Advertisement

Aberration Correctors, Monochromators, Spectrometers

  • Peter W. Hawkes
  • Ondrej L. Krivanek
Chapter
Part of the Springer Handbooks book series (SHB)

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.

Keywords

quadrupole correctors sextupole correctors monochromators in-column spectrometers post-column spectrometers geometrical aberrations chromatic aberrations parasitic aberrations 

It has been known since the early days of electron optics that the rotationally symmetric lenses employed in electron microscopes and similar instruments suffer from severe aberrations that cannot be eliminated by skillful lens design [13.1]. Immense effort has been devoted to finding lenses with small aberrations and devising aberration correctors . The theoretical proof that the two most important aberrations, the spherical and chromatic aberrations , cannot be eliminated required that several conditions be satisfied and, by relaxing one or another of these conditions, correctors could be designed. A near-exhaustive list was published by Scherzer [13.2] and reviews charting trends in thinking about aberration correction and progress in implementing correctors are to be found in [13.3], reprinted in [13.4], [13.5, 13.6]. These contain very full accounts of earlier attempts to correct aberrations with extensive reference lists, and the material presented there is not always reproduced here. In particular, a survey of attempts to build apochromats and aplanatic lenses by O. Scherzer, H. Rose and colleagues in Darmstadt is to be found in the article by Marko and Rose [13.7].

The types of correctors that seem most promising today are examined below but first, we describe the various kinds of aberrations and explain why they are important. We then look more closely at the aberration coefficients themselves, which leads naturally to a study of the correctors.

Some familiarity with basic electron optics is assumed here. In particular, the reader is expected to be acquainted with the paraxial properties of lenses and the cardinal elements that characterize them. The Handbook of Charged Particle Optics edited by J. Orloff [13.8] is recommended for readers who wish to brush up their knowledge of electron lenses, notably the chapters by Munro [13.9] (only in the first edition, 1997), Tsuno [13.10], Lencová [13.11] and Hawkes [13.12]. For very full accounts, see [13.13, 13.6].

13.1 Types of Aberrations

Geometrical aberrations act on electrons of the same energy whereas chromatic aberrations act on electrons of different energies. Mixed geometrical-chromatic aberrations arise beyond the lowest order of aberrations. The aberrations may be intrinsic, inseparable from the optics of the system as designed, or parasitic, arising from imperfections in machining, homogeneity of materials, alignment, etc.

13.1.1 Geometrical Aberrations

An ideal lens would provide point-to-point mapping of the structure of an object into an image, and such imaging is indeed predicted by the simplest approximate description of the effect of a lens on an electron beam, the paraxial approximation . This is valid provided that the electrons remain close to the optic axis (the axis of symmetry in the case of a round lens) and that the electron trajectory remains inclined at a small angle to this axis. When the electrons depart too far from the axis or the trajectories are inclined at a steeper angle, the paraxial approximation is perturbed by geometrical aberrations.

In the paraxial approximation, electrons (of charge \(-e\) and rest mass \(m_{0}\)) satisfy the linear, homogeneous, second-order differential equation
$$\frac{\mathrm{d}}{\mathrm{d}z}\left(\hat{\phi}^{1/2}x^{\prime}\right)+\frac{\gamma\phi^{\prime\prime}+\eta^{2}B^{2}}{4\hat{\phi}^{1/2}}x=0\;,$$
(13.1)
with an identical equation for \(y(z)\), in which \(\phi(z)\) is the axial electrostatic potential and \(B(z)\) is the magnetic induction on the optic axis, which coincides with the coordinate axis \(z\). Primes denote differentiation with respect to \(z\). The field and potential expansions are given in Appendix 13.1; for derivations of these, see [13.6, Chap. 7], where terms of higher order are included. Since the chapter by Rose [13.14] is a very relevant reference to much of the material presented here, the relation between his notation and that adopted here is also given. The relativistically corrected potential \(\hat{\phi}\) is given by
$$\hat{\phi}=\phi(1+\varepsilon\phi)\;,$$
(13.2)
where \(\varepsilon=e/2m_{0}c^{2}\approx{\mathrm{0.1}}\,{\mathrm{MV^{-1}}}\); \(\gamma=1+2\varepsilon\phi\) and \(\eta=(e/2m_{0})^{1/2}\approx{\mathrm{3\times 10^{5}}}\,{\mathrm{C^{1/2}{\,}kg^{-1/2}}}\). The distances \(x(z)\) and \(y(z)\) are the rotating coordinates routinely employed in the study of round magnetic lenses. The angle of rotation \(\chi(z)\) as we advance through the magnetic field is given by
$$\chi=\frac{\eta}{2}\int^{z}\frac{B(\zeta)}{\hat{\phi}^{1/2}}\mathrm{d}\zeta\;.$$
Such differential equations have the general solution
$$x(z)=Ax_{1}(z)+Bx_{2}(z)\;,$$
where \(x_{1}(z)\) and \(x_{2}(z)\) are linearly independent solutions of (13.1), and \(A\) and \(B\) are constants. It is frequently convenient to choose for \(x_{1}(z)\) and \(x_{2}(z)\) the solutions \(g(z)\) and \(h(z)\) that satisfy the boundary conditions
$$\begin{aligned}\displaystyle g(z_{\mathrm{o}})&\displaystyle=h^{\prime}(z_{\mathrm{o}})=1\;,\\ \displaystyle g^{\prime}(z_{\mathrm{o}})&\displaystyle=h(z_{\mathrm{o}})=0\;,\end{aligned}$$
(13.3)
where \(z_{\mathrm{o}}\) is the object plane, whereupon we have
$$\begin{aligned}\displaystyle x(z)&\displaystyle=x_{\mathrm{o}}g(z)+x_{\mathrm{o}}^{\prime}h(z)\;,\\ \displaystyle y(z)&\displaystyle=y_{\mathrm{o}}g(z)+y_{\mathrm{o}}^{\prime}h(z)\;.\end{aligned}$$
(13.4)
The cardinal elements of the lens, its focal lengths and the positions of its focal planes and principal planes , are defined with the aid of the rays \(g(z)\) and \(h(z)\). It is convenient to write \(u=x+\mathrm{i}y\), and it can readily be shown that
$$\begin{pmatrix}u\\ u^{\prime}\end{pmatrix}=\begin{pmatrix}(z_{\text{Fi}}-z)/f_{\mathrm{i}}&T_{12}\\ -1/f_{\mathrm{i}}&(z_{\mathrm{o}}-z_{\text{Fo}})/f_{\mathrm{i}}\end{pmatrix}\begin{pmatrix}u_{\mathrm{o}}\\ u_{\mathrm{o}}^{\prime}\end{pmatrix}\;.$$
(13.5)
If \(u\) is proportional to \(u_{\mathrm{o}}\) for all values of the gradient \(u_{\mathrm{o}}^{\prime}\), the matrix element \(T_{12}\) vanishes. The plane \(z\) is then said to be conjugate to \(z_{\mathrm{o}}\) and we write \(z=z_{\mathrm{i}}\). The matrix equation then collapses to
$$\begin{aligned}\displaystyle\begin{pmatrix}u_{\mathrm{i}}\\ u_{\mathrm{i}}^{\prime}\end{pmatrix}&\displaystyle=\begin{pmatrix}(z_{\text{Fi}}-z_{\mathrm{i}})/f_{\mathrm{i}}&0\\ -1/f_{\mathrm{i}}&(z_{\mathrm{o}}-z_{\text{Fo}})/f_{\mathrm{i}}\end{pmatrix}\begin{pmatrix}u_{\mathrm{o}}\\ u_{\mathrm{o}}^{\prime}\end{pmatrix}\\ \displaystyle&\displaystyle=\begin{pmatrix}M&0\\ -1/f_{\mathrm{i}}&f_{\mathrm{o}}/(f_{\mathrm{i}}M)\end{pmatrix}\begin{pmatrix}u_{\mathrm{o}}\\ u_{\mathrm{o}}^{\prime}\end{pmatrix}.\end{aligned}$$
(13.6)
Here, \(M\) denotes the magnification , \(g(z_{\mathrm{i}})=M\), \(f_{\mathrm{o}}\) and \(f_{\mathrm{i}}\) are the object and image focal lengths, and \(z_{\mathrm{Fo}}\) and \(z_{\mathrm{Fi}}\) are the object and image foci. (For more details regarding paraxial optics, see chapters 16 and 17 of [13.6].) In the case of purely magnetic round lenses and einzel lenses, \(f_{\mathrm{o}}=f_{\mathrm{i}}\) and we denote the focal length by \(f\).
The next higher approximation includes third-order terms in \(u_{\mathrm{o}}\) and \(u_{\mathrm{o}}^{\prime}\). Systems with a straight optic axis have no second-order terms, and the rotational symmetry about the optic axis restricts the permitted third-order terms to the following (an asterisk indicates the complex conjugate)
$$\begin{aligned}\displaystyle&\displaystyle\frac{u_{\mathrm{i}}-Mu_{\mathrm{o}}}{M}=Cu_{\mathrm{o}}^{\prime}({x_{\mathrm{o}}^{\prime}}^{2}+{y_{\mathrm{o}}^{\prime}}^{2})\quad\text{(spherical aberration)}\\ \displaystyle&\displaystyle+2(K+\mathrm{i}k)u_{\mathrm{o}}(x_{\mathrm{o}}^{\prime 2}+y_{\mathrm{o}}^{\prime 2})+(K-\mathrm{i}k)u_{\mathrm{o}}^{*}u_{\mathrm{o}}^{\prime 2}\quad\text{(coma)}\\ \displaystyle&\displaystyle+(A+\mathrm{i}a)u_{\mathrm{o}}^{2}u_{\mathrm{o}}^{\prime*}\quad\text{(field astigmatism)}\\ \displaystyle&\displaystyle+F(x_{\mathrm{o}}^{2}+y_{\mathrm{o}}^{2})u_{\mathrm{o}}^{\prime}\quad\text{(field curvature)}\\ \displaystyle&\displaystyle+(D+\mathrm{i}d)u_{\mathrm{o}}(x_{\mathrm{o}}^{2}+y_{\mathrm{o}}^{2})\quad\text{(distortion)}\;.\end{aligned} $$
(13.7)
The spherical aberration term is of particular concern since it does not vanish or even dwindle for object points close to or on the axis. This is the region that is imaged in high-resolution operation. We examine this aberration closely in a later section. The other terms are of decreasing interest for objective or probe-forming lenses, and moreover, the isotropic coma (next in importance after the spherical aberration) can be avoided. Lenses have a coma-free plane , the exact position of which is determined by the relative magnitudes of the spherical aberration and coma coefficients. In practice, for magnetic lenses, it falls within the lens field, upstream from the image focus (the diffraction plane) . Specifically, let us suppose that the aberrations are expressed in terms of ray position in the object plane, \(z=z_{\mathrm{o}}\), and some aperture plane, \(z=z_{\mathrm{a}}\). The paraxial solutions appearing in the aberration integrals will then be \(s(z)\) and \(t(z)\), which satisfy the conditions
$$\begin{aligned}\displaystyle s(z_{\mathrm{o}})&\displaystyle=t(z_{\mathrm{a}})=1\;,\\ \displaystyle s(z_{\mathrm{a}})&\displaystyle=t(z_{\mathrm{o}})=0\;.\end{aligned}$$
(13.8)
If another aperture position is selected, \(z=\bar{z}_{\mathrm{a}}\), the aberration coefficients will have exactly the same structure, but the appropriate paraxial solutions will now be \(\bar{s}(z)\) and \(\bar{t}(z)\)
$$\begin{aligned}\displaystyle\bar{s}(\bar{z}_{\mathrm{o}})&\displaystyle=\bar{t}(\bar{z}_{\mathrm{a}})=1\;,\\ \displaystyle\bar{s}(\bar{z}_{\mathrm{a}})&\displaystyle=\bar{t}(\bar{z}_{\mathrm{o}})=0\;.\end{aligned}$$
(13.9)
Since there can be only two linearly independent paraxial solutions, we must be able to write
$$\begin{aligned}\displaystyle\bar{s}(z)&\displaystyle=\alpha s(z)+\beta t(z)\;,\\ \displaystyle\bar{t}(z)&\displaystyle=\gamma s(z)+\delta t(z)\end{aligned}$$
(13.10)
and obviously \(\alpha=1\) and \(\gamma=0\); for \(\beta\) and \(\delta\), we have
$$\begin{aligned}\displaystyle\beta&\displaystyle=-\frac{s(\bar{z}_{\mathrm{a}})}{t(\bar{z}_{\mathrm{a}})}\;,\\ \displaystyle\delta&\displaystyle=\frac{1}{t(\bar{z}_{\mathrm{a}})}\;.\end{aligned}$$
(13.11)
It is then easy to show that the coma coefficient for \(z=\bar{z}_{\mathrm{a}},\,K(\bar{z}_{\mathrm{a}})\), is related to that for \(z=z_{\mathrm{a}},\,K(z_{\mathrm{a}})\), as follows
$$K(\bar{z}_{\mathrm{a}})\propto K(z_{\mathrm{a}})+\beta C_{\mathrm{s}}(z_{\mathrm{a}})\;.$$
(13.12)
The coma-free point is thus situated at the point for which
$$\frac{s(\bar{z}_{\mathrm{a}})}{t(\bar{z}_{\mathrm{a}})}=\frac{K(z_{\mathrm{a}})}{C_{\mathrm{s}}(z_{\mathrm{a}})}\;.$$
(13.13)
The distortion can be important in projector lenses, in which the trajectories may be relatively far from the axis while the gradient will be very small at high magnification. Field astigmatism (so-called to distinguish it from the parasitic paraxial astigmatism) and field curvature are mainly of importance in devices in which the size of the field is a major preoccupation, such as lithography .
Fig. 13.1a-c

Quadrupoles. (a) Quadrupole orientation. The paraxial equations of motion are uncoupled when the \(x{-}z\) and \(y{-}z\) planes coincide with the planes of symmetry of electrostatic quadrupoles and fall midway between the poles of magnetic quadrupoles. (b) Appearance of a hybrid magnetic–electrostatic quadrupole. (c) Formation of a line image in a magnetic quadrupole; the arrows show the directions of the currents in the windings. Reprinted from [13.6], with permission from Elsevier

For quadrupoles , the situation is slightly more complicated. A quadrupole usually consists of four electrodes or four magnetic poles, though many other configurations that create quadrupole fields exist [13.15, 13.16]. For convenience, we assume throughout that the quadrupole is disposed in such a way that the converging and diverging planes coincide with the \(x{-}z\) and \(y{-}z\) planes (Fig. 13.1a-c); no rotationally symmetric magnetic field is present, since the coordinate rotation associated with such a field would introduce an undesirable degree of complexity. The paraxial properties of the quadrupole are then characterized by equations of motion for \(x(z)\) and \(y(z)\) that are uncoupled but no longer identical. They are again linear, homogeneous, second-order differential equations and hence have solutions analogous to those for round lenses. Now, however, we have two sets of cardinal elements, one set for the \(x{-}z\) plane and a second set for the \(y{-}z\) plane. These equations take the form
$$\begin{aligned}\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(\hat{\phi}^{1/2}x^{\prime}\right)+\frac{\gamma\phi^{\prime\prime}-2\gamma p_{2}+4\eta Q_{2}\hat{\phi}^{1/2}}{4\hat{\phi}^{1/2}}x=0\;,\\ \displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\left(\hat{\phi}^{1/2}y^{\prime}\right)+\frac{\gamma\phi^{\prime\prime}+2\gamma p_{2}-4\eta Q_{2}\hat{\phi}^{1/2}}{4\hat{\phi}^{1/2}}y=0\end{aligned}$$
(13.14)
in which the functions \(p_{2}(z)\) and \(Q_{2}(z)\) characterize the potential and field distributions (Appendix 13.1). We can now write
$$\begin{aligned}\displaystyle \begin{pmatrix}x(z_{\mathrm{i}x})\\ x^{\prime}(z_{\mathrm{i}x})\end{pmatrix}&\displaystyle=\begin{pmatrix}(z_{\text{Fi}}^{(x)}-z_{\text{ix}})/f_{\mathrm{i}}^{(x)}&0\\ -1/f_{\mathrm{i}}^{(x)}&(z_{\mathrm{o}}-z_{\text{Fo}}^{(x)})/f_{\mathrm{i}}^{(x)}\end{pmatrix}\begin{pmatrix}x_{\mathrm{o}}\\ x_{\mathrm{o}}^{\prime}\end{pmatrix}\\ \displaystyle&\displaystyle=\begin{pmatrix}M_{x}&0\\ -1/f_{\mathrm{i}}^{(x)}&f_{\mathrm{o}}^{(x)}/(f_{\mathrm{i}}^{(x)}M_{x})\end{pmatrix}\begin{pmatrix}x_{\mathrm{o}}\\ x_{\mathrm{o}}^{\prime}\end{pmatrix}\end{aligned}$$
(13.15a)
and
$$\begin{aligned}\displaystyle \begin{pmatrix}y(z_{\mathrm{i}y})\\ y^{\prime}(z_{\mathrm{i}y})\end{pmatrix}&\displaystyle=\begin{pmatrix}(z_{\text{Fi}}^{({y})}-z_{\mathrm{i}y})/f_{\mathrm{i}}^{(\mathrm{y})}&0\\ -1/f_{\mathrm{i}}^{(y)}&(z_{\mathrm{o}}-z_{\text{Fo}}^{(y)})/f_{\mathrm{i}}^{(y)}\end{pmatrix}\begin{pmatrix}y_{\mathrm{o}}\\ y^{\prime}_{\mathrm{o}}\end{pmatrix}\\ \displaystyle&\displaystyle=\begin{pmatrix}M_{y}&0\\ -1/f_{\mathrm{i}}^{(y)}&f_{\mathrm{o}}^{(y)}/(f_{\mathrm{i}}^{(y)}M_{y})\end{pmatrix}\begin{pmatrix}y_{\mathrm{o}}\\ y^{\prime}_{\mathrm{o}}\end{pmatrix}\;.\end{aligned}$$
(13.15b)

Since the cardinal elements are different in the two planes, the planes conjugate to a given object plane will generally not coincide. The system is astigmatic, and if we operate two quadrupoles in tandem, theobject of the second member will be astigmatic. Clearly, if we require quadrupoles to produce a stigmatic image of an object, we must use quadrupole multiplets and must somehow arrange that the cardinal elements of the multiplet are the same in the two planes. One multiplet configuration is of particular importance. It can be shown by symmetry arguments that if the multiplet is geometrically symmetric and electrically antisymmetric about its center plane, the focal lengths in the \(x{-}z\) and \(y{-}z\) planes will be equal. It is then only necessary to satisfy one condition (coincidence of the focal planes) in order to render the multiplet stigmatic. Quadruplets with this property have been extensively studied and are known as Russian quadruplets , since their properties were first investigated by Dymnikov and Yavor [13.17] translated as [13.18], in the Ioffe Institute in Saint Petersburg (Fig. 13.2a-d).

Fig. 13.2a-d

The Russian quadruplet. The excitations are arranged as \(\text{Q4}=-\text{Q1},\text{Q3}=-\text{Q2}\). In the present example, \(\text{Q2}\approx-{\mathrm{0.5}}\text{Q1}\) (a) the quadruplet running symmetrically, object and image distances \(={\mathrm{35}}\,{\mathrm{mm}}\); (b) identical quadrupole excitations, object at \(-\infty\), image at \({\mathrm{23}}\,{\mathrm{mm}}\) after Q4; (c) schematic representation of the beam shape in \(x{-}z\) plane for asymmetric case (b); (d) schematic representation of the beam shape in \(y{-}z\) plane (d) illustrates the principal difficulty when focusing charged particle beams with strong quadrupoles: in the plane in which the last quadrupole is diverging, the beam in the preceding (converging) quadrupole has to be large, and this results in large aberration contributions

The aberrations of quadrupoles are more numerous than those of round lenses but they fall into the same categories: aperture aberrations (the analogues of the spherical aberration), comas, field astigmatisms and curvatures, and distortions. Here we consider only the aperture aberrations, since it is they that will be exploited in aberration correctors. At this point, we note that octopoles , which have eight electrodes or magnetic poles, also have quadrupole symmetry and should hence be included in the formalism for the aberrations. They have no linear effect and hence have no effect on the paraxial behavior.

Our interest here is exclusively with aberration correctors, and we therefore assume that any multiplets of quadrupoles and octopoles are stigmatic and that the magnifications in the two planes are likewise equal (\(M_{x}=M_{y}\)). In these circumstances, the additional terms arising from the aperture aberrations take the following form
$$\begin{aligned}\displaystyle\frac{\Updelta x}{M}&\displaystyle=x_{\mathrm{o}}^{\prime}(C_{x}x_{\mathrm{o}}^{\prime 2}+C_{xy}y_{\mathrm{o}}^{\prime 2})\;,\\ \displaystyle\frac{\Updelta y}{M}&\displaystyle=y_{\mathrm{o}}^{\prime}(C_{xy}x_{\mathrm{o}}^{\prime 2}+C_{y}y_{\mathrm{o}}^{\prime 2})\;.\end{aligned}$$
(13.16)
The other multipoles used for aberration correction are sextupoles (also known as hexapoles) . Like octopoles, they have no linear effect on charged particles, but unlike octopoles (and of course quadrupoles), their dominant effect is quadratic: the trajectories depend on products of two of the object coordinates \((x_{\mathrm{o}},y_{\mathrm{o}},x_{\mathrm{o}}^{\prime},y_{\mathrm{o}}^{\prime})\). Their primary aberrations are third order, like round lenses and quadrupoles, and their aperture aberration, the aberration that depends only on gradient and is independent of off-axis distance, has the same nature as the spherical aberration of round lenses. It is for this reason that sextupoles are potential correctors of \(C_{\mathrm{s}}\), but some way of eliminating the quadratic effects must be devised. We shall see how this is achieved in a later section. For extensive discussion of their optical properties, see [13.19].
Since the sextupoles will be used in conjunction with one or more round lenses, the paraxial solutions satisfy (13.1). We adopt a slightly different pair of solutions from those of (13.2), namely \(h(x)\), which, as before, satisfies
$$\begin{aligned}\displaystyle h(z_{\mathrm{o}})&\displaystyle=0\;,\\ \displaystyle h^{\prime}(z_{\mathrm{o}})&\displaystyle=1\end{aligned}$$
(13.17a)
and \(k(z)\),
$$\begin{aligned}\displaystyle k(z_{\mathrm{o}})&\displaystyle=1\;,\\ \displaystyle k(z_{k})&\displaystyle=0\;,\end{aligned}$$
(13.17b)
where\(z=z_{k}\) is the coma-free plane discussed earlier. The expression for a general ray becomes
$$u(z)=\Omega_{h}h(z)+\Omega_{k}k(z)$$
(13.18)
in which \(\Omega_{h}\) and \(\Omega_{k}\) are simply related to the usual object coordinates (for ample details, see [13.13], which is followed here closely). When this general ray passes through a sextupole lens, it will be deviated through a distance \(\Updelta u\), which has the form
$$\Updelta u=\bar{\Omega}_{h}^{2}u_{11}+\bar{\Omega}_{h}\bar{\Omega}_{k}u_{12}+\bar{\Omega}_{k}^{2}u_{22}\;,$$
(13.19)
with
$$\begin{aligned}\displaystyle&\displaystyle u_{11}(z)=\frac{1}{2}\left(h(z)\int_{z_{\mathrm{o}}}^{z}Sh^{2}k\,\mathrm{d}z-k(z)\int_{z_{\mathrm{o}}}^{z}Sh^{3}\,\mathrm{d}z\right),\\ \displaystyle&\displaystyle u_{12}(z)=\left(h(z)\int_{z_{\mathrm{o}}}^{z}Shk^{2}\,\mathrm{d}z-k(z)\int_{z_{\mathrm{o}}}^{z}Sh^{2}k\,\mathrm{d}z\right),\\ \displaystyle&\displaystyle u_{22}(z)=\frac{1}{2}\left(h(z)\int_{z_{\mathrm{o}}}^{z}Sk^{3}\,\mathrm{d}z-k(z)\int_{z_{\mathrm{o}}}^{z}Shk^{2}\,\mathrm{d}z\right).\end{aligned}$$
(13.20)
The function \(S(z)\) characterizes the field distribution in the sextupoles; in the most general case (in which both electrostatic and magnetic sextupoles may be present), \(S\) is given by
$$\begin{aligned}\displaystyle S(z)&\displaystyle=\frac{\exp(-3\mathrm{i}\chi)}{\phi_{\mathrm{o}}^{1/2}}\\ \displaystyle&\displaystyle\quad\,\times\left[\frac{(1+\varepsilon\phi)(p_{3}+\mathrm{i}q_{3})}{\phi^{1/2}}+\mathrm{i}\eta(P_{3}+\mathrm{i}Q_{3})\right],\end{aligned}$$
(13.21)
where \(\chi(z)\) characterizes the usual rotation in magnetic lenses.
All three contributions must vanish if the unwanted second-order effects of the sextupoles are to be eliminated; this can be achieved if the four integrals
$$\int_{z_{\mathrm{o}}}^{z_{\mathrm{e}}}S(z)h^{3-n}k^{n}\,\mathrm{d}z\;,\quad n=0,1,2\text{ and }3\;,$$
(13.22)
all vanish. The form of these conditions, in two of which \(h(z)\) is an even function while \(k(z)\) may be odd, and in the other two the situation is reversed, indicates that symmetry can be used to eliminate all four quantities. The simplest arrangement is that shown in Fig. 13.3. Here, one ray is symmetric within each sextupole, but antisymmetric about the center-plane of the combination; the other ray is antisymmetric about the midplane of each sextupole but symmetric about the center-plane of the whole combination. The members of the round-lens doublet have equal and opposite excitations, chosen in such a way that the centers of the sextupoles are conjugates, with magnification \(-1\). Since the sextupole strength is a free parameter, it can, as we shall see in Sect. 13.2.1 Sextupole Correctors, be used to cancel the spherical aberration of an adjoining round lens.
Fig. 13.3

The simplest sextupole arrangement. No second-order aberrations are introduced outside the corrector. After [13.14]

13.1.2 Chromatic Aberrations

Chromatic aberrations are a consequence of the rapid variation in electron lens strength with electron energy and lens excitation. The electron beam from the gun will inevitably have some energy spread and there will be some variation in the lens excitations, however carefully they have been stabilized. The result is a chromatic effect, characterized by chromatic aberration coefficients, that blurs the image. We can include this in the paraxial formalism by writing
$$\begin{aligned}\displaystyle\frac{u_{\mathrm{i}}-Mu_{\mathrm{o}}}{M}&\displaystyle=-\left[C_{\mathrm{c}}u_{\mathrm{o}}^{\prime}+(C_{\mathrm{D}}+\mathrm{i}C_{\theta})u_{\mathrm{o}}\right]\\ \displaystyle&\displaystyle\quad\,\times\left(\frac{\Updelta\phi_{0}}{\phi_{0}}-2\frac{\Updelta B_{0}}{B_{0}}\right)\\ \displaystyle&\displaystyle=-\left[C_{\mathrm{c}}u_{\mathrm{o}}^{\prime}+(C_{\mathrm{D}}+\mathrm{i}C_{\theta})u_{\mathrm{o}}\right]\Delta_{\mathrm{c}}\end{aligned}$$
(13.23)
in which \(C_{\mathrm{c}}\) is the (axial) chromatic aberration coefficient, \(C_{\mathrm{D}}\) is a measure of the chromatic aberration of magnification and \(C_{\theta}\) is the anisotropic distortion . The quantity \(\Updelta\phi_{0}\) is a measure of the potential variation corresponding to the energy spread of the beam, and \(\Updelta B_{0}\) represents any fluctuations in the field strength of the lens caused by variations of the current. (Here we are considering only magnetic lenses; a similar reasoning applies to electrostatic lenses.)

We thus have two types of aberrations: the chromatic aberration \(C_{\mathrm{c}}\), which is linear in gradient and hence does not vanish for object points close to the axis, just like the spherical aberration; and the complex chromatic aberration of magnification (\(C_{\mathrm{D}}+\mathrm{i}C_{\theta}\)), analogous to the distortion in that it is independent of gradient. We can therefore expect that the chromatic aberration, like the spherical aberration, will impose a limit on the resolution attainable in very high-resolution work. In practice, it defines an information limit, which, in instruments that have not been corrected for spherical aberration, is usually less severe than the limit imposed by spherical aberration . With the arrival of spherical aberration correctors, however, the situation is reversed, and it becomes imperative to correct the chromatic aberration as well, or find some way of rendering it innocuous. Incidentally, this is a return to the situation in the early days of electron microscopy, when chromatic effects dominated as a result of the relatively poor stabilization circuitry of the time.

