On the importance of fifth-order spherical aberration for a fully corrected electron microscope

doi:10.1016/j.ultramic.2005.09.004

Ultramicroscopy

Volume 106, Issues 4–5, March 2006, Pages 301-306

https://doi.org/10.1016/j.ultramic.2005.09.004 Get rights and content

Abstract

Next generation aberration correctors will not only eliminate the third-order spherical aberration, but also improve the information limit by correction of chromatic aberration. As a result of these improvements, higher order aberrations, which have largely been neglected in image analysis, will become important. In this paper, we concern ourselves with situations where sub-Å resolution can be achieved, and where the third-order spherical aberration is corrected and the fifth-order spherical aberration is measurable. We derive formulae to explore the maximum value of the fifth-order spherical aberration for directly interpretable imaging and discuss the optimum imaging conditions and their applicable range.

Introduction

For transmission electron microscopes (TEMs) with a conventional round magnetic objective lens, the third-order spherical aberration, $C_{3}$ is always positive and unavoidable [1]. For an uncorrected TEM, this aberration introduces a large phase shift in the specimen exit wave function and is the resolution limiting factor for direct interpretation of high resolution images for thin specimens [1]. Recently, with the successful implementation of aberration correctors [2], [3], [4], $C_{3}$ can be corrected, greatly improving the point resolution [1] (defined in Appendix A) and reducing the contrast delocalisation. Due to the optical design of the hexapole-type $C_{3}$ -corrector [2], the fifth-order spherical aberration, $C_{5}$ can be also adjusted using the round transfer lenses situated between the objective lens and the first hexapole element [5]. Without compensation, the $C_{5}$ value is no greater than 10 mm [4] in a 200 kV FEGTEM and with compensation, it can be adjusted to smaller positive values, or over-compensated to give negative values [5].

However, as with uncorrected instruments, the absolute information obtainable in a $C_{3}$ -corrected TEM is restricted by the effects of partial coherence. For field emission sources, the partial spatial coherence is not limiting due to the small effective source size, whereas the partial temporal coherence effect caused by the energy spread of electrons $Δ E$ , high voltage $Δ V$ and objective lens current $Δ I$ fluctuations, determines the information limit of the microscope. This temporal coherence effect is formalised, for thin specimens as a damping envelope function through a focal spread¹ [6], $Δ = C_{c} \sqrt{(Δ V / V)^{2} + 4 (Δ I / I)^{2} + (Δ E / E)^{2}}$ , where $C_{c}$ is the chromatic aberration coefficient of the objective lens. Hexapole-based correctors [2] also contribute a positive value to $C_{c}$ , and hence the overall value of $C_{c}$ is increased over that of an uncorrected lens, giving a poorer information limit compared to an uncorrected instrument.

Recently, much effort has been made to improve temporal coherence by reducing the instabilities in the high voltage and objective lens supplies [7]. In addition, a $C_{3}$ -corrected 200 kV TEM equipped with a monochromator has demonstrated a reduced focal spread due to a decrease in the energy spread to 0.15 eV FWHM [8]. Finally, proposals for chromatic aberration correctors may provide a smaller focal spread [9], [10], and with these correctors the information limit may be further improved to ca. 0.5 Å [11].

With these improvements to the information limit, higher order aberrations, which have been largely neglected in image analysis, become important [12]. In this paper, we concern ourselves with the next generation microscopes equipped with $C_{3}$ and $C_{c}$ correctors, where sub-Å resolution will be achieved, and in which $C_{3}$ is corrected and $C_{5}$ can be measured accurately and compensated. Accordingly we derive a formula to explore the maximum $C_{5}$ suitable for directly interpretable imaging, and discuss the optimum imaging conditions and their applicable range.

Section snippets

Phase contrast theories

Various phase contrast imaging conditions have been discussed previously [13], [14], [15], considering only the effects of defocus, $C_{1}$ and $C_{3}$ . In the present work, phase contrast imaging conditions are described with $C_{5}$ incorporated, with details of the derivations given in Appendix A Scherzer condition, Appendix B Chromatic aberration balanced conditions.

Discussion

For the $C_{5}$ limited condition discussed in Section 2.1, in which $C_{1}$ and $C_{3}$ are used to compensate residual $C_{5}$ , the corresponding point resolutions defined by Eq. (2) for accelerating voltages ranging from 80 to 300 kV are plotted in Fig. 1, assuming that $C_{5}$ is independent of the accelerating voltage. From this figure it is clear that for $C_{5}$ values less than 10 mm, the $C_{5}$ limited condition (Section 2.1) enables a point resolution better than 0.7 Å at 200 kV [17], and better than 1 Å at accelerating

Conclusion

Since the $C_{5}$ value of the current correctors is no greater than 10 mm, the contribution of $C_{5}$ is only important when the information limit is better than 0.7 Å at 200 kV, and 1 Å at above 80 kV. Hence the effect of $C_{5}$ can be ignored for the current generation of aberration corrected microscopes, whose information limit is typically worse than these limits. With the next generation of $C_{c}$ correctors, which offer the potential of further improvements in the information limit to ca. 0.5 Å, $C_{5}$ will not be

Acknowledgements

The authors would like to thank C. Dwyer for useful discussions. Financial support from the EPSRC and the Leverhulme Trust are gratefully acknowledged.

