Aberration Correction in Electron Optics with N- Fold Symmetric Line Currents

Transcription

1 Title Author(s) Aberration Correction in Electron Optics with N- Fold Symmetric Line Currents HOQUE, SHAHEDUL Citation Issue Date Text Version ETD URL DOI /69584 rights

2 Doctoral Dissertation Aberration Correction in Electron Optics with N-Fold Symmetric Line Currents Hoque Shahedul January 218 Graduate School of Engineering Osaka University

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4 Abstract The present thesis presents the results of research on novel aberration correctors free of magnetic materials using N-fold symmetric line currents (N-SYLC). The dissertation is divided into six chapters as follow. In Chapter 1, it is pointed out that conventional magnetic type aberration correctors suffer from problems such as hysteresis, non-linearity of excitation, magnetic saturation, inhomogeneity of materials, and so on, which make it impossible to have a robust system with high throughput and repeatability. Since the root causes of these difficulties are the physical properties of magnetic materials, we draw our attention to N -SYLC proposed recently as multipoles free of magnetic materials, and set the goal of the thesis to develop aberration corrector models using N-SYLC to solve the problems of conventional correctors. In Chapter 2, noting that N-SYLC generates 2N-pole field, a 3-SYLC doublet model is proposed based on the sextupole corrector of Rose et al. to correct 3rd order spherical aberration. It is analytically proved that aberration correction is possible in an idealized theoretical model, and its sensitivity can be controlled with certain geometrical parameters. By numerical calculations, it is shown that these results remain valid in a realistic model, and the sensitivity is commercially feasible for low to mid-energy electron microscopes. In Chapter 3, a model which is a superposition of 3-SYLC, 4-SYLC and 6-SYLC is proposed to correct the residual 5th order aberrations after the correction of 3rd order spherical aberration. Symmetry conditions for suppressing the generation of unwanted aberrations are derived analytically using an idealized theoretical model, and then proved to be valid in realistic models by numerical calculation. The correction sensitivity is also shown to be commercially feasible for low to mid-energy electron microscopes. In Chapter 4, we consider the superposition of N-SYLC and round lens fields, which we denote as in-lens N-SYLC model. It is shown that in-lens 3-SYLC can work as a 3rd order spherical aberration corrector if certain parameters are optimized, and the correction sensitivity can be higher than the model of Chapter 2. In Chapter 5, the required stability of the current supply units for N-SYLC is derived, and methods are devised to correct mechanical errors such as imperfections in size, shape and orientation of N- SYLC. Control systems for N-SYLC correctors are also outlined. In Chapter 6, the main results of the thesis are summarized, and future tasks are pointed out. i

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6 Table of Contents Introduction A brief history of aberration correction in electron microscopes Current and prospective fields of application of aberration correction Difficulties of present day multipole correctors Emergence of a new type of multipoles: N-SYLC Characteristics of N-SYLC and comparison with conventional multipoles Drawbacks of N-SYLC Objective of the present research An outlook of electron optical systems based on N-SYLC Structure of the thesis Spherical aberration correction Basics of N-SYLC Structures and multipole expansion of potential Designing N-SYLC Cs correction with 3-SYLC x3-SYLC doublet model for Cs correction Aberration analysis Computer simulations Estimation of maximum aperture angle for desired beam size Considering finite size cross-section of conducting wires Summary Higher order aberration correction Analysis of higher order aberrations in 3-SYLC doublet Strategy for higher order aberration correction x(3+4+6)-SYLC for higher order aberration correction Structure of 2x(3+4+6)-SYLC Conditions for cancelling unwanted aberrations Correcting sensitivity Computer simulation Model Model Model Discrepancies between analytical and simulation results Summary iii

7 Aberration correction with in-lens N-SYLC In-lens N-SYLC structure Aberration analysis of in-lens 3-SYLC Approximations for analytical calculations Paraxial trajectories Aberration calculation in Parameter dependence of the sign and value of C s Computer simulation Multi-winding ring coil model Simulation model Simulation results Summary Issues regarding implementation Consideration of Electrical and mechanical errors Stability of current supply unit Correcting mechanical errors N-SYLC structure with supplementary currents Azimuthally distributed structures N-SYLC corrector control system Summary Conclusion Acknowledgments References Publications Appendix iv

8 INTRODUCTION Aberration correction in electron microscopes is an extremely challenging problem, and it required more than a half century of research and development work before practically feasible correctors could be realized. However, the present-day correctors still have unresolved difficulties mainly due to the physical properties of magnetic materials. The objective of this thesis is to overcome these difficulties by proposing a novel scheme. In this chapter, we summarize the problems, present an outline of our scheme, and lay out the goals of the thesis A BRIEF HISTORY OF ABERRATION CORRECTION IN ELECTRON MICROSCOPES Electron microscopes first emerged with the promise of breaking the limit of and go much further beyond the resolution power of optical microscopes which is limited by the wavelength λ of visible light (λ is about 4 to 7 nm). During the dawn of quantum mechanics, de Broglie proposed in 1924 that particles have wavelike properties; the corresponding wavelength λ is related to the particle momentum p by λ = h p, where h is the Planck constant [1]. Depending on the energy, the electron wavelength can be shorter than visible light by a factor 1 5. However, when the first electron microscopes were built in the 193s, their resolution power was much lower, about 5 times what was predicted from electron wavelength [2,3]. This was due to not only the intrinsic aberrations of electrostatic or magnetic lenses, but also other parasitic aberrations and instabilities of the power supply units. Advancement in technologies related to power supply units, improvement in design of lenses and progress in manufacturing process increased the resolution power in the subsequent years. As electrical stabilities became better and parasitic aberrations of lenses were suppressed, the effect of intrinsic aberrations became more dominant, and by the late 194s, in the most advanced electron microscope of that time, the intrinsic aberrations were the main limiting factor on resolution [4]. The most dominant of these aberrations is the 3rd order spherical aberration, whose coefficient is denoted by C s. This aberration can be described as follows. If we try to focus electron beam with a rotationally symmetric lens or round lens, the beam is not focused to a point but suffer from a beam spread which can be expressed as r = C s r 3 o, where r o is the initial slope of the outermost trajectory at the object plane [5,6]. Although there are higher order aberrations and also chromatic aberrations, the 3rd order spherical aberration is the most dominant one. Hence, the ideal case would be to somehow cancel this C s, i.e., to make the overall C s of the system zero. However, O. Scherzer had already shown in 1936 that C s is always a positive finite value if the following conditions, denoted henceforth as Scherzer s conditions, are satisfied [7]. 1

