Aberration Correctors, Monochromators, Spectrometers

Abstract

After four decades of attempts to correct the primary spherical and chromatic aberrations of electron lenses that led to no improvement in resolution, success was at last achieved in the 1990s with both quadrupole-octupole and sextupole correctors. The successful correctors focused on three aspects of aberration correction: primary aberrations, parasitic aberrations , and overall stability. They quickly demonstrated resolution improvement in the microscopes they were built into, and in the early 2000s, they advanced the attainable resolution to $<{\mathrm{1}}\,{\mathrm{\AA{}}}$—a level not achievable by uncorrected electron microscopes. Subsequent generations of correctors included further multipoles and corrected aberrations up to the fifth order, enabling resolution of better than ${\mathrm{0.5}}\,{\mathrm{\AA{}}}$ to be reached at ${\mathrm{300}}\,{\mathrm{kV}}$ primary voltage, and around ${\mathrm{1}}\,{\mathrm{\AA{}}}$ at ${\mathrm{30}}\,{\mathrm{kV}}$. The effect of chromatic aberration was reduced by the use of hybrid quadrupoles or by incorporating a monochromator in the microscope column.

After a brief summary of the optics of multipoles, the various types of correctors are examined in detail: quadrupole–octopole correctors , which first improved the performance of a scanning electron microscope and, soon after, that of scanning transmission electron microscopes; and sextupole correctors , which first increased the resolving power of conventional (fixed-beam) transmission electron microscopes, and were later used in scanning transmission electron microscopes as well. Ways of combating chromatic aberration are then described, including mirror correctors employed in low-energy-electron and photoemission microscopes ( and PEEM ). A section is devoted to studies of aberrations beyond the third order and of parasitic aberrations.

Electron spectrometers and imaging filters are routine accessories of electron microscopes, and they too must be carefully designed, especially when attached to aberration-corrected instruments. A section covers these devices, and much of the reasoning also applies to monochromators. Separate paragraphs are devoted to post-column and in-column spectrometers and monochromators, and the attainable energy resolution is discussed. Practical aspects of the correction process are described, notably autotuning and aberration measurement. We conclude with a survey of current performance limits and comments on the problems to be overcome if further progress is to be made.

Appendix: Power Series Expansions of Electrostatic Potential and Vector Potential

The power series expansions used in the foregoing text are identical with those adopted in [13.6] and are reproduced here for the reader’s convenience. However, since the work of Rose is very frequently referred to, the relation between the notation used in his book [13.13] and in his chapter in Ernst and Rühle [13.368] and that employed here is also given:

• 1. Expansions for the electrostatic potential

  In [13.6], the electrostatic potential is written

  $$\begin{aligned}\displaystyle\Phi(x,y,z)&\displaystyle=\phi(z)-\frac{1}{4}(x^{2}+y^{2})\phi^{\prime\prime}(z)\\ \displaystyle&\displaystyle\quad\,+\frac{1}{64}(x^{2}+y^{2})^{2}\phi^{\text{(iv)}}\\ \displaystyle&\displaystyle\quad\,-xF_{1}(z)-yF_{2}(z)+\frac{1}{8}(x^{2}+y^{2})\\ \displaystyle&\displaystyle\qquad\,\times(xF_{1}^{\prime\prime}+yF_{2}^{\prime\prime})\\ \displaystyle&\displaystyle\quad\,+\frac{1}{2}(x^{2}-y^{2})p_{2}(z)+xyq_{2}(z)\\ \displaystyle&\displaystyle\quad\,-\frac{1}{24}(x^{2}+y^{2})(x^{2}-y^{2})p_{2}^{\prime\prime}\\ \displaystyle&\displaystyle-\frac{1}{12}(x^{2}+y^{2})xyq_{2}^{\prime\prime}\\ \displaystyle&\displaystyle-\frac{1}{6}(x^{3}-3xy^{2})p_{3}(z)\\ \displaystyle&\displaystyle+\frac{1}{6}(y^{3}-3x^{2}y)q_{3}(z)\\ \displaystyle&\displaystyle+\frac{1}{24}(x^{4}-6x^{2}y^{2}+y^{4})p_{4}(z)\\ \displaystyle&\displaystyle+\frac{1}{6}(x^{2}-y^{2})xyq_{4}(z)\;.\end{aligned}$$

