Aberration correction for low voltage optimized transmission electron microscopy

Method Article Aberration correction for low voltage optimized transmission electron microscopy Jaromír Ba9covský a,b a DELONG INSTRUMENTS a.s., Palackého t rída 3019/153 b, 612 00 Brno, Czech Republic b Institute of Physical Engineering, Faculty of Mechanical Engineering, Brno university of Technology, Technická 2896/2, 616 69 Brno, Czech Republic A B S T R A C T Further development of low voltage electron microscopy leads to an aberration correction of the device in order to improve its spatial resolution. The integration of a corrector to a desktop transmission electron microscope with exclusivelylow-voltagedesignseemstobeachallengingtask.The benefitsandpotentialoftheRose hexapolecorrector implemented to such a system are critically considered in this paper. The feasibility of miniaturized corrector suitable for desktop LVEM is especially discussed, including the aspect of corrector contribution to chromatic aberration that appears to be crucial. Optimal corrector parameters and resolution limits of such a system are proposed.  Improved spatial resolution  Spherical aberration correction  Permanent magnet transfer lenses © 2018 DELONG INSTRUMENTS a.s. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). A R T I C L E I N F O Method name: Aberration corrected low voltage transmission electron micrscopy Keywords: Low voltage transmission electron microscopy, Aberration correction, Hexapole corrector Article history: Received 16 October 2017; Accepted 17 August 2018; Available online 25 August 2018 Introduction In contrast to advanced light-optical microscopy techniques capable of resolution beyond the diffraction limit, which is considered the physical limit for classical microscopy, electron microscopy has not yet approached this physical limitation. Uncorrected conventional electron microscopy limited mainly by chromatic and spherical aberration is not able to achieve spatial resolution better than 50l [1]. E-mail address: [email protected] (J. Ba9covský). https://doi.org/10.1016/j.mex.2018.08.009 2215-0161/© 2018 DELONG INSTRUMENTS a.s. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). MethodsX 5 (2018) 1033–1047 Contents lists available at ScienceDirect MethodsX journal homepage: www.elsevier.com/locate/mex There are two combinable approaches to resolve smaller objects: reduction of the De Broglie wavelength and correction of aberrations. Although l itself is small enough even for low accelerating voltage, its reduction has a significant influence on system aberration. On the contrary, increasing accelerating voltage has high requirements for technical parameters of electronics, as well as dimensions of main microscope column and sample stability against radiation damage. The appropriate way to exploit the physical potential of the instruments is to correct the most serious aberrations. Basic principles of aberration correction have been known for 70 years [2], but practical implementation has been complicated by technological limitations for a very long time. The first generation of correctors called “proof of principle” confirmed the possibility to use rotational asymmetric multipole electronoptical components for aberration correction. However their own imperfections severely deteriorated image resolution. The rapid development of accuracy of mechanical manufacturing and stability of the electronics over the past decades has enabled the practical use of correctors in the most advanced electron microscopes [3]. The current commercially available corrected transmission microscopes work with accelerating voltage in the range of 100– 200 kV, but corrected systems with lower energy are still under development. Low voltage transmission electron microscopy (LVEM) uses an electron beam with an energy of 5– 30 kV. The reduction of radiation damage of the sample and contrast enhancement necessary for investigation of sensitive samples containing light atoms and weak bonds can be considered as the main advantage of such systems. Certain technical aspects also allow the LVEM to be constructed in a desktop design, in contrast to the common dimensions of conventional TEM/STEM devices. In this article the feasibility and potential of aberration corrections of desktop LVEM will be discussed with special attention to 5 kV LVEM5 [4] and 25 kV LVEM25 [5] optimized systems developed by Delong Instruments a.s. There are experimental projects focused on the development and construction of a suitable corrector designed for LVEM [6–8]. Namely The Sub-Ångstrom Low-Voltage Electron Microscopy (SALVE) project from the research group from Ulm University and the Japanese project of Delta corrector should be mentioned. Different approaches to the construction of a corrector were chosen by these groups, though both of them did use readjustment of