Ultramicroscopy 151 (2015) 191–198

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Catadioptric aberration correction in cathode lens microscopy R.M. Tromp a,b,n a b

IBM T.J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA Kamerlingh Onnes Laboratory, Leiden Institute of Physics, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands

art ic l e i nf o

a b s t r a c t

Article history: Received 13 August 2014 Received in revised form 23 September 2014 Accepted 25 September 2014 Available online 16 October 2014

In this paper I briefly review the use of electrostatic electron mirrors to correct the aberrations of the cathode lens objective lens in low energy electron microscope (LEEM) and photo electron emission microscope (PEEM) instruments. These catadioptric systems, combining electrostatic lens elements with a reflecting mirror, offer a compact solution, allowing simultaneous and independent correction of both spherical and chromatic aberrations. A comparison with catadioptric systems in light optics informs our understanding of the working principles behind aberration correction with electron mirrors, and may point the way to further improvements in the latter. With additional developments in detector technology, 1 nm spatial resolution in LEEM appears to be within reach. & 2014 Elsevier B.V. All rights reserved.

Keywords: Low energy electron microscopy aberration correction electron mirror optics

1. Introduction—Aberrations in light optics The limiting effect of chromatic and spherical aberrations in microscopy was realized soon after the invention of the microscope. Robert Hooke [1] wrote in his famous Micrographia: “The Glasses I used were of our English make, but though very good of the kind, yet far short of what might be expected, could we once find a way of making glasses elliptical, or of some more true shape; for though both microscopes, and telescopes, as they now are, will magnifie an object about a thousand thousand times bigger then it appears to the naked eye; yet the apertures of the object-glasses are so very small, that very few rays are admitted, and even of those few there are so many false, that the object appears dark and indistinct: and indeed these inconveniences are such, as seem inseparable from spherical glasses, even when most exactly made; but the way we have hitherto made use of for that purpose is so imperfect, that there may be perhaps ten wrought before one be made tolerably good, and most of those ten perhaps every one differing in goodness one from another, which is an argument, that the way hitherto used is, at least, very uncertain. So that these glasses have a double defect; the one, that very few of them are exactly true wrought; the other, that even of those that are best among them, none will admit a sufficient number of rayes to magnifie the object beyond a determinate bigness” (Fig. 1). Even much earlier, in Leonardo da Vinci’s notebooks [2] (Florence, ca. 1508), we find sketches of the deleterious effects of

n Correspondence address: IBM T.J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA.

http://dx.doi.org/10.1016/j.ultramic.2014.09.011 0304-3991/& 2014 Elsevier B.V. All rights reserved.

spherical aberrations in bringing a parallel beam to a focus using concave reflecting mirrors (Fig. 2). Chromatic correction in light optics has a long history. A lens system that corrects chromatic aberration at just two wavelengths (achromat) is composed of at least two different lens elements, as already demonstrated by justice of the peace and amateur astronomer Chester Moore Hall around 1730, using crown glass for the positive and flint glass for the negative lens element. (Interestingly, Isaac Newton, believing that color dispersion is the same for all refracting materials, had pronounced this to be impossible, and designed his telescope with chromatic-aberration-free mirrors.) An apochromat corrects chromatic aberration at three wavelengths, and was invented by optical instrument maker Peter Dollond in 1763. A full two centuries later, Maximilian Herzberger [3] introduced the superachromat, which corrects chromatic aberration over the full optical range from 400 to 1000 nm using just three optical elements. In 1866, two centuries after Hooke’s Micrographia, Carl Zeiss sold his 1000th optical microscope. However, continuing efforts to improve resolution by trial and error did not progress significantly. In the same year (1866) Zeiss met 26 year old Ernst Karl Abbe, and made him the director of research at the Zeiss Werke in Jena. In 1882, 31 year old materials scientist avant-la-lettre Friedrich Otto Schott, who had already been in contact with Abbe for several years, also moved to Jena. Abbe developed the wave-optical theoretical framework that would enable full aberration correction in light optics [4], while Schott invented and created the novel borosilicate glasses that would make it a reality. Now it became possible to correct both chromatic and spherical aberrations by judiciously combining achromats (or apochromats) in groups. The first corrected microscope objective was offered by the Zeiss company in 1886, some

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Fig. 1. From the Preface, Robert Hooke, Micrographia (London, 1665).

Fig. 2. Leonardo da Vinci, Florence, ca 1508. British Library Arundel MS 263, f.86v87; Copyright of the British Library.