  Rose [13.13, 13.14] uses the expansion

  $$\begin{aligned}\displaystyle&\displaystyle\varphi(w,z)\\ \displaystyle&\displaystyle\;=\Re\sum_{m=0}^{\infty}\,\sum_{l=0}^{\infty}(-)^{l}\frac{m!}{l!(m+l)!}\left(\frac{w\overline{w}}{4}\right)^{l}\overline{w}^{m}\Phi_{m}^{[2l]}(z)\end{aligned}$$

  in which $w=x+\mathrm{i}y$ and $[2l]$ signifies twofold differentiation with respect to $z$. An overbar indicates complex conjugate.

  This leads to the following correspondences

  $$\begin{aligned}\displaystyle\phi(z)&\displaystyle=\Phi_{0}(z)\;,\\ \displaystyle F_{1}(z)&\displaystyle=-\Phi_{1}^{\text{(r)}}\;,\quad F_{2}(z)=-\Phi_{1}^{\text{(i)}}\;,\\ \displaystyle\Phi_{1}&\displaystyle=-(F_{1}+\mathrm{i}F_{2})\;,\\ \displaystyle p_{2}(z)&\displaystyle=2\Phi_{2}^{\text{(r)}}\;,\quad q_{2}(z)=2\Phi_{2}^{\text{(i)}}\;,\\ \displaystyle\Phi_{2}&\displaystyle=\frac{1}{2}(p_{2}+\mathrm{i}q_{2})\;,\\ \displaystyle p_{3}(z)&\displaystyle=-6\Phi_{3}^{\text{(r)}}\;,\quad q_{3}(z)=-6\Phi_{3}^{\text{(i)}}\;,\\ \displaystyle\Phi_{3}&\displaystyle=\frac{1}{6}(p_{3}+\mathrm{i}q_{3})\;,\\ \displaystyle p_{4}(z)&\displaystyle=24\Phi_{4}^{\text{(r)}}\;,\quad q_{4}(z)=24\Phi_{4}^{\text{(i)}},\\ \displaystyle\Phi_{4}&\displaystyle=\frac{1}{24}(p_{4}+\mathrm{i}q_{4})\end{aligned}$$

