbbells with a separation of 0.142 nm. The simulated image with R7 of 9 mm also shows carbon atoms clearly and is consistent with the experimental image. The residual R7 of the image-forming system with the delta corrector is sufficiently small to observe single carbon atoms in graphene. Fig. 3. View largeDownload slide (a) Experimental TEM image of monolayer graphene at 60 kV. The measured sixth-order three-lobe R7 was 9 mm. The third- and fifth-order spherical aberrations were corrected to ~0. The energy width was reduced to 0.05 eV using a monochromator. The underfocused image was taken by Gatan OneView with an exposure time of 8 s with drift compensation. No filtering process was applied to the image. (b) Modulus of the Fourier transform of (a). Spots of 14.1 nm−1 = (0.071 nm)−1 correspond to (15 λ)−1. (c) Averaged image of small areas in (a). (d) Simulated image with R7 = 9 mm, 30°, and a defocus of −1.5 nm calculated using the software elbis [21]. Fig. 3. View largeDownload slide (a) Experimental TEM image of monolayer graphene at 60 kV. The measured sixth-order three-lobe R7 was 9 mm. The third- and fifth-order spherical aberrations were corrected to ~0. The energy width was reduced to 0.05 eV using a monochromator. The underfocused image was taken by Gatan OneView with an exposure time of 8 s with drift compensation. No filtering process was applied to the image. (b) Modulus of the Fourier transform of (a). Spots of 14.1 nm−1 = (0.071 nm)−1 correspond to (15 λ)−1. (c) Averaged image of small areas in (a). (d) Simulated image with R7 = 9 mm, 30°, and a defocus of −1.5 nm calculated using the software elbis [21]. Figure 4 shows the TEM image of monolayer tungsten disulfide WS2 at 15 kV. The power spectrum showed spots corresponding to 0.091 nm which is 9.2 times the electron wavelength (λ = 9.94 pm at 15 kV). These spots indicate that the information limit is sufficiently small to obtain atomic-resolution images at 15 kV. In the experimental image of WS2, an atomic distance of 0.182 nm was resolved. The simulated image, with a value of 16 mm for R7, did not show atomic-resolution image because the calculated phase shift exceeded the π/4 limit at 48 mrad which corresponds to 0.21 nm at 15 kV. R7 can be counterbalanced with R5 of 108 μm and A3 of 143 nm. For this condition, the π/4 limit expanded to 80 mrad, which corresponds to 0.12 nm. The simulated image using the counterbalance shows each atom clearly, as the one from the experiment. This result suggests that the atomic-resolution imaging at the ultra-low accelerating voltage of 15 kV essentially requires the wide flat-phase angle realized by a small value of R7 and its counterbalance with R5 and A3. Fig. 4. View largeDownload slide (a) Experimental TEM image of monolayer WS2 at 15 kV. The measured R7 was 16 mm. O4 and O6 were corrected to ~0. An energy width of 0.05 eV and an exposure time of 5 s were used. No filtering process was applied to the image. Various contrasts of the atoms appear in the image due to the specimen bend. (b) Modulus of the Fourier transform of (a). Spots of 11.0 nm−1 = (0.091 nm)−1 correspond to (9.2 λ)−1. White vertical stripes were caused by a characteristic of the camera. (c) Magnified image of indicated area in (a). (d) Simulated image with R7 = (16 mm, −8°), R5 = (108 μm, 52°), A3 = (143 nm, −8°) and (e) R7 = (16 mm, −8°). A defocus of −1.5 nm was used in both (d) and (e). Fig. 4. View largeDownload slide (a) Experimental TEM image of monolayer WS2 at 15 kV. The measured R7 was 16 mm. O4 and O6 were corrected to ~0. An energy width of 0.05 eV and an exposure time of 5 s were used. No filtering process was applied to the image. Various contrasts of the atoms appear in the image due to the specimen bend. (b) Modulus of the Fourier transform of (a). Spots of 11.0 nm−1 = (0.091 nm)−1 correspond to (9.2 λ)−1. White vertical stripes were caused by a characteristic of the camera. (c) Magnified image of indicated area in (a). (d) Simulated image with R7 = (16 mm, −8°), R5 = (108 μm, 