300 years after the invention of the compound microscope by Hans en Zacharias Janssen in Middelburg, The Netherlands. In the same year Zeiss sold his 10,000th microscope, testimony to the industrial powerhouse that he had founded. The key challenge in developing high quality lens systems for applications in microscopy, astronomy, and photography alike is to create an optical system that allows for a large aperture, i.e. low spherical aberration, with small chromatic aberration, across a large image field. With digital photography (and in the human eye) chromatic aberration can to some degree be corrected in post-processing. For spherical aberration this is much harder. In electron optics the problems are very similar, but even harder to solve.

2. Aberrations in electron optics In electron microscopy it was obvious from the beginning that aberrations limit resolution, just as in light microscopy. But nature offers even fewer handles to correct the problem. In cylindrically symmetric dioptric systems there are only positive focal length

lenses, and the aberration coefficients have fixed signs. (The only exception is the entrance aperture in the cathode objective lens, which is diverging.) And there is no choice of refractive indices as in light optical systems. Otto Scherzer discussed the effects of these ‘lens defects’ on resolution (Über einige Fehler von Elektronenlinsen, 1936) [5], and offered possible solutions [6] in 1947. The options are limited: 1—depart from cylindrical symmetry, i.e. use (combinations of) multipole optical elements to create aberrations coefficients that can correct the objective lens; 2—reverse the path of the electron beam, i.e. use an electron mirror; 3—use space charge on or near the optical axis; 4—use time varying fields in combination with a pulsed electron beam. All these options have been tried at one time or another [7]. In 1948, in one of the shortest papers ever to lead to a Nobel Prize in Physics (awarded in 1971), Dennis Gabor [8] introduced holography as an entirely different method to correct aberrations, in either light or electron optical systems. Indeed, holography has become an important tool, not just for aberration correction, but also for measurements of the spatial distribution of the phase of the wave field across the sample. The invention of holography was directly driven and motivated by the aberrations of the electron microscope. Gabor starts his 1948 paper as follows: “It is known that the spherical aberration of electron lenses sets a limit to the resolving power of electron microscopes at about 5 A. Suggestions for the correction of objectives have been made; but these are difficult in themselves, and the prospects of improvement are further aggravated by the fact that the resolution limit is proportional to the fourth root of the spherical aberration. Thus an improvement of the resolution by one decimal would require a correction of the objective to four decimals, a practically hopeless task.” He then goes on: “The new microscopic principle described below offers a way around this difficulty, as it allows one to dispense altogether with electron objectives.” Although it appeared a hopeless task in 1948, the required correction of the objective by 4 decimal places has happened, and resolution of 0.5 A has been realized. (Even if a task appears hopeless, we need not give up hope, although it may take several more generations of scientists and another 50 years for that hope to be fulfilled.) Most recently ptychography has emerged as one more alternative [9], reconstructing the image from diffraction information only, i.e. without the use of an image forming lens, using large redundancy and overlap in the data to solve the phase problem when inverting the diffraction data into a real-space image. Here, the spatial resolution is limited by the quality of, and the k-range included in the diffraction pattern. Ptychography, first demonstrated in light optical systems, is now also seeing application with X-rays and electrons. It is only in its infancy, and it certainly has a very bright future. But just as holography has not replaced microscopy, ptychography will likely find its own niche alongside the other techniques, complementing rather than killing its ancestors. Over the last twenty years, aberration correction has seen tremendous advances, in particular in transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM) where resolution was improved from
100 pm to about
50 pm. A very nice historical review of the long road to aberration correction in TEM was published by Peter Hawkes [7]. Here, the multipole aberration corrector designs by Harald Rose et al. [10] and by Ondrej Krivanek [11] deserve to be singled out, as they have become the basis for most of the commercial corrected microscopes today. Of course, Rose and Krivanek have also been pioneers in the development of electron energy filters which have greatly expanded the analytical capability of the electron microscope [12]. Spherical aberration correction became practical some 60 years after the invention of the electron microscope (a mere blink of the eyes in the field of optics), and many C3-corrected

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systems are in operation today. Correction of chromatic aberration still appears to be more difficult, although demonstrated in principle [13]. The stability of the corrected state becomes more difficult to maintain the higher the quality of correction [14], and even thermally excited electrical currents in the walls of the liner tubes in the microscope conspire to pose limits on the achievable resolution [15]. The breakthrough in aberration correction was enabled not only by better electron optical design, and superior mechanical fabrication techniques, but also by the routine availability of high performance computing systems at moderate costs. Without real-time, highly automated numerical evaluation of the state of the optical system, aberration correction would still be impossible today.