For quadrupoles, the chromatic aberrations are again different in the \(x{-}z\) and \(y{-}z\) planes
$$\begin{aligned}\displaystyle&\displaystyle\Updelta x_{\mathrm{i}}=(C_{\mathrm{c}x}x_{\mathrm{o}}^{\prime}+C_{Mx}x_{\mathrm{o}})\Delta_{\mathrm{c}}\;,\\ \displaystyle&\displaystyle\Updelta y_{\mathrm{i}}=(C_{\mathrm{c}y}y_{\mathrm{o}}^{\prime}+C_{My}y_{\mathrm{o}})\Delta_{\mathrm{c}}\end{aligned}$$
(13.24)
in which we have again represented the potential and field variation by \(\Delta_{\mathrm{c}}\) (13.23).

13.1.3 Parasitic Aberrations

Parasitic aberrations are, as their name suggests, not intrinsic defects of electron lenses. They result from imperfections of the construction or alignment of the lenses or of the entire instrument in question. The most serious is a (first-order) astigmatism , traditionally associated with imperfect circularity of the bore of the lens: in fact, Fourier analysis shows that this astigmatism will usually be the dominant parasitic effect, whatever the origin of the problem. It is as though a very weak quadrupole had been superimposed on the lens and the effect can be canceled by adding a weak quadrupole with the opposite strength. Such a device, which in practice has a more complex structure, is known as a stigmator, and several such correctors are routinely incorporated in commercial instruments. In high-resolution operation, other parasitic effects become noticeable, and ways of correcting or compensating for these are now known, as we shall see in Sect. 13.3.1. Attention has been concentrated on axial parasitic aberrations, and several classifications of these are in use. We list these in Table 13.1.

The dependence of the coefficients employed by Uhlemann and Haider [13.20] and Haider et al [13.21, 13.22, 13.23, 13.24] on angle can be recognized from the expression for the wave aberration
$$\begin{aligned}\displaystyle W&\displaystyle=\Re\left\{A_{0}\theta^{\ast}+\frac{1}{2}C_{1}\theta\theta^{\ast}+\frac{1}{2}A_{1}\theta^{\ast 2}\right.\\ \displaystyle&\displaystyle\quad\,\left.+B_{2}\theta^{2}\theta^{\ast}+\frac{1}{3}A_{2}\theta^{\ast 3}\right.\\ \displaystyle&\displaystyle\quad\,\left.+\frac{1}{4}C_{3}(\theta\theta^{\ast})^{2}+S_{3}\theta^{3}\theta^{\ast}+\frac{1}{4}A_{3}\theta^{\ast 4}\right.\\ \displaystyle&\displaystyle\quad\,\left.+B_{4}\theta^{3}\theta^{\ast 2}+D_{4}\theta^{4}\theta^{\ast}+\frac{1}{5}A_{4}\theta^{\ast 5}\right.\\ \displaystyle&\displaystyle\quad\,\left.+\frac{1}{6}C_{5}(\theta\theta^{\ast})^{3}+S_{5}\theta^{4}\theta^{\ast 2}+R_{5}\theta^{5}\theta^{\ast}+\frac{1}{6}A_{5}\theta^{\ast 6}\right.\\ \displaystyle&\displaystyle\quad\,\left.+B_{6}\theta^{4}\theta^{\ast 3}+D_{6}\theta^{5}\theta^{\ast 2}+F_{6}\theta^{6}\theta^{\ast}+\frac{1}{7}A_{6}\theta^{\ast 7}\right.\\ \displaystyle&\displaystyle\quad\,\left.+\frac{1}{8}C_{7}(\theta\theta^{\ast})^{4}+S_{7}\theta^{5}\theta^{\ast 3}+R_{7}\theta^{6}\theta^{\ast 2}+G_{7}\theta^{7}\theta^{\ast}\right.\\ \displaystyle&\displaystyle\quad\,\left.+\frac{1}{8}A_{7}\theta^{\ast 8}\right\}.\end{aligned} $$
(13.25a)
The corresponding expansions in the notation of Krivanek et al [13.25, 13.26] are as follows
$$\begin{aligned}\displaystyle W&\displaystyle=\frac{1}{n+1}\sum_{n}\sum_{m}\Re\left\{C_{n,m}\theta^{n+1}\exp(-\mathrm{i}m\varphi)\right\}\\ \displaystyle&\displaystyle=\frac{1}{n+1}\sum_{n}\sum_{m}\left(C_{n,m,a}\theta^{n+1}\cos m\varphi\right.\\ \displaystyle&\displaystyle\quad\,\left.+\,C_{n,m,b}\theta^{n+1}\sin m\varphi\right)\\ \displaystyle&\displaystyle=\frac{1}{n+1}\sum_{n}\sum_{m}\left(a^{2}+\beta^{2}\right)^{(n-m+1)/2}\\ \displaystyle&\displaystyle\quad\,\Re\left\{C_{n,m,a}(\alpha-\mathrm{i}\beta)^{m}+\mathrm{i}C_{n,m,b}(\alpha-\mathrm{i}\beta)^{m}\right\},\end{aligned}$$
(13.25b)
in which we have written
$$\theta=\alpha+\mathrm{i}\beta\text{ and }C_{n,m}=C_{n,m,a}+\mathrm{i}C_{n,m,b}\;.$$
(13.25c)
The notation employed by Sawada et al [13.27, 13.28, 13.29, 13.30] is similar to that of Uhlemann and Haider but not identical. The starting point is again
$$W=\Re\left(\sum_{m,p}\frac{1}{m+p}C_{mp}\theta^{p}\theta^{\ast m}\right).$$
Aberrations that contain only \(\theta^{\ast m}\) and not \(\theta\) are named \(m\)-fold astigmatisms , \(A_{2}\), \(A_{3}\), \(A_{4},\dots\), generated by
$$\Re\left(\frac{1}{2}A_{2}\theta^{\ast 2}\right),\Re\left(\frac{1}{3}A_{3}\theta^{\ast 3}\right),\Re\left(\frac{1}{4}A_{4}\theta^{\ast 4}\right),\dots$$
Aberrations in \((\theta\theta^{\ast})^{m}\) are denoted by \(O_{m}\); thus \(O_{2}\) represents defocus, \(O_{4}\) is third-order spherical aberration (otherwise \(C_{\mathrm{s}}\) or \(C_{3}\)) and \(O_{6}\) is a fifth-order spherical aberration.

Comas are labeled \(P_{m}:P_{3}\) is an axial coma, arising from \(\Re(P_{3}\theta\theta^{\ast{2}}/3)\), and \(P_{5}\) is a fourth-order axial coma, \(\Re(P_{5}\theta^{2}\theta^{\ast 3}/5)\).

The aberrations denoted by \(Q_{m}\) correspond to twofold symmetry and arise from
$$\Re(Q_{4}\theta\theta^{\ast 3}/4),\,\Re(Q_{6}\theta^{2}\theta^{\ast 4}),\dots$$

Terms in \(R_{m}\) correspond to threefold symmetry, \(\Re(R_{5}\theta\theta^{\ast 4}/5),\dots\), and terms in \(S_{m}\) to fourfold symmetry, \(\Re(S_{6}\theta\theta^{\ast 5}/6),\dots\)

We note that Krivanek's notation covers all possible axial aberrations, whereas a new symbol has to be introduced for each new order of aberrations in the notations of Haider and Sawada, rather like the Roman numbering system (I, X, C, M and V, L, D). The difference between the two types of notation becomes especially apparent when combination aberrations generated by the interaction of aberrations occurring in different parts of a multi-element system are considered.

A study of parasitic aberrations has been made by Krivanek [13.31], see also [13.32, 13.33, 13.34, 13.35, 13.36]. For measurement techniques, see Saxton [13.37, 13.38, 13.39, 13.40], Saxton et al [13.41] and Chand et al [13.42], as well as Ishizuka [13.43], Meyer et al [13.44, 13.45], Lupini [13.46], and Lupini et al [13.47]. The use of the Ronchigram is described in Sect. 13.3.2. The work of Zemlin on alignment is also relevant here [13.48, 13.49]; see also Krivanek [13.50]. Among the earlier literature on parasitic aberrations, we draw attention to the work of Glaser [13.51], Sturrock [13.52, 13.53], Archard [13.54], Glaser and Schiske [13.55], Der-Shvarts [13.56], Stoyanov [13.57] and Herrmann et al [13.58]. A long survey has been written by Yavor [13.59], and other references are listed in Hawkes [13.12]. The many publications on the stigmator are also relevant; see Chapter 31 of Hawkes and Kasper [13.6] for numerous references to these.

Table 13.1

Various notations for axial aberration coefficients

Aberration

Haider

Krivanek

Sawada

Shift

\(A_{0}\)

\(C_{0,1}\)

 

Defocus

\(C_{1}\)

\(C_{1,0}\)

\(O_{2}\)

Twofold astigmatism

\(A_{1}\)

\(C_{1,2}\)

\(A_{2}\)

Second-order axial coma

\(B_{2}\)

\((1/3)C_{2,1}\)

\(P_{3}\)

Threefold astigmatism

\(A_{2}\)

\(C_{2,3}\)

\(A_{3}\)

Third-order spherical aberration

\(C_{3}\)

\(C_{3,0}\)

\(O_{4}\)

Third-order star aberration

\(S_{3}\)

\((1/4)C_{3,2}\)

\(Q_{4}\)

Fourfold astigmatism

\(A_{3}\)

\(C_{3,4}\)

\(A_{4}\)

Fourth-order axial coma

\(B_{4}\)

\((1/5)C_{4,1}\)

\(P_{5}\)

Fourth-order three-lobe aberration

\(D_{4}\)

\((1/5)C_{4,3}\)

\(R_{5}\)

Fivefold astigmatism

\(A_{4}\)

\(C_{4,5}\)

\(A_{5}\)

Fifth-order spherical aberration

\(C_{5}\)

\(C_{5,0}\)

\(O_{6}\)

Fifth-order star aberration

\(S_{5}\)

\((1/6)C_{5,2}\)

\(Q_{6}\)

Fifth-order rosette aberration

\(R_{5}\)

\((1/6)C_{5,4}\)

 

Sixfold astigmatism

\(A_{5}\)

\(C_{5,6}\)

\(A_{6}\)

Sixth-order axial coma

\(B_{6}\)

\((1/7)C_{6,1}\)

\(P_{7}\)

Sixth-order three-lobe aberration

\(D_{6}\)

\((1/7)C_{6,3}\)

\(R_{7}\)

Sixth-order pentacle aberration

\(F_{6}\)

\((1/7)C_{6,5}\)

 

Sevenfold astigmatism

\(A_{6}\)

\(C_{6,7}\)

\(A_{7}\)

Seventh-order spherical aberration

\(C_{7}\)

\(C_{7,0}\)

\(O_{8}\)

Seventh-order star aberration

\(S_{7}\)

\((1/8)C_{7,2}\)

\(Q_{8}\)

Seventh-order rosette aberration

\(R_{7}\)

\((1/8)C_{7,4}\)

 

Seventh-order chaplet aberration

\(G_{7}\)

\((1/8)C_{7,6}\)

 

Eightfold astigmatism

\(A_{7}\)

\(C_{7,8}\)

\(A_{8}\)

13.2 Aberration Correction

In this section, we provide the theoretical basis for aberration correction and then describe correctors that have been successful in correcting spherical and chromatic aberration, as well as more complex correctors that correct additional aberrations. We also discuss practical aspects of aberration correction such as aberration autotuning, and its application to other electron-optical instruments such as monochromators and energy-loss spectrometers. We focus on correctors that made key advances in the aberration correction field and on correctors that led to practical success: those that provided better spatial resolution than uncorrected instruments. We stress the importance of combination aberrations, employed for correction of aberrations for which no direct controls are available, and we provide several examples of present-day performance made possible by aberration correction. A wider-ranging survey which included correctors that did not improve the resolution of the microscope they were built into was provided in the aberration correction chapter in Science of Microscopy, the predecessor of this volume [13.60]. Aberration correction has also been reviewed extensively by Hawkes [13.12, 13.61, 13.62] and Rose [13.63], with the last Hawkes publication providing a comprehensive list of papers published on aberration correction up to 2015; some later publications are listed in Chapter 41 of [13.6]. A general introduction to aberration-corrected transmission electron microscopy has been provided by Erni [13.64], and volume 153 in the Advances in Imaging and Electron Physics series has been dedicated to it [13.65]; the collection edited by Brydson [13.66] and the chapter by Sawada [13.27] are also relevant.

Spherical aberration is the dominant resolution-limiting aberration in uncorrected electron microscopes. It is characterized by the spherical aberration coefficient \(C_{\mathrm{s}}\), which can be expressed as an integral of the form
$$C_{\mathrm{s}}=\int_{z_{\mathrm{o}}}^{z}f[B(z),h(z)]\mathrm{d}z\;,$$
(13.26)
for magnetic lenses, in which, as usual, \(B(z)\) denotes the magnetic flux on the axis and \(h(z)\) is the particular solution of the paraxial ray equation  (13.1) that vanishes at the object plane. A similar formula, in which the axial potential distribution \({\phi}(z)\) replaces \(B(z)\), gives the spherical aberration coefficient of electrostatic lenses. Here we concentrate on magnetic lenses.

The integrand \(f[B(z),h(z)]\) can be written in different ways (see [13.6] for many of these and for a general formula from which all the others can be generated; forms particularly useful for programming are given by Lencová and Lenc [13.67] (for magnetic lenses) and [13.68, 13.69] as well as Lencová [13.11] (for electrostatic lenses)). In 1936, Otto Scherzer derived a nonnegative-definite form of the integrand, a sum of squared terms, from which it is clear that the sign of the coefficient cannot change. Scherzer’s formula was nonrelativistic, but Rose [13.70] (see corrigendum in [13.71]) has established a relativistic version, which essentially confirms Scherzer’s conclusion. Efforts to find field or potential distributions for which the integrand vanishes [13.72, 13.73] failed, as they were sure to do given the form of the integrand found by Scherzer. (Attempts to find a loophole nevertheless continue; see Nomura [13.74, 13.75], for example, refuted by Hawkes [13.76].) Tretner [13.77] later established bounds on the coefficient.

The practical effect of the aberrations can be estimated as follows. Optimizing the polepiece shape of a round objective lens can lower the coefficients of spherical aberration \(C_{\mathrm{s}}\) to about \(0.3\times\) the minimum focal length of the lens \(f_{\mathrm{min}}\) [13.78, 13.79], and the coefficient of chromatic aberration \(C_{\mathrm{c}}\) can be lowered to about \(\mathrm{0.8}\)\(f_{\text{min}}\). The latter can attain about \({\mathrm{0.5}}\,{\mathrm{mm}}\) for condenser-objective lenses with small polepiece gaps (of \({\mathrm{2}}\,{\mathrm{mm}}\) or less) operated at \({\mathrm{200}}\,{\mathrm{keV}}\) (or lower) primary energy, and \(C_{\mathrm{s}}\) values of \({\mathrm{0.3}}\,{\mathrm{mm}}\) at \({\mathrm{100}}\,{\mathrm{keV}}\) [13.79] and \({\mathrm{0.4}}\,{\mathrm{mm}}\) at \({\mathrm{200}}\,{\mathrm{keV}}\) have been achieved. The spatial resolution limits imposed by \(C_{\mathrm{s}}\) and \(C_{\mathrm{c}}\) are given by
$$d_{\mathrm{s}} =aC_{\mathrm{s}}^{1/4}\lambda^{3/4}\;,$$
(13.27a)
$$d_{\mathrm{c}} =b\left(\frac{\lambda C_{\mathrm{c}}\updelta E}{E_{0}}\right)^{1/2}\;,$$
(13.27b)
respectively [13.80, 13.81], where \(a\) is a constant that depends on the type of imaging (bright- or dark-field) and the type of illumination (coherent or incoherent), with values between \(\mathrm{0.43}\) (for incoherent annular dark-field imaging) and \(\mathrm{0.67}\) (for coherent axial bright-field imaging), \(b\) is a constant depending on the resolution criterion adopted, with a value around \(\mathrm{0.5}\), \(\lambda\) the electron wavelength, \(\delta E\) the energy spread of the primary beam and \(E_{\mathrm{o}}\) the primary energy. Inserting the best attainable values in the above expressions shows that without aberration correction, bright-field resolution of about \({\mathrm{2}}\,{\mathrm{\AA{}}}\) could be reached at \({\mathrm{200}}\,{\mathrm{keV}}\) and dark-field resolution of about \({\mathrm{2}}\,{\mathrm{\AA{}}}\) at \({\mathrm{100}}\,{\mathrm{keV}}\), and such performances had indeed been demonstrated in practice. At primary energies below \({\mathrm{100}}\,{\mathrm{keV}}\), the increased electron wavelength made \({\mathrm{2}}\,{\mathrm{\AA{}}}\) resolution unreachable, and thus precluded the possibility of resolving individual atoms, atomic columns or atomic planes in closely packed solids.

Prior to successful aberration correction, the quest for higher resolution primarily took the direction of reducing \(\lambda\) by going to higher primary energies. In \(1{-}1.5\,{\mathrm{MeV}}\) electron microscopes, this allowed \({\mathrm{1}}\,{\mathrm{\AA{}}}\) resolution to be reached [13.82]. But improving the performance in this way required large and expensive electron microscopes, and typically also produced unacceptably high radiation damage. Successful aberration correction provided a way round these difficulties.

The correction relies on the fact that Scherzer’s formula does not apply when any of the assumptions made in its derivation are not fulfilled. The assumptions are that the optical system must possess rotational symmetry, the excitation must be static, only dioptric operation is permissible (excluding an electron mirror mode), and in the case of electrostatic lenses, the potential distribution and its derivatives must be continuous. Object and image must both be real (not virtual). Scherzer himself showed [13.2] that by relaxing one or other of these conditions, a corrector could be devised.

Throughout the second half of the twentieth century, efforts were made to build and test the various types of correctors. These are described in detail in Septier [13.3] and Hawkes [13.5]; more recent information is included in Hawkes [13.12, 13.61, 13.65, 13.83] and Hawkes and Kasper [13.6, Chap. 41]. Departure from rotational symmetry has always seemed a promising approach, and as early as the 1950s, Seeliger [13.84, 13.85] attempted to put Scherzer's suggestion for exploiting the idea into practice; his work was pursued by Möllenstedt [13.86]. Archard [13.87] showed that relatively simple configurations would create the desired field distribution, and Burfoot [13.88] found a three-electrode geometry that would in principle be aberration-free. Numerous further experiments were conducted over the years, including a reassuring proof-of-principle \(C_{\mathrm{s}}\)-correction experiment by Deltrap [13.89] and a \(C_{\mathrm{c}}\)-correction experiment by Hardy [13.90]. However, subsequent attempts to use aberration correctors to improve the resolution beyond what was attainable in the best uncorrected electron microscopes were unsuccessful, and in the 1980s, aberration correction temporarily acquired an aura of an impossibly difficult problem.

Quadrupole correctors that can improve a microscope's resolution by correcting its geometrical aberrations consist of four or more quadrupoles and three or more octopoles, and sextupole correctors consist of two or more sextupoles as well as round lenses. The correctors principally correct the spherical aberration of the objective lens. Substantial contributions to the system’s total \(C_{\mathrm{s}}\) can also come from other parts of the optical system—for instance, the electron gun and condenser lenses in a probe-forming system using a large beam current, and hence a wide angular range beam extracted from the electron gun—and these contributions must be corrected as well. In addition to the strict stability requirements, every aberration-corrected system must also address parasitic aberrations described in Sect. 13.1.3 above, where the notation used later in this section is defined. The corrected optical system is invariably more complicated than the corresponding uncorrected system, and this opens up many new avenues for parasitic aberrations to be produced. At the same time, when aiming for higher resolution, the precision with which the aberrations must be nulled increases considerably.

Successful aberration-corrected optical systems must therefore accomplish three separate tasks: measure and correct the principal aberrations affecting the resolution, diagnose and eliminate parasitic aberrations, and improve the instrumental stabilities. This stage of development was first reached in the 1990s by the Darmstadt/Heidelberg group led by Harald Rose, Max Haider and Joachim Zach, and the Cambridge/Seattle group led by Ondrej Krivanek and Niklas Dellby.

As we shall see in the following sections, both types of multipole correctors are now in common use. The quadrupole–octopole corrector is used in scanning transmission electron microscopes ( s) and in conventional (fixed-beam) transmission electron microscopes ( s) when chromatic aberrations are corrected together with geometrical ones. The sextupole arrangement is incorporated in both CTEMs and STEMs. A valuable study of the performance of sextupole correctors was carried out by Haider et al [13.91] and more recently by Müller et al [13.92], and a description of various aspects of quadrupole–octopole corrector construction, including stability requirements, has been provided by Krivanek et al [13.25].

Other possibilities for correcting spherical aberration include the use of space charge  [13.93] or of phase plates , with both devices changing the phase of the electron wavefront so that \(C_{\mathrm{s}}\) is eliminated, or an optimal phase-contrast transfer function is obtained. Unfortunately, the space charge method has not proven to be reliable, and practical difficulties have so far prevented the phase plate method from reaching its theoretical potential [13.94]. Holography and wavefront reconstruction employing bright-field through-focus series data can also lead to the elimination of the influence of spherical aberration by post-processing of experimental data. In this chapter, however, we concentrate on electron-optical methods of correcting aberrations. The introduction of an electrode on the optic axis, and hence the use of conical beams , has been revived recently. We refer to Khursheed and Ang [13.95, 13.96] and Kawasaki et al [13.97, 13.98] for details.

Chromatic aberration correction also has a long history, though less effort has been devoted to it than to spherical aberration correction because, in the high-resolution imaging mode, developments in microscope design soon rendered the adverse effect of spherical aberration greater than that of chromatic aberration. This is obvious from an examination of the phase-contrast transfer function, which is a sinusoidal curve in the absence of any energy spread (chromatic effects or temporal partial coherence) and neglecting the nonvanishing source size. The sinusoidal curve is damped by an envelope function, representing the effect of energy spread, but the first zero of the sinusoidal curve (a measure of the limit of resolution determined by the spherical aberration) occurs well before the damping curve reduces it to an unacceptably small value (the so-called information limit). With the arrival of \(C_{\mathrm{s}}\) correctors, however, the situation has changed dramatically, and it is now of interest, indeed essential, to improve the information limit as well by reducing the undesirable effects of energy spread.

The (axial) chromatic aberration of electron lenses is characterized by a coefficient, \(C_{\mathrm{c}}\), which can be written as an integral of the form
$$C_{\mathrm{c}}=\int_{z_{\mathrm{o}}}^{z_{\mathrm{i}}}\frac{\eta^{2}B^{2}}{4\hat{\phi}_{0}}h^{2}\mathrm{d}z\;, $$
(13.28a)
for magnetic lenses with
$$\Delta=\frac{\Updelta\hat{\phi}_{0}}{\hat{\phi}_{0}}-2\frac{\Updelta B_{0}}{B_{0}}$$
and
$$C_{\mathrm{c}}=\hat{\phi}_{0}^{1/2}\int\frac{\gamma\phi^{\prime 2}(3+2\varepsilon\hat{\phi})}{8\hat{\phi}^{5/2}}h^{2}\mathrm{d}z\;,$$
(13.28b)

for electrostatic lenses. As in the case of\(C_{\mathrm{s}}\) (and, indeed, of all the aberration coefficients), the integrands in (13.28a) and (13.28b) can be written in different ways. The ones given here show immediately that \(C_{\mathrm{c}}\) is positive definite, and the best that can be hoped for in a round lens is a design for which \(C_{\mathrm{c}}\) is small.

Over the years, two approaches to the problem of avoiding the limitations imposed by the chromatic aberration of round lenses have emerged. One is a natural continuation of the efforts to reduce the energy spread of the beam emitted by the electron gun; by introducing a monochromator into the column, electrons with energies outside the chosen range can be excluded. There is of course a loss of beam current, but since the energy spread of the filtered beam can be made appreciably narrower than that of the original beam, this reduction in current may be acceptable. This solution is attractive not only in the imaging mode but also for electron energy-loss spectroscopy ( ) and energy-filtered transmission electron microscopy ( ), since it improves the energy resolution and hence the information content of EELS data.

From 2004 onward, monochromators have enabled substantially improved spatial resolution to be reached in aberration-corrected CTEM axial bright-field imaging (which is highly sensitive to resolution loss resulting from \(C_{\mathrm{c}}\) effects). More recently, a resolution improvement through the use of monochromator has also been demonstrated in the STEM, which is less sensitive to chromatic effects. Both developments are discussed in Sect. 13.3.3.

The alternative to using a monochromator is to devise a corrector of chromatic aberration. There are several ways in which such correctors can be conceived, involving the use of superimposed round lens and quadrupole fields, hybrid electrostatic–magnetic quadrupoles, or electron mirrors. An early suggestion by Scherzer [13.2] involved combining an electrostatic round lens and an electrostatic quadrupole in such a way that the overall chromatic aberration coefficient of the combination is negative. Such a device could then be used to correct the chromatic aberration of a round lens acting as an objective. This suggestion was taken up by Archard [13.99] and has subsequently been carefully investigated.

In 1961, Kel'man and Yavor showed that the chromatic aberration coefficient of a hybrid electrostatic–magnetic quadrupole can have either sign, depending on the relative strengths of the component quadrupoles, and hence that such hybrid lenses could be used to correct chromatic aberration [13.100], translated as [13.101]. The result was rediscovered by Septier [13.102] and generalized by Hawkes [13.103, 13.104]. Such hybrid quadrupoles appear in the latest designs of correctors, intended for the correction of both geometrical and chromatic aberrations.

Another possibility, pointed out by Rose [13.105], involves the use of a long Wien filter . This too has been studied carefully [13.106]. Attempts have also been made to redistribute the energy of the beam electrons in such a way that the energy spread is reduced. For details, see [13.107, 13.108, 13.109].

Finally, an electron mirror can be coupled to a round lens in such a way that the chromatic aberration is canceled. This system has led to important resolution improvements in low-energy-electron microscopy ( ) and photoemission electron microscopy ( ) of solid surfaces, see [13.110, 13.111, 13.112, 13.113, 13.114, 13.115, 13.116, 13.117, 13.118, 13.119, 13.120, 13.121, 13.122, 13.123]. Two designs including twin mirrors to correct the aberrations of a SEM are proposed by Dohi and Kruit [13.124].

13.2.1 Spherical Aberration Correctors

Quadrupole–Octopole Correctors

For the correction of spherical aberration, Scherzer proposed a sequence of cylindrical lenses and octopoles in his seminal paper of 1947 on ways of avoiding the consequences of his 1936 proof. Cylindrical lenses are the electron optical counterparts of glass lenses with cylindrical (as opposed to spherical) faces and are characterized by a round lens and a quadrupole potential distribution. It was soon realized [13.87] that quadrupole lenses could be used to advantage instead of cylindrical lenses, and the basic corrector configuration, which has remained essentially unaltered, soon emerged (Fig. 13.4): a sequence of three or preferably four quadrupole lenses, with an octopole situated at each of the line foci to cancel or over-correct the aperture aberrations in the \(x{-}z\) and \(y{-}z\) planes, together with a third octopole to complete the task of correction.

Fig. 13.4

The basic quadrupole corrector arrangement, showing four quadrupoles, with octopoles 1 and 2 situated at the two line foci. Octopole 3, which acts on a round beam, can be placed in Q1 or Q4, or in the middle of the system (separate from the quadrupoles), where it is more effective because the beam is wider

An important step forward was the introduction of the Russian quadruplet, which has geometrical symmetry and electrical antisymmetry about its midplane, as we have already mentioned. In common with all multiplets possessing these symmetry properties, such quadruplets have the same focal length in the \(x{-}z\) and \(y{-}z\) planes. For a given geometry, the positions of the foci in these planes can then be made to coincide by suitable choice of the two excitations, whereupon the quadruplet has the same overall paraxial behavior as a round lens. Sets of load curves , showing the appropriate excitations as a function of geometry, are available (see [13.15] for many such curves and [13.16]). Another interesting early contribution was made by Burfoot [13.88], who sought the (electrostatic) configuration with the smallest number of electrodes that would be free of spherical aberration. He established suitably shaped apertures in a three-electrode lens (a remarkable achievement in precomputer times), but concluded that the necessary tolerances could not be achieved in practice; a simpler way of attaining the same objective was proposed by Archard [13.125].