9 (i) (ii) (iii) (iv) (v) The lens fields are rotationally symmetric There are no space charges The lens fields are static The electrons are not reflected at some plane The lens fields and their derivatives are continuous Unfortunately, all typical objective lenses satisfy these conditions, which means that we cannot achieve overall C s = with a combination of these lenses. We need to break at least one of the above conditions. Condition (i) can be broken by multipole lenses [8], condition (ii) by placing electrode on the optic axis [9-13], condition (iii) by using oscillating fields, condition (iv) by electron mirror [14], and condition (v) by using thin film on the electron path [15-18]. However, placing electrode or thin film on electron path is difficult to design and realize, oscillating fields are difficult to control, and electron mirrors are susceptible to noise as there are points where electron speed becomes zero. Multipole lenses turned out to be practically most feasible and much research was carried out regarding aberration correction using multipole fields. Aberration correction with combination of quadrupole and octupole was first studied by Scherzer [8], and was subsequently investigated by Archard [19], Deltrap [2], Beck and Crewe [21], Koops et al [22], and others. Spherical aberration generation by sextupole field was first mentioned by Hawkes [23], and aberration correction by sextupole was studied by Beck [24,25], Rose [26,27], Crewe [28], Chen and Mu [29] and others. When the first multipole correctors were built, they could demonstrate that the principles of aberration correction work. However, they could not surpass the resolution that could be achieved by carefully designed objective lenses without any correctors. Several factors limited the improvement of resolution, or even increased of the overall aberrations of the system. These were instabilities of power supply units, and difficulties in beam alignment due to the disagreement of mechanical center and magnetic centers of the multipoles, which resulted in new parasitic aberrations [3]. Resolution improvement by multipole correctors deemed impractical until the 199s. Breakthroughs occurred in the 199s. By that time, development in engineering realized power supply units with sufficient stability and accuracy, and quality of magnetic materials and precision of the manufacturing process of multipoles were also improved. Techniques were also developed for real-time measurement of aberrations, which made it possible to make feedback to the correctors to achieve effective correction of unwanted aberrations generated by the correctors themselves. Sextupole-based correctors were studied and developed by Haider et al, and achieved sub-angstrom resolution [31-36]. These correctors have since been commercialized by CEOS [37]. In another trend, Quadrupole and Octupole-based correctors have been studied and developed by Krivanek et al., and commercialized by Nion Company [38-43]. These two companies have taken the leading roles in developing and commercializing aberration correctors, while some Japanese microscope manufacturers have also developed correctors for their own microscope columns. Some of the major projects for developing aberration correctors in recent years are TEAM [44], SALVE [45], SupeSTEM [46] and Triple C [47]. 2

10 1.2. CURRENT AND PROSPECTIVE FIELDS OF APPLICATION OF ABERRATION CORRECTION Electron microscopes are essential tools for observation and analysis in a wide variety of fields such as material science, nanotechnology, biology and life science, semiconductor device, and so on [48-5]. At present, electron microscopes are available with electron beam energy in the range of few kiloelectron-volts to megaelectron-volt order. The resolution also varies greatly from one model of microscope to another. In Figure 1.1 we show some important areas of applications of electron microscopes with respect to beam energy and resolution. So far aberration correctors have mainly been developed and successfully commercialized to achieve sub-angstrom resolution for STEM (Scanning Tunneling Electron Microscope) and TEM (Tunneling Electron Microscope) with relatively high beam energy (nearly or more than 1 kev). Beam Energy, E Low energy Few kev Mid energy E 1 kev High energy E 1 kev Ultra high voltage E MeV Resolution 1. nm 1. nm.1 nm.1 nm CD-SEM in Semiconductor device SEM in Surface observation, material analysis Cryo STEM in Biology, life sciences STEM/TEM in Material science, nanotechnology STEM/TEM in 3D, in situ observation SEM: Scanning Electron Microscope TEM: Tunneling Electron Microscope STEM: Scanning Tunneling Electron Microscope Figure 1.1. Various fields of application of electron microscopes (the images of the observation samples are taken from homepages of Hitachi High-Technologies Corp. and FEI company). SEM (Scanning Electron Microscope) with relatively low energy also plays very important roles, for example, in semiconductor device industry. SEM with energy of about 5 kev to few tens of kev is an essential tool to measure the semiconductor device size and its variation, and inspect defects for quality control. An example of such a tool is CD-SEM (Critical Dimension SEM), which is a highly automated SEM specialized in the measurement of semiconductor device patterns. Due to the scaling down of semiconductor devices (see Moore s Law [51]), critical dimension of semiconductor devices is approaching size less than 1 nm [52-6]. Variation in sizes and shapes of semiconductor devices (e.g., the line edge roughness (LER)) are needed to be controlled with precision that goes up as the devices become smaller. E. Solecky of IBM has estimated the CD-SEM resolution required to keep pace with the scaling down as shown in Figure 1.2 [59]. The required resolution is already in sub-nanometer scale (green curve), while the actual resolution is larger than 1 nm (red curve). The design of objective lenses in CD-SEM is already highly optimized, and for higher resolution we need to utilize aberration correctors. Similar is the case with SEM used for defect inspection in semiconductor devices (known 3