52°), A3 = (143 nm, −8°) and (e) R7 = (16 mm, −8°). A defocus of −1.5 nm was used in both (d) and (e). Discussion of the compensation of the sixth-order three-lobe aberration In the case of the advanced double-hexapole corrector, A6 is corrected by the combination aberration using the spherical aberration CsTL of transfer lenses. We first discuss below, the possibility to compensate R7 in the advanced double-hexapole corrector. The first term of Eq. (5) represents a portion of R7 generated only by the hexapole fields. The second term represents another portion of R7 generated as the combination aberration between the hexapole fields and CsTL. Since the focal length of the transfer lens is larger than half the thickness of the hexapole field (fTL > L/2), (L2−12fTL2) in the second term becomes negative. Therefore, both the first and second terms in Eq. (5) are negative and consequently both terms of R7 produce an aberration in the same direction. Therefore, R7 in the configuration of an advanced double-hexapole corrector is inevitable. For the case of the delta corrector, the direction of R7 generated by only the hexapole fields [the first term in Eq. (6)] is similar to that generated by the combination of the hexapole fields and CsTL [the second and third terms in Eq. (6)], and therefore cannot be canceled out either. In conclusion, R7 cannot be compensated in either the double-hexapole or delta correctors. To compensate R7 for wider uniform phase areas, different types of correctors might be required. Summary The aberration calculation showed that the sixth-order three-lobe aberration (R7) is the dominant residual aberration after the correction of the fifth-order aberration in the double-hexapole and the delta correctors. R7 is generally smaller in the delta corrector than in the advanced double-hexapole corrector. Counterbalancing R7 with the fourth-order three-lobe aberration (R5) and the 3-fold astigmatism (A3), the flat-phase angle was extended to more than 70 mrad, which was demonstrated experimentally in the probe- and image-forming systems. The small residual R7 and the effect of the counterbalance were experimentally identified in low-kV TEM imaging using samples of monolayer sheets of graphene and WS2. Supplementary data Supplementary data are available at Microscopy online. Acknowledgments The authors thank M. Mukai and T. Sasaki of JEOL for their cooperation in the development and installation of the microscope. The authors also thank Y.-C. Lin of AIST for the sample preparation and donation for the TEM observations. The authors acknowledge Y. Kondo for his careful reading of this manuscript. Funding This work was supported by the JST (Japan Science and Technology Agency), Research Acceleration Program (2012–2016). References 1 Haider M , Uhlemann S , Schwan E , Rose H , Kabius B , and Urban K ( 1998 ) Electron microscopy image enhanced . Nature 392 : 768 – 769 . Google Scholar CrossRef Search ADS 2 Erni R , Rossell M D , Kisielowski C , and Dahmen U ( 2009 ) Atomic-resolution imaging with a sub-50-pm electron probe . Phys. Rev. Lett. 102 : 096101 . Google Scholar CrossRef Search ADS PubMed 3 Sawada H , Tanishiro Y , Ohashi N , Tomita T , Hosokawa F , Kaneyama T , Kondo Y , and Takayanagi K ( 2009 ) STEM imaging of 47-pm-separated atomic columns by a spherical aberration-corrected electron microscope with a 300-kV cold field emission gun . J. Electron Microsc. 