In 1964, Deltrap showed that the spherical aberration of a test lens could be reduced by means of a quadrupole–octopole corrector, and thus confirmed that the principle of correction was sound. However, for the next three decades, all attempts to make a corrector capable of improving the performance of a well-designed objective failed; with hindsight, we can see that these disappointing failures were due to the natural complexity of the system and hence the need to control parasitic aberrations, as well as to the difficulty in satisfying the increased stability requirements. Progress was made in the Darmstadt project [13.126, 13.127, 13.128, 13.129, 13.130, 13.131, 13.132, 13.133, 13.134, 13.135, 13.136, 13.137, 13.138, 13.139, 13.140, 13.141, 13.142, 13.143] (see Scherzer [13.144] for a summary, and Marko and Rose [13.7] for a later account) and in the Chicago project [13.145, 13.146, 13.147, 13.148, 13.149, 13.150, 13.151]. The Darmstadt project demonstrated aberration correction including manual control of parasitic aberrations, but it did not demonstrate resolution improvement relative to the best uncorrected electron microscopes available at the time. The Chicago project introduced new concepts in aberration correction, such as the sextupole corrector, but it did not succeed in demonstrating aberration correction in practice. The tools necessary for the operation of such devices, including autotuning and electronics of sufficient stability, were not yet available.

In the early 1990s, Zach et al showed that a quadrupole/octopole \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector could improve the performance of scanning electron microscopes, and this finding continues to be exploited in commercial instruments [13.152, 13.153, 13.154, 13.155, 13.156, 13.157, 13.158, 13.159, 13.160, 13.161, 13.162]. For the transmission electron microscope, success came in 1997, when Krivanek and colleagues, working in the Cavendish Laboratory in Cambridge, built a corrector equipped with computer control, capable of making the many necessary adjustments rapidly and systematically [13.163, 13.164, 13.165]. This corrector was fitted to a STEM and enabled the size of the illumination aperture to be increased, which led to either an electron probe of decreased size with no loss of beam current, or an increase in the beam current with no loss of resolution.

The Cambridge corrector consisted of the basic quadrupoles and octopoles, all under computer control, together with other multipole fields designed to compensate for misalignments and parasitic aberrations in general. In a follow-up first-generation Nion aberration corrector , the quadrupoles and the strong octopoles were separated so that there would be no magnetic cross talk between them, in a 7-multipole arrangement. The correction half-angle of this corrector was limited to \({\mathrm{25}}\,{\mathrm{mrad}}\) by the magnitude of \(C_{5,4}\) (fourfold astigmatism of the fifth order), about \({\mathrm{10}}\,{\mathrm{cm}}\), and this gave a spatial resolution of \(\approx{\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{100}}\,{\mathrm{keV}}\) [13.166, 13.167, 13.168, 13.169, 13.170, 13.26] and \({\mathrm{0.78}}\,{\mathrm{\AA{}}}\) at \({\mathrm{300}}\,{\mathrm{keV}}\) [13.171, 13.172]. A second-generation Nion corrector employs 16 quadrupoles and three combined quadrupole–octopole elements, with additional multipoles to make all parasitic aberrations up to \(C_{5,6}\) adjustable [13.173, 13.174, 13.175, 13.176, 13.177]. The corrector itself consists of an alternating sequence of quadrupole quadruplets and quadrupole–octopole elements (Fig. 13.5). With this arrangement, the center-planes of the quadrupole–octopole elements are all conjugates, and four further quadrupoles situated between the corrector and the probe-forming lens transfer the image of the correcting planes to the vicinity of the coma-free plane of the probe-forming lens. In this way, the fifth-order geometrical aberrations of the combination of corrector and probe-forming lens can be eliminated. Imaging the correction planes into each other is also a feature of the Rose superaplanator and the related transmission electron aberration-corrected microscope (TEAM ) corrector, described in more detail in Sect. 13.2.3. Software adjusts the various components systematically. Another quadrupole–octopole corrector designed for an FEI STEM/TEM was described by Mentink et al [13.178].

Fig. 13.5

Schematic of the Nion quadrupole–octopole corrector, showing the axial and field trajectories. The coupling between the corrector and the objective lens is accomplished by four quadrupoles, which image the last octopole of the corrector into the coma-free plane of the objective lens, with different magnifications in \(x\) and \(y\). After [13.25]

Sextupole Correctors

Sextupoles were not among the correctors envisaged by Scherzer in his 1947 paper. In 1965, it was pointed out that the third-order aberrations, including of course the spherical aberration, of sextupoles have the same dependence on gradient in the object plane as that of a round lens [13.179]. However, the fact that the principal optical effect of sextupoles is not linear, as it is in round lenses and quadrupoles, but quadratic (second-order) seemed to rule out any hope of using them for aberration correction. It was not until 1979 that combinations of sextupoles and round lenses from which the quadratic effects had been eliminated by compensation were proposed [13.180], and subsequent developments have confirmed that such correctors are suitable for incorporation into transmission electron microscopes. As we have seen, the second-order effect of a sextupole is characterized by four terms of the form \(\int S(z)h^{3-n}k^{n}\mathrm{d}z\), in which \(S(z)\) represents the field distribution in the (electrostatic or magnetic) sextupole, and \(h(z)\), \(k(z)\) are two linearly independent solutions of the familiar paraxial equation for round lenses (these solutions collapse to straight lines in the absence of any round lens component). The integer \(n\) takes the four values 0, 1, 2 and 3. All four terms can be made to vanish by suitable choice of the symmetry of the configuration; the simplest is shown in Fig. 13.6. Before coupling such a device to a microscope objective, we must however ensure that the coma-free condition is satisfied. The (isotropic) coma-free plane of an objective is situated within the lens field and must hence be imaged onto the front focal plane of the round-lens doublet in the corrector by means of another doublet (Fig. 13.7). If it should be necessary to eliminate the anisotropic coma as well as the isotropic coma, an objective design in which two coils are used in tandem would have to be adopted [13.129]. Sextupole correction may be traced in the following articles (in addition to the early publications already cited) [13.181, 13.182, 13.183, 13.184, 13.185, 13.186, 13.187, 13.188, 13.189, 13.19, 13.190, 13.191, 13.192, 13.193, 13.194, 13.195, 13.196, 13.197, 13.198, 13.199, 13.200, 13.201, 13.202, 13.203, 13.204, 13.205, 13.206, 13.207, 13.208, 13.209, 13.21, 13.210, 13.211, 13.212, 13.213, 13.214, 13.215, 13.216, 13.217, 13.218, 13.219, 13.22, 13.220, 13.221, 13.222, 13.223, 13.224, 13.225, 13.226, 13.227, 13.228, 13.229, 13.23, 13.27, 13.30, 13.91, 13.92].

Fig. 13.6

Sextupole correctors. Course of the second-order fundamental rays in the corrector shown in Fig. 13.3. After [13.14]

Fig. 13.7

Sextupole correctors. Correction of coma as well as spherical aberration requires a more complex system. The transfer doublet between the objective lens and the corrector allows for the fact that the coma-free plane lies within the magnetic field of the objective. After [13.14]

Two alternative ways of generating the sextupole fields are also of interest. Following a suggestion of Nishi et al [13.230], Hoque et al [13.231] have designed a corrector in which the fields are generated by wires that follow the optic axis; see Nishi et al [13.232] and Hoque et al [13.233] for details. Janzen has observed that the regular sextupole correctors are too bulky for use in miniature microscopes and multicolumn arrays and has therefore inquired whether the necessary fields could be generated by noncircular apertures such as octagons, hexagons and squares [13.234]. A promising design is described by Janzen et al [13.235].

13.2.2 Chromatic Aberration Correctors

All-Electrostatic Correctors

Multipole correctors consisting of electrostatic elements only can correct both the chromatic and spherical aberration, provided that their fields satisfy the so-called Scherzer condition [13.2]. Configurations in which this condition is satisfied have been found by Weißbäcker and Rose [13.236, 13.237, 13.238] and by Maas and coworkers [13.239, 13.240, 13.241, 13.242]. In the studies of Weißbäcker and Rose, several configurations were examined, in which the complexity increased with the practical usefulness of the corrector. The configurations were described in detail in the corresponding chapter of Science of Microscopy [13.60]. The symmetry conditions are arranged in such a way that the chromatic aberration and the coma vanish, while the spherical aberration is corrected by means of octopoles. In the complementary investigations of Henstra, Maas, and Mentink and coworkers, a configuration consisting of nine elements was explored.

Although the correction principles are sound, electrostatic-only multipole correctors have not yet found practical application in electron optics. They are, however, highly promising for ion-optical applications, as electrostatic focusing is more suitable for slow ions than magnetic focusing, and their use has been demonstrated by Bajo et al [13.243].

Quadrupole \(C_{\mathrm{s}}/C_{\mathrm{c}}\) Correctors

Quadrupole lenses consisting of four electrodes and four magnetic poles situated midway between the electrodes have the power of correcting the chromatic aberration of a round lens. They must of course be part of a suitable configuration, such as the one used by Hardy [13.90], the Darmstadt \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector [13.127, 13.135], and the Zach and Haider \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector for a SEM [13.154, 13.155]. The simpler configurations produce large field and fifth-order combination aberrations, but this can be overcome by two types of more complex arrangements. The first of these relies on precise imaging of the correction planes into each other and then into the coma-free plane of the objective lens, which avoids combination fifth-order spherical aberration \(C_{5}\) and decreases field aberrations. This is the arrangement used by the superaplanator corrector due to Rose, whose ideas have led to the TEAM and PICO correctors , [13.21, 13.22, 13.222] and by the Nion \(C_{3}/C_{5}\) corrector [13.175, 13.176, 13.177]. In the TEAM corrector [13.13], two symmetric quadrupole quintuplets and three (or more) octopoles correct spherical and chromatic aberration as well as \(C_{5}\). The quadrupole fields are symmetric with respect to the center-plane of each quintuplet; conversely, the whole (double-quintuplet) unit exhibits antisymmetry about its midplane (Fig. 13.8).

Fig. 13.8

The TEAM corrector. Anamorphic images of the diffraction plane are formed at the center planes of the multipole quintuplets, 1 and 2. N denotes the nodal planes. After [13.13]

The second type of arrangement distributes the correction action into several planes and makes sure that higher-order combination aberrations produced by incorrectly imaged correction elements cancel each other. This can be accomplished by using pairs of identically excited correcting elements (identical in theory, small deviations are likely to be needed in practice); these elements are located at equal distances from the corrector midplane, with trajectories that are symmetric or antisymmetric about this plane. This maps all the effective correction planes, encapsulating the summed effects of the correcting elements, into the midplane of the corrector, and the midplane is then imaged into the vicinity of the coma-free plane of the objective lens by coupling lenses. Three types of aberrations are corrected internally in a quadrupole–octopole corrector (\(C_{3,0,}C_{3,2}\) and \(C_{3,4}\)), and the effective correction planes for all of them are projected into the corrector midplane. This type of corrector was also proposed by Rose, in versions with five, seven or eight multipoles consisting of quadrupoles and octopoles and trajectories that are symmetric or antisymmetric about the central plane [13.128, 13.129, 13.192]. A modified eight-multipole version has been implemented as the SALVE corrector  [13.244, 13.245], with additional controls for parasitic aberrations.

The external appearance of the SALVE corrector and the ray diagram of its multipole part (with round coupling lenses not included) are shown in Fig. 13.9a,b. Multipoles MO3 and M13 contain the hybrid electrostatic-magnetic quadrupoles, plus octopoles and other multipoles. The corrector’s height and weight are about \({\mathrm{50}}\%\) of those of the PICO \(C_{\mathrm{s}}/C_{\mathrm{c}}\) corrector, but it reaches the same goals, and is less affected by instabilities and Johnson–Nyquist noise (discussed in Sect. 13.2.5).

Fig. 13.9a,b

The SALVE corrector. (a) The external appearance of the corrector, (b) the electron-optical trajectories through the corrector. Reprinted with permission from [13.244]. Copyright 2016 by the American Physical Society

Very high stability of the excitations is essential for all correctors, especially \(C_{\mathrm{s}}/C_{\mathrm{c}}\) ones. When aiming for \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) resolution at \(200{-}300\,{\mathrm{keV}}\), or \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) and below, better than one part in \(\mathrm{10^{7}}\) stability is needed for the regular optical elements, and about two parts in \(\mathrm{10^{8}}\) or better for the hybrid quadrupoles that perform the \(C_{\mathrm{c}}\) correction [13.21]. Fortunately, such stabilities are now available with modern electronics. Attention must also be paid to Johnson–Nyquist noise , which can affect the performance of \(C_{\mathrm{s}}/C_{\mathrm{c}}\) correctors.

An All-magnetic Corrector

An \(\Upomega\)-shaped all-magnetic corrector has been developed for use at high voltages [13.246].

Wien Filters and Correction

In an attempt to design a corrector that is reasonably easy to align and consists of as few separate elements as possible, Rose [13.105] has also examined the properties of an inhomogeneous Wien filter, and his ideas were followed up by Mentink et al [13.247] and Steffen et al [13.248]. No practical corrector based on these ideas appears to have been built to date, and the theoretical designs were reviewed in detail in [13.60].

Mirror Correctors

Several schemes have been proposed to compensate for the aberrations of round lenses by introducing an electron mirror (see [13.249, 13.250, 13.251] for early work on electron mirrors) into the optical system [13.252, 13.253, 13.254]. Aberration correction has now reached the stage at which simultaneous correction of all the aberrations that are likely to impair the performance of the instrument in question must be envisaged: correction of individual aberrations without considering the effect of the remainder is no longer sufficient. For this reason, we focus here on the scheme devised by Preikszas and Rose [13.255] and surveyed by Hartel et al [13.116], intended for the SMART project at BESSY II and also adopted for PEEM3 at the Lawrence Berkeley National Laboratory [13.256]. This differs from earlier schemes, notably that of Rempfer [13.110], Shao and Wu [13.257], Rempfer and Mauck [13.111] and Rempfer et al [13.258], in that the beam splitter, the role of which is to separate the beam incident on the mirror from the beam emerging from it, is now nondispersive. A four-electrode mirror, such as that shown in Fig. 13.10a, offers enough degrees of freedom to adjust the focal length and the spherical and chromatic aberration coefficients satisfactorily [13.259, 13.260]. As an example, Preikszas and Rose [13.255] show that the spherical and chromatic aberration coefficients can be chosen anywhere inside the shaded region in Fig. 13.10b for a fixed position of the Gaussian image plane. Such a mirror could be combined with a dispersion-free magnetic beam splitter as shown in Fig. 13.11a,ba. Figure 13.11a,bb shows the device incorporated in the spectromicroscope for all relevant techniques ( ). In practice, the SMART system has demonstrated \({\mathrm{2.6}}\,{\mathrm{nm}}\) spatial resolution in LEEM mode, using a landing energy of \({\mathrm{15}}\,{\mathrm{eV}}\) [13.261]. Another very successful mirror-type LEEM/PEEM corrector has been built at IBM T.J. Watson laboratories [13.120, 13.121, 13.122] and is shown in Fig. 13.12. It allows a variety of operating modes that include energy-filtered imaging, and it has achieved \({\mathrm{2}}\,{\mathrm{nm}}\) edge definition. A similar design has been described by Mankos and Shadman [13.262]. Dohi and Kruit [13.124] have suggested that, by using microelectromechanical systems (MEMS) technology to fabricate very small mirrors, a compact SEM corrector could be built.

Fig. 13.10

(a) Section of a tetrode mirror. The potentials applied to the electrons determine the focal length and the chromatic and spherical aberration coefficients. (b) For a fixed image plane, the aberration coefficients of the tetrode mirror lie within the shaded zone. After [13.255]

Fig. 13.11a,b

Incorporation of the tetrode mirror of Fig. 13.16 into a complete system. The beam separator is free of dispersion. (a) Basic configuration. (b) The entire layout of the SMART (spectromicroscope for all relevant techniques). (a) After [13.255] (b) reprinted from [13.116], with permission from Elsevier

Fig. 13.12

Schematic diagram of a LEEM system with \(C_{\mathrm{c}}/C_{\mathrm{s}}\) correction optics. The edges of orange boxes around the magnetic prism arrays ( ) show the positions of the symmetrically located diffraction planes. Intermediate image planes are located on the MPA diagonals (orange diagonals). Optical elements shown in purple are electrostatic, the rest are electromagnetic. Reprinted from [13.120], with permission from Elsevier

13.2.3 Correction of Aberrations Beyond Third Order

Two complex correctors were proposed by Rose in the early 2000s that are capable of correcting the spherical and chromatic aberrations, as well as other primary aberrations such as field curvature and astigmatism that can be harmful once the axial aberrations have been brought under control [13.13, 13.14, 13.263, 13.264, 13.265]. These are called superaplanator and ultracorrector . The first of these, discussed briefly in the context of \(C_{\mathrm{c}}/C_{\mathrm{s}}\) correctors above, is suitable for transmission electron microscopes, while the second is intended for lithography where a wide field of view is required. A simplified version of the superaplanator , called an achroplanator  [13.222], has been used in the \(C_{\mathrm{s}}/C_{\mathrm{c}}\) TEAM and PICO projects [13.63]. The results from these projects and from SALVE can be found in several publications [13.244, 13.245, 13.266, 13.267, 13.268, 13.269, 13.270, 13.271, 13.272].

It has been known since the work of Shao [13.273] that when threefold astigmatism \(C_{2,3}\) is canceled by opposing the astigmatisms created by two sextupoles, sixfold astigmatism \(C_{5,6}\) remains as the strongest uncorrected aberration. The magnitude of \(C_{5,6}\) can be reduced by decreasing the length of the two sextupoles and increasing their excitation [13.274], but a complete cancellation of \(C_{\mathrm{5,6}}\) is not possible without adding a 12-pole \(C_{5,6}\) stigmator to the two-sextupole design. Another way around this difficulty is possible with three-sextupole corrector designs, as these provide an additional degree of freedom that allows both \(C_{2,3}\) and \(C_{5,6}\) to be nulled simultaneously [13.218].

Attaining optimal performance with correctors of higher-order aberrations means that parasitic aberrations of all orders up to the highest one being corrected must be kept under strict control. We discuss the work in this field in Sect. 13.3.1 below.

13.2.4 Aberration Correction in Monochromators, Electron Spectrometers and Imaging Filters

The task of a monochromator is to reduce the energy spread of the illuminating beam, typically by producing an energy spectrum of electrons emitted by the electron source and projecting the spectrum onto an energy-selecting slit which defines the passband of energies that is admitted into the rest of the electron microscope column. The task of an electron energy-loss spectrometer is to produce an energy spectrum and project it onto a detector, typically with adjustable magnification, and to read it out efficiently. An imaging filter performs two related tasks: (a) it projects an energy-loss spectrum onto an energy-selecting slit, admits a passband of energies through the slit, and transforms the selected spectrum into an achromatic and largely distortion-free image on the final detector, or (b) it projects the spectrum onto the final detector, the slit having been withdrawn or opened wide. The three types of instruments are therefore closely related, and the same electron-optical principles apply to all of them.

An important issue concerns how the spectrometer (or imaging filter) is coupled electron-optically to the rest of the microscope column. Two solutions are possible:
  1. 1.

    Image coupling , in which a very small image of the illuminated area of the sample is projected into the spectrometer entrance object plane, where a small crossover is needed, with a magnification (relative to the magnification on the sample) of typically around 10\(\times\), or

     
  2. 2.

    Diffraction coupling , in which a very small diffraction pattern, with a camera length of the order of a few micrometers, is projected into the spectrometer entrance object plane.

     

The spectrometer energy-disperses the entrance object into a spectrum and images it onto the final detector, whereas the imaging filter energy-disperses the entrance object into a spectrum and images it onto the energy-selecting slit. Monochromators use the image-coupling mode, with a small image of the source dispersed into an energy spectrum imaged onto the monochromator slit.

It is useful to remember that a diffraction pattern appears on the final viewing screen as well as at the spectrometer (or imaging filter) entrance aperture in every CTEM operating in the image-coupling mode, and an image of the sample appears in these places in the diffraction-coupling mode. Planes containing reciprocally related images succeeding each other is the usual order of the day in electron microscopes, and it is important not to lose sight of the reciprocal relationship.

Only the image-coupling mode is used to couple to EEL spectrometers in STEMs. Good coupling efficiency for energy-loss events is obtained with EELS entrance apertures slightly larger than the STEM bright-field cone , and this means that electrons scattered elastically to higher angles (typically greater than about \({\mathrm{80}}\,{\mathrm{mrad}}\)) can be directed at the same time to a high-angle annular dark-field ( ) detector. Structural information on the sample can thus be obtained from the HAADF detector at the same time as the EEL spectrum, and simultaneous collection of the two types of signals has become the standard procedure in STEM. Alternatively, a pixelated two-dimensional () detector can be inserted in the pre-spectrometer plane, temporarily obscuring the EELS entrance. This is used for recording convergent-beam Ronchigrams for STEM autotuning, as well as diffraction patterns obtained with a more parallel beam.

Figure 13.13 shows a schematic cross section of a STEM column containing a ground-potential monochromator, probe-forming optics, and a spectrometer. In all three of these microscope components, the object being imaged is the electron source, i. e., the virtual crossover that is located inside the field emission tip and from which the electrons appear to emanate. The correction of aberrations affecting the monochromator spectrum, sample-level probe and the final EEL spectrum has the same aim: decreasing the size of the imaged object as much as possible. The correction inside the monochromator compensates for aberrations introduced by the electron gun, the monochromator coupling lenses and the monochromator optics; the probe-formation corrector compensates for the aberrations of the objective lens, the condenser lenses and any remaining aberrations of the monochromator; and the spectrometer-level correction compensates for aberrations of the spectrometer as well as aberrations of post-sample optics including the post-sample part of the objective lens.

Fig. 13.13

Schematic diagram of the electron trajectories through the Nion ultrahigh-energy resolution monochromatized EELS-STEM (U-HERMES). All the beam crossovers contain an image of the electron source; the most important crossovers are numbered: \(1=\) cold field emission gun ( ) virtual crossover; \(2=\) monochromator slit crossover; \(3=\) sample-level crossover; \(4=\) EEL spectrum crossover

A major difference between the correction carried out in the monochromator and the spectrometer and the correction of the sample-level probe is that electron spectrometers, imaging filters and monochromators use prisms that have inherent second-order aberrations that are not present in optical systems with round symmetry. The prisms employ transverse magnetic or electrostatic fields to disperse electrons in energy, and they bend the electron beam. (Wien filters constitute an important exception: they disperse the beam using combined transverse magnetic and electrostatic fields working in opposition in such a way that there is no overall bend.)

Optical systems with transverse fields can be mirror-symmetric about their \(x{-}z\) (dispersion) plane, but cannot be rotationally symmetric. Second-order aberrations are usually very strong in uncorrected versions of such systems, and sextupoles or curved prism faces, which produce sextupole effects, have been employed to correct second-order aberrations of prisms for some time [13.275, 13.276, 13.277]. Inclining the prism entrance and exit faces so that they are not exactly normal to the beam produces quadrupole moments, and these too have been used to adjust the prism’s first-order focusing properties.

The quadrupole and sextupole strengths produced by modifying the prism faces are not adjustable, and in more recent designs, inclined and curved prism faces have been abandoned in favor of flat prism faces at \(90^{\circ}\) combined with multipoles next to them that produce adjustable quadrupole and sextupole moments [13.278]. This makes it possible to fine-tune the aberrations exactly as needed. It also avoids the danger of the deviations from the ideal trajectory growing catastrophically on passing through a system containing many prisms with inclined and curved faces and no way of fine-tuning the quadrupole and sextupole strengths, a phenomenon that has been known to render an imaging filter using four prisms—making a total of eight inclined and curved prism faces—inoperable. Extension to higher-order correction through the use of higher-order multipoles is straightforward, and recent versions of spectrometers and imaging filters include correction of aberrations up to the fifth order.

The correction principles are the same as those used in aberration correctors for round beams, with three exceptions:
  1. 1.

    In imaging filters, the aberrations of the spectrum and distortions of the energy-filtered image typically need to be corrected

     
  2. 2.

    When the electron beam becomes dispersed in energy, mixed chromatic–geometrical aberrations arise and need to be corrected

     
  3. 3.

    Aberrations in the spectrum in the direction perpendicular to the energy dispersion are typically of limited interest and are often not corrected.

     
Sometimes, however, correction is required, notably when a spectrum has to be squeezed onto a narrow detector. Furthermore, the standard notation for the aberrations of spectrometers and imaging filters is tailored to a matrix algebra approach for calculating them (e. g., Egerton [13.279], where a full account of spectrometers, imaging filters and monochromators is to be found); it is not the same as the notation typically used for systems with a straight optic axis.

We now describe the various families of these devices in turn. Spectrometers and imaging filters can be situated either at the end of the microscope column, or incorporated in the column itself, and we consider the two types separately.

Post-Column Spectrometers and Imaging Filters

Whereas in-column filters use multiple prisms to create energy dispersion, post-column spectrometers typically use a single magnetic prism for this purpose. Post-column spectrometer configurations using multiple prisms are also possible and can provide important advantages [13.280]. But the benefits of using multiple prisms in post-column spectrometers must be weighed against the extra complexity, and the multiple-prism solution has not gained wide acceptance.

The single prism deflects the electron beam typically through \(90^{\circ}\), although other angles can be used as well. A single prism with straight faces without extra multipoles provides no aberration correction and cannot deliver a sufficiently large range of scattering angles to the spectrometer. It also fails to provide spectra with sufficiently high dispersion to be well matched to the channel size of a parallel detector. The optics of post-column spectrometers has therefore been improved through the addition of various optical elements. Adding a quadrupole-based magnification system to a second-order-corrected prism allowed spectra to be detected in parallel with independently adjustable dispersion and width of the spectrum [13.281]. Adding a second prism and three flexible dodecapole elements, in which rotatable dipole, quadrupole, sextupole and octopole fields plus a nonrotatable 12-pole field could be generated under computer control, allowed spectrum aberrations up to third order to be canceled [13.280]. Adding sextupoles to a quadrupole-based imaging filter allowed second-order distortions of energy-filtered images (or diffraction patterns) to be canceled [13.282, 13.283]; adding octopoles to this system allowed third-order aberrations to be canceled as well [13.284], and adding further multipoles allowed full third-order and partial fourth-order correction [13.285].

The Gatan imaging filter (GIF ) Quantum [13.286] uses multipole elements consisting of 12-poles in an arrangement similar to the scheme of independent computer control of individual pole excitations introduced by Haider [13.280, 13.287]. It provides rotatable dipole, quadrupole, sextupole, octopole and 10-pole fields, plus a nonrotatable dodecapole field. A 12-pole element capable of generating these multipoles precedes the prism, and seven further such elements are situated after the prism, followed by fast deflectors and a fiber-optically coupled CCD or CMOS, or a direct-exposure CMOS 2-D detector. The disposition of these components and of the energy-selecting slit is shown in Fig. 13.14. The ability to use very large entrance apertures—up to \({\mathrm{9}}\,{\mathrm{mm}}\) diameter—for good energy resolution has been demonstrated for this spectrometer [13.286], as well as an energy resolution of \({\mathrm{12}}\,{\mathrm{meV}}\), measured as the full width at half-maximum ( ) of the zero-loss peak in a spectrum acquired in \({\mathrm{2}}\,{\mathrm{ms}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) primary energy [13.288].

Fig. 13.14

The Gatan Quantum imaging filter. D denotes dodecapole. Reprinted from [13.286], with permission from Elsevier

The Nion Iris spectrometer , designed for ultrahigh-energy resolution [13.289], has three layers of multipoles of up to 16-poles in front of the prism and nine layers of multipoles after the prism. Used together with the Nion ground-potential monochromator described later in this section, and employing two linkage schemes that make the spectrometer insensitive to instabilities in the microscope's high voltage and in the prism current, it has been able to attain energy resolution (FWHM of the zero-loss peak) of \({\mathrm{6}}\,{\mathrm{meV}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) primary energy [13.290], and \({\mathrm{4.2}}\,{\mathrm{meV}}\) at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.291].