11 as Review SEM or Defect Review SEM depending on the manufacturer). However, no aberration correctors are commercially available at present for these microscopes. 1. CD-SEM Resolution Trend (Reproduced from Eric Solecky [59]) Resolution (nm) ITRS Node (nm) Figure 1.2. Resolution requirement of CD-SEM. From the above discussions, we conclude that quality control in semiconductor industry can be a new prospective field for aberration correction. The target resolution power is better than 1 nm, for which spherical aberration correction is necessary. Moreover, higher order aberration corrector might also be required in the following cases where the aperture angle on the sample is large. First, for high S/N (signal to noise ratio) imaging we require higher probe current, which is realized by larger aperture angle. Second, lower beam energy is desirable for certain materials, for example ArF resist, to minimize irradiation damage [61-63]. Larger aperture angles are required at lower energy to suppress the effect of diffraction to maintain sub-nanometer resolution. In these cases, higher order aberrations also need to be corrected DIFFICULTIES OF PRESENT DAY MULTIPOLE CORRECTORS In Section 1.1 we touched upon some difficulties faced in the early years of developing aberration correctors with multipoles, namely, stability of power supply units and problems with alignment. Although they have been overcome to great extent to realize sub-angstrom resolution in high energy STEM, the conventional multipole correctors still suffer from difficulties inherent to their structures. Conventional multipole correctors comprise of multipoles which are made of magnetic material. An example is shown in Figure 1.3. It has 12 pole pieces, and wire winding at the base of each pole piece for current flow. The optic axis is the z-axis at the center. The current flow generates magnetic field in the pole pieces, and from their tips magnetic flux reaches the optic axis to generate strong multipole fields. Electrons traveling along the optic axis are deflected by these multipole fields. 4

12 Wires for electric current Pole piece Figure 1.3. Conventional magnetic multipole with pole pieces made of magnetic material (Photo source: CEOS GmbH homepage [37]). A very important characteristic of magnetic materials is that they have hysteresis, which makes it practically impossible to have repeatability when the operation conditions change. It requires tuning each time the conditions change, and hence no predetermined control parameters can be implemented successfully. Magnetic materials by their very nature absorbs magnetic fields. Aberration correctors comprise of several steps of multipoles, i.e., they are usually multiplet of multipoles. Adjacent poles in the same step affect each other, and there is usually crosstalk between adjacent steps, which makes magnetic field excitation with electric current in conventional multipoles essentially a non-linear effect. It is difficult to generate multipole fields without undesired effects, and thus maintain proper alignment of multipole fields and the electron beam. Usually highly complicated control system is required, and even that may not always yield satisfactory results. Inhomogeneity of magnetic materials was one of the reasons behind mismatch of mechanical center and magnetic centers of multipoles in the early years of corrector development. Much progress has been made in the production techniques of magnetic materials to mitigate this; however, the problem is not fully resolved, and to this day inhomogeneity is an issue not only in multipoles, but also in round objective lenses. The above issues make it quite impossible to implement the conventional aberration correctors in electron microscopes like CD-SEM. This is because these are in-line tools, i.e., they are used in the semiconductor process line for quality control at different phases of device production. It is a wellknown fact that high precision, yield and throughput are crucial in semiconductor industry, which means that CD-SEM and other defect inspection and review SEM should satisfy the following conditions. (1) They need to have high throughput to support high yield of semiconductor device fabrication. (2) They need to be highly stable over a long period of time. (3) They need to have high repeatability when the operation conditions change. For example, in CD-SEM, conditions such as beam energy, aperture angle, beam deflection angle vary depending on 5

13 the types of semiconductor patterns. When these conditions change, the aberrations of the objective lens change, and the corrector should have high repeatability to correct them without being tuned each time. (4) All tools in a semiconductor fabrication facility need to have the same characteristics [64]. (5) Measurements with sub-nanometer precision is highly sensitive to the beam profile, which in turn depends on all kinds of aberrations. Therefore, it is essential to be able to control the aberration corrector with high precision. The above conditions are extremely difficult to meet with aberration correctors made of conventional magnetic multipoles. This made us look for an alternative to the conventional multipoles EMERGENCE OF A NEW TYPE OF MULTIPOLES: N-SYLC The problems of the conventional aberration correctors discussed in the last section originate from the magnetic materials of pole pieces. In principle, we can overcome the difficulties if we do not use magnetic materials. Such an electron optical element was proposed by H. Ito et al [65, 66]. It consists of N number of wires carrying electric current which are parallel to the optic axis and have N-fold symmetry about the axis. An example for N=3 is shown in Figure 1.4. Here z-axis is the optic axis and three infinitely long line currents (each I) are placed parallel to and at equal distance R from the z-axis, and at equal azimuth angle 2π from each other. H. Ito showed that, in case of infinitely long 3 wires, the magnetic field density at distance r from the optic axis due to the currents is proportional to r N 1 if r is small. He argued that this is in fact 2N-pole field, and hence can be utilized to correct aberrations. Subsequently, R. Nishi et al. used the nomenclature wire correctors to denote such arrangements, and studied the magnetic potentials generated by them to show that the lowest order component of the potential is indeed 2 N -pole [67]. They also noticed some differences with conventional multipoles in the higher order terms of the potential. They mentioned that the case N=2, which generates quadrupole field, should correct chromatic aberration, while N=3, which generates sextupole field, should correct 3rd order spherical aberration. However, detailed study of the wires correctors was yet to be done, and aberration correction was still to be demonstrated either analytically or by numerical calculation. Another concern has been the sensitivity; absence of pole pieces unlike conventional multipoles led to doubt whether they would have enough sensitivity to be practical and commercially feasible. I R I I Figure 1.4. Three infinitely long wires with current I arranged in threefold symmetry about z-axis. 6