References (22)

M.A. O’Keefe et al.
Ultramicroscopy
(2001)
B. Freitag et al.
Ultramicroscopy
(2005)
P.W. Hawkes
Nucl. Instrum. Meth. A
(2004)
H. Rose
Nucl. Instrum. Meth. A
(2004)
M. Lentzen et al.
Ultramicroscopy
(2002)
M. Lentzen
Ultramicroscopy
(2004)
J.L. Hutchison et al.
Ultramicroscopy
(2005)
O. Scherzer
J. Appl. Phys.
(1949)
M. Haider et al.
Optik
(1995)
M. Haider et al.
J. Electron Microsc.
(1998)

F. Hosokawa et al.

J. Electron Microsc.

(2003)

Cited by (49)

Principles of Electron Optics, Volume 3: Fundamental Wave Optics
2022, Principles of Electron Optics, Volume 3: Fundamental Wave Optics
Exploration of the resolving power for a magnetic lens using Glaser field distribution
2020, Optik
Citation Excerpt :
Recently, this defect could be eliminated (or reduced) using unconventional lenses that may build up according to different techniques. Among most of them are ir-rotational lenses run out of using rotational symmetry [5], mono-chromator technology [6], considering the fifth-order correction [7], Foil Corrector lenses [8] and time varying magnetic field [9]. Although, these lenses could aberration less images, they implied a somehow theoretical complication and so a sensible experimental attention must take into account.
Resolution estimation for magnetic lenses have been investigated using Glaser Bell-Shaped model. The adopted model used to approximates the magnetic field distribution for the proposed lens. Indeed, the field distribution is formulated in terms of three optimization parameters namely the half width, field beak value and lens length. Four values for each of these parameters selected to check the resolution behavior, where the other two parameter maintained fixed. Results reveals that the resolution get better enhancement whenever the field excitation increases. This behavior attained whenever the variation in the optimization variable leads to increases the field refracting power. Besides, the slope of the approximated model correspondingly influence the resolution behavior. In addition, results shows that resolution always becomes better for long and strong magnetic lenses.
Aberration correction for low voltage optimized transmission electron microscopy
2018, MethodsX
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 exclusively low-voltage design seems to be a challenging task. The benefits and potential of the Rose hexapole corrector 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
A quantitative method for measuring small residual beam tilts in high-resolution transmission electron microscopy
2018, Ultramicroscopy
Citation Excerpt :
iii) The proposed method applies only to the case of measuring small residual beam tilts (in the order of 0.1°), being supplementary to the rotation center alignment and coma-free alignment procedure in the HRTEM instrument. It should be noted that in the analysis above, only the defocus and the Cs constant are considered in the imaging formation theory with a small beam tilt, though other aberrations can be important for imaging both in aberration corrected TEM and in uncorrected TEM [13,22,49–51]. Nevertheless, it can be proved that apart from the defocus, other aberrations, if assumed constant, have no impact on the proposed method for measuring the beam tilt.
In a transmission electron microscope, electron illumination beam tilt, or the degree of deviation of electron beam from its optical axis, is an important parameter that has a significant impact on image contrast and image interpretation. Although a large beam tilt can easily be noticed and corrected by the standard alignment procedure, a small residual beam tilt is difficult to measure and, therefore, difficult to account for quantitatively. Here we report a quantitative method for measuring small residual beam tilts, including its theoretical schemes, numerical simulation testing and experimental verification. Being independent of specimen thickness and taking specimen drifts into account in measurement, the proposed method is supplementary to the existing “rotation center” and “coma-free” alignment procedures. It is shown that this method can achieve a rather good accuracy of 94% in measuring small residual beam tilts of about 0.1° or less.
Artefacts in geometric phase analysis of compound materials
2015, Ultramicroscopy
The geometric phase analysis (GPA) algorithm is known as a robust and straightforward technique that can be used to measure lattice strains in high resolution transmission electron microscope (TEM) images. It is also attractive for analysis of aberration-corrected scanning TEM (ac-STEM) images that resolve every atom column, since it uses Fourier transforms and does not require real-space peak detection and assignment to appropriate sublattices. Here it is demonstrated that, in ac-STEM images of compound materials with compositionally distinct atom columns, an additional geometric phase is present in the Fourier transform. If the structure changes from one area to another in the image (e.g. across an interface), the change in this additional phase will appear as a strain in conventional GPA, even if there is no lattice strain. Strategies to avoid this pitfall are outlined.
Recording low and high spatial frequencies in exit wave reconstructions
2013, Ultramicroscopy
Aberration corrected Transmission Electron Microscope (TEM) images can currently resolve information at significantly better than 0.1 nm. Aberration corrected imaging conditions seek to optimize the transfer of high-resolution information but in doing so they prevent the transfer of low spatial frequency information. To recover low spatial frequency information, aberration corrected images must be acquired at a large defocus which compromises high spatial frequency information transfer. In this paper we present two a posteriori solutions to this problem in which the information bandwidth in an exit wave reconstruction is increased. In the first we reconstruct the electron exit wavefunction from two focal series datasets, with different, uniform focal steps, experimentally demonstrating that the width of the transfer interval can be extended from 0.2 nm⁻¹ (∼5 nm) to better than 10 nm⁻¹ (0.1 nm). In the second we outline the use of a focal series recorded with a non-uniform focal step to recover a wider range of spatial frequencies without the need for a large number of images. Using simulated data we show that using this non-uniform focal step the spatial frequency interval for a five image data set may be increased to between 0.25 nm⁻¹ (4 nm) and 8.3 nm⁻¹ (0.12 nm) compared to between 0.74 nm⁻¹ (1.4 nm) and 8.3 nm⁻¹ (0.12 nm) for the standard focal series geometry.

View all citing articles on Scopus

View full text

On the importance of fifth-order spherical aberration for a fully corrected electron microscope

Abstract

Introduction

Section snippets

Phase contrast theories

Discussion

Conclusion

Acknowledgements

Ultramicroscopy

Ultramicroscopy

Nucl. Instrum. Meth. A

Nucl. Instrum. Meth. A

Ultramicroscopy

Ultramicroscopy

Ultramicroscopy

J. Appl. Phys.

Optik

J. Electron Microsc.

J. Electron Microsc.