A key stability-enhancing measure used by the Iris spectrometer is that it produces the magnetic field distributions needed for dipoles, quadrupoles, sextupoles, octopoles, etc. by serially connecting coil windings distributed over different poles. Here, the number of turns on each pole ensures that the desired multipole is created. For a multipole \(m\) (\(m=1\) for a dipole, \(m=2\) for a quadrupole, \(m=3\) for a sextupole, \(m=4\) for an octopole, etc.), and an angle \(\varphi\) between the mirror symmetry plane and the \(j\)-th pole, we have
$$\begin{aligned}\displaystyle&\displaystyle u_{m}(j)=u_{m0}\sin m\varphi(j)\;,\\ \displaystyle&\displaystyle v_{m}(j)=v_{m0}\cos m\varphi(j)\;,\end{aligned}$$
(13.29)
for the numbers of turns \(u_{m}(j)\) and \(v_{m}(j)\) that need to be wound on pole \(j\) for the principal multipole \(m\) and the orthogonal (skew) multipole, respectively. \(u_{m0}\) and \(v_{m0}\) are the numbers of turns that would be wound on coils for which the sin or cos terms were equal to 1. (A negative number of turns means \(|u|\) or \(|v|\) turns connected with the opposite polarity.) Although the serially connected multipole wiring scheme is physically more complicated than the one pole \(=\) one winding \(=\) one power supply scheme, it has an important advantage: a change in the power supply producing the field of an individual multipole only changes the strength of that multipole. In this way, when there is an electrical instability in strong multipoles of higher order than dipoles, strong parasitic deflections are not generated. By comparison, in the independently powered pole scheme, a change in a single pole excitation changes all multipole strengths including the dipole ones. Electron-optical systems tolerate small changes in high-order focusing properties more readily than small changes in dipole strengths, with the result that, for power supplies of similar performance, the serially linked multipole scheme gives better overall stability. The serially linked multipole scheme was employed in the first multipole-using spectrometers and imaging filters [13.281, 13.282, 13.283]. A more sophisticated version is employed in quadrupole–octopole aberration correctors [13.163], in which minimizing parasitic dipole noise is essential for achieving good stability of the electron probe at the sample, where probe jitter of more than \({\mathrm{0.1}}\,{\mathrm{\AA{}}}\) can be very objectionable.

In-Column Analyzers

In-column imaging filters are placed just before the final projector lens (or lenses) in an electron microscope. They typically perform \(1:1\) imaging of an entrance object into an exit image, and they also produce an energy spectrum in a plane containing a removable energy-selecting slit. Round lenses placed between the objective lens and the filter allow it to operate either in a diffraction-coupled mode, in which there is a diffraction pattern in the energy-selecting plane and an image of the sample in an achromatic plane typically within the filter, or an image-coupled mode, in which the energy-selecting plane contains an energy-dispersed image and the achromatic plane a diffraction pattern. The post-filter lenses image either an energy-filtered image (or diffraction pattern) or the energy-loss spectrum onto the viewing screen or the camera of the microscope.

The first in-column imaging filter was the CastaingHenry analyzer  [13.292], in which a prism first deflects the electron beam through \(90^{\circ}\), an electrostatic mirror reflects the beam, and the beam re-enters the prism and is deflected through a further \(90^{\circ}\) and thus continues along the original optic axis. The optics of this device is well understood and set out in full by Metherell [13.293], reprinted in [13.294]. For accelerating voltages of a hundred kilovolts or more, it was clearly desirable to avoid using a mirror; by introducing extra prisms, all-magnetic devices were designed by Senoussi [13.295, 13.296]. These led to the family of \(\Upomega\)-filters and \(\upalpha\)-filters (so called from their resemblance to the Greek letters, e. g., [13.297, 13.298, 13.299, 13.300, 13.301]). Such filters suffer from the usual aberrations of prisms and the primitive models have evolved into more sophisticated devices, in which numerous supplementary multipoles cancel harmful intrinsic and parasitic aberrations.

Imaging filters are closely related to monochromators, the principal difference arising from the type of symmetry employed. The filters need to be able to display the spectrum on the final detector, and this means that the energy-deviating ray cannot be symmetric about the filter midplane, since this would result in the energy dispersion in the midplane being canceled at the filter's exit. (This type of symmetry is very suitable for monochromators, but not for imaging filters.) The energy-deviating ray in imaging filters is therefore neither symmetric nor antisymmetric about the midplane, but arranged such that the dispersion keeps on growing as the ray progresses through the filter. At the same time, axial and field rays are typically arranged to be either mirror-symmetric or point-symmetric (also called antisymmetric) about the midplane, and this is used for canceling key aberrations.

Both \(\upalpha\) and \(\Upomega\) configurations are possible, and there are a great many different possible embodiments—Table 1 in [13.301] lists more than \(\mathrm{1000}\) different solutions! Filters of comparable complexity have comparable properties, and they can be greatly improved by incorporating extra aberration-canceling multipoles [13.299]. Two broad classes of both the \(\upalpha\) and \(\Upomega\) filters can be distinguished: A-type filters, in which the filter midplane contains a stigmatic (doubly focused) image, and B-type filters, in which the midplane contains an astigmatic image—actually an image in one direction and a diffraction pattern in the perpendicular direction.

\(\Upomega\) and \(\upalpha\) in-column filters are shown schematically in Fig. 13.15a-c. In each case, the device is mechanically symmetric about the midplane, and the axial trajectories (blue) and field trajectories (green) are either mirror-symmetric or point-symmetric about the midplane. It is readily seen that the polarity of the magnets in \(\upalpha\) filters alternates in each half of the device so that the beam returns to the optic axis with no overall deflection. In the \(\upalpha\) filters, the polarity is the same throughout and the beam has undergone a deflection of \(2\uppi\) when it returns to the optic axis.

Fig. 13.15a-c

The \(\Upomega\) and \(\upalpha\) filters. (a) A-type \(\Upomega\) filter. (b) B-type \(\Upomega\) filter. (c) B-type \(\upalpha\) filter After [13.301, 13.302]

Aberration correction is accomplished by multipoles located in planes in which the first-order properties of the beam are substantially different: different widths in the dispersion and nondispersion directions, different amounts of energy dispersion. For the best performance, it can be advantageous to add an extra symmetry element to the basic \(\Upomega\) design. The result is the mandoline filter  [13.303, 13.304], in which the 60 and \(120^{\circ}\) planes are also symmetry planes. In reality, the symmetries described above are never perfectly respected and one role of the correction elements is to compensate for the ensuing parasitic aberrations. Figure 13.16 shows the mandoline filter in detail, including the numerous correction multipoles. Here, the first and fourth prisms of the original \(\Upomega\) configuration collapse to a single prism, as in the Castaing–Henry arrangement. The action of the prisms and multipoles is explained in great detail by Rose [13.13].

Fig. 13.16

The mandoline filter . After [13.13]

Monochromators

Monochromators can be grouped into two broad classes:
  1. 1.

    Dispersing-only monochromators that do not cancel the energy dispersion at their exit, leading to a larger (energy-dispersed) source image and a consequent loss of brightness [13.305].

     
  2. 2.

    Dispersing–undispersing monochromators that use more complicated optical arrangements, in which the energy selection is typically accomplished in the midplane of the instrument, and the energy dispersion in the beam leaving the monochromator is canceled [13.304, 13.306, 13.307, 13.308].

     
The second scheme largely preserves the original source size and thus also the beam brightness \(B_{\mathrm{n}}\) (brightness normalized by the energy width of the source).

A useful way to view a monochromator is as a spectrometer in which the detector is replaced by an energy-selecting slit. The slit selects a range of pass energies (as it does in imaging filters and did in earlier serial spectrometers), and the post-slit optics sends the monochromatized beam into the rest of the electron-optical column. Simple monochromators that do not cancel the energy dispersion at their exit are equivalent to a single spectrometer; dispersing–undispersing monochromators are equivalent to two spectrometers arranged back-to-back, the first one producing the energy dispersion and the second one canceling it.

An early electrostatic monochromator design was described by Plies [13.309] and Huber et al [13.310]. Other early monochromator models are described in Chapter 52 of [13.6]. Although many monochromators use the \(\Upomega\) or \(\upalpha\) configuration, several designs based on the Wien filter have also been proposed and developed. The principle is easily understood: in an ideal Wien filter, transverse electrostatic and magnetic fields with the same axial distribution deflect all electrons except those whose energy satisfies the Wien condition. Those that do not approximately satisfy this condition can easily be intercepted, leaving a beam with a narrowed energy range. Practical studies have been presented by Tiemeijer and colleagues, who designed the monochromators for the TEAM instruments [13.311, 13.312, 13.313, 13.314, 13.315]. All these designs lengthen the microscope column and, in an attempt to overcome this inconvenience, Mook and Kruit [13.316] have tested a fringe-field design, in which the fringing fields are dominant, the main filter being only \({\mathrm{4}}\,{\mathrm{mm}}\) long. A double Wien filter arrangement is employed in aberration-corrected JEOL microscopes [13.308, 13.317, 13.318]. The optics of the four designs in practical use is illustrated in Fig. 13.17a-d.

Fig. 13.17a-d

Monochromator optics. (a) Single Wien filter ; (b) double Wien filter; (c) electrostatic \(\Upomega\)-type device; (d) magnetic \(\upalpha\)-type device. Open circles: source images brown circles: line images of the source. ML: multipole element, adapted from [13.319], by permission of Oxford University Press

An important decision in monochromator design concerns the placement of the monochromator in the microscope column. Most monochromators are located inside the electron gun, before the accelerator. They act on electrons of a few keV in energy, which helps them achieve energy dispersion values of the order of \(10{-}30\,{\mathrm{\upmu{}m/eV}}\) in a compact design. A recently introduced ground-potential monochromator placed in the probe-forming column after a set of round magnetic lenses coupling the source image produced by the gun into the monochromator [13.288, 13.307] is an exception to this rule. A cross section of this monochromator is shown in Fig. 13.18. The monochromator employs an \(\alpha\) design, and has three magnetic prisms and 16 multipoles. The energy-selecting slit is situated in the midplane. The energy dispersion it achieves just beyond its primary prism is only \({\mathrm{2}}\,{\mathrm{\upmu{}m/eV}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) primary energy, but it magnifies the spectrum with quadrupole optics to give dispersions between \(\mathrm{30}\) and \({\mathrm{100}}\,{\mathrm{\upmu{}m/eV}}\) at its slit. Aberration correction up to third order is performed by sextupoles and octopoles in the multipole layers indicated in the figure. Higher-order correction is at present not needed—the placement of the monochromator in the optical column is such that the imaging properties demanded from it are similar to the requirements made on round condenser lenses, which are not aberration-corrected. Moreover, in a well-aligned STEM, small aberration contributions that originate in the condenser lenses and the monochromator are readily corrected by the probe corrector. Another factor that makes aberration correction in monochromators less demanding than in spectrometers is that the beam passing through the monochromator has not yet been broadened and shifted to higher angles by scattering in the sample.

Fig. 13.18

Cross section of Nion monochromator . The optical layers are numbered: \(\mathrm{Q}=\) quadrupole, \(\mathrm{S}=\) sextupole, \(\mathrm{O}=\) octopole. The energy separation is shown schematically by different colored beams; after passing through the slit, the beam is monochromatized. MOA: monochromator aperture; VOA: virtual objective aperture adapted from [13.288], by permission of Oxford University Press

Yet another design has been investigated by Mankos et al [13.320]. Here the beam is deflected towards an electron mirror and then returned to the original optic axis, as in the Castaing–Henry device. A knife edge intercepts some of the slower electrons when they approach the mirror and some of the faster ones as they leave it.

Attainable Energy Resolution

The fundamental limit on the attainable energy resolution in the spectrum produced at the final detector in spectrometers or at the energy-selecting slit in imaging filters and monochromators is given by the size of the beam crossover produced in the detector (slit) plane, divided by the energy dispersion. This means that for the best energy resolution, the dispersion should be made as large as possible, and the size of the crossover as small as possible. The dispersion needs to be maximized for the primary spectrum produced by the prism—increasing the dispersion by post-prism magnifying lenses increases the size of the final crossover just as much as it increases the dispersion, and therefore does not change the attainable energy resolution. (Practical exceptions to this rule arise when the spectrum magnification needs to be increased so that the resolution is not limited by the point spread function of a parallel detector, or unsharp edges of an energy-selecting slit.)

The primary energy dispersion can be made larger by using a larger-bending-radius prism, or a multiple-prism arrangement. However, practical considerations such as the prism weight, needed space, mechanical stability and sensitivity to stray magnetic fields typically place an upper limit on the optimal prism size. This often leaves making the crossover smaller as the most practical path to improved energy resolution.

The size of the spectrum crossover is limited by three principal influences, just like the size of the sample-level electron probe: the source size, the diffraction limit imposed by the angles used in the crossover formation, and instabilities. In nonmonochromatized electron microscopes, the energy resolution in a well-functioning spectrometer is typically dictated by the energy spread of the electron source. The addition of a monochromator makes it possible to reduce the width of the energy distribution of the source considerably. The next important limit on the energy resolution is then typically posed by instabilities, especially instabilities of the high voltage of the microscope and of the prism supply current.

These instabilities can be rendered unimportant in two different ways:
  1. 1.

    By putting the monochromator and spectrometer on the same high voltage, as was done by Boersch et al [13.321], eventually producing an energy resolution of \({\mathrm{3}}\,{\mathrm{meV}}\) at \({\mathrm{30}}\,{\mathrm{keV}}\) (while analyzing a sample area about \({\mathrm{30}}\,{\mathrm{\upmu{}m}}\) large; [13.322]), and by Terauchi et al [13.323], producing an energy resolution of \({\mathrm{12}}\,{\mathrm{meV}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) (with the beam in vacuum, \({\mathrm{25}}\,{\mathrm{meV}}\) when the beam passed through a sample),or

     
  2. 2.

    By putting an all-magnetic monochromator and an all-magnetic spectrometer both at ground potential and running the same current through all the prisms of the system by linking them in series [13.25, 13.288, 13.307]. This solution has given \({\mathrm{4.2}}\,{\mathrm{meV}}\) energy resolution at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.291] compatible with a \({\mathrm{1}}\,{\mathrm{\AA{}}}\) diameter monochromatized electron probe.

     
The diffraction limit has not yet been reached, but it will become dominant as the energy resolution approaches \(1{-}2\,{\mathrm{meV}}\). The path to better energy resolution in monochromators and spectrometers will then be the same as the path leading to improved spatial resolution: the correction of higher-order aberrations, so that the angular range can be increased and the diffraction limit made smaller, in parallel with great attention being paid to minimizing instabilities.

13.2.5 Johnson–Nyquist Noise

Johnson–Nyquist noise is an instability that can limit the spatial resolution of electron-optical systems, discovered and analyzed by Uhlemann et al [13.324]. The underlying cause is random electron motions [13.325, 13.326] that occur at finite temperatures in conductors and cause fluctuations in the background magnetic field. The fluctuations change with time and produce random deflections of electrons traveling through the instrument, and can be strong enough to worsen the attainable spatial resolution. For the image broadening (image spread) \(\langle\sigma^{2}\rangle\), we have [13.324]
$$\langle\sigma^{2}\rangle\propto TL\left(\frac{\varnothing}{R}\right)^{2}.$$
(13.30)
where \(T\) is the temperature of the microscope’s metallic drift tube, \(L\) the length of the region through which a beam whose axial trajectories have a diameter \(\varnothing\) passes, and \(R\) is the radius of the drift tube (or the radius of the aperture through a round lens or a multipole, if there is no separate drift tube). (Axial trajectories in this context mean partial trajectories originating (or arriving, in the STEM) at the same place on the sample and corresponding to different angles.) The broadening is added to the usual resolution-limiting factors such as the diffraction limit and the source size in the STEM case or detector point spread function in the TEM case. Equation (13.30) shows that the effect is especially important when there is a beam with wide axial trajectories propagating in extended lengths of narrow drift tubes, or, in the absence of a drift tube, narrow openings surrounded by the poles of multipoles, or small-bore round lenses. These kinds of trajectories are difficult to avoid in \(C_{\mathrm{c}}\) correctors such as the TEAM/PICO one, in which the electrostatic fields present in the hybrid electrostatic–magnetic quadrupoles are weak and require wide axial trajectories extending through the length of the quadrupoles, as well as narrow openings through the quadrupoles. In the SALVE corrector, the resolution loss has been reduced by a careful choice of axial trajectories.

In magnetic-only correctors of geometrical aberrations, in which strong multipole fields allow the diameter of the axial trajectories to be smaller than in \(C_{\mathrm{c}}\) correctors, Johnson–Nyquist noise has not yet emerged as a problem. But the noise may raise its head once more if the spatial resolution is improved significantly beyond present levels, and it may also become important in ultrahigh-energy resolution spectrometers and monochromators, in which broad beams are needed to achieve useful energy dispersion.

13.3 Practical Aspects of Corrector Operation

13.3.1 Correction of Parasitic Aberrations

When the primary aberrations have been corrected and instabilities brought under sufficient degree of control, the next important resolution-limiting influence is likely to be parasitic aberrations. It is important to remember that the tolerance to higher-order aberrations increases as \(1/\alpha^{(n+1)}\), where \(\alpha\) is the illumination half-angle (in the STEM) or collection half-angle (in the TEM), and \(n\) is the order of the aberration [13.23, 13.25, 13.327], and also that higher-order aberrations can be partially compensated by lower-order aberrations of the same azimuthal multiplicity [13.25, 13.327]. The angle \(\alpha\) is typically about \({\mathrm{0.05}}\,{\mathrm{rad}}\) or less, and this means that whereas first-order aberrations (defocus and twofold astigmatism) need to be set with a precision of a few nanometers or less for better than \({\mathrm{1}}\,{\mathrm{\AA{}}}\) resolution, fourth-order aberrations of around \({\mathrm{100}}\,{\mathrm{\upmu{}m}}\) and fifth-order aberrations of \({\mathrm{1}}\,{\mathrm{mm}}\) are not likely to spoil even resolution of \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) at \({\mathrm{200}}\,{\mathrm{keV}}\) primary energy [13.25]. These numbers apply to the STEM case, in which high spatial frequencies are transferred into the electron probe via pairwise interference of electron wave subcomponents. In the CTEM case, three different beams (the unscattered beam and \(\pm g\) beams) have to be combined with the phase differences accurately preserved, which means that the objective aperture angles need to be 1.6\(\times\) as large to attain the same spatial resolution as in the STEM. This means that the tolerances to higher-order aberrations are \((n+1)^{1.6}\) times more strict in CTEM [13.23] than in STEM.

Nulling parasitic aberrations with precision sufficient for the increasingly large aperture angles used by present-day aberration correction remains a substantial challenge, both for the autotuning algorithms that measure them and for the controls that eliminate them. One useful strategy is to align the corrector so that the beam passes close to the optical centers of all its stages, which usually minimizes the parasitic aberrations. But if a thorough alignment of this kind does not reduce one or more parasitic aberrations as much as needed, or the alignment that minimizes one particular aberration causes other aberrations to rise unacceptably, then a procedure able to change that particular aberration separately from all the other aberrations is needed. This kind of situation typically arises when the correction half-angles are pushed beyond \(40{-}50\,{\mathrm{mrad}}\), as becomes possible when principal aberrations up to the fifth order are nulled, and chromatic aberration effects are rendered unimportant by means of a \(C_{3}{-}C_{5}{-}C_{\mathrm{c}}\) corrector or a \(C_{3}{-}C_{5}\) corrector used together with a monochromator set to produce a sufficiently narrow energy passband.

With the exception of individual multipoles addressing individual aberrations (e. g., a 12-pole multipole acting on a round beam to change \(C_{5,6}\)), modification of individual parasitic aberrations relies on using combination aberrations , which are capable of producing a wide variety of effects. The general rule [13.177, 13.25, 13.327] is that when an optical element \(u\) produces an aberration \(C_{nu,mu}\) that is mis-projected (not imaged in focus) into a subsequent element \(v\) producing aberration \(C_{nv,mv}\), the principal combination aberrations that arise are
$$C_{N1,M1}=C_{nu+nv-1,|mu-mv|}\;,$$
(13.31a)
and
$$C_{N2,M2}=C_{nu+nv-1,mu+mv}\;,$$
(13.31b)

with the second aberration appearing only when \(mu+mv\leq nu+nv\). Table 13.2 shows how different aberrations produced in separate multipoles of an aberration corrector combine to produce new aberrations. Some combination aberrations are of limited interest; e. g., combining defocus \(C_{1,0}\) with \(C_{n,m}\) produces \(C_{n,m}\) as the combination aberration, and these have been omitted from the table. Line 2 shows that displacing a round beam (changing \(C_{0,1}\) of element \(u\)) in a sextupole \(v\) producing \(C_{2,3}\) leads to twofold astigmatism \(C_{1,2}\), and this is typically used for twofold stigmation in sextupole correctors. Line 3 shows that displacing the beam in a \(C_{3,0}\)-producing element such as the objective lens allows axial coma \(C_{2,1}\) to be adjusted, and line 5 shows that displacing a round beam in an excited octopole leads to adjustable threefold astigmatism \(C_{2,3}\), which is used to adjust \(C_{2,3}\) in quadrupole-octopole correctors not containing any sextupoles.

Table 13.2

Selected combination aberrations which can be used for controlling parasitic aberrations. The combination aberrations listed on lines shown in bold (2, 3, 5, 15 and 19) are in common use. Lines 16–21 and 23, shown in italics (or bold for line 19), are used in the improved Nion corrector for complete control of parasitic aberrations of fourth and fifth order

Line

\(nu\)

\(mu\)

\(nv\)

\(mv\)

\(N1\)

\(M1\)

\(N2\)

\(M2\)

Principal effect

1

0

1

2

1

1

0

1

2

\(C_{1,0}\); \(C_{1,2}\)

2

0

1

2

3

1

2

\(\mathbf{C_{1,2}}\)

3

0

1

3

0

2

1

\(\mathbf{C_{2,1}}\)

4

0

1

3

2

2

1

2

3

\(C_{2,1}\); \(C_{2,3}\)

5

0

1

3

4

2

3

\(\mathbf{C_{2,3}}\)

6

0

1

4

1

3

0

3

2

\(C_{3,0}\); \(C_{3,2}\)

7

0

1

4

3

3

2

3

4

\(C_{3,2}\); \(C_{3,4}\)

8

0

1

4

5

3

4

\(C_{3,4}\)

9

0

1

5

0

4

1

\(C_{4,1}\)

10

2

1

2

1

3

0

3

2

\(C_{3,0}\); \(C_{3,2}\)

11

2

1

2

3

3

2

3

4

\(C_{3,2}\); \(C_{3,4}\)

12

2

1

3

0

4

1

\(C_{4,1}\)

13

2

1

3

2

4

1

4

3

\(C_{4,1}\); \(C_{4,3}\)

14

2

1

3

4

4

3

4

5

\(C_{4,3}\); \(C_{4,5}\)

15

2

3

2

3

3

0

\(\mathbf{C_{3,0}}\)

16

2

3

3

0

4

3

\(C_{4,3}\)

17

2

3

3

2

4

1

4

5

\(C_{4,1}\); \(C_{4,5}\)

18

2

3

3

4

4

1

\(C_{4,1}\)

19

3

0

3

0

5

0

\(\mathbf{C_{5,0}}\)

20

3

0

3

2

5

2

\(C_{5,2}\)

21

3

0

3

4

5

4

\(C_{5,4}\)

22

3

2

3

2

5

0

5

4

\(C_{5,0}\); \(C_{5,4}\)

23

3

2

3

4

5

2

5

6

\(C_{5,2}\); \(C_{5,6}\)

24

3

4

3

4

5

0

\(C_{5,0}\)

Using the above principles to eliminate second- and third-order parasitic aberrations is straightforward, but the situation grows more complicated for nulling parasitic fourth- and fifth-order aberrations. Lines 17 and 18 of the table show that combining a \(C_{2,3}\)-producing sextupole with a \(C_{3,2}\)-producing octopole (an octopole acting on an elliptical beam) gives adjustable \(C_{4,1}\) and \(C_{4,5}\), whereas combining \(C_{2,3}\) with a \(C_{3,4}\)-producing octopole (one acting on a round beam) gives adjustable \(C_{4,1}\). Both controls become possible when weak sextupoles are incorporated in quadrupole stages of a \(C_{3}/C_{5}\) quadrupole–octopole corrector, and this has been done in the improved Nion \(C_{3}/C_{5}\) corrector. The corrector is therefore able to null \(C_{4,1}\) and \(C_{4,5}\) independently. In similar ways, independent controls for all parasitic aberrations up to \(C_{5,6}\) (and higher) can be devised. In the case of the Nion corrector, this approach has increased the maximum half-angle of correction to beyond \({\mathrm{50}}\,{\mathrm{mrad}}\) ([13.177] and Sect. 13.3.3). Correction angles greater than \({\mathrm{50}}\,{\mathrm{mrad}}\) have also been achieved by the JEOL Delta corrector [13.328] and the CEOS SALVE corrector [13.244, 13.245].

13.3.2 Autotuning

In the STEM, aberrations up to third order produce distinct signatures in Ronchigrams (shadow images or far-field convergent-beam diffraction patterns) of amorphous materials [13.25, 13.46]. Figure 13.19a-d shows how twofold astigmatism, axial coma and threefold astigmatism affect the appearance of the Ronchigram. Skilled operators are therefore able to tune correctors up to third order by optimizing the appearance of Ronchigrams, and this can be used on third-order corrected STEMs in lieu of autotuning. However, requiring operators to perform nontrivial tuning of the microscope manually would restrict the number of users, and automated microscope tuning (autotuning) is therefore nearly always provided. Furthermore, when the correction progresses up to fifth-order axial aberrations, 25 different aberrations (\(C_{1,0}\) to \(C_{5,6b}\)) need to be measured and set to acceptably small values. Many of them produce effects that are difficult to distinguish visually in a Ronchigram: second- and fourth-order threefold astigmatisms (\(C_{2,3}\) and \(C_{4,3}\)) both produce a star of Mercedes similar to Fig. 13.19a-dd. Tuning such aberrations is therefore not a task that can be done manually, and precise and fast autotuning is essential.

Fig. 13.19a-d

Ronchigrams for different aberrations (a) correctly adjusted, (b) twofold astigmatism, (c) axial coma, (d) threefold astigmatism from [13.25] reprinted by permission of the publisher (Taylor & Francis Ltd., http://www.tandfonline.com)

Different autotuning strategies have been reviewed by Erni [13.64], and we provide only a brief account here. The tuning methods can be divided into two broad classes: image-based and Ronchigram-based. Image-based methods developed prior to practical aberration correction, for instance as reviewed by Krivanek and Fan [13.33], can be readily employed in aberration-corrected transmission electron microscopes as well. They are based on either beam-tilt-induced image shift or beam-tilt-induced changes in apparent defocus and astigmatism. The second method analyzes diffractogram tableaux of amorphous samples recorded for different values of beam tilt [13.48, 13.50]. Unlike shift-based methods, the diffractogram method is not greatly affected by changes in image appearance for different beam tilts or by small sample drift. This results in better tuning precision, and the method has become the most widely used for CTEM aberration correctors. Its usual implementation divides each experimental diffractogram into azimuthal segments, cross-correlates the intensity profile of the segments with theoretical profiles computed for different defocus values and thereby determines defocus values in different azimuthal directions, and then analyzes the azimuthal variation in each diffractogram to determine its apparent defocus and astigmatism [13.32]. The variation in these values with the induced beam tilt is then used to determine the aberration coefficients. In principle, any number of aberrations can be quantified, provided that there is a sufficient number of experimental diffractograms corresponding to different beam tilts.

For the STEM, the above methods are applicable as well, thanks to the principle of reciprocity linking STEM and TEM imaging [13.329, 13.330, 13.331, 13.332]. The image-based techniques simply record bright-field images for different detector tilts [13.164, 13.26]. Images for different beam tilt values can be collected in the STEM in parallel by recording images corresponding to different detection angles in the Ronchigram plane. By reciprocity, this corresponds to collecting CTEM images for different beam tilts of the incident illumination. The parallel recording can be implemented using multiple detectors (or a pixelated detector with fast readout) to monitor the intensities at different detection angles as the probe is scanned across the sample. This can lead to a substantial increase in collection efficiency and hence shorter collection time for the required data than in CTEM.

An even more efficient approach is analyzing experimental Ronchigrams directly, without acquiring whole scanned images. An out-of-focus Ronchigram is a shadow image of the sample that is distorted in characteristic ways by different aberrations, and the distortion can be quantified by analyzing how the local magnification varies from place to place in the Ronchigram [13.25, 13.46]. This can be seen in Fig. 13.19a-d, especially in Fig. 13.19a-dc and Fig. 13.19a-dd.