3. Mirror systems The use of catadioptric systems, i.e. systems that combine reflective optics (catoptrics) with refractive optics (dioptrics) goes back at least to the early 1800s, with the use of a silver-coated negative curvature lens patented (English patent No. 3718, Feb. 12, 1814) by William Francis Hamilton. His patent claims: “(…) I do make the curvatures of the surfaces of the meniscus a (i.e. the entrance lens to the telescope, see Fig. 3) such that the aberrations from sphericity, and from the unequal or chromatic refraction of light produced by the lens a, shall be corrected or nearly so, as to practise by the contrary refraction effected in passing to the remoter or silvered surface of b (i.e. the silver-coated negative curvature lens at the far side of the telescope), and back again towards the common axis from the last-mentioned lens or glass, at

Fig. 3. Hamilton telescope, from the 1814 patent. Lens a (center) admits light into the telescope and converges it. The back surface of the negative lens b (far right) is silvered, and reflects light back towards a, (partially) correcting spherical and chromatic aberration. Red and violet rays are drawn.

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or near which axis a correct image will be formed”. Unfortunately, the application is rather short on detail. Says Hamilton: “I have not considered it to be needful to explain the general principles of optical theory upon which the improvements herein set forth are founded and established, because the said general principles are well known to opticians, who will easily apply the same unto the said improvement.” Or maybe he preferred the keep the competition in the dark a little longer (so to speak). While not utilized in electron optics until recently (with the exception of the Castaing-Henry energy filter [16]), mirror systems have played an important role in astronomy. ‘Hamiltonian’ telescopes are still popular with amateur astronomers today. Alternatively, the aberrations of a concave primary mirror can be corrected with the aberrations of a convex mirror, as in the famed and temporarily ill-fated Hubble Space telescope. In microscopy the same can be accomplished with a Schwarzschild objective. Long focal length, compact and light catadioptric lens systems are also used in photography, where the characteristic ring-shaped bokeh can give rise to interesting and appealing effects. Catadioptric systems also find application in the lithography tools used for semiconductor manufacturing. In general, mirror surfaces can be generated by rotation of a two-dimensional conic section about its axis of symmetry. The conic section is characterized by its eccentricity, ε, and the curvature of the mirror at its apex, κ. When used as a focusing element, the properties of the mirror depend on this eccentricity. As shown in Fig. 2, the spherical aberration coefficient C3 of a spherical mirror (ε ¼ 0) is positive. Using dimensionless units where all distances are scaled to the radius of curvature of the mirror at its apex, the transverse 3rd order spherical aberration for a parallel beam of light impinging upon a concave mirror (i.e. the lateral displacement from the optical axis, dR, of the reflected ray in the paraxial focal plane) is given by dR ¼C3y3 ¼½(1 ε2)y3, where y is the distance from the incident ray to the optical axis [17]. For an ellipsoid (0 o ε o1) C3 is positive; when ε ¼1 (paraboloid) C3 ¼0. For ε 41 (hyperboloid) C3 turns negative. Thus, if we continuously change the eccentricity of the mirror from spherical to paraboloid to hyperboloid, the spherical aberration coefficient changes continuously from positive to zero to negative. Fig. 4 shows a comparison between three mirrors with equal curvature at the apex. To the left is a spherical mirror, in the center a parabolic mirror, and to the right a hyperbolic mirror with ε ¼√2. The change in spherical aberration is clearly apparent, even as the changes in the shapes of the mirrors appear to be small. Below the parabolic mirror (center) we overlay the shapes of the spherical and hyperbolic mirrors. Small changes in shapes can have big

Fig. 4. Ray diagrams for spherical, parabolic, and hyperbolic mirrors with identical paraxial focal lengths. The eccentricities, as well as the dimensionless spherical aberrations, C3 ¼ ½ (1  ε [2]), are indicated. Below the parabolic mirror we overlay the shapes of the spherical and hyperbolic mirrors for comparison. The hyperbolic mirror is flatter.