14 We draw our attention to the wire correctors as a candidate for aberration correctors free of magnetic materials, and study their magnetic potentials and aberration characteristics in detail. We use a new nomenclature to denote them: N-fold symmetric line currents, or, N-SYLC in short [68]. N- SYLC can be realized, for example, by means of toroidal coils. We show an example of 3-SYLC realized by toroidal coils in Figure 1.5, where, currents parallel to the z-axis through the inner portion of the coils at distance R (denoted as inner radius) correspond to the currents in Figure 1.4. Optic axis Electron trajectory x I I Inner radius R I Coil current y z Figure 1.5. Realization of 3-SYLC by toroidal coils. N -SYLC should in principle be free of the drawbacks of conventional multipole correctors mentioned in Section 1.3 and will be a suitable candidate for new correctors for electron microscopes, especially for in-line systems in the semiconductor industry CHARACTERISTICS OF N-SYLC AND COMPARISON WITH CONVENTIONAL MULTIPOLES Detailed description of different possible N -SYLC structures, along with expressions of the potentials are given in Chapter 2. Here, we discuss some important characteristics of N-SYLC: they are analytically solvable, additive in natures, free of magnetic saturation, and their higher order terms are determined by the distribution of line currents. These are discussed below. Analytically solvable: Magnetic fields in N-SYLC are generated in vacuum directly by the line currents according to the Biot-Savart law, without using any magnetic materials, and hence are analytically solvable. The magnetic fields by the toroidal coils of Figure 1.5 can be determined analytically as closed-form solution. On the other hand, fields by the conventional multipoles cannot be calculated analytically since they involve magnetic materials. 7

15 Additive in nature: N-SYCL are, in principle, free of cross-talk due to the absence of magnetic materials. As a result, even when different combinations of currents are superimposed in each of the toroidal coils of an N- SYLC (which may be necessary for correcting mechanical errors as described in Chapter 5), the resultant magnetic field will be simple addition of fields due to each current. This simple rule of addition also holds in an N-SYLC multiplet where there are several steps of N-SYLC: the excitation current of an N-SYLC does not affect the adjacent one. Therefore, N-SYLC will be free of crosstalk that makes the designing and controlling of correctors with conventional multipoles extremely difficult. This was one of the factors behind the failure of early aberration correctors [69]. In N-SYLC, excitation characteristics are predictable with high accuracy, which should make the designing and controlling of correctors with N-SYLC much simpler. Free of magnetic saturation: By principle, N -SYLC is free of magnetic saturation. On the other hand, conventional multipoles suffer from saturation when the current is too high, and/or the poles are thin. This might be a difficulty in generating higher order multipole fields with sufficient strength. For example, with a structure like Figure 1.3, sextupole fields can be generated with high enough intensity to correct 3rd order spherical aberration. However, not enough intensity for dodecapole field can be achieved due to magnetic saturation for 5th order aberration correction. Higher order components controllable by current distribution: The most dominant component in N -SYLC is 2 N -pole field. The higher order multipole components are determined by the distribution and number of the line currents (see Chapter 2 for further details). In case of conventional multipoles, higher order terms depend on the shape of the tip of the poles. In theory, in conventional multipoles, higher order terms can be suppressed to achieve pure multipole field by appropriate pole shape [7]. However, in practice, this is difficult to achieve due to error of manufacturing process, and, if implemented, the production yield of aberration correctors will get low and the cost high. Controlling higher order terms in N-SYLC should be easier to implement and less costly, since it requires only to arrange several toroidal coils with appropriate symmetry DRAWBACKS OF N-SYLC Despite various advantages of N-SYLC, there are also some drawbacks. First, conventional multipoles are typically highly sensitive since they concentrate magnetic field density lines inside magnetic materials. It is also possible to allow for high (few hundreds of) ampere-turns by making the winding number of conducting wires large. In case of N-SYLC, there is no such concentrating effect, and large number of line currents is difficult to implement to increase the ampere-turn due to limitation in space. Again, to prevent overheating, the current through a single wire should not be too high. Hence, to achieve enough sensitivity, the inner radius R in Figure 1.5 should be short, and the length along the optics axis long, so that the current through a single wire is less than 1 A, which is already proven to be safe in other typical electron optical elements. Another way to overcome heating is using superconducting coils. In that case, N -SYLC will have high sensitivity to outperform 8