Quantifying the Ronchigram distortions, and hence the aberrations, is accomplished using the expression
$$ \mathbf{M}_{i}=\frac{D}{\lambda}\begin{pmatrix}(\partial^{2}\chi(\boldsymbol{\theta}))/(\partial\theta_{x}^{2})&(\partial^{2}\chi(\boldsymbol{\theta}))/(\partial\theta_{x}\partial\theta_{y})\\ (\partial^{2}\chi(\boldsymbol{\theta}))/(\partial\theta_{y}\partial\theta_{x})&(\partial^{2}\chi(\boldsymbol{\theta}))/(\partial\theta_{y}^{2})\end{pmatrix}^{\!-1}\!,$$
(13.32)
where \(\mathbf{M}_{i}\) is the local magnification at a point \(i\) in a Ronchigram, characterized as a \(2\times 2\) matrix to account for different magnifications in different Ronchigram directions, \(D\) is the camera length of the Ronchigram, \(\lambda\) the electron wavelength, \(\chi\) the aberration function, \(\boldsymbol{\theta}\) the angle at which the ray arriving in the Ronchigram at position \(i\) traversed the sample, and \(\theta_{x}\) and \(\theta_{y}\) the \(x\) and \(y\) components of \(\boldsymbol{\theta}\) [13.168, 13.46].

To determine the local magnification experimentally on a general sample whose features are not known a priori, the probe is moved by a small amount in the \(x\) and \(y\) directions, and Ronchigrams are recorded for each probe displacement ([13.168], see Chap.  17). Cross-correlating small Ronchigram subareas then shows how far that part of the Ronchigram moved as a result of the probe shift, and this allows the local magnification of the Ronchigram to be determined. In practice, more Ronchigrams than the minimum of three (no probe displacement, displacement in \(x\), displacement in \(y\)) are recorded for each autotuning run, so as to be able to account for spurious effects such as sample drift and other instabilities. But even so, high-quality Ronchigram data can be typically acquired in \(<{\mathrm{5}}\,{\mathrm{s}}\), and complete geometrical aberration analysis up to the fifth order can be performed in \(<{\mathrm{10}}\,{\mathrm{s}}\). The aberrations are then tuned using various controls whose effects on the aberrations have been calibrated, and a new aberration diagnosis is performed. The procedure is repeated until all aberration coefficients being tuned are within acceptable ranges. When amorphous samples giving suitable contrast are used in a well-performing STEM, fifth-order tuning can typically be completed in about \({\mathrm{30}}\,{\mathrm{s}}\). In practice, day-to-day changes in fourth- and fifth-order aberrations are much smaller than tuning tolerances, and fourth- and fifth-order tuning is therefore not performed as a daily adjustment. Third-order tuning is typically performed at the start of the day, followed by second-order tuning, which can be done in \({\mathrm{10}}\,{\mathrm{s}}\), whenever a substantially new sample area is selected. Related Ronchigram-based autotuning techniques have been developed by Sawada et al [13.29, 13.333], and image-based autotuning for adjustment of defocus and other low-order aberrations using crystalline samples has been demonstrated by Lazar et al [13.334].

13.3.3 Current Performance Limits

When spherical aberration is corrected in an electron microscope, other aberrations begin to limit the resolution. The resolution that becomes attainable in the STEM as different types of aberrations are corrected is depicted in Fig. 13.20 as a function of the primary energy of the microscope. The theoretical curves are computed for probe currents equal to \({\mathrm{10}}\%\) (solid lines) and \({\mathrm{50}}\%\) (dashed lines) of the coherent current \(I_{\mathrm{c}}\), using the expressions given in Krivanek et al [13.81]. \(I_{\mathrm{c}}\) is the probe current value for which the probe is \({\mathrm{50}}\%\) coherent and \({\mathrm{50}}\%\) incoherent, and it is a property of the gun only: it does not depend on the aberrations of the system, or on its primary energy. It is also is described more fully in Krivanek et al [13.81].

Fig. 13.20

Attainable probe size for different degrees of correction and different STEM probe currents (solid: \(I_{\mathrm{p}}={\mathrm{0.1}}I_{\mathrm{c}}\), dashed: \(I_{\mathrm{p}}={\mathrm{0.5}}I_{\mathrm{c}}\)) as a function of primary energy

The purple curves in the upper right part of the graph correspond to the uncorrected, low-\(C_{\mathrm{s}}\) STEMs marketed by several manufacturers (FEI, Hitachi, JEOL and VG) before aberration correction became commercially available. This type of microscope was able to form \({\mathrm{2}}\,{\mathrm{\AA{}}}\) probes at \({\mathrm{100}}\,{\mathrm{keV}}\), \({\mathrm{1.4}}\,{\mathrm{\AA{}}}\) probes at \({\mathrm{200}}\,{\mathrm{kV}}\), and smaller probes still at \({\mathrm{300}}\,{\mathrm{keV}}\). Third-order aberration correction improved the performance to the blue curves, and allowed \({\mathrm{1}}\,{\mathrm{\AA{}}}\) probes to be reached at \({\mathrm{100}}\,{\mathrm{keV}}\). Further improvements came when fifth-order aberrations were either corrected or sufficiently minimized, with probes about \({\mathrm{0.5}}\,{\mathrm{\AA{}}}\) in size becoming possible at \({\mathrm{200}}\,{\mathrm{keV}}\), \({\mathrm{0.47}}\,{\mathrm{\AA{}}}\) (and later \({\mathrm{0.405}}\,{\mathrm{\AA{}}}\)) at \({\mathrm{300}}\,{\mathrm{keV}}\) [13.219, 13.335, 13.336], and \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{60}}\,{\mathrm{keV}}\) [13.337]. Chromatic aberration \(C_{\mathrm{c}}\) then became the new limit, especially at lower primary energies, and it could be overcome either by correcting \(C_{\mathrm{c}}\) or decreasing the energy spread of the beam \(\updelta E\). Only moderate monochromatization is needed in the STEM: for instance, \({\mathrm{1}}\,{\mathrm{\AA{}}}\) spatial resolution can be achieved at \({\mathrm{30}}\,{\mathrm{keV}}\) with an energy spread of \(\approx{\mathrm{100}}\,{\mathrm{meV}}\), as is shown below.

It is useful to mention that in the CTEM, the unscattered beam parallel to the optic axis interferes with scattered beams passing some distance from the axis, and this makes the phase changes that are imparted to the interfering beams by defocus spread (caused by \(C_{\mathrm{c}}\) combined with energy spread) very different. As a result, CTEM imaging is more sensitive to \(C_{\mathrm{c}}\) effects than STEM imaging, in which much of the interference is between partial beams passing at equal angles to the optic axis, and therefore insensitive to defocus spread ([13.338], Chap.  2). The heightened \(C_{\mathrm{c}}\) sensitivity and the relatively wide energy spread of Schottky guns used in CTEM \((\updelta E\approx{\mathrm{700}}\,{\mathrm{meV}}\)) made the use of monochromatization to improve the CTEM spatial resolution fairly routine from 2004 onward, before fourth- and fifth-order aberrations were eliminated [13.208, 13.209, 13.339, 13.340, 13.341]. With correction half-angles presently increasing to \({\mathrm{50}}\,{\mathrm{mrad}}\) and above thanks to improved geometrical aberration correction, monochromatization will need to decrease the energy spread of the illuminating beam in the CTEM to below \({\mathrm{30}}\,{\mathrm{meV}}\) [13.244, 13.245] and even smaller. This is not readily achievable, for two reasons: the medium- and long-term stabilities of the microscope high voltage are typically worse than \({\mathrm{30}}\,{\mathrm{mV}}\), and decreasing the energy width of the illumination from \({\mathrm{700}}\,{\mathrm{meV}}\) (typical of Schottky sources) to \({\mathrm{30}}\,{\mathrm{meV}}\) would reduce the illumination intensity more than 20\(\times\). Correction of chromatic aberration offers a way around this difficulty.

When the chromatic limit is overcome, seventh-order principal aberrations and sixth-order parasitic aberrations are expected to become dominant. This limit, shown by the green lines in Fig. 13.20, indicates that when aberrations including fifth-order and chromatic ones are eliminated, \({\mathrm{0.3}}\,{\mathrm{\AA{}}}\) STEM probe sizes should become possible at \({\mathrm{200}}\,{\mathrm{kV}}\), \({\mathrm{0.45}}\,{\mathrm{\AA{}}}\) at \({\mathrm{100}}\,{\mathrm{kV}}\), and \({\mathrm{1}}\,{\mathrm{\AA{}}}\) at \({\mathrm{20}}\,{\mathrm{kV}}\). These three primary energies correspond to electron wavelengths \(\lambda\) of \(\mathrm{2.51}\), \(\mathrm{3.7}\) and \({\mathrm{8.59}}\,{\mathrm{pm}}\), and this means that the probe size, which is closely related to resolution, divided by the wavelength (\(d/\lambda\)) will be approaching about 12 when such performance is reached. This is a far cry from optical microscopies, for which resolutions \(d<\lambda\) or even \(d\ll\lambda\) have been available for some time. But it is a major improvement on the pre-aberration-correction days, in which the best-performing electron microscopes were reaching \(d/\lambda\) values of only \(60{-}100\).

Figures 13.2113.23a-d demonstrate how this kind of performance is approached in practice. Figure 13.21 shows graphene and monolayer \(\mathrm{MoS_{2}}\) imaged with the \(C_{\mathrm{s}}/C_{\mathrm{c}}\) SALVE corrector in the bright-field phase-contrast mode at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.244]. Transfer to spatial frequencies close to \({\mathrm{1}}\,{\mathrm{\AA{}^{-1}}}\) has been reached in both cases, giving \(d/\lambda={\mathrm{15}}\). The contrast modulation is about \({\mathrm{6}}\%\) in the graphene image, and about \({\mathrm{40}}\%\) in the \(\mathrm{MoS_{2}}\) image, owing to the presence of heavier atoms and to the fact that two \(\mathrm{S}\) atoms are projected on top of each other.

Fig. 13.21

Graphene (A) and monolayer \(\mathrm{MoS_{2}}\) (B) imaged with the \(C_{\mathrm{s}}/C_{\mathrm{c}}\) SALVE corrector in the bright-field phase-contrast mode at \({\mathrm{30}}\,{\mathrm{keV}}\). A1, B1, Diffractograms in bright atom contrast, field of view \(40\times{\mathrm{40}}\,{\mathrm{nm^{2}}}\); A2, B2, raw images; A3, B3, magnified images in which the atomic structure can be recognized; A4, B4, averaged images, which agree well with the simulated images A5, B5. In the latter, the scale bars represent \({\mathrm{0.5}}\,{\mathrm{nm}}\). A6, B6, line profiles through A4, A5, illustrating the good agreement. Reprinted with permission from [13.244]. Copyright 2016 by the American Physical Society

Figure 13.22a,b shows a pair of out-of-focus \((\Updelta z\approx{\mathrm{500}}\,{\mathrm{nm}})\) Ronchigrams extending to \({\mathrm{50}}\,{\mathrm{mrad}}\) half-angle, recorded at \({\mathrm{30}}\,{\mathrm{keV}}\) in a monochromatized STEM using an amorphous sample [13.342]. The Ronchigram on the left was recorded with the energy-selecting slit retracted and thus admitting the full beam produced by the microscope’s cold-field emission gun ( ), with a measured energy spread of \({\mathrm{320}}\,{\mathrm{meV}}\) (full width at half-maximum \(=\) FWHM; this value is typical for a CFEG). The Ronchigram on the right was obtained with the energy-selecting slit inserted, restricting the energy spread \(\updelta E\) to \({\mathrm{110}}\,{\mathrm{meV}}\). The unmonochromatized Ronchigram shows blurring in the radial direction for angles larger than about \({\mathrm{20}}\,{\mathrm{mrad}}\). The blurring is due to the defocus spread that occurs when the full range of energies of the CFEG are used: the Ronchigram magnifications are different for the different defocus values, which causes blurring near the outer edges of the Ronchigram. The blur-free center is the voltage axis of the microscope, and the virtual objective aperture ( ) used to limit the range of angles contributing to the probe was centered on it. Had the Ronchigram been recorded close to zero defocus, it would have been largely featureless, and the blurring due to the energy spread would not have been visible. The blurring might have even improved the appearance of the Ronchigram by fuzzing out any features intruding into the flat phase (sweet spot) part of the Ronchigram, thereby giving a misleading impression regarding the perfection of the implemented tuning.

Fig. 13.22a,b

Ronchigrams extending to \({\mathrm{50}}\,{\mathrm{mrad}}\) half-angle recorded at \({\mathrm{30}}\,{\mathrm{kV}}\) for (a) no monochromatization (\(\delta E={\mathrm{320}}\,{\mathrm{meV}}\)), and (b) monochromatization (\(\delta E\approx{\mathrm{110}}\,{\mathrm{meV}}\))

Figure 13.23a-d shows a medium-angle annular dark-field ( ) image recorded at \({\mathrm{30}}\,{\mathrm{kV}}\) in the monochromatized condition illustrated by Fig. 13.23a-db [13.342]. Single atoms of carbon are well resolved and the image noise is not excessive, despite the beam current being only about \({\mathrm{10}}\,{\mathrm{pA}}\). The highest spatial frequency transferred is \(({\mathrm{107}}\,{\mathrm{pm}})^{-1}\), similar to the phase-contrast image shown in Fig. 13.21, but the modulation for monolayer graphene is \({\mathrm{50}}\%\), as shown by a line profile taken along the dashed line in the as-acquired image. The modulation increase is mostly due to different contrast mechanisms: phase-contrast imaging for the earlier figure and medium-angle annular dark-field (MAADF) imaging for the present one. The MAADF image contrast modulation is similar to the modulation achieved in graphene and monolayer boron nitride (BN ) in an unmonochromatized \({\mathrm{60}}\,{\mathrm{keV}}\) STEM in 2010 [13.337, 13.343]. The energy width was \({\mathrm{110}}\,{\mathrm{meV}}\) and the system \(C_{\mathrm{c}}\) \({\mathrm{0.96}}\,{\mathrm{mm}}\), of which \({\mathrm{0.6}}\,{\mathrm{mm}}\) was due to the objective lens, and the rest was due to the electron gun, condenser lenses, monochromator and the aberration corrector. The \(C_{\mathrm{c}}\)-limited resolution, computed using (13.27b) above (with \(b={\mathrm{0.5}}\)), was therefore only \({\mathrm{84}}\,{\mathrm{pm}}\), and allowed strong transfer of \({\mathrm{107}}\,{\mathrm{pm}}\) spatial frequency even in the presence of other probe-broadening effects such as finite source size.

Fig. 13.23a-d

MAADF image of graphene recorded at \({\mathrm{30}}\,{\mathrm{kV}}\) with the monochromatized beam, (a) as recorded, (b) Fourier filtered, (c) Fourier transform, (d) line profile through image (a) courtesy of Dr. N. Dellby

Defining reproducible resolution criteria is an important subject (see Chap.  12), with many prior contributions, e. g., Sato [13.344]. It is useful to remember that the MAADF image contributions arising from different energy electrons add up mostly incoherently, and that, as discussed for example by Nellist [13.345] and Krivanek et al [13.346], the resultant probe shape for such an addition can be a small narrow maximum surrounded by an intense broad tail. This may be why the \({\mathrm{107}}\,{\mathrm{pm}}\) spacing was observed in unmonochromatized MAADF STEM images obtained at \({\mathrm{30}}\,{\mathrm{keV}}\) with an energy spread of \({\mathrm{0.4}}\,{\mathrm{eV}}\) and a \(C_{\mathrm{c}}\) coefficient of \({\mathrm{0.61}}\,{\mathrm{mm}}\) [13.328], for which (13.27b) predicts \(d_{\mathrm{c}}={\mathrm{120}}\,{\mathrm{pm}}\). The intensity of the tail can be evaluated from the intensity distribution of the experimental image and it can be subtracted from the images; this is necessary for quantitative comparisons of the image intensities of different atoms [13.337]. The main effect of the tail is therefore to increase the image background level and also its statistical noise.

Because the structure and thickness of every single-layer graphene sample that is free of defects is exactly the same as that of any other graphene sample, using graphene as a test object to obtain a quantitative evaluation of microscope performance can be highly reproducible. This makes graphene an ideal test object for evaluating the relative performance of different microscopes. Comparing the quality of the images produced by the three techniques that have achieved atomic resolution in graphene at \({\mathrm{30}}\,{\mathrm{keV}}\) [13.244, 13.328, 13.342] gives the edge to monochromatized STEM (Fig. 13.23a-d), but this is of course likely to change in the future with further progress in aberration correction.

Figure 13.24 illustrates another important aspect of aberration-corrected electron microscopy. It shows a zero-loss peak with an FWHM of \({\mathrm{5}}\,{\mathrm{meV}}\), acquired at \({\mathrm{30}}\,{\mathrm{keV}}\) primary energy with a monochromator that is corrected to the third order and an EELS that is corrected up to the fifth order. The peak was acquired with an acquisition time of \({\mathrm{100}}\,{\mathrm{ms}}\), (i. e., spanning six periods of North American mains). The excellent energy resolution was possible only because the STEM–EELS system used stabilization schemes that made the EELS system insensitive to fluctuations in the high voltage and in the prism current [13.288, 13.307]. The monochromator was briefly described in Sect. 13.2.4 above. The energy-loss spectrometer is a recently developed one, with aberration correction up to the fifth order and several other features that optimize the attainable energy resolution [13.289]. At \({\mathrm{60}}\,{\mathrm{keV}}\), the zero-loss peak FWHM in this system broadens to \({\mathrm{6}}\,{\mathrm{meV}}\), but even so, EELS in the electron microscope, which began exploring vibrational signals only in 2014 [13.347, 13.348], is now able to resolve vibrational peaks due to C–C, C–N and C–O bonds in proteins and other biological materials [13.349]. When carried out using the aloof EELS technique [13.350], which avoids radiation damage even in highly sensitive materials such as water [13.351] and ice, this development promises to open new possibilities for sensitive and damage-free analysis of biological and other organic materials at a spatial resolution of a few tens of nanometers.

Fig. 13.24

Monochromated EELS zero-loss peak, \({\mathrm{30}}\,{\mathrm{keV}}\), \({\mathrm{100}}\,{\mathrm{ms}}\) acquisition time

Vibrational EELS in the EM has recently also demonstrated isotopic sensitivity, whereby deuterium is distinguished from hydrogen and \(\mathrm{{}^{13}C}\) from \(\mathrm{{}^{12}C}\), by energy shifts in vibrational spectra [13.349, 13.351]. Labeling biological compounds with different isotopes and analyzing cells that have taken them up may allow metabolic pathways to be traced in an electron microscope, at nanometer-level resolutions.

13.4 Concluding Remarks

Aberrations correction has progressed considerably since Science of Microscopy was published. Correction of geometrical aberrations up to the third order allows the spatial resolution to be improved some \(2{-}3\times\) relative to uncorrected electron microscopes, and several hundred such instruments have now been installed in laboratories around the world. The next major resolution-limiting influence, chromatic aberration, can be overcome by \(C_{\mathrm{s}}/C_{\mathrm{c}}\) correctors or by monochromatizing the illuminating beam. Combined with correction of geometrical aberrations up to the fifth order, these strategies have enabled aberration-corrected imaging to achieve a resolution of \(15\lambda\), a very substantial improvement on the resolution of \(60\lambda{-}100\lambda\) achievable with uncorrected instruments.

With a suitable test sample, the resolution achieved by aberration correction is typically similar to the STEM probe size, and the two terms are sometimes used interchangeably. However, there is an important distinction: the probe size defines the best spatial resolution for incoherent signals such as high angle dark field (HAADF) imaging and EELS and EDXS maps. For coherent signals such as bright field imaging, the resolution can be improved beyond the probe size limit by ptychographic reconstruction techniques (Chap.  17) [13.352, 13.353]. Using these techniques, probe-limited resolution of about \({\mathrm{1}}\,{\mathrm{\AA}}\) has recently been improved to a resolution of less than \({\mathrm{0.4}}\,{\mathrm{\AA}}\) at \({\mathrm{80}}\,{\mathrm{keV}}\), i.e. about \(10\lambda\) [13.354].

Aberration-corrected monochromators and electron energy-loss spectrometers have improved the energy resolution in electron microscopes to better than \({\mathrm{10}}\,{\mathrm{meV}}\). This has improved the quality of EELS results substantially, and has opened up a new research field: vibrational spectroscopy combined with high spatial resolution. The spatial resolution can be greatly enhanced by using dark-field EELS, in which the forward-scattered electrons (which can undergo delocalized scattering events that result in blurred image information) are excluded [13.355]. This approach has now allowed Hage et al [13.356] to reach \({\mathrm{2}}\,{\mathrm{\AA}}\) resolution in vibrational images of h-BN using dark field EELS. The forward-scattered electrons arise by dipole scattering, which is strong in polar materials, but only a weak second-order effect in non-polar materials such as Si. Taking advantage of the absence of dipole scattering in non-polar materials, Venkatraman et al [13.357] reached similar (\(\approx{\mathrm{2}}\,{\mathrm{\AA}}\)) resolution in vibrational images of Si without employing dark field EELS—by simply making sure that large angle scattering events, which carry high spatial resolution information, contributed to the images.

Aiming in the opposite direction, vibrational spectroscopy that admits the forward-scattered electrons and positions a narrow electron beam just outside the sample ensures that only long-distance interactions take place between the fast electrons and the sample. The technique is called aloof spectroscopy and, in a first for electron microscopy, it avoids radiation damage by limiting the amount of energy that can be deposited in the sample by the fast electrons [13.350].

Aloof spectroscopy is not the only way of avoiding radiation damage. Following the example of diffracting then destroy with ultrashort pulses from x-ray lasers [13.358, 13.359], the new field of ultrafast electron microscopy (Chap.  8) may allow the microscopist to outrun radiation damage by recording image or diffraction data before damage has had time to occur, as has been suggested by Egerton et al and Spence et al [13.360, 13.361, 13.362, 13.363]. But outrunning radiation damage in the electron microscope remains to be demonstrated in practice, whereas minimizing the damage by aloof positioning of the electron beam is now a practical technique.

Two other recent trends should be pointed out. First, aberration-corrected STEM enjoys four fundamental advantages over CTEM:
  1. 1.

    The resolution is higher in a STEM than in a CTEM with equivalent optical parameters.

     
  2. 2.

    The interpretation of incoherent dark-field STEM images is more intuitive than the interpretation of coherent bright-field CTEM images.

     
  3. 3.

    The STEM can analyze the sample composition by spectroscopic techniques, often at the same time as recording image data.

     
  4. 4.

    STEM is compatible with ptychographical reconstruction techniques, which can significantly enhance the spatial resolution and improve the signal-to-noise ratio in images recorded with limited electron doses.

     
As a consequence, aberration-corrected STEMs have recently become the instrument of choice for materials science. In electron microscopy for structural biology, on the other hand, which needs phase-contrast imaging, primary energies of the order of \({\mathrm{300}}\,{\mathrm{keV}}\) and spatial resolution of only \(2{-}3\,{\mathrm{\AA{}}}\), uncorrected CTEM used for bright-field imaging is the primary tool (Chap.  4).

Second, as the STEM resolution improves as a result of advances in aberration correction and the bright-field disk becomes larger and larger, an increasing fraction of the scattered electrons contributes to the signal within the bright-field disk rather than outside it, making dark-field techniques less efficient. The electrons scattered into the bright-field disk owe their origin to a wide variety of contrast mechanisms, and the corresponding signals can be used for phase-contrast imaging, magnetic imaging and several other types of imaging experiments. An electron Ronchigram is of course a diffraction pattern recorded with a highly convergent probe, typically out of focus. When the illumination is made more parallel, the Ronchigram becomes a regular diffraction pattern with Bragg disks, and recording such patterns from every image point is another way of capturing a large amount of information about the sample. See Chap.  17 and the earlier reviews of these techniques by Rodenburg [13.352] and Zuo and Tao [13.364]. Their complete blossoming is facilitated by detectors able to record the 2-D diffraction pattern at every image pixel, with high DQE for single electron detection, high dynamic range and high speed. Recently introduced detectors record diffraction patterns \({\mathrm{128}}\times{\mathrm{128}}\) to \({\mathrm{256}}\times{\mathrm{256}}\) in size at a rate of about 1000 patterns per second [13.365, 13.366, 13.367], and faster detectors with a greater number of pixels are being planned. The recent improvement in the attainable spatial resolution [13.354] is an important outcome of these developments. Moreover, the efficiency of STEM phase-contrast imaging (the amount of useful information extracted per each incident electron) is likely to rival that of CTEM phase-contrast imaging.

In summary, aberration correction has progressed to a new development phase: with the utility of probe and image correctors firmly established, it has moved on to the exploration of new contrast mechanisms and even signals that are new to electron microscopy. Electron microscopists have also realized that aberration correction may be useful in areas other than simple improvement of the spatial or energy resolution. An example is in situ microscopy, where ample space is needed between the polepieces of the objective lens; the coefficients of spherical and chromatic aberration are then inevitably large. By correcting these aberrations, in situ microscopy should become a high-resolution technique. In the life sciences, chromatic correction will make it possible to work with thicker specimens. The overall development of aberration correction—from a technique once dismissed as too complicated to become practical, to a useful tool focused on improving spatial resolution and subsequently to a multifaceted tool making many different kinds of contributions—has exceeded expectations more than once. The development is by no means over, and many fundamental surprises are likely waiting in the wings.