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consequences: due to a measurement error during polishing, the eccentricity of the Hubble primary mirror was too large by
0.6%, causing a spherical aberration that was a factor 6 larger than designed. It took three years and some seven hundred million dollars to fix the problem. It may be anticipated that a suitably designed hyperbolic mirror should be able to correct the positive 3rd order spherical aberration of the objective lens. Unlike mirrors in light optics, which don’t have chromatic aberrations since specular reflections do not depend on wavelength, electron optical mirrors do have chromatic aberrations as the reflection of the electron does not happen at a physical surface, but at an equipotential plane in front of a physical surface. Different energy electrons reflect from different equipotential planes with different values of κ and ε, and therefore different imaging properties. As the electron penetrates deeper, the equipotential planes are more strongly curved, i.e. the focal length is shorter, offsetting the negative chromatic aberration of the objective lens. This important distinction between light optical and electron optical mirrors makes it possible to correct chromatic aberrations with the latter. Already in 1936, Alfred Recknagel [18] recognized that the electron trajectories in a hyperbolic field (parabolic field strength along the axis) can be found analytically. The properties of hyperbolic lenses and mirrors were further investigated by Reinhold Rüdenberg [19] (the first person to be awarded patents for the invention of the electron microscope, although without any recorded effort towards its realization) in 1948. A detailed analysis of a hyperbolic dipole mirror as an aberration corrector in electron optics was carried out in 1990 by Gertrude Rempfer [20]. The optical properties can be analyzed analytically, and she was the first to show experimentally [21] that such a mirror is indeed capable of correcting chromatic and spherical aberrations. However, such a simple system, with only one adjustable potential, also has serious limitations for practical application in a cathode lens instrument.

4. Aberrations of the cathode lens The aberrations of the cathode lens as used in all LEEM/PEEM experiments [22] are composed of the contributions of the strong electrostatic immersion field, and of the electrostatic or magnetic imaging element following the immersion field. The uniform electrostatic field forms a virtual image behind the sample at unit magnification, at a spacing between the virtual image and the sample surface which is equal to the sample-anode spacing, L. The aberrations of the uniform field can be obtained analytically to any order [23]. (As there is no optical axis, there are no off-axis aberrations.) They depend strongly on the energy with which the electrons leave the sample surface, E0, and on the energy gained by acceleration from sample to anode, E. With the virtual image (at E0) transferred 1:1 to a real space image (at E þE0) by a perfect lens, the chromatic and spherical aberration coefficients in the real image plane are given by Cc ¼ –C3 ¼  L(E/E0)1/2, while the 3rd rank chromatic and 5th order spherical aberrations (limiting resolution when Cc ¼C3 ¼0) are given by Ccc ¼C5 ¼¼ L(E/E0)3/2. All aberrations depend strongly on the take-off energy E0, which is varied routinely and frequently in LEEM and PEEM experiments. They all diverge as E0 approaches zero. For PEEM experiments this is rather troubling, as most experiments are performed with (secondary) electrons with energies well below 10 eV, and the energy distribution can be quite broad. Two additional complications arise: the anode is not a perfectly flat metallic plate. Rather, it has a small circular opening to allow passage of the electrons into the rest of the optical system. This aperture acts as a diverging lens [24] with focal length f ¼  4L(E þE0)/E, shifting the virtual image from z¼  L to z ¼  1/3L at M¼ 2/3, adding its own aberrations in

the process. The electrostatic or magnetic lens transferring the virtual image from z ¼  1/3L into a real image at positive z, acting on electrons with energy E þE0, adds yet more aberrations. In practice, the aberrations of the aperture lens are much smaller than those of the uniform field, but the aberrations of the realimage-forming lens are not [23]. The electron mirror system must correct the combined aberrations of the cathode lens, i.e. the sum of the aberrations of the uniform field and of the real-imageforming lens (plus any additional transfer optics between objective lens and electron mirror), and these combined aberrations depend strongly on E0. Due to the addition of the aberrations of the realimage-forming lens, the spherical and chromatic aberrations in the real image have a different dependence on E0, unlike the aberrations of the uniform field only, so that they don’t track each other 1:1 as E0 is changed. In other words, a correction system must be able to vary Cc and C3 independently over a large parameter range. In a diode mirror, at fixed focal length (set by adjusting the potential applied to the mirror electrode) Cc and C3 are fixed [20]. Such a simple system, while it may be designed to correct either Cc or C3 at a given energy E0 and magnification M, fails to correct the cathode objective lens over a range of E0, nor can the mirror easily be adjusted to account for small sample-tosample variations in magnification. Nonetheless, the diode mirror was incorporated in a PEEM instrument, where the takeoff energy E0 is constant, and a spatial resolution of 5.4 nm was demonstrated experimentally [25]. In general, however, we will need to independently control 3 mirror parameters: focal length f, Cc and C3. We must therefore have at least 3 independent knobs to turn, i.e. have at least 3 independently adjustable potentials, and therefore at least 3 cylindrically symmetric electrodes, plus a ground electrode. It is not obvious a priori that such a four-element electron mirror, combining an electron mirror with refractive lens elements is sufficient to obtain independent control over these three critical parameters, nor is it obvious that it can provide control over a sufficiently large parameter range. Fortunately, it is possible to fulfill these essential requirements with the simple arrangement of Fig. 5.