  and we have written

  $$\Phi_{m}=\Phi_{m}^{\text{(r)}}+\mathrm{i}\Phi_{m}^{\text{(i)}}\;.$$

• 2. Expansions for the vector potential

  In [13.6], the components of the vector potential are written

  $$\begin{aligned}\displaystyle A_{x}&\displaystyle=-\frac{y}{2}\left(B-\frac{1}{8}(x^{2}+y^{2})B^{\prime\prime}\right)\\ \displaystyle&\displaystyle\quad+\frac{1}{4}(x^{2}-y^{2})B_{2}^{\prime}-\frac{1}{48}(x^{2}-y^{2})(x^{2}+y^{2})B_{2}^{\prime\prime\prime}\\ \displaystyle&\displaystyle\quad-\frac{1}{2}xyB_{1}^{\prime}+\frac{1}{24}xy(x^{2}+y^{2})B_{1}^{\prime\prime\prime}\\ \displaystyle&\displaystyle\quad-\frac{1}{12}(x^{3}-3xy^{2})Q_{2}^{\prime}-\frac{1}{12}(y^{3}-3x^{2}y)P_{2}^{\prime}\\ \displaystyle&\displaystyle\quad+\frac{1}{48}(x^{4}-6x^{2}y^{2}+y^{4})Q_{3}^{\prime}-\frac{1}{12}(x^{2}-y^{2})xyP_{3}^{\prime}\;,\\ \displaystyle A_{y}&\displaystyle=\frac{x}{2}\left(B-\frac{1}{8}(x^{2}+y^{2})B^{\prime\prime}\right)\\ \displaystyle&\displaystyle\quad+\frac{1}{4}(x^{2}-y^{2})B_{1}^{\prime}-\frac{1}{48}(x^{2}-y^{2})(x^{2}+y^{2})B_{1}^{\prime\prime\prime}\\ \displaystyle&\displaystyle\quad+\frac{1}{2}xyB_{2}^{\prime}-\frac{1}{24}xy(x^{2}+y^{2})B_{2}^{\prime\prime\prime}\\ \displaystyle&\displaystyle\quad-\frac{1}{12}(x^{3}-3xy^{2})P_{2}^{\prime}+\frac{1}{12}(y^{3}-3x^{2}y)Q_{2}^{\prime}\\ \displaystyle&\displaystyle\quad+\frac{1}{48}(x^{4}-6x^{2}y^{2}+y^{4})P_{3}^{\prime}+\frac{1}{12}(x^{2}-y^{2})xyQ_{3}^{\prime}\;,\\ \displaystyle A_{z}&\displaystyle=-xB_{2}(z)_{1}-yB_{1}(z)+\frac{1}{8}(x^{2}+y^{2})\\ \displaystyle&\displaystyle\qquad\,\times(xB_{2}^{\prime\prime}-yB_{1}^{\prime\prime})\\ \displaystyle&\displaystyle\quad+\frac{1}{2}(x^{2}-y^{2})Q_{2}(z)-xyP_{2}(z)-\frac{1}{24}(x^{2}+y^{2})\\ \displaystyle&\displaystyle\qquad\,\times(x^{2}-y^{2})Q_{2}^{\prime\prime}+\frac{1}{12}(x^{2}+y^{2})xyP_{2}^{\prime\prime}\\ \displaystyle&\displaystyle\quad-\frac{1}{6}(x^{3}-3xy^{2})Q_{3}(z)-\frac{1}{6}(y^{3}-3x^{2}y)P_{3}(z)\\ \displaystyle&\displaystyle\quad+\frac{1}{24}(x^{4}-6x^{2}y^{2}+y^{4})Q_{4}(z)-\frac{1}{6}(x^{2}-y^{2})\\ \displaystyle&\displaystyle\qquad\,\times xyP_{4}(z)\;.\end{aligned}$$

  In Rose [13.13, 13.14], the same gauge is adopted, and the components of the vector potential are given by

  $$\begin{aligned}\displaystyle&\displaystyle\begin{aligned}\displaystyle A=A_{x}+\mathrm{i}A_{y}&\displaystyle=\sum_{m=0}^{\infty}\,\sum_{l=0}^{\infty}\frac{(-)^{l}}{2\mathrm{i}}\frac{m!}{l!(m+1+l)!}\\ \displaystyle&\displaystyle\quad\times\left(\frac{w\overline{w}}{4}\right)^{l}w^{m+1}\bar{\Psi}_{m}^{[2l+1]}(z)\;,\end{aligned}\\ \displaystyle&\displaystyle\begin{aligned}\displaystyle A_{z}&\displaystyle=\Im\sum_{m=0}^{\infty}\,\sum_{l=0}^{\infty}(-)^{l}\frac{m!}{\!(m+l)!}\\ \displaystyle&\displaystyle\quad\times\left(\frac{w\overline{w}}{4}\right)^{l}\overline{w}^{m}\Psi_{m}^{[2l]}(z)\;.\end{aligned}\end{aligned}$$

  The correspondence is now

  $$\begin{aligned}\displaystyle\Psi_{1}&\displaystyle=-(B_{1}+\mathrm{i}B_{2})\;,\\ \displaystyle\Psi_{2}&\displaystyle=\frac{1}{2}(P_{2}+\mathrm{i}Q_{2})\;,\\ \displaystyle\Psi_{3}&\displaystyle=-\frac{1}{6}(P_{3}+\mathrm{i}Q_{3})\;,\\ \displaystyle\Psi_{4}&\displaystyle=\frac{1}{24}(P_{4}+\mathrm{i}Q_{4})\;.\end{aligned}$$

Higher-order terms in the expansions are given in Hawkes and Kasper [13.6], and the correspondences can be obtained straightforwardly.