16 conventional multipoles, which eventually suffer from saturation when the ampere-turn is too high. Such an N-SYLC will be expensive to build and will not be suitable for low to mid-cost microscopes, but will be suitable for very high energy electron microscopes, which are likely to afford high cost and sophistication of the system. In the present thesis, aiming systems of moderate costs, we will optimize the inner radius and length of N-SYLC rather than considering superconducting coils. Another possible drawback of N-SYLC is the effect of mechanical errors in size and shapes. The exact cross-section of the wires for the line currents will affect the generation of multipole fields, and high degree of uniformity is likely to be required. However, the effect of errors in size and shape of the toroidal coils can be corrected by supplementary coils (see Chapter 5), and we hope that similar techniques may be effective in compensating for non-uniformity in wire cross-section OBJECTIVE OF THE PRESENT RESEARCH The present research is the first undertaking to carry out a detailed study of N-SYLC as aberration correctors. The goal here is to understand the basic properties of N-SYLC and construct the theoretical foundation of aberration correction with N-SYLC. We will primarily be concerned with geometrical aberration correction in low to mid-energy electron microscopes, and our objective is to investigate the following. (a) 3rd order spherical aberration correction with N-SYLC (b) Higher order aberration correction with N-SYLC (c) Overlap of N-SYLC and round lens field (d) Considering issues regarding implementation In (a), we consider correcting 3rd order spherical aberration (coefficient C s ) correction with N- SYLC. As already mentioned in Section 1.1, C s correction with multipole fields has been studied extensively. We will consider 3-SYLC doublet model following the sextupole doublet model of Rose et al [27]. Since the principle component of the potential of 3-SYLC is sextupole, 3-SYLC should be able to correct C s. However, the sensitivity or required current for C s correction is a matter of concern, and so is the effects of higher order aberrations. We will be interested in finding ways to make the 3-SYLC doublet sensitive enough so that the required current is less than 1 AT. We will also find conditions (for example, initial slopes of electron trajectory) for which the effect of the higher order terms is small enough to allow beam size less than 1 nm at the corrector image plane after C s correction. In (b), after C s correction, we consider the correction of the higher order aberrations, namely 5th order sixfold astigmatism and 5th order spherical aberration correction. This will enable us to increase the aperture angle, which decreases the effect of diffraction to yield higher resolution, and also allow higher probe current for better S/N (signal to noise ratio). Current methods for correcting 5th order spherical aberration adjust the trajectory of paraxial rays according to the C s of the objective lens. However, C s changes as observation conditions such as working distance, beam energy, beam tilt angle on the sample change. Hence, it will be difficult to implement the conventional methods in cases where these conditions change frequently, for example, in a tool like CD-SEM. We will be interested in devising a method in which the paraxial rays can be kept fixed. As for sixfold astigmatism, although 9

17 using higher order multipole field is the simplest solution to correct it theoretically, it cannot be utilized in practice in conventional multipoles due to magnetic saturation of pole pieces. We will be interested in implementing this simplest solution of using higher order multipole by N-SYLC, since it is free of magnetic saturation. To make the model practically feasible, our aim will again be excitation currents less than 1 AT for aberration correction. In (c), we will explore the possibility of immersing N-SYLC in round lens fields with the aim of making a compact and simple structure. We call this in-lens N-SYLC. Such a structure can only be contemplated with N-SYLC and not with conventional multipoles, since the former is free of magnetic material, by virtue of which it will be unaffected by surrounding magnetic fields. However, it is not immediately evident if such an overlap model can still function as a corrector. Our aim is to find the conditions under which it can correct 3rd order spherical aberrations and will also have practically feasible sensitivity. To achieve the above goals, we use two steps of analysis and investigation. First, we consider simplified and idealized theoretical models that are analytically solvable, and by analyzing the aberrations we derive the necessary conditions. The magnetic fields of N-SYLC are solved using the Biot-Savart law, and aberrations are calculated by solving the differential equations of motion of electrons by means of successive approximation (see, for example [71]). Second, we check the validity and effects of the conditions in realistic models by means of numerical calculations, and estimate the final beam size and correction sensitivity. For numerical calculation we use a software called the LANTERN Lens developed jointly by MEBS [72]. A summary of the LANTERN lens software is given in Appendix A.3. Finally, in (d), we consider such important issues in designing a system with N-SYLC as electrical stability and mechanical errors. We will derive the required accuracy and stability of the current supply units of N-SYLC. As for mechanical errors, we will device methods to correct the resultant parasitic aberrations by means of appropriate arrangement of supplementary line currents, and thereby make the required mechanical precision less severe than conventional multipoles. In this thesis, we leave the correction of chromatic aberration as future project. At present we assume that chromatic aberration is taken care of by well-designed monochromators AN OUTLOOK OF ELECTRON OPTICAL SYSTEMS BASED ON N-SYLC In the last section, we have described the goals of the present thesis. As a first project on N-SYLC as aberration correctors, we are content here with aberration correction in low to mid-energy electron microscopes. However, N-SYLC can be the ultimate aberration correctors for electron microscopes of all energy ranges as explained below. For relatively low to mid-energy electron microscopes with moderate cost, it will have enough sensitivity with no or simple cooling system. It will have the advantages of being free of hysteresis and having high speed response, which will make it particularly suitable for a new aberration corrector for high throughput electron microscopes used in the semiconductor industry for quality control (see Section 1.2 and 1.3). Moreover, due to the higher efficiency of design and simpler control system, it should also be a better candidate for corrector for any mid to high-energy microscopes in general compared to conventional multipoles. 1