References

  1. O. Scherzer: Über einige Fehler von Elektronenlinsen, Z. Phys. 101, 593–603 (1936)CrossRefGoogle Scholar
  2. O. Scherzer: Sphärische und chromatische Korrektur von Elektronenlinsen, Optik 2, 114–132 (1947)Google Scholar
  3. A. Septier: The struggle to overcome spherical aberration in electron optics, Adv. Opt. Electron Microsc. 1, 204–274 (1966)Google Scholar
  4. A. Septier: The struggle to overcome spherical aberration in electron optics, Adv. Imaging Electron Phys. 202, 75–147 (2017)Google Scholar
  5. P.W. Hawkes: Methods of computing optical properties and combating aberrations for low-intensity beams, Adv. Electron. Electron Phys. Suppl. 13A, 45–157 (1980)Google Scholar
  6. P.W. Hawkes, E. Kasper: Principles of Electron Optics, 2nd edn. (Academic Press, Kidlington 2018)Google Scholar
  7. M. Marko, H. Rose: The contributions of Otto Scherzer (1909–1982) to the development of the electron microscope, Microsc. Microanal. 16, 366–374 (2010)CrossRefGoogle Scholar
  8. J. Orloff (Ed.): Handbook of Charged Particle Optics, 2nd edn. (CRC, Boca Raton 2009)Google Scholar
  9. E. Munro: Computational techniques for design of charged particle optical systems. In: Handbook of Charged Particle Optics, ed. by J. Orloff (CRC, Boca Raton 1997) pp. 1–76Google Scholar
  10. K. Tsuno: Magnetic lenses for electron microscopy. In: Handbook of Charged Particle Optics, 2nd edn., ed. by J. Orloff (CRC, Boca Raton 2009) pp. 129–159Google Scholar
  11. B. Lencová: Electrostatic lenses. In: Handbook of Charged Particle Optics, 2nd edn., ed. by J. Orloff (CRC, Boca Raton 2009) pp. 161–208Google Scholar
  12. P.W. Hawkes: Aberrations. In: Handbook of Charged Particle Optics, 2nd edn., ed. by J. Orloff (CRC, Boca Raton 2009) pp. 209–340Google Scholar
  13. H. Rose: Geometrical Charged-Particle Optics, 2nd edn. (Springer, Heidelberg 2012)CrossRefGoogle Scholar
  14. H. Rose: Advances in electron optics. In: High-Resolution Imaging and Spectrometry of Materials, ed. by F. Ernst, M. Rühle (Springer, Berlin 2003) pp. 189–270CrossRefGoogle Scholar
  15. P.W. Hawkes: Quadrupoles in Electron Lens Design, Adv. Electron. Electron Phys. Suppl. 7 (Academic Press, Cambridge 1970)Google Scholar
  16. L.A. Baranova, S.Ya. Yavor: The optics of round and multipole electrostatic lenses, Adv. Electron. Electron Phys. 76, 1–207 (1989)Google Scholar
  17. A.D. Dymnikov, S.Ya. Yavor: Four quadrupole lenses as an analogue of an axially symmetric system, Zh. Tekh. Fiz. 33, 851–858 (1963)Google Scholar
  18. A.D. Dymnikov, S.Ya. Yavor: Four quadrupole lenses as an analogue of an axially symmetric system, Sov. Phys. Tech. Phys. 8, 639–643 (1963)Google Scholar
  19. H. Rose: Theory of electron-optical achromats and apochromats, Ultramicroscopy 93, 293–303 (2002)CrossRefGoogle Scholar
  20. S. Uhlemann, M. Haider: Residual wave aberrations in the first spherical aberration corrected transmission electron microscope, Ultramicroscopy 72, 109–119 (1998)CrossRefGoogle Scholar
  21. M. Haider, H. Müller, S. Uhlemann, J. Zach, U. Loebau, R. Hoeschen: Prerequisites for a Cc/Cs-corrected ultrahigh-resolution TEM, Ultramicroscopy 108, 167–178 (2008)CrossRefGoogle Scholar
  22. M. Haider, P. Hartel, U. Loebau, R. Hoeschen, H. Müller, S. Uhlemann, F. Kahl, F. Zach: Progress on the development of a Cc/Cs corrector for TEAM, Microsc. Microanal. 14(Suppl. 2), 800–801 (2008)CrossRefGoogle Scholar
  23. M. Haider, H. Müller, S. Uhlemann: Present and future hexapole aberration correctors for high resolution electron microscopy, Adv. Imaging Electron Phys. 153, 43–120 (2008)CrossRefGoogle Scholar
  24. M. Haider, S. Uhlemann, J. Zach: Upper limits for the residual aberrations of a high-resolution aberration-corrected STEM, Ultramicroscopy 81, 163–175 (2000)CrossRefGoogle Scholar
  25. O.L. Krivanek, N. Dellby, M.F. Murfitt: Aberration correction in electron microscopy. In: Handbook of Charged Particle Optics, 2nd edn., ed. by J. Orloff (CRC, Boca Raton 2009) pp. 601–640Google Scholar
  26. O.L. Krivanek, N. Dellby, A.R. Lupini: Towards sub-Å electron beams, Ultramicroscopy 78, 1–11 (1999)CrossRefGoogle Scholar
  27. H. Sawada: Aberration correction in STEM. In: Scanning Transmission Electron Microscopy of Nanomaterials, ed. by N. Tanaka (Imperial College Press, London 2015) pp. 283–305Google Scholar
  28. H. Sawada, T. Sannomiya, F. Hosokawa, T. Nakamichi, T. Kaneyama, T. Tomita, Y. Kondo, T. Tanaka, Y. Oshima, Y. Tanishiro, K. Takayanagi: Measurement method of aberration from Ronchigram by autocorrelation function, Ultramicroscopy 108, 1467–1475 (2008)CrossRefGoogle Scholar
  29. H. Sawada, M. Watanabe, E. Okunishi, Y. Kondo: Auto-tuning of aberrations using high-resolution STEM images by auto-correlation function, Microsc. Microanal. 17(Suppl. 2), 1308–1309 (2011)CrossRefGoogle Scholar
  30. H. Sawada, F. Hosokawa, T. Sasaki, T. Kaneyama, Y. Kondo, K. Suenaga: Aberration correctors developed under the Triple C project, Adv. Imaging Electron Phys. 168, 297–336 (2011)CrossRefGoogle Scholar
  31. O.L. Krivanek: Three-fold astigmatism in high-resolution transmission electron microscopy, Ultramicroscopy 55, 419–433 (1994)CrossRefGoogle Scholar
  32. O.L. Krivanek, G.Y. Fan: Complete HREM autotuning using automated diffractogram analysis, Proc. Annu. Meet. EMSA 50(1), 96–97 (1992)Google Scholar
  33. O.L. Krivanek, G.Y. Fan: Application of slow-scan charge-coupled device (CCD) cameras to on-line microscope control, Scanning Microsc. Suppl. 6, 105–114 (1992)Google Scholar
  34. O.L. Krivanek, M.L. Leber: Three-fold astigmatism: An important TEM aberration. In: Proc. 51st Annu. Meet. Microsc. Soc. Am, ed. by G.W. Bailey, C.L. Rieder (San Francisco Press, San Francisco 1993) pp. 972–973Google Scholar
  35. O.L. Krivanek, M.L. Leber: Autotuning for 1 Å resolution. In: Proc. 13th Int. Conf. Electron Microsc., Paris, Vol. 1, ed. by B. Jouffrey, C. Colliex, J.-P. Chevalier, F. Glas, P.W. Hawkes, D. Hernandez-Verdun, J. Schrevel, D. Thomas (Editions de Physique, Les Ulis 1994) pp. 157–158Google Scholar
  36. O.L. Krivanek, P.A. Stadelmann: Effect of three-fold astigmatism on high-resolution electron micrographs, Ultramicroscopy 60, 103–113 (1995)CrossRefGoogle Scholar
  37. W.O. Saxton: Observation of lens aberrations for very high resolution electron microscopy I: Theory, J. Microsc. (Oxford) 179, 201–213 (1995)CrossRefGoogle Scholar
  38. W.O. Saxton: Simple prescriptions for measuring three-fold astigmatism, Ultramicroscopy 58, 239–243 (1995)CrossRefGoogle Scholar
  39. W.O. Saxton: A new way of measuring aberrations, Ultramicroscopy 81, 41–45 (2000)CrossRefGoogle Scholar
  40. W.O. Saxton: Observation of lens aberrations for very high resolution electron microscopy. II. Simple expressions for optimal estimates, Ultramicroscopy 151, 168–177 (2015)CrossRefGoogle Scholar
  41. W.O. Saxton, G. Chand, A.I. Kirkland: Accurate determination and compensation of lens aberrations in high resolution EM. In: Proc. 13th Int. Conf. Electron Microsc., Paris, Vol. 1, ed. by B. Jouffrey, C. Colliex, J.-P. Chevalier, F. Glas, P.W. Hawkes, D. Hernandez-Verdun, J. Schrevel, D. Thomas (Editions de Physique, Les Ulis 1994) pp. 203–204Google Scholar
  42. G. Chand, W.O. Saxton, A.I. Kirkland: Aberration measurement and automated alignment of the TEM. In: Electron Microscopy and Analysis, ed. by D. Cherns (Institute of Physics, Bristol 1995) pp. 297–300Google Scholar
  43. K. Ishizuka: Coma-free alignment of a high-resolution electron microscope with three-fold astigmatism, Ultramicroscopy 55, 407–418 (1994)CrossRefGoogle Scholar
  44. R. Meyer, A.I. Kirkland, W.O. Saxton: A new method for the determination of the wave aberration function for high resolution TEM. I. Measurement of the symmetric aberrations, Ultramicroscopy 92, 89–109 (2002)CrossRefGoogle Scholar
  45. R. Meyer, A.I. Kirkland, W.O. Saxton: A new method for the determination of the wave aberration function for high resolution TEM. II. Measurement of the antisymmetric aberrations, Ultramicroscopy 99, 115–123 (2004)CrossRefGoogle Scholar
  46. A.R. Lupini: The electron Ronchigram. In: Scanning Transmission Electron Microscopy. Imaging and Analysis, ed. by S.J. Pennycook, P.D. Nellist (Springer, New York 2011) pp. 117–161CrossRefGoogle Scholar
  47. A.R. Lupini, M. Chi, S. Jesse: Rapid aberration measurement with pixelated detectors, J. Microsc. (Oxford) 263, 43–50 (2016)CrossRefGoogle Scholar
  48. F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann: Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms, Ultramicroscopy 3, 49–60 (1978)CrossRefGoogle Scholar
  49. F. Zemlin: A practical procedure for alignment of a high resolution electron microscope, Ultramicroscopy 4, 241–224 (1979)CrossRefGoogle Scholar
  50. O.L. Krivanek: EM contrast transfer functions for tilted illumination imaging. In: Proc. 9th Int. Electron Microsc. Congr, Vol. 1, ed. by J.M. Sturgess (Microscopical Society of Canada, Toronto 1978) pp. 168–169Google Scholar
  51. W. Glaser: Über elektronenoptische Abbildung bei gestörter Rotationssymmetrie, Z. Phys. 120, 1–15 (1942)CrossRefGoogle Scholar
  52. P.A. Sturrock: The aberrations of magnetic electron lenses due to asymmetries. In: Proc. First Conf. Electron Microsc., 1949, ed. by A.L. Houwink, J.B. Le Poole, W.A. Le Rütte (Hoogland, Delft 1950) pp. 89–93Google Scholar
  53. P.A. Sturrock: The aberrations of magnetic electron lenses due to asymmetries, Philos. Trans. Royal Soc. A 243, 387–429 (1951)Google Scholar
  54. G.D. Archard: Magnetic electron lens aberrations due to mechanical defects, J. Sci. Instrum. 30, 352–358 (1953)CrossRefGoogle Scholar
  55. W. Glaser, P. Schiske: Bildstörungen durch Polschuhasymmetrien bei Elektronenlinsen, Z. Angew. Phys. 5, 329–339 (1953)Google Scholar
  56. G.V. Der-Shvarts: Influence of imperfections of the rotational symmetry of the focusing fields on the resolution of the magnetic objectives of electron microscopes, Zh. Tekh. Fiz. 24, 859–870 (1954)Google Scholar
  57. P.A. Stoyanov: The effect of departures of the geometrical shape of the polepieces of an objective from circular symmetry on the resolving power of the electron microscope, Zh. Tekh. Fiz. 25, 625–635 (1955)Google Scholar
  58. K.-H. Herrmann, P. Schiske, F. Zemlin: Einfluß und Nachweis des dreizähligen Astigmatismus in der Hochauflösungsmikroskopie, Mikroskopie 32, 235 (1976)Google Scholar
  59. M.I. Yavor: Methods for calculation of parasitic aberrations and machining tolerances in electron optical systems, Adv. Electron. Electron Phys. 86, 225–281 (1993)CrossRefGoogle Scholar
  60. P.W. Hawkes: Aberration correction. In: Science of Microscopy, ed. by P.W. Hawkes, J.C.H. Spence (Springer, New York 2007) pp. 696–750CrossRefGoogle Scholar
  61. P.W. Hawkes: Aberration correction past and present, Philos. Trans. Royal Soc. A 367, 3637–3664 (2009)CrossRefGoogle Scholar
  62. P.W. Hawkes: The correction of electron lens aberrations, Ultramicroscopy 156, A1–A64 (2015)CrossRefGoogle Scholar
  63. H. Rose: Historical aspects of aberration correction, J. Electron Microsc. 58, 77–85 (2009)CrossRefGoogle Scholar
  64. R. Erni: Aberration-corrected Imaging in Transmission Electron Microscopy: An Introduction, 2nd edn. (Imperial College Press, London 2015)CrossRefGoogle Scholar
  65. P.W. Hawkes (Ed.): Aberration-corrected electron microscopy, Adv. Imaging Electron Phys. 153 (2008)Google Scholar
  66. R. Brydson (Ed.): Aberration-Corrected Analytical Electron Microscopy (Wiley and the Royal Microscopical Society, Chichester, Oxford 2011)Google Scholar
  67. B. Lencová, M. Lenc: Computation of multi-lens focusing systems, Nucl. Instrum. Methods Phys. Res. A 298, 45–55 (1990)CrossRefGoogle Scholar
  68. B. Lencová, M. Lenc: Computation of properties of electrostatic lenses, Optik 97, 121–126 (1994)Google Scholar
  69. B. Lencová, M. Lenc: Third order geometrical and first order chromatic aberrations of electrostatic lenses, Optik 105, 121–128 (1997)Google Scholar
  70. H. Rose: Über die Korrigierbarkeit von Linsen für schnelle Elektronen, Optik 26, 289–298 (1967)Google Scholar
  71. D. Preikszas, H. Rose: Procedures for minimizing the aberrations of electromagnetic compound lenses, Optik 100, 179–187 (1995)Google Scholar
  72. W. Glaser: Über ein von sphärischer Aberration freies Magnetfeld, Z. Phys. 116, 19–33 (1940)CrossRefGoogle Scholar
  73. A. Recknagel: Über die sphärische Aberration bei elektronenoptischer Abbildung, Z. Phys. 117, 67–73 (1941)CrossRefGoogle Scholar
  74. S. Nomura: Aberration-free electron microscope composed of round lenses. In: Proc. 8th Asia–Pac. Conf. Electron Microsc., Kanazawa (2004) pp. 34–35Google Scholar
  75. S. Nomura: Design of an apochromatic TEM composed of usual round lenses. In: Proc. 14th Eur. Microsc. Congr., Aachen, Vol. 1, ed. by M. Luysberg, K. Tillmann (Springer, Berlin 2008) pp. 41–42Google Scholar
  76. P.W. Hawkes: Can the Nomura lens be free of spherical aberration?, J. Microsc. (Oxford) 234, 325 (2009)CrossRefGoogle Scholar
  77. W. Tretner: Existenzbereiche rotationssymmetrischer Elektronenlinsen, Optik 16, 155–184 (1959)Google Scholar
  78. W.D. Riecke: Practical lens design. In: Magnetic Electron Lenses, ed. by P.W. Hawkes (Springer, Berlin-Heidelberg 1982) pp. 163–357CrossRefGoogle Scholar
  79. T. Yanaka, A. Yonezawa, A. Oosawa, T. Iwaki, S. Suzuki, O. Nakamura, M. Watanabe: Development of ultra-high resolution analytical electron microscope ISI-EM-002A, Proc. Annu. Meet. EMSA 41, 312–313 (1982)Google Scholar
  80. O. Scherzer: The theoretical resolution limit of the electron microscope, J. Appl. Phys. 20, 20–29 (1949)CrossRefGoogle Scholar
  81. O.L. Krivanek, M.F. Chisholm, N. Dellby, M.F. Murfitt: Atomic-resolution STEM at low primary energies. In: Scanning Transmission Electron Microscopy: Imaging and Analysis, ed. by S.J. Pennycook, P.D. Nellist (Springer, Berlin-Heidelberg 2011) pp. 613–656Google Scholar
  82. H. Ichinose, H. Sawada, E. Takuma, M. Osaki: Atomic resolution HVEM and environmental noise, J. Electron Microsc. 48, 887–889 (1999)CrossRefGoogle Scholar
  83. P.W. Hawkes: Signposts in electron optics, Adv. Imaging Electron Phys. 123, 1–28 (2002)CrossRefGoogle Scholar
  84. R. Seeliger: Versuche zur sphärischen Korrektur von Elektronenlinsen mittels nicht rotationssymmetrischer Abbildungselemente, Optik 5, 490–496 (1949)Google Scholar
  85. R. Seeliger: Die sphärische Korrektur von Elektronenlinsen mittels nicht-rotationssymmetrischer Abbildungselemente, Optik 8, 311–317 (1951)Google Scholar
  86. G. Möllenstedt: Elektronenmikroskopische Bilder mit einem nach O. Scherzer sphärisch korrigierten Objektiv, Optik 13, 209–215 (1956)Google Scholar
  87. G.D. Archard: Requirements contributing to the design of devices used in correcting electron lenses, Brit. J. Appl. Phys. 5, 294–299 (1954)CrossRefGoogle Scholar
  88. J.C. Burfoot: Correction of electrostatic lenses by departure from rotational symmetry, Proc. Phys. Soc. B 66, 775–792 (1953)CrossRefGoogle Scholar
  89. J.M.H. Deltrap: Correction of spherical aberration with combined quadrupole–octopole units. In: Proc. 3rd Eur. Reg. Conf. Electron. Microsc., Vol. A, ed. by M. Titlbach (Czechoslovak Academy of Sciences, Prague 1964) pp. 45–46Google Scholar
  90. D.F. Hardy: Combined Magnetic and Electrostatic Quadrupole Electron Lenses, Ph.D. Thesis (University of Cambridge, Cambridge 1967)Google Scholar
  91. M. Haider: Towards sub-ångstrom point resolution by correction of spherical aberration. In: Proc. 12th Eur. Congr. Electron Microsc, Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp. I.145–I.148Google Scholar
  92. H. Müller, S. Uhlemann, P. Hartel, M. Haider: Aberration-corrected optics: From an idea to a device, Phys. Procedia 1, 167–178 (2008)CrossRefGoogle Scholar
  93. D. Gabor: The Electron Microscope (Hulton, London 1945)Google Scholar
  94. M. Linck, P.A. Ercius, J.S. Pierce, B.J. McMorran: Aberration corrected STEM by means of diffraction gratings, Ultramicroscopy 182, 36–43 (2017)CrossRefGoogle Scholar
  95. A. Khursheed, W.K. Ang: On-axis electrode aberration correctors for scanning electron/ion microscopes, Microsc. Microanal. 21, 106–111 (2015)CrossRefGoogle Scholar
  96. A. Khursheed, W.K. Ang: Annular focused electron/ion beams for combining high spatial resolution with high probe current, Microsc. Microanal. 22, 948–954 (2016)CrossRefGoogle Scholar
  97. T. Kawasaki, T. Ishida, M. Tomita, T. Kodama, T. Matsutani, T. Ikuta: Development of a new electrostatic Cs-corrector consisted of annular and circular electrodes. In: Proc. 16th Eur. Microsc. Congr., Lyon, Vol. 1, ed. by O. Stéphan, M. Hÿtch, B. Satiat-Jeunemaître, C. Venien-Bryan, P. Bayle-Guillemaud, T. Epicier (Wiley-VCH, Weinheim 2016) pp. 430–432Google Scholar
  98. T. Kawasaki, R. Yoshida, T. Kato, T. Nomaguchi, T. Agemura, T. Kodama, M. Tomita, T. Ikuta: Development of compact Cs/Cc corrector with annular and circular electrodes, Microsc. Microanal. 23(Suppl. 1), 466–467 (2017)CrossRefGoogle Scholar
  99. G.D. Archard: A possible chromatic aberration system for electron lenses, Proc. Phys. Soc. B 68, 817–829 (1955)CrossRefGoogle Scholar
  100. V.M. Kel'man, S.Ya. Yavor: Achromatic quadrupole electron lenses, Zh. Tekh. Fiz. 31, 1439–1442 (1961)Google Scholar
  101. V.M. Kel'man, S.Ya. Yavor: Achromatic quadrupole electron lenses, Sov. Phys. Tech. Phys. 6, 1052–1054 (1961)Google Scholar
  102. A. Septier: Lentille quadrupolaire magneto-électrique corrigée de l’aberration chromatique. Aberration d’ouverture de ce type de lentilles, C. R. Acad. Sci. Paris 256, 2325–2328 (1963)Google Scholar
  103. P.W. Hawkes: The paraxial chromatic aberrations of quadrupole systems. In: Proc. 3rd Eur. Reg. Conf. Electron Microsc., Vol. A, ed. by M. Titlbach (Czechoslovak Academy of Sciences, Prague 1964) pp. 5–6Google Scholar
  104. P.W. Hawkes: The paraxial chromatic aberrations of rectilinear orthogonal systems, Optik 22, 543–551 (1965)Google Scholar
  105. H. Rose: Inhomogeneous Wien filter as a corrector for the chromatic and spherical aberration of low-voltage electron microscopes, Optik 84, 91–107 (1990)Google Scholar
  106. K. Tsuno, D. Ioanoviciu, G. Martínez: Aberration corrected Wien filter as a monochromator of high spatial and high energy resolution electron microscopes, Microsc. Microanal. 9(Suppl. 2), 944–945 (2003)CrossRefGoogle Scholar
  107. G. Schönhense, H. Spiecker: Correction of chromatic and spherical aberration in electron microscopy utilizing the time structure of pulsed electron sources, J. Vac. Sci. Technol. B 20, 2526–2534 (2002)CrossRefGoogle Scholar
  108. A. Khursheed: A method of dynamic chromatic correction in low-voltage scanning electron microscopes, Ultramicroscopy 103, 255–260 (2005)CrossRefGoogle Scholar
  109. G. Schönhense, H. Spiecker: Chromatic and spherical correction using time-dependent acceleration and lens fields, Microsc. Microanal. 9(Suppl. 3), 34–35 (2003)CrossRefGoogle Scholar
  110. G.F. Rempfer: A theoretical study of the hyperbolic electron mirror as a correcting element for spherical and chromatic aberration in electron optics, J. Appl. Phys. 67, 6027–6040 (1990)CrossRefGoogle Scholar
  111. G.F. Rempfer, M.S. Mauck: Correction of chromatic aberration with an electron mirror, Optik 92, 3–8 (1992)Google Scholar
  112. R. Fink, M.R. Weiss, E. Umbach, D. Preikszas, H. Rose, R. Spehr, P. Hartel, W. Engel, R. Degenhardt, R. Wichtendahl, H. Kuhlenbeck, W. Erlebach, K. Ihmann, R. Schlögl, H.-J. Freund, A.M. Bradshaw, G. Lilienkamp, T. Schmidt, E. Bauer, G. Benner: SMART: A planned ultrahigh-resolution spectromicroscope for BESSY II, J. Electron Spectrosc. Relat. Phenom. 84, 231–250 (1997)CrossRefGoogle Scholar
  113. R. Wichtendahl, R. Fink, H. Kuhlenbeck, D. Preikszas, H. Rose, R. Spehr, P. Hartel, W. Engel, R. Schlögl, H.-J. Freund, A.M. Bradshaw, G. Lilienkamp, T. Schmidt, E. Bauer, G. Benner, E. Umbach: SMART: An aberration-corrected XPEEM/LEEM with energy filter, Surf. Rev. Lett. 5, 1249–1256 (1998)CrossRefGoogle Scholar
  114. H. Müller, D. Preikszas, H. Rose: A beam separator with small aberrations, J. Electron Microsc. 48, 191–204 (1999)CrossRefGoogle Scholar
  115. P. Hartel, D. Preikszas, R. Spehr, H. Rose: Performance of the mirror corrector for an ultrahigh-resolution spectromicroscope. In: Proc. 12th Eur. Congr. Electron Microsc, Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp. I.153–I.154Google Scholar
  116. P. Hartel, D. Preikszas, R. Spehr, H. Müller, H. Rose: Mirror corrector for low-voltage electron microscopes, Adv. Imaging Electron Phys. 120, 41–133 (2002)CrossRefGoogle Scholar
  117. D. Preikszas, P. Hartel, R. Spehr, H. Rose: SMART electron optics. In: Proc. 12th Eur. Congr. Electron Microsc, Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp. I.181–I.184Google Scholar
  118. W. Wan, J. Feng, H.A. Padmore, D.S. Robin: Simulation of a mirror corrector for PEEM, Nucl. Instrum. Methods Phys. Res. A 519, 222–229 (2004)CrossRefGoogle Scholar
  119. Y.K. Wu, D.S. Robin, E. Forest, R. Schleuter, S. Anders, J. Feng, H. Padmore, D.H. Wei: Design and analysis of beam separator magnets for third generation aberration compensated PEEMs, Nucl. Instrum. Methods Phys. Res. A 519, 230–241 (2004)CrossRefGoogle Scholar
  120. R.M. Tromp, J.B. Hannon, A.W. Ellis, W. Wan, A. Berghaus, O. Schaff: A new aberration-corrected, energy-filtered LEEM/PEEM instrument. I. Principles and design, Ultramicroscopy 110, 852–861 (2010)CrossRefGoogle Scholar
  121. R.M. Tromp, J.B. Hannon, W. Wan, A. Berghaus, O. Schaff: A new aberration-corrected, energy-filtered LEEM/PEEM instrument II. Operation and results, Ultramicroscopy 127, 25–39 (2013)CrossRefGoogle Scholar
  122. R.M. Tromp: Measuring and correcting aberrations of a cathode objective lens, Ultramicroscopy 111, 273–281 (2011)CrossRefGoogle Scholar
  123. H. Rose, D. Preikszas: Outline of a versatile corrected LEEM, Optik 92, 31–44 (1992)Google Scholar
  124. H. Dohi, P. Kruit: Design for an aberration corrected scanning electron microscope using miniature electron mirrors, Ultramicroscopy 189, 1–23 (2018)CrossRefGoogle Scholar
  125. G.D. Archard: An unconventional electron lens, Proc. Phys. Soc. B 72, 135–137 (1958)CrossRefGoogle Scholar
  126. M. Reichenbach, H. Rose: Entwurf eines korrigierten magnetischen Objektivs, Optik 28, 475–487 (1968)Google Scholar
  127. H. Rose: Berechnung eines elektronenoptischen Apochromaten, Optik 32, 144–164 (1970)Google Scholar
  128. H. Rose: Abbildungseigenschaften sphärisch korrigierter elektronenoptischer achromate, Optik 33, 1–24 (1971)Google Scholar
  129. H. Rose: Elektronenoptische Aplanate, Optik 34, 285–311 (1971)Google Scholar
  130. B. Bastian, K. Spengler, D. Typke: An electric–magnetic octupole element to correct spherical and chromatic aberrations of electron lenses, Optik 33, 591–596 (1971)Google Scholar
  131. W. Pöhner: Ein in dritter und fünfter Ordnung sphärisch korrigierter elektronenoptischer Aplanat, Optik 45, 443–454 (1976)Google Scholar
  132. W. Pöhner: Ein sphärisch korrigierter elektronenoptischer Apochromat, Optik 47, 283–297 (1977)Google Scholar
  133. W. Bernhard, H. Koops: Kompensation der Farbabhängigkeit der Vergrößerung und der Farbabhängigkeit der Bilddrehung eines Elektronenmikroskops, Optik 47, 55–64 (1977)Google Scholar
  134. H. Koops, G. Kuck, O. Scherzer: Erprobung eines elektronenoptischen Achromators, Optik 48, 225–236 (1977)Google Scholar
  135. H. Koops: Aberration correction in electron microscopy. In: Proc. 9th Int. Congr. Electron Microsc., Vol. 3, ed. by J.M. Sturgess (Microscopical Society of Canada, Toronto 1978) pp. 185–196Google Scholar
  136. H. Koops: Erprobung eines chromatisch korrigierten elektronenmikroskopischen Objektives, Optik 52, 1–18 (1978)Google Scholar