conventional high voltage microscope for low energy. These solutions for LVEMs are not able to completely profit from the low energy of electrons. Our goal is to investigate of the possible implementation of a compact corrector to the existing Delong LVEM desktop microscope with an exclusively low-voltage design. Correctors Image resolution is mainly influenced by a spherical and chromatic aberration of the objective lens. Correction of these unwanted phenomenons is the first step to improve imaging ability. The effects of other terms in the aberration polynomial are less important. The degree of importance of each aberration is a relevant issue determining the appropriate method of correction. According to Scherzer's proposals, successful corrections of unavoidable primary spherical and chromatic aberrations of round lenses are based on multipole elements. Various configurations of multipoles with different correction abilities have been proposed [2]. The most successful basic concepts are hexapole and quadrupole–octupole correctors for spherical aberration and a quadrupole–octupole complex corrector for spherical and chromatic aberrations. All concepts have advantages and disadvantages. Their relevancies as a function of electron-optical system parameters are crucial for applicability of the particular correction system. The hexapole corrector of spherical aberration The simplest design of a hexapole corrector consists of two hexapole elements separated by a round-lens doublet. The system of the two hexapoles is inconvenient not only because of its increasing correction power, but it also reduces the intrinsic hexapole astigmatism – a predominant hexapole aberration [9,10]. A detailed scheme with required positioning of each component can be found in [11]. 1034 J. Ba9covský / MethodsX 5 (2018) 1033–1047 For the purpose of coma elimination and minimization of the secondary 5th order spherical aberration, enhanced by the combination of a hexapole with the objective lens, there has to be another round lens doublet situated in front of the corrector [12]. This kind of corrector is not able to solve the problem of chromatic aberration. On the contrary, the transfer lenses increase the total chromatic aberration of the electron-optical system. The effect is remarkable especially with the minimization of the corrector for a low-voltage system. The consequences and reasons for this problem to occur will be discussed later. The quadrupole–octupole corrector The quadrupole–octupole corrector of spherical and chromatic aberrations consists of 4 quadrupoles and 2 octupoles in minimal configuration. However arrangements with a different complexity have been published. The most advanced design, called ultracorrector, was published by Rose [13]. This complex corrector is assembled by combining two identical multipole multiplets, each consisting of seven quadrupoles and seven octopoles [14]. The universal ultracorrector should be able to compensate for the aberrations up to the third rank inclusively. The possibility of correction of chromatic aberration is an undisputed advantage of such systems, but the complexity of the multipole arrangement leads to some problems regarding mechanical alignment. Due to the higher number of multipole elements, the size of the complete device is significantly larger than the hexapole version. It would be negligible, though, when considering conventional transmission microscope with massive main column. It can, however, be important for smaller desktop LVEMs. The comparable size of the corrector and the main column can lead to some additional mechanical vibrations, which further deteriorate the final resolution. That is all due to the increase in size of the microscope column. Parameters of current LVEM 5 & LVEM 25 This paper pays special attention to low voltage microscopes produced by Delong Instruments co. The calculations were made using the parameters of these devices. For clarity, the important parameters as specified by the manufacturer are listed below (Tables 1 and 2) Table 2 Declared experimental resolution limit of uncorrected LVEM 5 and LVEM 25 [4,5]. LVEM 5 LVEM 5 LVEM 25 LVEM 25 LVEM 25 5 keV 5 keV 10 keV 15 keV 25 keV TEM STEM STEM STEM TEM 2 [nm] 2.5 [nm] 1 [nm] 1.3 [nm] 1[nm] Table 1 Relevant theoretically calculated parametres of LVEM5 and LVEM25. [4] [5]. Mode Accelerating voltage [kV] C s [mm] C c [mm] LVEM 5 TEM 5 0.64 0.89 STEM 5 0.64 0.89 LVEM 25 TEM 25 1.03 1.05 STEM 15 0.80 0.85 STEM 10 0.64 0.72 J. Ba9covský / MethodsX 5 (2018) 1033–1047 1035 Estimated resolution of corrected LVEM The total resolution is determined by the combination of all aberrations, but it is necessary to use only the most important ones to arrive at a relevant estimate. In the case of an uncorrected system the following have to be considered: the primary spherical aberration, the chromatic aberration