18 As for the sophisticated systems of very high energy electron microscopes, conventional multipoles will face the difficulty of magnetic saturation depriving them of enough sensitivity. In that case, N-SYLC utilizing superconducting wires to overcome heating can be the solution for aberration correction. The in-lens N-SYLC mentioned in Section 1.7 can open the possibility of much flexible design of electron optical systems. For example, dipole field can easily be superimposed onto an in-lens N-SYLC immersed in the objective lens to have in-lens deflector simultaneously with a corrector. Thus, we can have a compact and miniaturized system for electron beam probe formation for SEM or STEM. In case of a high energy electron microscope, we can consider making use of superconductor not only for N- SYLC, but also for objective lens by using superconducting ring coils. Thus, we can have a system which is completely free of hysteresis, inhomogeneity and magnetic saturation, and have high speed response. This can open the door to a whole new paradigm for electron microscope design STRUCTURE OF THE THESIS We give an outline to the structure of the dissertation below. In Chapter 2, we first derive the magnetic potential of N-SYLC and show how to realize N-SYLC with toroidal coils. Then we introduce the 3-SYLC doublet model for 3rd order spherical aberration correction and address the topic (a) of Section 1.7. In Chapter 3, we address the topic (b) of Section 1.7. We start by investigating the residual higher order aberration after 3rd order spherical aberration correction, and based on our analysis, propose an N -SYLC quadruplet (4 step) model consisting of 3-SYLC, 4-SYLC and 6-SYLC to correct axial geometrical aberrations up to 5th order. Conditions for suppressing unwanted aberrations are derived, and the final beam size and sensitivities of each type of N-SYLC are estimated. In Chapter 4, to address the topic (c) of Section 1.7, we first describe the structure of an in-lens N- SYLC, in which N-SYLC is overlapped with round lens fields. Moreover, we consider using ring coils as round lenses to make the model completely free of magnetic materials. The basic characteristics of the model are derived, and the final beam size after aberration correction and sensitivity is estimated. In Chapter 5, to address the topic (d) of Section 1.7, we discuss the issues regarding electrical and mechanical tolerance, and give a sketch of the control system for N-SYLC aberration correctors. Finally, in the last chapter, we summarize the results of each chapter, and discuss some issues for future research. 11

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20 SPHERICAL ABERRATION CORRECTION [68] In this chapter, we first describe the structures and potentials of N-SYLC, and show how to design them. Then we present a 3-SYLC doublet model for 3rd order spherical aberration (coefficient C s ) correction. We analytically derive the aberrations of 3-SYLC to prove that it can indeed correct C s, and show how its sensitivity can be optimized by adjusting geometrical parameters of the model. Then, we perform computer simulation to demonstrate C s correction, and estimate the final beam size and the required excitation current for correction BASICS OF N-SYLC STRUCTURES AND MULTIPOLE EXPANSION OF POTENTIAL We first consider an infinitely long line current I parallel to z-axis, which is also the optic axis, at distance R and azimuth angle φ as shown in Figure 2.1. The magnetic potential due to this line current can be shown to be a linear combination of multipole fields as given by Equation (2.1). R φ I Figure 2.1. Infinitely long line current along the optic axis. Ψ Single = μ I r n sin(nθ + nφ). nrn n=1 (2.1) Here, μ is the magnetic permeability of vacuum, (r, θ) are polar coordinates related to Cartesian coordinates as ( = r cos θ, = r sin θ) (see Appendix A.1 for a derivation of the potential). If we place two line currents with twofold symmetry, i.e., combine the cases of φ = and of Equation (2.1), we obtain twofold symmetric line currents or 2-SYLC, which is shown in Figure 2.2(a). The most dominant component of potential near the optic axis, which we also call the principle component, is quadrupole. Similarly, a combination of φ =, 2π, 2π is 3-SYLC (which is already 3 3 shown in Figure 1.4), and φ =, π,, 3π is 4-SYLC, with principle components sextupole and octupole, 2 2 respectively. The structures are shown in Figure 2.2(b) and (c). N-SYLC for other integer values of N can be derived in similar fashion. 13

21 I I I I I I I R I R R I (a) (b) (c) Figure 2.2. Examples of N-SYLC: (a) 2-SYLC, (b) 3-SYLC, (c) 4-SYLC. The magnetic potentials are straightforward to derive from Equation (2.1) and are as follow. For 2-SYLC: Ψ 2 SYLC = μ I r 2n n=1 sin( nθ) nr2n (2.2) For 3-SYLC: Ψ 3 SYLC = μ I r 3n n=1 sin( nθ) nr3n (2.3) For 4-SYLC: Ψ 4 SYLC = μ I r 4n n=1 sin(4nθ) nr4n (2.4) The general form of N-SYLC potential is given in Equation (2.5), in which the principle component is 2N-pole. Thus, N-SYLC can be considered as an electron optical element that produces 2N-pole field without any magnetic material. Ψ N SYLC = μ I r Nn n=1 sin(nnθ) nrnn (2.5) We show a variation of the N-SYLC structures in Figure 2.3, in which line currents of opposite sign are place in between the existing line currents. These also have 2, 3, and 4-fold symmetry as before, and we denote them as 2x2-SYLC, 2x3-SYLC and 2x4-SYLC, respectively, to differentiate from those in Figure 2.2. Their magnetic potentials are given in Equation (2.6) to (2.8). The principle components are still quadrupole, sextupole and octupole, respectively. However, there are fewer higher order terms in this case because some components are cancelled out due to the symmetry. For example, the higher order terms in 3-SYLC after sextupole are dodecapole, 18-pole, 24-pole and so on, while in 2x3-SYLC 14