5. Catadioptric aberration correction A detailed design of a Cc/C3 corrected cathode lens instrument [26] was first presented by Preikszas and Rose, forming the basis of the pioneering SMART instrument [27]. A further analysis of this mirror was given by Wan et al. for the PEEM-3 instrument [28], showing that Cc depends primarily on the mirror potential V1, while C3 depends primarily on V3  V2. Mirror focusing is accomplished by adjusting V2. (A similar design is used in the AC-LEEM

Fig. 5. Catadioptric aberration corrector for LEEM/PEEM.

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designed by Elmitec GmbH, but since no details are available, we cannot comment further on this instrument.) Experimentally, the focusing properties can be measured by placing a TEM grid in the object/image plane of the mirror. As the electrons pass through this plane twice, once on the path towards and then again on the path away from the mirror, we see the shadow of the TEM grid twice. The mirror is focused when both shadows are focused. Experimental maps of the first order focusing properties of the mirror (i.e. V2 as a function of V3  V2 and V1) are in excellent agreement with theoretical calculations, requiring only minor static offset adjustments (few parts in 1000) of the high voltage power supplies (Fig. 6). With this excellent agreement of the first order properties, we have also shown experimentally that f, Cc and C3 can be adjusted independently of each other [29,30]. Thus, Cc and C3 can be changed independently over a sufficiently wide parameter range to cover the E0-dependence of the cathode lens aberrations, at fixed mirror focal length. In fact, as E0 is changed,

Fig. 6. First order properties of the mirror system shown in Fig. 5, measured on three different microscopes. For the upper curve V1 ¼  1200 V (relative to the beam potential of  15000 V). For the lower curve it is  1800 V, with 100 V steps between curves. The focusing potential V2 is shown as a function of V3  V2. Black lines: theory.

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the electron mirror can be programmed to automatically track the aberrations of the objective lens. The actual mirror parameter range is sufficiently large [28,29] so that C3 can be changed from positive (ellipsoid), to zero (paraboloid) to negative (hyperboloid). Thus, we can think of this four electrode catadioptric element as a continuously deformable mirror, where the eccentricity is delicately controlled by the applied potentials. In the designs used by SMART, PEEM-3, and the IBM LEEM the metallic mirror surface itself is spherical. However, the electrons do not reflect from this physical surface, but from an equipotential plane well in front of the mirror electrode. The lens elements in front of the mirror electrode (V2 and V3, Fig. 5) retard the electrons, opening up the beam angles in front of the mirror. In combination with V1, these dioptric elements also serve to shape the field in front of the mirror, and thereby its eccentricity. The electrons are reflected by the mirror field, and then returned to the object/image plane at unit magnification, correcting for objective lens aberrations the process. Being able to set the aberrations of the mirror to zero (parabolic condition) is very useful, as it allows us to inspect and measure the properties of the uncorrected cathode lens (defocus, C3, astigmatism, tilt) by itself, as we will see below. In order to correct the objective lens, we must be able to measure its aberrations. In transmission microscopy, the use of Zemlin tableaus (TEM) or ronchigrams (STEM) is well established. Unfortunately, in LEEM/PEEM suitable samples that could utilize these methods are absent. Note that the diffraction rings observed in an amorphous carbon film occur at spacings that are far beyond the projected resolution of a Cc/C3 corrected LEEM. In addition, the LEEM instrument works in reflection, not transmission, and we only interact with the few outermost atomic layers of the sample. To measure Cc we can record [31] the excitation of the magnetic part of the cathode lens in Hg-light-excited PEEM as we change the sample potential E. As E0 is fixed, this measurement is only sensitive to the chromatic aberration of the magnetic part of the lens, while the dominant chromatic aberration of the uniform field is invisible. In a LEEM experiment, as we change sample potential E, E þE0 is held constant, so the chromatic aberration of the magnetic part of the cathode lens is invisible, and we measure only the Cc of the electrostatic field. The combination of these two measurements gives us the E0-dependent Cc of the full cathode lens. To measure C3, we have found it most convenient [32] to illuminate a diffracting sample with a small spot (o100 nm diameter) electron beam in LEEM mode. The image formed by each diffracted beam intersects the image plane at a distance from the origin which depends on defocus f, tilt τ, astigmatism C12, and spherical aberration C3 (Fig. 7a).

Fig. 7. (a) Diffracted beams reflected from a surface illuminated by a microspot electron beam intersect the image plane (right) at a distance from the optical axis that depends on defocus, spherical aberration, astigmatism, and tilt. (b) Real space microdiffraction pattern obtained on Si(1 1 1)(7  7) at 5.4 eV electron energy. Due to spherical aberration the distance from the diffracted beams to the center increases non-linearly.