  137. H. Koops, W. Bernhard: An objective lens for an electron microscope with compensated axial chromatic aberration. In: Proc. 9th Int. Congr. Electron Microsc., Vol. 1, ed. by J.M. Sturgess (Microscopical Society of Canada, Toronto 1978) pp. 36–37Google Scholar
  138. W. Pejas: Magnetische Abschirmung eines korrigierten Elektronenmikroskops, Optik 50, 61–72 (1978)Google Scholar
  139. G. Kuck: Erprobung eines elektronenoptischen Korrektivs für Farb- und Öffnungsfehler, Ph.D. Thesis (Technical University of Darmstadt, Darmstadt 1979)Google Scholar
  140. W. Bernhard: Erprobung eines sphärisch und chromatisch korrigierten Elektronenmikroskops, Optik 57, 73–94 (1980)Google Scholar
  141. G. Fey: Elektrische Versorgung eines elektronenoptischen Korrektivs, Optik 55, 55–65 (1980)Google Scholar
  142. H. Hely: Messungen an einem verbesserten korrigierten Elektronenmikroskop, Optik 60, 353–370 (1982)Google Scholar
  143. H. Hely: Technologische Voraussetzungen für die Verbesserung der Korrektur von Elektronenlinsen, Optik 60, 307–326 (1982)Google Scholar
  144. O. Scherzer: Limitations for the resolving power of electron microscopes. In: Proc. 9th Int. Congr. Electron Microsc., Vol. 3, ed. by J.M. Sturgess (Microscopical Society of Canada, Toronto 1978) pp. 123–129Google Scholar
  145. A.V. Crewe, D. Cohen, P. Meads: A multipole element for the correcting of spherical aberration. In: Proc. 4th Reg. Conf. Electron. Microsc., Vol. I, ed. by D.S. Bocciarelli (Tipographia Poliglotta Vaticana, Rome 1968) p. 183Google Scholar
  146. M.G.R. Thomson: The primary aberrations of a quadrupole corrector system, Optik 34, 528–534 (1972)Google Scholar
  147. V. Beck, A.V. Crewe: A quadrupole octupole corrector for a 100 kV STEM, Proc. Annu. Meet. EMSA 32, 426–427 (1974)Google Scholar
  148. V. Beck, A.V. Crewe: Progress in aberration correction in a STEM, Proc. Annu. Meet. EMSA 34, 578–579 (1976)Google Scholar
  149. V. Beck: Experiments with a quadrupole–octupole corrector in a STEM, Proc. Annu. Meet. EMSA 35, 90–91 (1977)Google Scholar
  150. A.V. Crewe: Is there a future for the STEM? In: Proc. 9th Int. Congr. Electron. Microsc., Vol. 3, ed. by J.M. Sturgess (Microscopical Society of Canada, Toronto 1978) pp. 197–204Google Scholar
  151. A.V. Crewe: The work of Albert Victor Crewe on the scanning transmission electron microscope and related topics, Adv. Imaging Electron Phys. 159, 1–61 (2009)CrossRefGoogle Scholar
  152. J. Zach: Design of a high-resolution low-voltage scanning electron microscope, Optik 83, 30–40 (1989)Google Scholar
  153. J. Zach: Aspects of aberration correction in LVSEM. In: Proc. 12th Eur. Congr. Electron Microsc., Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp. I.169–I.172Google Scholar
  154. J. Zach, M. Haider: Correction of spherical and chromatic aberration in a low-voltage SEM, Optik 98, 112–118 (1995)Google Scholar
  155. J. Zach, M. Haider: Aberration correction in a low voltage SEM by a multipole corrector, Nucl. Instrum. Methods Phys. Res. A 363, 316–325 (1995)CrossRefGoogle Scholar
  156. K. Honda, S. Uno, N. Nakamura, M. Matsuya, B. Achard, J. Zach: An automatic geometrical aberration correction system of scanning electron microscopes. In: Proc. 13th Eur. Microsc. Congr., Antwerp, Vol. I, ed. by D. van Dyck, P. van Oostveldt (Belgian Society for Microscopy, Liège 2004) pp. 43–44Google Scholar
  157. K. Honda, S. Uno, N. Nakamura, M. Matsuya, J. Zach: An automatic geometrical aberration correction system of scanning electron microscopes. In: Proc. 8th Asia–Pac. Conf. Electron Microsc., Kanazawa, ed. by N. Tanaka, Y. Takano, H. Mori, H. Seguchi, S. Iseki, H. Shimada, E. Simamura (8APEM Publication Committee, Uchinada 2004) pp. 44–45Google Scholar
  158. H. Kazumori, K. Honda, M. Matsuya, M. Date, C. Nielsen: Field emission SEM with a spherical and chromatic aberration corrector, Microsc. Microanal. 10(Suppl. 2), 1370–1371 (2004)CrossRefGoogle Scholar
  159. H. Kazumori, K. Honda, M. Matuya, M. Date: Field emission SEM with a spherical and chromatic aberration corrector. In: Proc. 8th Asia–Pac. Conf. Electron. Microsc., Kanazawa, ed. by N. Tanaka, Y. Takano, H. Mori, H. Seguchi, S. Iseki, H. Shimada, E. Simamura (8APEM Publication Committee, Uchinada 2004) pp. 52–53Google Scholar
  160. S. Uno, K. Honda, N. Nakamura, M. Matsuya, B. Achard, J. Zach: An automated aberration correction method in scanning electron microscopes. In: Proc. 13th Eur. Microsc. Congr., Antwerp, Vol. I, ed. by D. van Dyck, P. van Oostveldt (Belgian Society for Microscopy, Liège 2004) pp. 37–38Google Scholar
  161. S. Uno, K. Honda, N. Nakamura, M. Matsuya, J. Zach: Probe shape extraction and automatic aberration correction in scanning electron microscopes. In: Proc. 8th Asia–Pac. Conf. Electron Microsc., Kanazawa, ed. by N. Tanaka, Y. Takano, H. Mori, H. Seguchi, S. Iseki, H. Shimada, E. Simamura (8APEM Publication Committee, Uchinada 2004) pp. 46–47Google Scholar
  162. S. Uno, K. Honda, N. Nakamura, M. Matsuya, J. Zach: Aberration correction and its automatic control in scanning electron microscopes, Optik 16, 438–448 (2005)CrossRefGoogle Scholar
  163. O.L. Krivanek, N. Dellby, A.J. Spence, A. Camps, L.M. Brown: Aberration correction in the STEM. In: Proc. EMAG 1997, ed. by J.M. Rodenburg (Institute of Physics, Bristol 1997) pp. 35–39Google Scholar
  164. O.L. Krivanek, N. Dellby, A.J. Spence, R.A. Camps, L.M. Brown: On-line aberration measurement and correction in STEM, Microsc. Microanal. 3(Suppl. 2), 1171–1172 (1997)Google Scholar
  165. O.L. Krivanek, N. Dellby, L.M. Brown: Spherical aberration corrector for a dedicated STEM. In: Proc. EUREM-11, 11th Eur. Conf. Electron. Microsc., Dublin 1996, Vol. I, ed. by CESEM (CESEM, Brussels 1998) pp. 352–353Google Scholar
  166. O.L. Krivanek, N. Dellby, A.R. Lupini: STEM without spherical aberration, Microsc. Microanal. 5(Suppl. 2), 670–671 (1999)Google Scholar
  167. O.L. Krivanek, N. Dellby, A.R. Lupini: Advances in Cs-corrected STEM. In: Proc. 12th Eur. Congr. Electron Microsc., Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp.  I.149–I.150Google Scholar
  168. N. Dellby, O.L. Krivanek, P.D. Nellist, P.E. Batson, A.R. Lupini: Progress in aberration-corrected scanning transmission electron microscopy, J. Electron Microsc. 50, 177–185 (2001)Google Scholar
  169. P.E. Batson, N. Dellby, O.L. Krivanek: Sub-ångstrom resolution using aberration corrected optics, Nature 418, 617–620 (2002)CrossRefGoogle Scholar
  170. P.E. Batson: Aberration correction results in the IBM STEM instrument, Ultramicroscopy 96, 239–249 (2003)CrossRefGoogle Scholar
  171. P.D. Nellist, N. Dellby, O.L. Krivanek, M.F. Murfitt, Z. Szilagyi, A.R. Lupini, S.J. Pennycook: Towards sub 0.5 angstrom beams through aberration corrected STEM. In: Proc. EMAG 2003, ed. by S. McVitie, D. McComb (Institute of Physics Publishing, Bristol and Philadelphia 2004) pp. 159–164Google Scholar
  172. P.D. Nellist, M.F. Chisholm, N. Dellby, O.L. Krivanek, M.F. Murfitt, Z.S. Szilagyi, A.R. Lupini, A. Borisevich, W.H. Sides, S.J. Pennycook: Direct sub-angstrom imaging of a crystal lattice, Science 305, 1741 (2004)CrossRefGoogle Scholar
  173. O.L. Krivanek, P.D. Nellist, N. Dellby, M.F. Murfitt, Z. Szilagyi: Towards sub-0.5 Å beams, Ultramicroscopy 96, 229–237 (2003)CrossRefGoogle Scholar
  174. O.L. Krivanek, G.J. Corbin, N. Dellby, M. Murfitt, K. Nagesha, P.D. Nellist, Z. Szilagyi: Nion UltraSTEM: A new STEM for sub-0.5 Å imaging and sub-0.5 eV analysis. In: Proc. 13th Eur. Microsc. Congr., Antwerp, Vol. I, ed. by D. van Dyck, P. van Oostveldt (Belgian Society for Microscopy, Liège 2004) pp. 35–36Google Scholar
  175. N. Dellby, O.L. Krivanek, M.F. Murfitt, P.D. Nellist: Design and testing of a quadrupole/octupole C3/C5 aberration corrector, Microsc. Microanal. 11(Suppl. 2), 2130–2131 (2005)Google Scholar
  176. N. Dellby, O.L. Krivanek, M.F. Murfitt: Optimized quadrupole-octupole C3/C5 aberration corrector for STEM, Phys. Procedia 1, 179–183 (2008)CrossRefGoogle Scholar
  177. N. Dellby, G.J. Corbin, Z. Dellby, T.C. Lovejoy, Z.S. Szilagyi, M.F. Chisholm, O.L. Krivanek: Tuning high order geometric aberrations in quadrupole-octupole correctors, Microsc. Microanal. 20(Suppl. 3), 928–929 (2014)CrossRefGoogle Scholar
  178. S.A.M. Mentink, M.J. van der Zande, C. Kok, T.L. van Rooy: Development of a Cs corrector for a Tecnai 20 FEG STEM/TEM. In: Proc. EMAG 2003, ed. by S. McVitie, D. McComb (Institute of Physics Publishing, Bristol 2004) pp. 165–168Google Scholar
  179. P.W. Hawkes: The geometrical aberrations of general electron optical systems, I and II, Philos. Trans. Royal Soc. A 257, 479–552 (1965)CrossRefGoogle Scholar
  180. V.D. Beck: A hexapole spherical aberration corrector, Optik 53, 241–255 (1979)Google Scholar
  181. H. Rose: Correction of aperture aberrations in magnetic systems with threefold symmetry, Nucl. Instrum. Methods 187, 187–199 (1981)CrossRefGoogle Scholar
  182. A.V. Crewe: Studies on sextupole correctors, Optik 57, 313–327 (1980)Google Scholar
  183. A.V. Crewe: A system for the correction of axial aperture aberrations in electron lenses, Optik 60, 271–281 (1982)Google Scholar
  184. A.V. Crewe, D. Kopf: A sextupole system for the correction of spherical aberration, Optik 55, 1–10 (1980)Google Scholar
  185. A.V. Crewe, D. Kopf: Limitations of sextupole correctors, Optik 56, 391–399 (1980)Google Scholar
  186. M. Haider, W. Bernhardt, H. Rose: Design and test of an electric and magnetic dodecapole lens, Optik 63, 9–23 (1982)Google Scholar
  187. M. Haider, G. Braunshausen, E. Schwan: Correction of the spherical aberration of a 200 kV TEM by means of a hexapole-corrector, Optik 99, 167–179 (1995)Google Scholar
  188. M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban: Electron microscopy image enhanced, Nature 392, 768–769 (1998)CrossRefGoogle Scholar
  189. M. Haider, H. Rose, S. Uhlemann, B. Kabius, K. Urban: Towards 0.1 nm resolution with the first spherically corrected transmission electron microscope, J. Electron Microsc. 47, 395–405 (1998)CrossRefGoogle Scholar
  190. M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban: A spherical-aberration-corrected 200 kV transmission electron microscope, Ultramicroscopy 75, 53–60 (1998)CrossRefGoogle Scholar
  191. H. Rose: Outline of a spherically corrected semiaplanatic medium-voltage TEM, Optik 85, 19–24 (1990)Google Scholar
  192. H. Rose: Correction of aberrations, a promising method for improving the performance of electron microscopes. In: Proc. 10th Eur. Congr. Electron Microsc., Vol. 1, ed. by A. Ríos, J.M. Arias, L. Megías-Megías, A. López-Galindo (Secretariado de Publicaciones de la Universidad de Granada, Granada 1992) pp. 47–48Google Scholar
  193. H. Rose: Correction of aberrations – past, present and future, Microsc. Microanal. 8(Suppl. 2), 6–7 (2002)CrossRefGoogle Scholar
  194. M. Haider, S. Uhlemann: Seeing is not believing: Reduction of artefacts by an improved point resolution with a spherical aberration corrected 200 kV transmission electron microscope, Microsc. Microanal. 3(Suppl. 2), 1179–1180 (1997)Google Scholar
  195. M. Haider: Correctors for electron microscopes: Tools or toys for scientists? In: Proc. 11th Eur. Conf. Electron Microsc., Dublin 1996, Vol. I (CESEM, Brussels 1998) pp. 363–364Google Scholar
  196. M. Haider: Current and future developments in order to approach a point resolution of dpr ∼ 0.5 Å with a TEM, Microsc. Microanal. 9(Suppl. 2), 930–931 (2003)Google Scholar
  197. M. Foschepoth, H. Kohl: Amplitude contrast—A way to obtain directly interpretable high-resolution images in a spherical-aberration-corrected transmission electron microscope, Phys. Status Solidi (a) 166, 357–366 (1998)CrossRefGoogle Scholar
  198. S. Uhlemann, M. Haider, E. Schwan, H. Rose: Towards a resolution enhancement in the corrected TEM. In: Proc. EUREM-11, 11th Eur. Conf. Electron Microsc., Dublin 1996, Vol. I (CESEM, Brussels 1998) pp. 365–366Google Scholar
  199. K. Urban, B. Kabius, M. Haider, H. Rose: A way to higher resolution: Spherical-aberration correction in a 200 kV transmission electron microscope, J. Electron Microsc. 48, 821–826 (1999)CrossRefGoogle Scholar
  200. H. Müller, S. Uhlemann, M. Haider: Benefits and possibilities for Cc-correction in TEM/STEM, Microsc. Microanal. 8(Suppl. 2), 12–13 (2002)CrossRefGoogle Scholar
  201. H. Müller, M. Haider, P. Hartel, S. Uhlemann, J. Zach: Improved aberration correctors for the conventional and the scanning transmission electron microscope, Recent Trends Charged Part. Opt. Surf. Phys. Instrum. 12, 39–40 (2010)Google Scholar
  202. H. Müller, I. Maßmann, S. Uhlemann, P. Hartel, J. Zach, M. Haider: Aplanatic imaging systems for the transmission electron microscope, Nucl. Instrum. Methods Phys. Res. A 645, 20–27 (2011)CrossRefGoogle Scholar
  203. H. Müller, I. Maßmann, S. Uhlemann, P. Hartel, J. Zach, M. Haider: Practical aspects of an aplanatic transmission electron microscope, Recent Trends Charged Part. Opt. Surf. Phys. Instrum. 13, 47–48 (2012)Google Scholar
  204. B. Kabius, M. Haider, S. Uhlemann, E. Schwan, K. Urban, H. Rose: First application of a spherical-aberration corrected transmission electron microscope in materials science, J. Electron Microsc. 51, S51–S58 (2002)CrossRefGoogle Scholar
  205. M. Lentzen, B. Jahnen, C.L. Jia, A. Thust, K. Tillmann, K. Urban: High-resolution imaging with an aberration-corrected transmission electron microscope, Ultramicroscopy 92, 233–242 (2002)CrossRefGoogle Scholar
  206. H. Liu, E. Munro, J. Rouse, X. Zhu: Simulation methods for multipole imaging systems and aberration correctors, Ultramicroscopy 93, 271–291 (2002)CrossRefGoogle Scholar
  207. G. Benner, A. Orchowski, M. Haider, P. Hartel: State of the first aberration-corrected, monochromatized 200 kV FEG-TEM, Microsc. Microanal. 9(Suppl. 3), 938–939 (2003)CrossRefGoogle Scholar
  208. G. Benner, E. Essers, M. Matijevic, A. Orchowski, P. Schlossmacher, A. Thesen: Performance of monochromized and aberration-corrected TEMs, Microsc. Microanal. 10(Suppl. 2), 108–109 (2004)CrossRefGoogle Scholar
  209. G. Benner, M. Matijevic, A. Orchowski, P. Schlossmacher, A. Thesen, M. Haider, P. Hartel: Sub-ångstrom and sub-eV resolution with the analytical SATEM, Microsc. Microanal. 10(Suppl. 3), 6–7 (2004)CrossRefGoogle Scholar
  210. L.Y. Chang, F.R. Chen, A.I. Kirkland, J.J. Kai: Calculations of spherical aberration-corrected imaging behaviour, J. Electron Microsc. 52, 359–364 (2003)CrossRefGoogle Scholar
  211. F. Hosokawa, T. Tomita, M. Naruse, T. Honda, P. Hartel, M. Haider: A spherical aberration-corrected 200 kV TEM, J. Electron Microsc. 52, 3–10 (2003)CrossRefGoogle Scholar
  212. F. Hosokawa, T. Sannomiya, H. Sawada, T. Kaneyama, Y. Kondo, M. Hori, S. Yuasa, M. Kawazoe, Y. Nakamichi, T. Tanishiro, N. Yamamoto, K. Takayanagi: Design and development of Cs correctors for 300 kV TEM and STEM. In: Proc. 16th Int. Microsc. Conf., Sapporo, Vol. 2, ed. by H. Ichinose, T. Sasaki (2006) p. 582Google Scholar
  213. F. Hosokawa, H. Sawada, Y. Kondo, K. Takayanagi, K. Suenaga: Development of Cs and Cc correctors for transmission electron microscopy, Microscopy 62, 23–41 (2013)CrossRefGoogle Scholar
  214. C.L. Jia, M. Lentzen, K. Urban: Atomic-resolution imaging of oxygen in perovskite ceramics, Science 299, 870–873 (2003)CrossRefGoogle Scholar
  215. H. Sawada, T. Tomita, T. Kaneyama, F. Hosokawa, M. Naruse, T. Honda, P. Hartel, M. Haider, N. Tanaka, C.J.D. Hetherington, R.C. Doole, A.I. Kirkland, J.L. Hutchison, J.M. Titchmarsh, D.J.H. Cockayne: Cs corrector for imaging, Microsc. Microanal. 10(Suppl. 2), 978–979 (2004)Google Scholar
  216. H. Sawada, T. Tomita, M. Naruse, T. Honda, P. Hartel, M. Haider, C.J.D. Hetherington, R.C. Doole, A.I. Kirkland, J.L. Hutchison, J.M. Titchmarsh, D.J.H. Cockayne: Cs corrector for illumination, Microsc. Microanal. 10(Suppl. 2), 1004–1005 (2004)CrossRefGoogle Scholar
  217. H. Sawada, T. Tomita, M. Naruse, T. Honda, P. Hartel, M. Haider, C.J.D. Hetherington, R.C. Doole, A.I. Kirkland, J.L. Hutchison, J.M. Titchmarsh, D.J.H. Cockayne: 200 kV TEM with Cs correctors for illumination and imaging. In: Proc. 8th Asia–Pac. Conf. Electron Microsc., Kanazawa, ed. by N. Tanaka, Y. Takano, H. Mori, H. Seguchi, S. Iseki, H. Shimada, E. Simamura (8APEM Publication Committee, Uchinada 2004) pp. 20–21Google Scholar
  218. H. Sawada, T. Sasaki, F. Hosokawa, S. Yuasa, M. Terao, M. Kawazoe, T. Nakamichi, T. Kaneyama, Y. Kondo, K. Kimoto, K. Suenaga: Correction of higher order geometrical aberration by triple 3-fold astigmatism field, J. Electron Microsc. 58, 341–347 (2009)CrossRefGoogle Scholar
  219. H. Sawada, Y. Tanishiro, N. Ohashi, T. Tomita, F. Hosokawa, T. Kaneyama, Y. Kondo, K. Takayanagi: 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 (2009)CrossRefGoogle Scholar
  220. H. Sawada, T. Sasaki, F. Hosokawa, S. Yuasa, M. Terao, M. Kawazoe, T. Nakamichi, T. Kaneyama, Y. Kondo, K. Kimoto, K. Suenaga: Higher-order aberration corrector for an image-forming system in a transmission electron microscope, Ultramicroscopy 110, 958–961 (2010)CrossRefGoogle Scholar
  221. M. Haider, M. Müller, P. Hartel: Present state and future trends of aberration correction. In: Proc. 8th Asia–Pac. Conf. Electron Microsc., Kanazawa, ed. by N. Tanaka, Y. Takano, H. Mori, H. Seguchi, S. Iseki, H. Shimada, E. Simamura (8APEM Publication Committee, Uchinada 2004) pp. 16–17Google Scholar
  222. M. Haider, P. Hartel, H. Müller, S. Uhlemann, J. Zach: Current and future aberration correctors for the improvement of resolution in electron microscopy, Philos. Trans. Royal Soc. A 367, 3665–3682 (2009)CrossRefGoogle Scholar
  223. P. Hartel, H. Müller, S. Uhlemann, M. Haider: Residual aberrations of hexapole-type Cs–correctors. In: Proc. 13th Eur. Microsc. Congr. Antwerp, Vol. I, ed. by D. van Dyck, P. van Oostveldt (Belgian Society for Microscopy, Liège 2004) pp. 41–42Google Scholar
  224. J.M. Titchmarsh, D.J.H. Cockayne, R.C. Doole, C.J.D. Hetherington, J.L. Hutchison, A.I. Kirkland, H. Sawada: A versatile double aberration-corrected, energy-filtered HREM/STEM for materials science. In: Proc. 13th Eur. Microsc. Congr., Antwerp, Vol. I, ed. by D. van Dyck, P. van Oostveldt (Belgian Society for Microscopy, Liège 2004) pp. 27–28Google Scholar
  225. J.L. Hutchison, J.M. Titchmarsh, D.J.H. Cockayne, R.C. Doole, C.J.D. Hetherington, A.I. Kirkland, H. Sawada: A versatile double aberration-corrected energy filtered HREM/STEM for materials science, Ultramicroscopy 103, 7–15 (2005)CrossRefGoogle Scholar
  226. H. Sawada: Ronchigram and geometrical aberrations in STEM. In: Scanning Transmission Electron Microscopy of Nanomaterials, ed. by N. Tanaka (Imperial College, London 2015) pp. 461–485Google Scholar
  227. S. Morishita, Y. Kohno, F. Hosokawa, K. Suenaga, H. Sawada: Evaluation of residual aberrations in higher-order geometrical aberration correctors, Microscopy 67, 156–163 (2017)CrossRefGoogle Scholar
  228. T. Sasaki, S. Morishita, Y. Kohno, M. Mukai, K. Kimotao, K. Suenaga: Performance of low-kV aberration-corrected STEM with delta-corrector and CFEG in ultrahigh vacuum environment, Microsc. Microanal. 23(Suppl. 1), 468–469 (2017)CrossRefGoogle Scholar
  229. G. Benner, E. Essers, B. Huber, A. Orchowski: Design and first results of SESAM, Microsc. Microanal. 9(Suppl. 3), 66–67 (2003)CrossRefGoogle Scholar
  230. R. Nishi, H. Ito, S. Hoque: Wire corrector for aberration corrected electron optics. In: Proc. 18th Int. Microsc. Conf., Prague (2014), IT-1-P2984Google Scholar
  231. S. Hoque, H. Ito, R. Nishi, A. Takaoka, E. Munro: Spherical aberration correction with threefold symmetric line currents, Ultramicroscopy 161, 74–82 (2016)CrossRefGoogle Scholar
  232. R. Nishi, S. Hoque, H. Ito, A. Takaoka: Higher order aberration analysis and optimization of N-SYLC spherical aberration corrector by differential algebra method, Kenbikyo 52(Suppl. 1), 23 (2017)Google Scholar
  233. S. Hoque, H. Ito, A. Takaoka, R. Nishi: Axial geometrical aberration correction up to 5th order with N-SYLC, Ultramicroscopy 182, 68–80 (2017)CrossRefGoogle Scholar
  234. R. Janzen: Concept for electrostatic correctors for reduction of aberrations within miniaturized columns. In: Proc. MC-2011, Kiel (2011) IM1.p104Google Scholar
  235. R. Janzen, S. Burkhardt, P. Fehlner, T. Späth, M. Haider: The SPANOCH method: A key to establish aberration correction in miniaturized (multi)column systems? In: Proc. Microsc. Conf., Regensburg, Vol. 1, ed. by R. Rachel, J. Schröder, R. Witzgall, J. Zweck (2013) pp. 107–108Google Scholar
  236. C. Weißbäcker, H. Rose: Electrostatic correction of the chromatic and spherical aberration of charged particle lenses. In: Proc. 12th Eur. Congr. Electron Microsc., Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp.  I.157–I.158Google Scholar
  237. C. Weißbäcker, H. Rose: Electrostatic correction of the chromatic and the spherical aberration of charged-particle lenses I, J. Electron Microsc. 50, 383–390 (2001)CrossRefGoogle Scholar
  238. C. Weißbäcker, H. Rose: Electrostatic correction of the chromatic and the spherical aberration of charged-particle lenses, II, J. Electron Microsc. 51, 45–51 (2002)CrossRefGoogle Scholar
  239. D.J. Maas, A. Henstra, M.P.C.M. Krijn, S.A.M. Mentink: Electrostatic correction in LV-SEM, Microsc. Microanal. 6(Suppl. 2), 746–747 (2000)Google Scholar
  240. D.J. Maas, S. Henstra, M. Krijn, S. Mentink: Electrostatic aberration correction in low-voltage SEM, Proceedings SPIE 4510, 205–217 (2001)CrossRefGoogle Scholar
  241. D. Maas, S. Mentink, A. Henstra: Electrostatic aberration correction in low-voltage SEM, Microsc. Microanal. 9(Suppl. 3), 24–25 (2003)CrossRefGoogle Scholar
  242. A. Henstra, M.P.C.M. Krijn: An electrostatic achromat. In: Proc. 12th Eur. Congr. Electron Microsc., Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp.  I.155–I.156Google Scholar
  243. K. Bajo, S. Itose, M. Matsuya, M. Ishihara, K. Uchino, M. Kudo, I. Sakaguchi, H. Yurimoto: High spatial resolution imaging of helium isotope by TOF-SNMS, Surf. Interface Anal. 48, 1190–1193 (2016)CrossRefGoogle Scholar
  244. M. Linck, P. Hartel, S. Uhlemann, F. Kahl, H. Müller, J. Zach, M. Haider, M. Niestadt, M. Bischoff, J. Biskupek, Z. Lee, T. Lehnert, F. Börr-nert, H. Rose, U. Kaiser: Chromatic aberration correction for atomic resolution TEM Imaging from 20 to 80 kV, Phys. Rev. Lett. 117, 076101 (2016)Google Scholar
  245. M. Linck, P. Hartel, S. Uhlemann, F. Kahl, H. Müller, J. Zach, J. Biskupek, M. Niestadt, U. Kaiser, M. Haider: Status of the SALVE-microscope: Cc-correction for atomic-resolution TEM imaging at 20 kV. In: Proc. 16th Eur. Microsc. Congr., Lyon, Vol. 1, ed. by D.J. Stokes, W.M. Rainforth (2016) pp. 314–315Google Scholar
  246. H. Rose, A. Nejati, H. Müller: Cc/Cs-corrector compensating for the chromatic aberration and the spherical aberration of electron lenses, Ultramicroscopy 203, 139–144 (2019)Google Scholar
  247. S.A.M. Mentink, T. Steffen, P.C. Tiemeijer, M.P.C.M. Krijn: Simplified aberration corrector for low-voltage SEM. In: Proc. EMAG 1999, ed. by C.J. Kiely (Institute of Physics Publishing, Bristol 1999) pp. 83–86Google Scholar
  248. T. Steffen, P.C. Tiemeijer, M.P.C.M. Krijn, S.A.M. Mentink: Correction of chromatic and spherical aberration using a Wien filter. In: Proc. 12th Eur. Congr. Electron Microsc, Vol. III, ed. by P. Tománek, R. Kolařík (Czechoslovak Society for Electron Microscopy, Brno 2000) pp. I.151–I.152Google Scholar
  249. G. Hottenroth: Über Elektronenspiegel, Z. Phys. 103, 460–462 (1936)CrossRefGoogle Scholar
  250. G. Hottenroth: Untersuchungen über Elektronenspiegel, Ann. Phys. (Leipzig) 30, 689–712 (1937)CrossRefGoogle Scholar
  251. A. Recknagel: Zur Theorie des Elektronenspiegels, Z. Phys. 104, 381–394 (1937)CrossRefGoogle Scholar