22 they are 18-pole, 3-pole and so on. The higher order terms in 2x3-SYLC can practically be ignored. This is true for 2xN-SYLC in general, the potential of which is given by Equation (2.9). Thus, we see that the higher order terms can be controlled by line current distribution in N-SYLC structure, whereas in conventional multipoles they depend on the shape of pole pieces (see the discussion in Section 1.5). I I I I I I I I I I I 4 I R R R 4 I I I I I I (a) (b) (c) Figure 2.3. Examples of 2xN-SYLC: (a) 2x2-SYLC, (b) 2x3-SYLC, (c) 2x4-SYLC. For 2x2-SYLC: For 2x3-SYLC: For 2x4-SYLC: For 2xN-SYLC: Ψ 2 2 SYLC = μ I r 2(2n 1) n=1 Ψ 2 3 SYLC = μ I r 3(2n 1) n=1 Ψ 2 4 SYLC = μ I r 4(2n 1) n=1 Ψ 2 N SYLC = μ I r N(2n 1) n=1 sin{ ( n 1)θ} nr2(2n 1) sin{ ( n 1)θ} nr3(2n 1) sin{4( n 1)θ} nr4(2n 1) sin{n( n 1)θ} nrn(2n 1) (2.6) (2.7) (2.8) (2.9) DESIGNING N-SYLC In the above discussion, we have considered infinitely long line currents for N-SYLC. A finite size N-SYLC can be approximated, for example, by toroidal coils, and a 3-SYLC comprising of toroidal coils is shown in Figure 1.5. In this section, we justify this approximation as follows. We start with a single toroidal coil as shown in Figure 2.4, where, the z-axis is taken to be the optic axis. AB and CD are parallel to the z-axis, while BC and DA are perpendicular to it. Current through the coil is I, and AB is at distance R from the optic axis. We denote L = AB as the coil length. The length of the perpendicular portion is h. The angle between the positive direction and BC is φ. 15

23 Optic axis Electron trajectory x R B h C φ I L z A y D Current Figure 2.4. Toroidal coil as a building block of N-SYLC., and z components of magnetic field densities B x, B y and B z due to current I can be calculated analytically as closed-form solutions according to the Biot-Savart law. For φ =, L=4 mm, R=2 mm and h=55 mm, B x and B z at (, ) = (,1 m) are shown in Figure 2.5. We see that, in the proximity of the optic axis, B z is negligible compared to B x (similarly to B y ). In B x (and B y ), the most significant contribution is from the portion AB; contribution from CD is much smaller if R h. Thus, the toroidal coil approximates line current of Figure 2.1 for finite length. Arranging toroidal coils with N-fold symmetry about the optic axis approximates N-SYLC. The toroidal coils can be supported by solid structure made of resin or ceramic, which is shown in Figure 2.6(a). Another way to design N-SYLC is using metal rod strong enough to support themselves, rather than toroidal coils of wires, as shown in Figure 2.6 (b). Toroidal coils are conducting wires (e.g., Cu wires) which have finite cross-section. For simplicity, we discard the details of cross-section and consider only ideal line currents (i.e., zero cross-section) to investigate aberration correction. The effect of finite diameter is discussed in Section 2.4, where we argue that if all the wires have the same cross-section, the effect is equivalent to line currents with zero cross-section positioned at the centers of the wires C s CORRECTION WITH 3-SYLC 3rd order spherical aberration correction with sextupole field has been studied extensively, and successfully developed into commercially available correctors. The correction principle is as follows. Sextupole field produces 2nd order threefold symmetric astigmatism, which in turn produces negative C s (see Appendix A.2 for details). The basic structure of the conventional corrector proposed by Rose et al. has two magnetic sextupoles connected by a pair of transfer lenses, which image the first sextupole onto the second one with opposite sign, so that the unwanted threefold symmetric astigmatism cancels out. We start with this basic structure, with 3-SYLC or 2x3-SYLC as sextupole elements. In this section, we will show that 3-SYLC do indeed produces negative C s. We also demonstrate that 3-SYLC can have enough sensitivity to be commercially feasible. 16

24 Optic axis Optic axis Coil length L.5 Coil length L.25 B x μ I 4-5 B z μ I z (mm) (a) z (m) (b) Figure 2.5. Magnetic field density of the toroidal coil of Figure 2.4 at (x, y) = (,1 m): (a) x component of magnetic field density, (b) z component of Magnetic field density. Resin or ceramic structure Metal rod (a) (b) Figure 2.6. Two examples of 3-SYLC design: (a) toroidal coils supported by resin or ceramic structure, (b) metal rods. 17

25 X3-SYLC DOUBLET MODEL FOR C s CORRECTION We consider a 2x3-SYLC doublet model based on the conventional sextupole doublet model. The reason for considering 2x3-SYLC rather than 3-SYLC is that the latter generates dodecapole field as higher order component (n = in Equation (2.3)), which produces unwanted 5th order aberration. This limit the improvement of beam size when the radius of 3-SYLC is made shorter for higher sensitivity (see [68] for details). On the other hand, 2x3-SYLC is free of the dodecapole component (n = in Equation (2.7)), hence does not have such drawback. We show the schematic of the 2x3-SYLC doublet in Figure 2.7. Two 2x3-SYLCs are connected by a pair of rotationally symmetric lenses (also called round lenses) RL 2 and RL 3. z o is the object plane of the corrector, and z i is the image plane. A round lens RL 1 is placed in front of the 2x3-SYLC of step 1 to form telescopic beam, and another round lens RL 4 is placed after the 2x3-SYLC of step 2 to focus the beam at z i. In principle, the last round lens can function as the objective lens, in which case the corrector image plane z i coincides with the sample under observation. However, in most common cases, there will be a separate objective lens after the corrector. The paraxial trajectories are also shown in Figure 2.7, which are trajectories with the following boundary conditions. G(z o ) = 1, G (z o ) = ; H(z o ) =, H (z o ) = 1. (2.1) In the following section, we analyze the aberrations for electron trajectories r(z) = r o H(z), where r o is the initial radial slope ABERRATION ANALYSIS To simplify the analytical calculation, we make the following approximations. (i) (ii) (iii) We assume that the fields of the rotationally symmetric lenses do not overlap with the N- SYLC fields. We use sharp cut-off fringing field (SCOFF) approximation for the N-SYLC fields. The fields are taken to be non-zero only in the region the N-SYLC, as shown in Figure 2.8. In the region of N-SYLC, we take the fields to be that for infinitely long line currents. Henceforth we refer to condition (i) as no-overlap condition, while (ii) and (iii) as ideal N-SYLC conditions. Under the above approximations, we analyze the aberrations in the model shown in Figure 2.7. We assume that the fields by the round lenses are adjusted such that G and H trajectories are as shown in the figure. We will be concerned with the aberrations due to 2x3-SYLC, and since round lens fields and 2x3-SYLC fields do not overlap, we only need explicit expression for the fields by 2x3-SYLC, which is immediately obtained from Equation (2.7). The principle component is sextupole μ I r 3 π nr3 sin θ, after which are 18-pole and higher order terms. Since we consider aberrations at most of order 5, we ignore the higher order terms. Aberrations by the sextupole field can be calculated analytically and the results are as follow (see Appendix A.2 for details). 18