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A single so-called Real-Space microspot Low Energy Electron Diffraction (RS-μLEED) pattern (Fig. 7b), obtained at (arbitrary) positive defocus Δf, contains sufficient information to quantitatively determine Δf, C12, C3, and τ, provided that there are enough diffracted beams in the pattern. The Si(1 1 1)-(7  7) LEED pattern at 5.4 eV contains 73 diffracted beams, more than sufficient for an accurate measurement. Since the angles of the diffracted beams are known with high accuracy (given by the electron energy E þE0, and the lattice constant of the sample), angular calibrations do not enter into the analysis, removing a significant source of error. The measurement is simple, and numerical analysis quick, so that C3 can be measured [30] as function of the mirror settings, and the point of correction found with good accuracy (Fig. 8). This is essential in obtaining a suitably corrected state that allows for optimum resolution. In LEEM, where the energy spread of the cold field emission gun can be kept below 300 meV, chromatic aberration is a minor factor in determining spatial resolution, and for 1 nm spatial resolution it must be corrected [14] to an accuracy of
10%, with resolution depending on C1/2 for a C3 corrected state. For a Cc c corrected state, resolution varies with C1/4 (as Gabor remarked to 3 his dismay), so correction of C3 must be significantly more accurate, i.e. a few percent. As E0 is changed (as it is routinely in LEEM experiments) the electron mirror must track the E0-dependent objective lens aberrations with similar accuracy. In the case of LEEM, resolution is determined by a tradeoff between diffraction limit and 5th order spherical aberration C5. The opposite is true in PEEM, where energy spread can range from a few to tens of eV. In that case, resolution is limited not by C5, but by chromatic aberration. There are two important factors. First, the electron mirror can only correct the second rank aberration Cc at one electron energy E0. However, remember that for the uniform field Cc ¼  L(E/E0)1/2, i.e. at low takeoff energies Cc diverges sharply. If we correct Cc at 5 eV, it will not be corrected at 1 eV, or at 10 eV. If an image is obtained with unfiltered secondary electrons over an energy range of 20 eV or so, it will be important to correct the

cathode lens at a value of E0 that also takes into account the electron intensity distribution as a function of energy. Unfortunately, this distribution always peaks at energies of just a few eV where Cc varies sharply. Second, higher rank chromatic aberrations quickly become dominant over 5th order spherical aberration, even at higher electron energies [33,34]. In most cases, the optimum aperture angle in PEEM will be significantly smaller than in LEEM [33]. Still, improvement in both resolution and transmission is very significant compared to uncorrected instruments. An important advantage of the mirror correction systems used in LEEM/PEEM as compared to the multipole optics used in (S)TEM is that the catadioptric system is extremely compact and simple. To correct both Cc and C3 only three potentials must be controlled with an accuracy of
10 ppm. Of course, we also need transfer optics from the cathode lens to the electron mirror, and the electron path retraces itself, necessitating the use of doublefocusing magnetic prism arrays to spatially separate the incoming and returning beams [35]. But even so, the number of optical elements, the number of associated power supplies, and the absolute stability required for each of these power supplies is more modest for the mirror system than for the multipole systems used in (S)TEM. Operation of electron mirrors for electron beams up to 20 keV electron energy has proven to be non-problematic. As TEM instruments are now under development [36] for lowdamage operation at energies as low as 30 keV, it becomes opportune to ask at what point catadioptric aberration correction becomes a viable alternative in low voltage TEM. At Leiden University we are presently developing a TEM system that takes this question to the extreme limit. At very low electron energies (o30 eV), the electron mean free path again increases strongly [37], as the electron energy loss processes operative at higher electron energies are no longer active. Thus, at 5 eV the electron mean free path is about the same as at 30 keV. Incorporating an electron source inside the sample holder of the LEEM/PEEM instrument enables us to perform TEM experiments at few eV energies (eV-TEM [38]), taking advantage of the Cc/C3 correction optics already included in the LEEM instrument. At these ultralow energies radiation damage is virtually eliminated. With a spatial resolution of 1 nm such a damage-free eV-TEM instrument will be of interest for the study of numerous biological specimens, including cell membranes, DNA, small proteins, etc.

6. Outlook

Fig. 8. Combined spherical aberration coefficient (cathode lens þ mirror) as function of the mirror setting. The cathode lens is corrected at a mirror setting of about  2680 m. The slope is equal to 1/M4, where M is the magnification of the image in the object plane of the mirror, i.e. M ¼9.08, close to the design value of 8.5. The magnification can be adjusted by adjusting L, the distance between sample and anode. E0 ¼ 5.4 eV.