58 : 357 – 361 . Google Scholar CrossRef Search ADS 4 Sawada H , Shimura N , Hosokawa F , Shibata N , and Ikuhara Y ( 2015 ) Resolving 45-pm-separated Si-Si atomic columns with an aberration-corrected STEM . Microscopy 64 : 213 – 217 . Google Scholar CrossRef Search ADS PubMed 5 Sasaki T , Sawada H , Hosokawa F , Sato Y , and Suenaga K ( 2014 ) Aberration-corrected STEM/TEM imaging at 15 kV . Ultramicroscopy 145 : 50 – 55 . Google Scholar CrossRef Search ADS PubMed 6 Morishita S , Mukai M , Suenaga K , and Sawada H ( 2016 ) Atomic resolution imaging at an ultralow accelerating voltage by a monochromatic transmission electron microscope . Phys. Rev. Lett. 117 : 153004 . Google Scholar CrossRef Search ADS PubMed 7 Akashi T , Takahashi Y , Tanigaki T , Shimakura T , Kawasaki T , Furutsu T , Shinada H , Müller H , Haider M , Osakabe N , and Tonomura A ( 2015 ) Aberration corrected 1.2-MV cold field-emission transmission electron microscope with a sub-50-pm resolution . Appl. Phys. Lett. 106 : 074101 . Google Scholar CrossRef Search ADS 8 Suenaga K , Sato Y , Liu Z , Kataura H , Okazaki T , Kimoto K , Sawada H , Sasaki T , Omoto K , Tomita T , Kaneyama T , and Kondo Y ( 2009 ) Visualizing and identifying single atoms using electron energy-loss spectroscopy with low accelerating voltage . Nat. Chem. 1 : 415 – 418 . Google Scholar CrossRef Search ADS PubMed 9 Krivanek O , Dellby N , and Lupini A ( 1999 ) Towards sub-Å electron beams . Ultramicroscopy 78 : 1 – 11 . Google Scholar CrossRef Search ADS 10 Sawada H , Sasaki T , Hosokawa F , Yuasa S , Terao M , Kawazoe M , Nakamichi T , Kaneyama T , Kondo Y , Kimoto K , and Suenaga K ( 2009 ) Correction of higher order geometrical aberration by triple 3-fold astigmatism field . J. Electron Microsc. 58 : 341 – 347 . Google Scholar CrossRef Search ADS 11 Kabius B , Hartel P , Haider M , Müller H , Uhlemann S , Loebau U , Zach J , and Rose H ( 2009 ) First application of Cc-corrected imaging for high-resolution and energy-filtered TEM . J. Electron Microsc. 58 : 147 – 155 . Google Scholar CrossRef Search ADS 12 Linck M , Hartel P , Uhlemann S , Kahl F , Muller H , Zach J , Haider M , Niestadt M , Bischoff M , Biskupek J , Lee Z , Lehnert T , Borrnert F , Rose H , and Kaiser U ( 2016 ) Chromatic aberration correction for atomic resolution TEM imaging from 20 to 80 kV . Phys. Rev. Lett. 117 : 076101 . Google Scholar CrossRef Search ADS PubMed 13 Sawada H , Tomita T , Naruse M , Honda T , Hambridge P , Hartel P , Haider M , Hetherington C , Doole R , and Kirkland A ( 2005 ) Experimental evaluation of a spherical aberration-corrected TEM and STEM . J. Electron Microsc. 54 : 119 – 121 . 14 Müller H , Uhlemann S , Hartel P , and Haider M ( 2006 ) Advancing the Hexapole Cs-corrector for the scanning transmission electron microscope . Microsc. Microanal. 12 : 442 – 455 . Google Scholar CrossRef Search ADS PubMed 15 Sawada H , Hosokawa F , Sasaki T , Kaneyama T , Kondo Y , and Suenaga K ( 2011 ) Chapter 6 Aberration correctors developed under the triple C project. In: Hawkes P (ed.) Advances in Imaging and Electron Physics Vol 168 : pp. 297 – 336 ( Academic Press , San Diego). 16 Krivanek O , Dellby N , Keyse R , Murfitt M , Own C , and Szilagyi Z ( 2008 ) Chapter 3 Advances in aberration-corrected scanning transmission electron microscopy and electron energy-loss spectroscopy. In: Hawkes P (ed.) Advances in Imaging and Electron Physics Vol 153 : pp. 121 – 160 ( Academic Press , New York ). 17 Hosokawa F , Sawada H , Kondo Y , Takayanagi K , and Suenaga K ( 2013 ) Development of Cs and Cc correctors for transmission electron microscopy . Microscopy 62 : 23 – 41 . Google Scholar CrossRef Search ADS PubMed 18 Beck V ( 1979 ) A hexapole spherical aberration corrector . Optik 4 : 241 – 255 . 19 Mukai M , Kim J S , Omoto K , Sawada H , Kimura A , Ikeda A , Zhou J , Kaneyama T , Young N P , Warner J H , Nellist P D , and Kirkland A I ( 2014 ) The development of a 200 kV monochromated field emission electron source . Ultramicroscopy 140 : 37 – 43 . Google Scholar CrossRef Search ADS PubMed 20 Morishita S , Mukai M , Suenaga K , and Sawada H ( 2016 ) Resolution enhancement in transmission electron microscopy with 60-kV monochromated electron source . Appl. Phys. Lett. 108 : 013107 . Google Scholar CrossRef Search ADS 21 elbis, BioNet Laboratory Inc., http://www.bio-net.co.jp/SP-elbis_e/index.html. © The Author(s) 2018. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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MicroscopyOxford University Press

Published: Feb 21, 2018

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