and the diffraction limit. The contributions of the partial aberration discs d s , d c and d d can be summarized to a total aberration disc d: d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 2 s þ d 2 c þ d 2 d q: ð:1Þ It can be specified in more detail: d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 C s a 3 i   2 þ k 2 DE E 0 C c a i   2 þ 0:61l a i   2 s; ð:2Þ where standard notation is used [15]. The experimental spatial resolutions of the current LVEM 5 and LVEM 25 are given in Table 2. The declared values are in good agreement with theoretical models using the standard evaluation method. From Eq. (.1) it is obvious that the optimal solution of a corrected system is not significantly limited by one major contribution, but all primary contributions should be comparable. The degree of importance of each aberration has been studied in various system conditions to determine which aberration is limiting and thus should be corrected. The most manifesting aberrations of common electron optical devices are the chromatic and the geometric (especially the primary spherical aberration) ones, their influences on the size of the paraxial space (determined by aperture angle) are competing with the influence of the diffraction limit. An optimal aperture angle therefore exists. The correct choice of the aperture angle is necessary to maintain the best possible resolution. Optimal aperture angles a optim have been calculated in the range of LVEM accelerating voltage. From Fig. 1 it can be seen that for specific a set of aberration coefficients there is preferred accelerating energy. This conclusion can also be obtained directly from the aberration integrals, where the integrands are dependent on the accelerating voltage. Fig. 1. Optimal accelerating energy for a specific set of aberration coefficients (LVEM 5) C s ¼ 0:64 mm, C c ¼ 0:89 mm, D E ¼ 0:6 eV. 1036 J. Ba9covský / MethodsX 5 (2018) 1033–1047 The results for all above mentioned modes of interest with a typical energy spread DE of Schottky cathode and specific aberration coefficients (Table 2) are presented in Table 3. The change of the optimal aperture angle for typical energy spreads of Schottky and CFE cathodes is 0.2–1.5 mrad. The change between Schottky and CFE is significantly bigger considering the lower electron energy, all because of higher importance of the chromatic aberration. The effect of the monochromatisation degree is continuously demonstrated in Fig. 2. It can be clearly observed that the lower energy spread leads to a larger a optim and thus to a larger paraxial space. The integral aberration discs were calculated with a optim according to Eq. (.2). It can be considered to be the resolution limit of an uncorrected system. The calculations were done for a Gaussian imaging plane (i.e. k 1 ¼ 1, k 2 ¼ 1), where the screen is placed approximately (Table 4). A closer look at the partial aberration discs is helpful in order to understand the behavior of the resolution limit. The data prove that the decrease in the accelerating voltage makes the chromatic aberration more severe. The diameter of the integral aberration disc for the 5 kV LVEM is reduced by Table 3 Optimal setting of aperture angles obtained by minimisation of the integral aberration disc. D E [eV] LVEM 5 LVEM 25 LVEM 25 LVEM 25 5 keV 10 keV 15 keV 25 keV 0.6 (Schottky) 9.5 [mrad] 10.2 [mrad] 9.4 [mrad] 8.3 [mrad] 0.3 (CFE) 11.0 [mrad] 10.6 [mrad] 9.6 [mrad] 8.5 [mrad] Fig. 2. Model parameters C s ¼ 0:64 mm, C c ¼ 0:89 mm, E ¼ 5 keV. Table 4 Integral aberration discs for a optim in the Gaussian imaging plane D E ¼ 0:6 eV. LVEM 5 LVEM 25 LVEM 25 LVEM 25 5 keV 10 keV 15 keV 25 keV d [nm] 1.5 0.92 0.79 0.67 J. Ba9covský / MethodsX 5 (2018) 1033–1047 1037 CFE to app. 80% when using Schottky cathode. In the case of 25 kV with other identical parameters (LVEM5), the integral aberration disc became smaller by only 4% (5). Wave aberration theory Geometrical aberrations can also be described by a phase shift x between an ideal nonaberrated system and a model system with a considered set of aberrations: xðaÞ ¼2p l1 2Dfa 2 þ1 4C 3 a 4 þ1 6C 5 a 6 þ1 8C 7 a 8 þ . . .  : ð:3Þ A different notation is used in Eq. (.3). The subscript i of spherical aberration coefficient C i denotes an order of aberration. A phase shift is also dependent on aperture angle a and the wavelength of used particles l. For the purpose of this paper the polynom will be reduced, because of the correction ability of the hexapole corrector, which is able to achieve active correction of the 3rd order and the partial compensation of the 5th order. Furthermore quadrupole–octupole corrector is able to correction up to 5th order spherical aberration (system limited by the C 7 ). Thus only spherical aberration up to the 7th order is considered. To achieve the best possible resolution, it is necessary to find a compromise between the geometrical aberrations and the diffraction limit. To maintain the required image quality, the phase shift has to be below a certain value. The Rayleigh criterion is used as a commonly accepted phase shift. x ¼ p=2 i.e. P-V (peak to valley) is equal l=4. A higher phase shift causes low intensity of the 0th order maximum of the diffraction pattern. It results in an increase of the intensity of the higher order maximums and de facto to a loss of the image contrast. A quantitative evaluation of the image quality was done by Strehl ratio: S ¼ e ð2pRMSÞ 2 ; ð:4Þ where RMS Root-Mean-Square, a parametre which describes the wavefront, is defined: RMS ¼P  V 4:5: ð:5Þ Although the definition of the spatial resolution itself is a tricky task and different approaches are described in literature and used by different manufacturers, the generally accepted value of Strehl ratio to maintain reasonable image quality is S  0:8 [16,17]. For the purpose of this study the value is set to S ¼ 0:88. The goal of eliminating the undesirable effect of aberrations is to achieve a given limit of a phase shift at the greatest possible aperture angle. The behaviour of the polynomial function (.3) determines the existence of local extremes. The required form of the polynomial, which means oscillating phase shift below a certain predefined limit until the last local extreme appears, can be created by the proper choice of coefficients (Fig. 3). Formulas for the most appropriate coefficients C i , defocus, the corresponding resolution and the aperture angle in different conditions were derived analytically (Table 6) by Intaraprasonk et al. [18] Table 5 The aberration discs for energy spread typical for Schottky and CFE cathodes (0.6 eV, 0.3 eV). Aberration coefficients were used from LVEM5 and LVEM25 [4,5]. 5 keV 25 keV D E 0 0.6 eV 0.3 eV 0.6 eV 0.3 eV d s [nm] 0.27 0.43 0.30 0.31 d c [nm] 1.01 0.59 0.21 0.10 d d [nm] 1.11 0.95 0.56 0.55 d [nm] 1.52 1.20 0.67 0.64 1038 J. Ba9covský / MethodsX 5 (2018) 1033–1047 and Chang et al. [19]. It should be emphasized that it is not optimal to set coefficients to zero, because the aberrations of a lower order are capable of reducing the influence of the higher order aberrations, which then cannot be principally corrected by the chosen type of corrector. According to the optimal setting of coefficients, the significance of different order correction of the spherical aberration is shown in Fig. 4. The new calculated resolution limits, depending on the value of the coefficient of limiting aberration, can be clearly seen. The significant effect of active correction of the primary spherical aberration can be observed. The influence of the additional correction of the secondary spherical aberration (the 5th order) theoretically improves the attainable resolution, but requirements for stability of the corrector's power supplies can be too high. In addition, it is necessary to take into account the influence of the chromatic aberration, which will be discussed later. The discussed model is able to find the optimal setting of the aberration coefficients by a corrector to reach the biggest aperture angle, meeting the chosen requirements of the phase shift. The proper choice of coefficients enables to extension of the aperture angle to the interval after the last local phase shift minimum, as is shown in Fig. 3. Optimal aperture angles of all LVEM5 and LVEM25 imaging modes were calculated for a system limited by the 3rd and the 5th order spherical aberration (Table 7). The aperture angles would clearly Fig. 3. left: The phase shift behaviour with an appropriate choice of a coefficient (corrected system). x ð a Þ stays below a certain limit in the biggest possible interval of aperture angle. right: In case of improperly selected coefficients or uncorrected system, the phase shift will exceed (red dot) the limit already with lower a . (For interpretation of the references to color in text/this figure legend, the reader is referred to the web version of the article.) Table 6 Formulas for an optimal setting of the aberration coefficients by the corrector and parameters of systems corrected for the primary (limited by the 5th order) spherical aberration resp. corrected for the secondary (limited by the 7th order) spherical aberration. Term 1=b means limiting fraction of wavelength l with tolerable effect on the image quality [18]. Order of correction 3 5 Optimal C 3 4 3 2b l C 2 5   1=3 10 2 l C 7 b   1=2 Optimal C 5 – 6 2 l C 3 7 b   1=4 D f3 9 4b 2 l 2 C 5   1=3 4 8 l 3 C 7 b 3   1=4 a max 96 l bC 5   1=6 2 2 l bC 7   1=8 Resolution 0:61 b 96 C 5 l 5   1=60:61 2b 2 C 7 l 7   1=8 J. Ba9covský / MethodsX 5 (2018) 1033–1047 1039 enlarge due to the correction. The aperture angle of an uncorrected microscope is 0.01 rad [4,5]. A correction of the primary spherical aberration promises to improve this value by at