The correction of Cc and C3 in cathode lens instruments by catadioptric elements has reached a certain level of maturity. It has been shown conclusively that these methods work [25,29,30,39,40]. In LEEM, spatial resolution has improved from an uncorrected 4 nm to a corrected
1.5 nm. In PEEM, spatial resolution of
5 nm has been shown, a factor 2–4 better than with uncorrected instruments. Fig. 9 shows the theoretically achievable resolution for both PEEM and LEEM [33]. For PEEM we distinguish UV-excited PEEM, where electrons have very low kinetic energy near 0 eV, with spreads of 1.5 and 3 eV, and X-ray excited PEEM where we use a broad band of secondary electrons with an energy spread of 20 eV, or an energy-filter-selected slice of 2 eV width. Black points are without correction, red points with. For PEEM we see a very significant improvement in resolution (to about 3 nm for the most favorable cases), and an increase in transmission (proportional to the square of the aperture angle) by up to a factor 10. For LEEM we find an improvement in resolution by more than a factor 3. Issues of alignment of the instrument (not a trivial task when the electron beam is deflected over a total angle of 7201 from gun to detector) have been largely solved [30]. The properties and

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Fig. 9. Theoretical resolution in PEEM and LEEM. Black points: uncorrected. Red points: Cc and C3 corrected. For PEEM we distinguish UV-PEEM (electron energy near 0 eV) and X-PEEM (secondary electrons with broad energy distribution). For LEEM we show results for an amplitude object (circles) and for strong- and weakphase objects (triangles and squares). In all cases, the aberration-corrected instrument gives strongly improved resolution and much larger apertures angles, i.e. increased transmission. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

performance of the 4 element electron mirror are well understood, and experimental characterization is in close agreement with theory [30]. A growing number of corrected instruments are installed in user laboratories, producing useful, even exciting experimental results. So where do we go from here? One open issue, which has not been addressed in any systematic fashion, concerns the shape of the mirror electrode. The most widely used shape is spherical (ε ¼0), while the Rempfer-designed mirror is hyperbolic (with fixed ε ¼√(3/2) ) as it allows for analytical treatment. The spherical tetrode system has a very large operating range, but it is not clear that it is also the optimum system. We find it very useful that the aberrations can be set to zero (ε ¼1), which is impossible with the hyperbolic diode. A theoretical study of the properties of mirrors over a range of eccentricities between spherical and paraboloid would be very desirable, as it may be possible to further minimize higher order/ rank aberrations and thus improve spatial resolution. Of course, additional electrodes can be added to obtain control over higher order aberrations, at least in principle. For example, adding another lens electrode at potential V4 may give control over C5, at the expense of increased experimental complexity (the 2D curves in Fig. 6 now become 3D surfaces). However, at the present time we are still well removed from the theoretically predicted optimum resolution of
0.5 nm. Therefore it seems that other problems (stability, accuracy) need to be resolved first, before it makes sense to increase complexity even more. Finally, we have not mentioned the role of the image detector. The standard channelplate/CCD detector used in most LEEM/PEEM systems today has a spatial resolution of
140 μm over a 40 mm diameter detector, corresponding to
300 resolution elements across the image [41]. These detectors have good sensitivity, adjustable gain, but also low dynamic range. They are subject to aging, particularly in those areas of the detector that are frequently exposed to intense diffraction patterns. Improved versions of this detector developed by SPECS [42] achieve a resolution of
70 μm, or 600 resolution elements. Several years ago we tested the suitability of the CERNdeveloped Medipix-2 detector for use in LEEM/PEEM experiments [43]. This CMOS-based detector showed superior signal/noise ratio and dynamic range. However, the small pixel count (256  256), the large pixel size (65 μm), and very limited availability of this detector has prevented its deployment beyond initial testing. Recently, direct electron CMOS detectors with much larger pixel counts (3k  4k, 4k  5k), and much smaller pixel size (6 μm) have become commercially available for use in TEM instruments. We have installed a prototype backthinned, back-illumination detector fabricated by

197

Fig. 10. Dark field image of Si(0 0 1)(2  1) at E0 ¼ 3.5 eV obtained with Direct Electron42 CMOS detector. 1600  1200 crop from full 3k  4k image. Field of view 8.6 mm. Upper left inset: LEED pattern at 10.8 eV. The broad ‘blob' to the right of the (0,0) beam is caused by secondary electrons dispersed away from the optical axis by the 901 magnetic prism array separating incoming and outgoing electron beams.

