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In electron microscopes (EMs), sphericalaberration (C_{s}) correction leads electrons from the probeforming lens to be better utilized with an increased convergence angle. This results in several advantages:
i) A reduction of the probe diameter,
ii) An increase in the probe current density because of the better utilization of the available angular spread of electrons from the source (increase of the numerical aperture of the lens),
iii) A better defined probe due to the inclusion of the high spatial frequency components of the electron waves,
iv) A reduced depth of focus.
The magnetic lenses are only convergent because they are in rotational symmetry. However, the absence of divergent lenses prevents spherical aberration correction. Spherical aberration correctors combine multipoles and
rotational symmetry lenses. In other words, the major steps forward in the C_{s} correction in highresolution electron microscopy (HREM) have been enabled by breaking the rotational symmetry of electron lenses using electromagnetic multipoles. However, so far no applicable corrector can compensate for both the chromatic and spherical aberration without introducing obvious offaxial aberrations.
There is no convenient way of correcting spherical aberrations. Fortunately, two methods can normally be used for spherical aberration correction at high accelerating voltages in modern TEMs, SEMs, and STEMs:
i) Using magnetic hexapoles (also called sextupoles) with transfer round lenses [1] in both TEM and STEM system.
Without introducing other multipole elements, the spherical aberration correction only by hexapoles can be done. In this case, the correcting system consists of round lenses and hexapoles only. Therefore, the Gaussian optics is exclusively determined by the round lenses because the hexapole fields do not affect the paraxial region. In addition, asymmetric dodecapoletype spherical aberration correctors, developed by Hosokawa et al. [4], are also widely utilized in STEM and TEM.
ii) Combining four magnetic quadrupoles and at least three octupoles in STEM system [2] due to its offaxis aberrations. For instance, in aberrationcorrected EMs (electron microscopes) a combination of hexapole or octupole lenses are used in a aberration corrector which lacks rotational (circular) symmetry and thus doesn’t have to have a positive spherical aberration like conventional, round magnetic lenses. The overall spherical aberration of the objective lens and this corrector can be minimized by the operator, which can significantly improve the point to point resolution by choosing a smaller spherical aberration. Furthermore, the corrector has also made it possible to use negative spherical aberrations giving bright atoms on a dark background, which actually enhances the contrast in the images compared to the images taken with a large, positive spherical aberration.
The first type has the advantage that the hexapole fields need only a stability of about 10 ppm, which is two orders of magnitude less than that required for the quadrupole fields of the multipole correctors, while it has the disadvantage that it cannot compensate for the chromatic aberration.
With the introduction of a C_{s}corrector [3], the dependence of spatial coherence on beam convergence reduces significantly and may even be eliminated completely at C_{s} = zero. In that case, the point resolution is equal to the information limit.
Objective lens in EMs normally uses a round magnetic lens. The round magnetic lens cannot form a concave lens and thus, spherical aberration cannot be corrected by any combination of cylindrically symmetric round magnetic lenses. According to Scherzer theorem, spherical and chromatic aberrations can in principle be corrected if at least one of the four conditions in Scherzer theorem has been broken, while in the most successful practices the corrections have been obtained by breaking rotational symmetry.
As discussed before, the physical separation between the objective lens where the third order spherical aberration (C_{s}, or called C_{3,0}) is introduced and the corrector where this aberration is removed induces a higher order spherical aberration, which is fifth order spherical aberration (C_{5,0}) as indicated in green in Table 4579. However, the separation can be minimized by placing the corrector as close as possible to the objective lens, and by keeping the length of the corrector to a minimum. Table 4449 gives examples of aberration coefficients before and after C_{s} corrections.
Table4449. Examples of aberration coefficients before and after C_{s} corrections.
Accelerating voltage (keV) 
Status 
C_{s }(mm) 
C_{5}(mm) 
Δf_{Scherzer}(nm) 
300 
Before 
1.15 
0 
 56.2 
After 
0.03 
100 

Note that the C_{s} and coma corrections cannot further improve the resolution when reaching 0.5 Å at voltages ≤ 200 kV due to chromatic aberration.
These aberrations can practically be automatically compensated through a softwarecontrolled feedback loop [5] after the aberrations are determined by measurements.
Any robust and efficient tuning of a C_{s} corrector system employs software to characterize the lens aberrations and to use the obtained information to correct the aberrations prior to image recording. The commercial products today can essentially reduce C_{s} to zero, and even allow it to go negative. Two approaches are currently available to identify and measure the aberrations based on PC control:
i) Using the diffraction information shown in the ringlike patterns below each phase contrast image that are acquired with the incident beam aligned along the optic axis. By tilting the beam off the optic axis, lens aberrations become apparent in the Fourier transforms of phase contrast images. A set of such patterns obtained at various beam tilts around the optic axis is called a "Zemlin tableaus". The distortions of the circular patterns in the Zemlin tableaus are used for characterizing the aberration. This method is wellsuited for HRTEM imaging.
ii) Using "Ronchigrams" somewhat similar to convergent beam electron diffraction. However, unlike the CBED pattern, a Ronchigram is obtained using a large aperture angle, so the diffraction disks overlap considerably. Phase interferences of the different beams occur in the Ronchigram and vary with focus. In this method, a set of Ronchigrams acquired at different focus settings can characterize the lens aberrations.
In C_{s}corrected EMs, due to the absence of spherical aberration it is not possible anymore to correct the residual axial coma by tilting the illumination beam. In this case an appropriate coma compensator is needed to eliminate the coma.
With the stateoftheart spherical aberration correction, the current generation of STEMs has an achievable point resolution at 50 – 80 pm levels, satisfying the possible atomicscale analysis for most materials. However, the residual higher order aberrations will still be one of the limiting factors, albeit at a higher spatial resolution. [6]
Note that, in reality, it is hard to correct the aberrations under lowdose conditions. The typical method for obtaining low dose is to change the parameters of the electron gun. However, this modifies the electron optics in the column. For a STEM with a C_{s} corrector, this also causes a misalignment of the corrector. Unfortunately, it is hard to realign the column as well as the corrector under lowdose conditions since the auto tuning software or even manual adjustment normally needs high signaltonoise images and/or diffractograms to converge precisely.
[1] Haider, M., Braunshausen, G. and Schwan, E. 1995. Correction of the spherical aberration of a 200 kV TEM
by means of a hexapolecorrector, Optik, 99, 167–179.
[2] Krivanek, O. L., Dellby, N., and Lupin, A. R. 1999. Towards subÅ electron beams, Ultramicroscopy, 78,
1–11.
[3] Rose, H., 1990, Outline of a spherically corrected semiaplanatic mediumvoltage transmission electron microscope, Optik, 85, 19–24.
[4] Hosokawa F, Sannomiya T, Sawada H, Kaneyama T, Kondo Y, Hori M,
Yuasa S, Kawazoe M, Nakamichi T, Tanishiro Y, Yamamoto N, and
Takayanagi K (2006) Design and development of C_{s} correctors for
300 kV TEM and STEM. IMC 16: 582.
[5] Takeo Sasaki, Hidetaka Sawada, Fumio Hosokawa, Yuji Kohno, Takeshi Tomita, Toshikatsu Kaneyama, Yukihito Kondo, Koji Kimoto, Yuta Sato, and Kazu Suenaga, Performance of lowvoltage STEM/TEM with delta corrector and cold field emission gun, Journal of Electron Microscopy 59(Supplement): S7–S13 (2010).
[6] Haider M. Ultramicroscopy 2000;81:163–75.