least three times and with the correction of the 5th order spherical aberration, it would be possible to use an aperture angle approximately eight times higher as compared with the current systems. The resolution of the corrected system was calculated with the aperture angles from the table above (Table 7). The results are shown in Table 8. Because of significant enlargement of the optimal aperture angle, the chromatic aberration disc is also bigger. To maintain the diameter of the chromatic Fig. 4. The image shows the dependence of an expected resolution limit on the aberration coefficient of a limiting order of the spherical aberration for a different order of an aberration correction. Non-corrected system limited by the primary spherical aberration (green), correction of the 3rd order spherical aberration i.e. limited by the 5th order of spherical aberration (green) and correction of the 5th order spherical aberration i.e. limited by the 7th order of spherical aberration (red). E 0 ¼ 5 kV (upper fig.), E 0 ¼ 25 kV (lower fig.). (For interpretation of the references to color in text/this figure legend, the reader is referred to the web version of the article.) Table 7 Optimal aperture angles of the corrected system: correction up to the 3rd order a 3 (C 5 ¼ 100 mm), correction up to the 5th order a 5 (C 7 ¼ 1000 mm) S ¼ 0:88. 5 keV 10 keV 15 keV 25 keV a 3 [rad] 0.040 0.038 0.037 0.035 a 5 [rad] 0.083 0.079 0.077 0.075 1040 J. Ba9covský / MethodsX 5 (2018) 1033–1047 aberration disc comparable with the spherical one and therefore take full advantage of the correction potential of the spherical aberration, it is necessary to increase the degree of monochromatisation. Using a new aperture angle and keeping the energy spread without any reduction leads to deterioration of the chromatic aberration disc to d c5 ¼ 8:8 nm in case of LVEM5 with correction of the 5th order spherical aberration. It has been proved that, in order to achieve some meaningful values of d c comparable with d s , it is necessary to reduce the energy spread to DE ¼ 0:08 eV for LVEM 25 and DE ¼ 0:04 eV for LVEM5 (correction of the 3rd order spherical aberration). The energy spread requirements for the 5th order correction are even higher. In this case, it should not exceed the value DE ¼ 0:02 eV. The available technology of monochromators is currently capable of fulfilling these conditions in a very limited way, because the energy spread below DE ¼ 0:1 eV is very difficult to achieve [20]. The chromatic aberration thus still poses a very serious problem. The estimated resolution of the C s (C 3 ) corrected systems with a monochromator and a CFE gun is shown in Table 9. Chromatic aberration of systems corrected by a hexapole corrector The previous part of this article dealt with the problem of a spherical aberration correction. There is, however, also a chromatic aberration that can be significantly severe for LVEM, as is shown in Section “Estimated resolution of corrected LVEM”. Unfortunately as was mentioned in the introduction, chromatic aberration is correctable only by a more intricate quadrupole–octupole corrector. A hexapole corrector is not able to correct or even improve the manifestation of this aberration. On the contrary, rotationally symmetric elements, which are part and parcel of a hexapole corrector, add their own contributions to the chromatic aberration of the whole electron-optical system. The minimum configuration of the sextupole corrector consists of a telescopic round lens doublet (transfer lens) and two sextupoles placed in front of and behind the doublet, according to the Rose arrangement [11]. In order to create an aplanat, it is necessary to transfer the coma-free nodal point N 1 to the comafree point of the objective. For this purpose another doublet of the transfer lens has to be used. From the chromatic aberration point of view, only the transfer lenses have to be investigated. The contribution of a separate transfer round lens will be derived then. The chromatic aberration is usually described by a coefficient C c defined, in case of a magnetic rotationally symmetric lens, by integral [21,22]: C c ¼h 2 4UZ z i z o BðzÞ 2 h 2 dz: ð:6Þ Table 8 The resolution of the system with a correction of the 3rd order d 3 (C 5 ¼ 100 mm) and the 5th order spherical aberration d 5 (C 7 ¼ 1000 mm) neglecting the influence of the chromatic aberration. S ¼ 0:88. 5 keV 10 keV 15 keV 25 keV d 3 [nm] 0.26 0.20 0.17 0.13 d 5 [nm] 0.13 0.09 0.08 0.06 Table 9 Computed resolution for systems with reduced energy spread and accordingly optimized aperture angles. The typical energy spread of D E ¼ 0:1 eV is used in case of monochromated system (d monochrom ) and D E ¼ 0:3 for cold-field emission gun d CFE . 5 keV 10 keV 15 keV 25 keV d CFE [nm] 1.06 0.56 0.45 0.34 d monochrom [nm] 0.61 0.32 0.26 0.20 J. Ba9covský / MethodsX 5 (2018) 1033–1047 1041