Direct Electron [44] in our aberration corrected LEEM system at IBM. The detector is retractable, and is positioned above the improved SPECS channelplate/CCD detector. The spacing between the detectors is such that they both record the same field of view. The DE detector is water-cooled, and the detector circuitry is placed inside the vacuum system, where we maintain a pressure in the detector chamber of
5  10  7 Torr, i.e. still adequate for channelplate operation. Differential pumping between the detector chamber and the rest of the system effectively isolates the detector from the sample and cold FE gun chambers. In initial testing we observe a spatial resolution of
4 pixels, resulting in
1000 resolution elements along the 4k direction of the detector. We can record images at
4  larger field of view with the DE detector than with the standard channelplate/ CCD detector while obtaining similar resolution in the image. Thus, the information content is more than 10 times greater in the DE image, at the same field of view. For 1 nm spatial resolution, we will be able to operate the DE detector at a field of view of about 1 μm, while the standard channelplate detector would tolerate a field of view of at most 0.28 μm. At present, it appears that the DE detector may have a detection efficiency that is significantly smaller than the channelplate detector. However, further thinning of the detector chip is likely to remedy this situation. A combination of improved channelplate/CCD detector (with 70 μm detector resolution across a 40 mm field [42]) and a 4k  5k DE CMOS detector (which would increase the field of view by another 67%) appears to be a nearly ideal detector setup for future aberration corrected LEEM/PEEM instruments. A 1600  1200 pixel dark field image of a Si(0 0 1)(2  1) surface at 3.5 eV (cropped from a full image size of 3k  4k) is shown in Fig. 10, revealing sample details that would not be observable at this field of view with any other detector at this time. The inset shows the low energy electron diffraction pattern at 10.8 eV.

7. Conclusion Over the last 10 years catadioptric aberration correction in cathode lens microscopes has advanced from a somewhat exotic and daring electron optics experiment to a sound and proven technology that is actively being deployed in commercial instruments in laboratories around the world. In LEEM, spatial resolution of 2 nm is now routinely achievable, an improvement of a factor

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2 over the best uncorrected results. A further factor 2 improvement is still possible, with new detector technologies playing a key role. In PEEM, a similar improvement in resolution has been achieved, from 10 nm to 5 nm, and there is still room for further gains. A simultaneous increase in transmission by a factor 10 is particularly important for PEEM, where signal intensity is often an issue, and where photon flux (at synchrotron rings) cannot be increased much further due to space charge issues [45]. I finish with one more comparison between electron optics and light optics. The superachromat developed by Herzberger [3] corrects chromatic aberrations over a wavelength range from 400 to 900 nm, a photon energy range ΔE/E of order 1. In TEM the range is much smaller: ΔE/E is at most 10  4. Even so, chromatic aberration correction is not easy [13]. For PEEM ΔE0/ E0 is again of order 1, with strongly varying chromatic aberration across the energy spectrum, presenting a difficult challenge to the microscope designer. While chromatic aberration correction with catadioptrics is very much easier than with multipole optics, we unfortunately don’t have the different curvatures and different glasses to play with that so much favor light optics. Is developing a superachromat for PEEM really a hopeless task? Acknowledgements I thank Prof. Harald Rose for his encouragement during this work, and Ondrej Krivanek for many stimulating discussions (particularly on the issue of stability). I also thank Jim Hannon and Weishi Wan for unwavering support and patience. Finally, Liang Jin and Robert Bilhorn at Direct Electron, and Oliver Schaff at SPECS developed and provided me with the novel detectors discussed in this paper. References [1] Robert Hooke, Micrographia: or Some Physiological Descriptions of Minute Bodies Made My Magnifying Glasses with Observations and Inquiries Thereupon; London; 1665. [2] Leonardo Da Vinci, Florence ca. 1508. Copyright British Library, Arundel MS 263, f.86v-87. Reproduced with permission from the British Library. [3] M. Herzberger, N.R. McClure, Appl. Opt. 2 (1963) 553–560. [4] E. Abbe, J. R. Micr. Soc. 3 (1883) 790–812 (2 (1882) 300-309; 2 (1882) 460473). [5] O. Scherzer, Zeitschrift für Physik 101 (1936) 593–600 (Pages). [6] O. Scherzer, Optik 2 (1947) 114–132. [7] P.W. Hawkes, Biol. Cell 93 (2001) 432–439. [8] D. Gabor, Nature 161 (1948) 777–778. [9] W. Hoppe, Adv. Imaging Electron. Phys 150 (2008) 87–184. [10] See H. Rose, Adv. Imaging Electron. Phys 153 (2008) 1–37. [11] P.E. Batson, N. Dellby, O.L. Krivanek, Nature 418 (2002) 617–620.

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