  252. V.K. Zworykin, G.A. Morton, E.G. Ramberg, J. Hillier, A.W. Vance: Electron Optics and the Electron Microscope (Wiley, New York 1945)Google Scholar
  253. E.G. Ramberg: Aberration correction with an electron mirror, J. Appl. Phys. 20, 183–186 (1949)CrossRefGoogle Scholar
  254. E. Kasper: Die Korrektur des Öffnungs- und Farbfehlers im Elektronenmikroskop durch Verwendung eines Elektronenspiegels mit überlagertem Magnetfeld, Optik 28, 54–64 (1968)Google Scholar
  255. D. Preikszas, H. Rose: Correction properties of electron mirrors, J. Electron Microsc. 46, 1–9 (1997)CrossRefGoogle Scholar
  256. H. Rose, P. Hartel, D. Preikszas: Outline of the mirror corrector for SMART and PEEM3, Microsc. Microanal. 10(Suppl. 3), 28–29 (2004)CrossRefGoogle Scholar
  257. Z. Shao, X.D. Wu: A study on hyperbolic mirrors as correctors, Optik 84, 51–54 (1990)Google Scholar
  258. G.F. Rempfer, D.M. Desloge, W.P. Skoczylas, O.H. Griffith: Simultaneous correction of spherical and chromatic aberrations with an electron mirror, Microsc. Microanal. 3, 14–27 (1997)CrossRefGoogle Scholar
  259. Z. Shao, X.D. Wu: Adjustable four-electrode electron mirror as an aberration corrector, Appl. Phys. Lett. 55, 2696–2697 (1989)CrossRefGoogle Scholar
  260. Z. Shao, X.D. Wu: Properties of a four-electrode adjustable electron mirror as an aberration corrector, Rev. Sci. Instrum. 61, 1230–1235 (1990)CrossRefGoogle Scholar
  261. T. Schmidt, H. Marchetto, P.L. Lévesque, U. Groh, F. Maier, D. Preikszas, P. Hartel, R. Spehr, G. Lilienkamp, W. Engel, R. Fink, E. Bauer, H. Rose, E. Umbach, H.-J. Freund: Double aberration correction in a low-energy electron microscope, Ultramicroscopy 110, 1358–1361 (2010)CrossRefGoogle Scholar
  262. M. Mankos, K. Shadman: A monochromatic, aberration-corrected, dual-beam low energy electron microscope, Ultramicroscopy 130, 13–28 (2013)CrossRefGoogle Scholar
  263. H. Rose: Outline of an ultracorrector compensating for all primary chromatic and geometrical aberrations of charged-particle lenses, Microsc. Microanal. 9(Suppl. 3), 32–33 (2003)CrossRefGoogle Scholar
  264. H. Rose: Outline of an ultracorrector compensating for all primary chromatic and geometrical aberrations of charged-particle lenses, Nucl. Instrum. Methods Phys. Res. A 519, 12–27 (2004)CrossRefGoogle Scholar
  265. H. Rose: Prospects for aberration-free microscopy, Ultramicroscopy 103, 1–6 (2005)CrossRefGoogle Scholar
  266. C. Kisielowski, B. Freitag, M. Bischoff, H. van Lin, S. Lazar, G. Krippels, P. Tiemeijer, M. van der Stam, S. von Harrach, M. Stekelenburg, M. Haider, S. Uhlemann, H. Müller, P. Hartel, B. Kabius, D. Miller, I. Petrov, E.A. Olson, T. Donchev, E.A. Kenik, A.R. Lupini, J. Bentley, S.J. Pennycook, I.M. Anderson, A.M. Minor, A.K. Schmid, T. Duden, V. Radmilovic, Q.M. Ramasse, M. Watanabe, R. Erni, E.A. Stach, P. Denes, U. Dahmen: Detection of single atoms and buried defects in three dimensions by aberration-corrected electron microscope with 0.5 Å information limit, Microsc. Microanal. 14, 469–477 (2008)CrossRefGoogle Scholar
  267. U. Dahmen, R. Erni, V. Radmilovic, C. Kisielowski, M.-D. Rossell, P. Denes: Background, status and future of the transmission electron aberration-corrected microscope project, Philos. Trans. Royal Soc. A 367, 3795–3808 (2009)CrossRefGoogle Scholar
  268. U. Kaiser, A. Chuvilin, J. Meyer, J. Biskupek: Microscopy at the bottom. In: Proc. Microsc. Conf. MC-2009, Vol. 3, ed. by W. Grogger, F. Hofer, P. Pölt (Verlag der Technischen Universität, Graz 2009) pp. 1–6Google Scholar
  269. U. Kaiser, J. Biskupek, J.C. Meyer, J. Leschner, L. Lechner, H. Rose, M. Stöger-Pollach, A.N. Khlobystov, P. Hartel, H. Müller, M. Haider, S. Eyhusen, G. Benner: Transmission electron microscopy at 20 kV for imaging and spectroscopy, Ultramicroscopy 111, 1239–1246 (2011)CrossRefGoogle Scholar
  270. H. Rose, U. Kaiser: Prospects and first results of sub-angstroem low-voltage electron microscopy–the SALVE project, Recent Trends Charged Part. Opt. Surf. Phys. Instrum. 13, 65–66 (2012)Google Scholar
  271. H. Müller, M. Linck, P. Hartel, F. Kahl, J. Zach, S. Uhlemann, J. Biskupek, F. Börrnert, Z. Lee, M. Mohn, U. Kaiser, M. Haider: Correction of the chromatic and spherical aberration in low-voltage transmission electron microcopy, Recent Trends Charged Part. Opt. Surf. Phys. Instrum. 15, 38–39 (2016)Google Scholar
  272. U. Kaiser: Adv. Imaging Electron Phys (2019) in preparationGoogle Scholar
  273. Z. Shao: On the fifth order aberration in a sextupole corrected probe forming system, Rev. Sci. Instrum. 59, 2429–2437 (1988)CrossRefGoogle Scholar
  274. H. Müller, S. Uhlemann, P. Hartel, M. Haider: Advancing the hexapole Cs—Corrector for the scanning transmission electron microscope, Microsc. Microanal. 12, 442–455 (2006)CrossRefGoogle Scholar
  275. H. Rose, W. Pejas: Optimization of imaging magnetic filters free of second-order aberrations, Optik 54, 235–250 (1979)Google Scholar
  276. H. Shuman: Correction of the second-order aberrations of uniform field magnetic sectors, Ultramicroscopy 5, 45–53 (1980)CrossRefGoogle Scholar
  277. O.L. Krivanek, P.R. Swann: An advanced electron energy loss spectrometer. In: Quantitative Microanalysis with High Spatial Resolution, ed. by G.W. Lorimer, M.H. Jacobs, P. Doig (Metals Society, London 1981) pp. 136–140Google Scholar
  278. H. Rose, D. Krahl: Electron optics of imaging energy filters. In: Energy-Filtering Transmission Electron Microscopy, ed. by L. Reimer (Springer, Berlin 1995) pp. 43–149CrossRefGoogle Scholar
  279. R.F. Egerton: Electron Energy-Loss Spectroscopy in the Electron Microscope (Springer, Berlin-Heidelberg 2011)CrossRefGoogle Scholar
  280. M. Haider: A corrected double-deflection electron spectrometer equipped with a parallel recording system, Ultramicroscopy 28, 190–200 (1989)CrossRefGoogle Scholar
  281. O.L. Krivanek, C.C. Ahn, R.B. Keeney: Parallel detection electron spectrometer using quadrupole lenses, Ultramicroscopy 22, 103–115 (1987)CrossRefGoogle Scholar
  282. O.L. Krivanek, A.J. Gubbens, N. Dellby: Developments in EELS instrumentation for spectroscopy and imaging, Microsc. Microanal. Microstruct. 2, 315–332 (1991)CrossRefGoogle Scholar
  283. O.L. Krivanek, A.J. Gubbens, N. Dellby, C.E. Meyer: Design and first applications of a post-column imaging filter, Microsc. Microanal. Microstruct. 3, 187–199 (1992)CrossRefGoogle Scholar
  284. A.J. Gubbens, H.A. Brink, M.K. Kundmann, S.L. Friedman, O.L. Krivanek: A post-column imaging energy filter with a 20482 pixel slow-scan CCD camera, Micron 29, 81–87 (1988)CrossRefGoogle Scholar
  285. H.A. Brink, M.M.G. Barfels, R.P. Burgner, B.N. Edwards: A sub-50 meV spectrometer and energy filter for use in combination with 200 kV monochromated (S)TEMs, Ultramicroscopy 96, 367–384 (2003)CrossRefGoogle Scholar
  286. A.J. Gubbens, M. Barfels, C. Trevor, R. Twesten, P. Mooney, P. Thomas, N. Menon, B. Kraus, C. Mao, B. McGinn: The GIF Quantum, a next generation post-column imaging energy filter, Ultramicroscopy 110, 962–970 (2010)CrossRefGoogle Scholar
  287. N.E. Webster, M. Haider, H. Houf: Design and construction of a multipole element control unit, Rev. Sci. Instrum. 59, 999–1001 (1988)CrossRefGoogle Scholar
  288. O.L. Krivanek, T.C. Lovejoy, N. Dellby, R.W. Carpenter: Monochromated STEM with a 30 meV-wide, atom-sized electron probe, Microscopy 62, 3–21 (2013)CrossRefGoogle Scholar
  289. T.C. Lovejoy, G.C. Corbin, N. Dellby, M.V. Hoffman, O.L. Krivanek: Advances in ultra-high energy resolution STEM–EELS, Microsc. Microanal. 24(Suppl. 1), 446–447 (2018)CrossRefGoogle Scholar
  290. O.L. Krivanek, G.C. Corbin, N. Dellby, M. Hoffman, T.C. Lovejoy: Monochromator and spectrometer design for ultra-high energy resolution EELS. In: Proc. 19th Int. Microsc. Congr., Sydney (2018) pp. 1530–1531Google Scholar
  291. O.L. Krivanek, N. Dellby, J.A. Hachtel, J.-C. Idrobo, M.T. Hotz, B. Plotkin-Swing, N.J. Bacon, A.L. Bleloch, G.J. Corbin, M.V. Hoffman, C.E. Meyer, T.C. Lovejoy: Progress in ultrahigh energy resolution EELS, Ultramicroscopy 203, 60–67 (2019)Google Scholar
  292. R. Castaing, L. Henry: Filtrage magnétique des vitesses en microscopie électronique, C. R. Acad. Sci. Paris B 255, 76–86 (1962)Google Scholar
  293. A.J.F. Metherell: Energy analyzing and energy selecting electron microscopes, Adv. Opt. Electron Microsc. 4, 263–360 (1971)Google Scholar
  294. A.J.F. Metherell: Energy analyzing and energy selecting electron microscopes, Adv. Imaging Electron Phys. 204, 147–230 (2017)CrossRefGoogle Scholar
  295. S. Senoussi: Etude d'un Dispositif de Filtrage des Vitesses Purement Magnétique Adaptable à un Microscope Électronique à Très Haute Tension Thèse de 3e Cycle (Univ. Paris, Orsay 1971)Google Scholar
  296. S. Senoussi, L. Henry, R. Castaing: Etude d’un analyseur filtre de vitesses purement magnétique adaptable aux microscopes électroniques très haute tension, J. Microsc. (Paris) 11, 19 (1971)Google Scholar
  297. H. Rose, E. Plies: Entwurf eines fehlerarmen magnetischen Energie-Analysators, Optik 40, 336–341 (1974)Google Scholar
  298. G. Zanchi, J.-P. Perez, J. Sevely: Adaptation of a magnetic filtering device on a one megavolt electron microscope, Optik 43, 495–501 (1975)Google Scholar
  299. H. Rose: Aberration correction of homogeneous magnetic deflection systems, Optik 51, 15–38 (1978)Google Scholar
  300. S. Lanio, H. Rose, D. Krahl: Test and improved design of a corrected imaging magnetic energy filter, Optik 73, 56–68 (1986)Google Scholar
  301. S. Lanio: High-resolution imaging magnetic filters with simple structure, Optik 73, 99–107 (1986)Google Scholar
  302. S. Kujawa, D. Krahl: Comparison between A-type and B-type imaging Ω-filters. In: Proc. 14th Int. Congr. Electron Microsc., Cancún, Vol. 1, ed. by H.A. Calderón Benavides, M.J. Yacamán (Institute of Physics Publishing, Bristol 1998) pp. 241–242Google Scholar
  303. S. Uhlemann, H. Rose: The MANDOLINE filter—A new high-performance imaging filter for sub-eV EFTEM, Optik 96, 163–178 (1994)Google Scholar
  304. E. Essers, G. Benner, T. Mandler, S. Meyer, D. Mittmann, M. Schnell, R. Höschen: Energy resolution of an omega-type monochromator and imaging properties of the mandoline filter, Ultramicroscopy 110, 971–980 (2010)CrossRefGoogle Scholar
  305. P.C. Tiemeijer, J.H.A. van Lin, B.H. Freitag, A.F. de Jong: Monochromized 200 kV (S)TEM, Microsc. Microanal. 8(Suppl. 2), 70–71 (2002)CrossRefGoogle Scholar
  306. F. Kahl: Design eines Monochromators für Elektronenquellen, Ph.D. Thesis (Technical University of Darmstadt, Darmstadt 1999)Google Scholar
  307. O.L. Krivanek, J.P. Ursin, N.J. Bacon, G.J. Corbin, N. Dellby, P. Hrncirik, M.F. Murfitt, C.S. Own, Z.S. Szilagyi: High-energy-resolution monochromator for aberration-corrected scanning transmission electron microscopy/electron energy-loss spectroscopy, Philos. Trans. Royal Soc. A 367, 3683–3697 (2009)CrossRefGoogle Scholar
  308. M. Mukai, E. Okunishi, M. Ashino, K. Omoto, T. Fukuda, A. Ikeda, K. Somehara, T. Kaneyama, T. Saitoh, T. Hirayama, T. Ikuhara: Development of a monochromator for aberration-corrected scanning transmission electron microscopy, Microscopy 64, 151–158 (2015)CrossRefGoogle Scholar
  309. E. Plies: Proposal for an electrostatic energy filter and a monochromator. In: Proc. 9th Int. Congr. Electron Microsc., Toronto, Vol. 1, ed. by J.M. Sturgess (Microscopical Society of Canada, Toronto 1978) pp. 50–51Google Scholar
  310. A. Huber, J. Bärtle, E. Plies: Initial experiences with an electrostatic Ω-monochromator for electrons, Nucl. Instrum. Methods Phys. Res. A 519, 320–324 (2004)CrossRefGoogle Scholar
  311. P.C. Tiemeijer: Operation modes of a TEM monochromator. In: Proc. EMAG 1999, ed. by C.J. Kiely (Institute of Physics Publishing, Bristol 1999) pp. 191–194Google Scholar
  312. P.C. Tiemeijer: Measurement of Coulomb interactions in an electron beam monochromator, Ultramicroscopy 78, 53–62 (1999)CrossRefGoogle Scholar
  313. P.C. Tiemeijer, M. Bischoff, B. Freitag, C. Kisielowski: Using a monochromator to improve the resolution in TEM to below 0.5 Å. Part I: Creating highly coherent monochromated illumination, Ultramicroscopy 114, 72–81 (2012)CrossRefGoogle Scholar
  314. P.C. Tiemeijer, M. Bischoff, B. Freitag, C. Kisielowski: Using a monochromator to improve the resolution in TEM to below 0.5 Å. Part II: Application to focal series reconstruction, Ultramicroscopy 118, 35–43 (2012)CrossRefGoogle Scholar
  315. B. Freitag, P.C. Tiemeijer: Sub 30 meV energy resolution in the monochromated Themis transmission electron microscope. In: Proc. MC-2017, Lausanne (2017) p. 467Google Scholar
  316. H.W. Mook, P. Kruit: On the monochromatisation of high brightness sources for electron microscopy, Ultramicroscopy 78, 43–51 (1999)CrossRefGoogle Scholar
  317. M. Mukai, J.S. Kim, K. Omoto, H. Sawada, A. Kimura, A. Ikeda, J. Zhou, T. Kaneyama, N.P. Young, J.H. Warner, P.D. Nellist, A.I. Kirkland: The development of a 200 kV monochromated field emission electron source, Ultramicroscopy 140, 37–43 (2018)CrossRefGoogle Scholar
  318. M. Mukai, K. Omoto, T. Sasaki, Y. Kohno, S. Morishita, A. Kimura, A. Ikeda, K. Somehara, H. Sawada, K. Kimoto, K. Suenaga: Design of a monochromator for aberration-corrected low-voltage (S)TEM. In: Proc. 18th Int. Microsc. Conf., Prague (2014), IT-1-P-2578Google Scholar
  319. K. Kimoto: Practical aspects of monochromators developed for transmission electron microscopy, Microscopy 63, 337–344 (2014)CrossRefGoogle Scholar
  320. M. Mankos, K. Shadman, V. Kolařík: Novel electron monochromator for high resolution imaging and spectrometry, J. Vac. Sci. Technol. B 34, 06KP01 (2016)CrossRefGoogle Scholar
  321. H. Boersch, J. Geiger, W. Stickel: Das Auflösungsvermögen des elektrostatisch-magnetischen Energieanalysators für schnelle Elektronen, Z. Phys. 180, 415–424 (1964)CrossRefGoogle Scholar
  322. J. Geiger: Inelastic electron scattering with energy losses in the meV-region, Proc. Annu. Meet. EMSA 39, 182–185 (1981)Google Scholar
  323. M. Terauchi, M. Tanaka, K. Tsuno, M. Ishida: Development of a high-energy resolution electron energy-loss spectroscopy microscope, J. Microsc. (Oxford) 194, 203–209 (1999)CrossRefGoogle Scholar
  324. S. Uhlemann, H. Müller, P. Hartel, J. Zach, M. Haider: Thermal magnetic field noise limits resolution in transmission electron microscopy, Phys. Rev. Lett. 111, 046101 (2013)CrossRefGoogle Scholar
  325. J. Johnson: Thermal agitation of electricity in conductors, Phys. Rev. 32, 97–109 (1928)CrossRefGoogle Scholar
  326. H. Nyquist: Thermal agitation of electric charge in conductors, Phys. Rev. 32, 110–113 (1928)CrossRefGoogle Scholar
  327. O.L. Krivanek, N. Dellby, R.J. Keyse, M.F. Murfitt, C.S. Own, Z.S. Szilagyi: Advances in aberration-corrected scanning transmission electron microscopy and electron energy-loss spectroscopy, Adv. Imaging Electron Phys. 153, 121–160 (2008)CrossRefGoogle Scholar
  328. H. Sawada, T. Sasaki, F. Hosokawa, K. Suenaga: Atomic-resolution STEM imaging of graphene at low voltage of 30 kV with resolution enhancement by using large convergence angle, Phys. Rev. Lett. 114, 166102 (2015)CrossRefGoogle Scholar
  329. J.M. Cowley: Image contrast in a scanning transmission electron microscope, Appl. Phys. Lett. 15, 58–59 (1969)CrossRefGoogle Scholar
  330. E. Zeitler, M.G.R. Thomson: Scanning transmission electron microscopy, I, Optik 31, 258–280 (1970)Google Scholar
  331. E. Zeitler, M.G.R. Thomson: Scanning transmission electron microscopy, II, Optik 31, 359–366 (1970)Google Scholar
  332. F.F. Krause, A. Rosenauer: Reciprocity relations in transmission electron microscopy: A rigorous derivation, Micron 92, 1–5 (2017)CrossRefGoogle Scholar
  333. H. Sawada, M. Watanabe, I. Chiyo: Ad hoc auto-tuning of aberrations using high-resolution STEM images by autocorrelation function, Microsc. Microanal. 18, 705–710 (2012)CrossRefGoogle Scholar
  334. S. Lazar, P. Tiemeijer, S. Henstra, T. Dennemans, J. Ringnalda, B. Freitag: High performance in low voltage HR-STEM applications enabled by fast automatic tuning of the combination of a monochromator and probe Cs-corrector, Microsc. Microanal. 22(Suppl. 3), 980–981 (2016)CrossRefGoogle Scholar
  335. R. Erni, M.D. Rossell, C. Kisielowski, U. Dahmen: Atomic-resolution imaging with a sub-50-pm electron probe, Phys. Rev. Lett. 102, 096101 (2009)CrossRefGoogle Scholar
  336. S. Morishita, R. Ishikawa, Y. Kohno, H. Sawada, N. Shibata, Y. Ikuhara: Attainment of 40.5 pm spatial resolution using 300 kV scanning transmission electron microscope equipped with fifth-order aberration corrector, Microscopy 67, 46–50 (2018)CrossRefGoogle Scholar
  337. O.L. Krivanek, M.F. Chisholm, V. Nicolosi, T.J. Pennycook, G.J. Corbin, N. Dellby, M.F. Murfitt, C.S. Own, Z.S. Szilagyi, M.P. Oxley, S.T. Pantelides, S.J. Pennycook: Atom-by-atom structural and chemical analysis by annular dark-field electron microscopy, Nature 464, 571–574 (2010)CrossRefGoogle Scholar
  338. P.D. Nellist, S.J. Pennycook: Subångstrom resolution by underfocused incoherent transmission electron microscopy, Phys. Rev. Lett. 81, 4156–4159 (1998)CrossRefGoogle Scholar
  339. B. Freitag, S. Kujawa, P.M. Mul, P.C. Tiemeijer, E. Snoeck: First experimental proof of spatial resolution improvement in a monochromized and Cs-corrected TEM. In: Proc. 8th Asia–Pac. Conf. Electron Microsc., Kanazawa, ed. by N. Tanaka, Y. Takano, H. Mori, H. Seguchi, S. Iseki, H. Shimada, E. Simamura (8APEM Publication Committee, Uchinada 2004) pp. 18–19Google Scholar
  340. B. Freitag, S. Kujawa, P.M. Mul, P.C. Tiemeijer: First experimental proof of spatial resolution improvement in a monochromized and Cs-corrected TEM, Microsc. Microanal. 10(Suppl. 3), 4–5 (2004)CrossRefGoogle Scholar
  341. B. Freitag, S. Kujawa, P.M. Mul, J. Ringnalda, P.C. Tiemeijer: Breaking the spherical and chromatic aberration barrier in transmission electron microscopy, Ultramicroscopy 102, 209–214 (2005)CrossRefGoogle Scholar
  342. A.L. Bleloch, N.J. Bacon, N. Dellby, T.C. Lovejoy, C. Shi, O.L. Krivanek: Overcoming the STEM chromatic aberration resolution limit by monochromatization. In: Proc. IMC19, Sydney (2018) pp. 948–949Google Scholar
  343. O.L. Krivanek, N. Dellby, M.F. Murfitt, M.F. Chisholm, T.J. Pennycook, K. Suenaga, V. Nicolosi: Gentle STEM: ADF imaging and EELS at low primary energies, Ultramicroscopy 110, 935–945 (2010)CrossRefGoogle Scholar
  344. M. Sato: Resolution. In: Handbook of Charged Particle Optics, 2nd edn., ed. by J. Orloff (CRC, Boca Raton 2009) pp. 391–435Google Scholar
  345. P.D. Nellist: Scanning transmission electron microscopy. In: Science of Microscopy, Vol. 2007, ed. by P.W. Hawkes, J.C.H. Spence (Springer, Berlin-Heidelberg 2007) pp. 65–132CrossRefGoogle Scholar
  346. O.L. Krivanek, G.J. Corbin, N. Dellby, B.F. Elston, R.J. Keyse, M.F. Murfitt, C.S. Own, Z.S. Szilagyi, J.W. Woodruff: An electron microscope for the aberration-corrected era, Ultramicroscopy 108, 179–195 (2008)CrossRefGoogle Scholar
  347. O.L. Krivanek, T.C. Lovejoy, N. Dellby, T. Aoki, R.W. Carpenter, P. Rez, E. Soignard, J. Zhu, P.E. Batson, M. Lagos, R.F. Egerton, P.A. Crozier: Vibrational spectroscopy in the electron microscope, Nature 514, 209–212 (2014)CrossRefGoogle Scholar
  348. T. Miyata, M. Fukuyama, A. Hibara, E. Okunishi, M. Mukai, T. Mizoguchi: Measurement of vibrational spectrum of liquid using monochromated scanning transmission electron microscopy–electron energy loss spectroscopy, Microscopy 63, 377–382 (2014)CrossRefGoogle Scholar
  349. J.A. Hachtel, J. Huang, I. Popovs, S. Jansone-Popova, J.K. Keum, J. Jakowski, T.C. Lovejoy, N. Dellby, O.L. Krivanek, J.C. Idrobo: Identification of site-specific isotopic labels by vibrational spectroscopy in the electron microscope, Science 363, 525–528 (2019)Google Scholar
  350. P. Rez, T. Aoki, K. March, D. Gur, O.L. Krivanek, N. Dellby, T.C. Lovejoy, S.G. Wolf, H. Cohen: Damage-free vibrational spectroscopy of biological materials in the electron microscope, Nat. Commun. 7, 10945 (2016)CrossRefGoogle Scholar
  351. J.R. Jokisaari, J. Hachtel, X. Hu, A. Mukherjee, C. Wang, A. Konecna, J. Aizpurua, T.C. Lovejoy, N. Dellby, O.L. Krivanek, J.-C. Idrobo, R.F. Klie: Vibrational spectroscopy of liquid water at high spatial resolution, Adv. Mater. 30, 1802702 (2018)Google Scholar
  352. J.M. Rodenburg: Ptychography and related diffractive imaging methods, Adv. Imaging Electron Phys. 150, 87–184 (2008)CrossRefGoogle Scholar
  353. A.M. Maiden, J.M. Rodenburg: An improved ptychographical phase retrieval algorithm for diffractive imaging, Ultramicroscopy 109, 1256–1262 (2009)Google Scholar
  354. Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M.W. Tate, J. Park, S.M. Gruner, V. Elser, D.A. Muller: Electron ptychography of 2D materials to deep sub-ångström resolution, Nature 559, 343–349 (2018)Google Scholar
  355. C. Dwyer, T. Aoki, P. Rez, S.L.Y. Chang, T.C. Lovejoy, O.L. Krivanek: Electron-beam mapping of vibrational modes with nanometer spatial resolution, Phys. Rev. Lett. 117, 256101 (2016)CrossRefGoogle Scholar
  356. F.S. Hage, D.M. Kepaptsoglou, Q.M. Ramasse, L.J. Allen: Phonon spectroscopy at atomic resolution, Phys. Rev. Lett. 122, 016103 (2019)Google Scholar
  357. K. Venkatraman, B.D.A. Levin, K. March, P. Rez, P.A. Crozier: Vibrational spectroscopy at atomic resolution with electron impact scattering, https://arxiv.org/abs/1812.08895 (2018)
  358. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, J. Hadju: Potential for biomolecular imaging with femtosecond x-ray pulses, Nature 406, 752–757 (2000)CrossRefGoogle Scholar
  359. H.N. Chapman, P. Fromme, A. Barty, et al.: Femtosecond x-ray protein nanocrystallography, Nature 470, 73–77 (2011)CrossRefGoogle Scholar
  360. R.F. Egerton: Outrun radiation damage with electrons?, Adv. Struct. Chem. Imaging 1, 5–15 (2015)CrossRefGoogle Scholar
  361. R.F. Egerton, R.K. Li, Y. Zhu: Diffract-before-destroy with electrons? In: Proc. 18th Int. Microsc. Conf., Prague, ed. by P. Hozak (2014), IT-8-O-1901Google Scholar
  362. J.C.H. Spence: Outrunning damage: Electrons vs. x-rays—Timescales and mechanisms, Struct. Dyn. 4, 044027 (2017)CrossRefGoogle Scholar
  363. J.C.H. Spence, G. Subramanian, P. Musumeci: Hollow cone illumination for fast TEM, and outrunning damage with electrons, J. Phys. B 48, 214003 (2015)CrossRefGoogle Scholar
  364. J.-M. Zuo, J. Tao: Scanning electron nanodiffraction and diffraction imaging. In: Scanning Transmission Electron Microscopy: Imaging and Analysis, ed. by S.J. Pennycook, P.D. Nellist (Springer, Berlin-Heidelberg 2011) pp. 393–427CrossRefGoogle Scholar
  365. M.W. Tate, P. Purohit, D. Chamberlain, K.X. Nguyen, R. Hovden, C.S. Chang, P. Deb, E. Turgut, J.T. Heron, D.G. Schlom, D.C. Ralph, G.D. Fuchs, K.S. Shanks, H.T. Philipp, D.A. Muller, S.M. Gruner: High dynamic range pixel array detector for scanning transmission electron microscopy, Microsc. Microanal. 22, 237–249 (2016)CrossRefGoogle Scholar
  366. M. Krajnak, D. McGrouther, D. Maneuski, V. O'Shea, S. McVitie: Pixelated detectors and improved efficiency for magnetic imaging in STEM differential phase contrast, Ultramicroscopy 165, 42–50 (2016)CrossRefGoogle Scholar
  367. J.A. Mir, R. Clough, R. MacInnes, C. Gough, R. Plackett, I. Shipsey, H. Sawada, I. MacLaren, R. Ballabriga, D. Maneuski, V. O'Shea, D. McGrouther, A.I. Kirkland: Characterisation of the Medipix3 detector for 60 and 80 keV electrons, Ultramicroscopy 182, 44–53 (2017)CrossRefGoogle Scholar
  368. F. Ernst, M. Rühle (Eds.): High-resolution Imaging and Spectrometry of Materials (Springer, Berlin 2003)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CEMES-CNRSToulouseFrance
  2. 2.Nion CoKirkland, WAUSA

Personalised recommendations