26 f f f f f f f f RL 1 RL 2 RL 3 RL 4 4 z o z i z Corrector object plane R L 2x3-SYLC 2x3-SYLC Step 1 Step 2 Corrector image plane H trajectory G trajectory Figure x3-SYLC doublet model. We consider electron trajectories r(z) = r o H(z). Length of N-SYLC I Line current of N-SYLC Flux density Flux density = z 1 R Flux density, constant Flux density = z 2 z Figure 2.8. SCOFF approximation for N-SYLC fields. The lowest order aberration by 2x3-SYLC is threefold astigmatism (coefficient A 2 ), which cancels out due to the symmetry of the model. The next higher order aberration is negative 3rd order spherical aberration. We denote the aberration coefficient as C s 2x3 SYLC, which is as follows. C s 2x3 SYLC = M { ( where, I is the excitation current of each 2x3-SYLC, ημ 2 I R 3 Φ ) f 4 L 3 }, (2.11) η = e m, e is the absolute value of electron charge, m is electron rest-mass, μ is magnetic permeability in vacuum, eφ is electron energy, 19

27 M is the total magnification (in case of Figure 2.7, M = 1), f is the focal length of the round lenses (same for all the four lenses), L is the length of 2x3-SYLC (taken to be the same for both the 2x3-SYLC in step 1 and 2), and the total length of the corrector is L = 8f. If the spherical aberration coefficient of the four round lenses RL 1, RL 2, RL 3 and RL 4 in total is C s RL, and of the objective lens C s Obj. at the corrector image plane, then the condition for correction is as follows. C s total = C s Obj. + C s RL + C s 2x3 SYLC =. (2.12) We define the sensitivity of C s correction as the required 2x3-SYLC current I c to satisfy the above condition. Hence, the sensitivity depends on the beam energy and other corrector parameters as follows. Sensitivity: I c Φ 1 2L 2 R 3 L 3 2. (2.13) This shows how the sensitivity can be optimized by adjusting the geometrical dimensions of the corrector such as R, L and L COMPUTER SIMULATIONS In the analytical calculation, we have assumed no-overlap condition and Ideal N -SYLC conditions for simplicity, and derived the basic characteristics of the 2x3-SYLC doublet model. In this section, we check if these results hold in a realistic case, i.e., when we discard the no-overlap condition and Ideal N-SYLC conditions. For this purpose, we perform computer simulation by the LANTERN Lens software. We show the simulation model in Figure 2.9, and list the values of the parameters in Table As 2x3-SYLC we use toroidal coils, and as round lenses ring-shaped coils (denoted henceforth simply as ring coils). The z-coordinate of each element is indicated in Figure 2.9. The coordinates of the toroidal coils are the z-coordinates of the center of toroidal coils. It is to be noted that adjacent ring coils have currents of opposite signs, which is indicated by arrows in opposite directions. This ensures that the rotation of the paraxial trajectories is the same in step 1 and step 2. The reason for using ring coils as rotationally symmetric lenses is twofold. First, like N-SYLC, magnetic field by ring coils is analytically solvable, which makes the results of numerical calculations highly accurate. Second, we will consider using ring coils in a real corrector because it has the advantage of being free of magnetic materials. This will allow us to overlap N-SYLC and round lenses for simplification of the model, which is discussed in Chapter 4. As explained in Appendix A.3, in the LANTERN lens software, the fields due to ring coils and N- SYLC are determined analytically according to the Biot-Savart law, i.e., the solutions include terms of 2

28 all orders. As a result, electron trajectories, which are determined by numerical calculation (ray tracing) using 5th order Runge-Kutta method, include aberrations of all order Corrector object plane R R z o L Toroidal coils as 2x3-SYLC (here only three coils are shown for simplicity. In 2x3-SYLC, there will be at least six coils) L Ring coils as rotationally symmetric lens 1 Corrector image plane z z i Figure 2.9. Simulation model of 2x3-SYLC doublet. Table 2.1. Values of the parameters of the simulation model Parameter Toroidal coil length of each 2x3-SYLC Inner radius of each 2x3-SYLC Outer radius of each 2x3-SYLC Radius of each ring coil Focal length of each ring coils: f Length of the system: L = 8f Beam energy: eφ Value 35 mm 2, 2.5, 3 mm 65 mm 1 mm 25 mm 2 mm 5. kev Initial position: r o = ( o) 2 + ( o) 2 Initial slope: r o = ( o ) 2 + ( o ) 2 2, 4, 6, 8, 1 mrad Azimuth angles at the corrector object plane, 2, 4,, 358 degrees: 18 angles for each r o 21

29 The ring coil current required to focus the paraxial trajectories is 5 AT. An example of calculation of a single ray is shown in Figure 2.1(a). At the image plane, it starts from the 𝑧 axis in the 𝑧- plane with initial slope o. It is rotated by the first round lens by a certain angle, and rotated again in the opposite direction by the second round lens by the same amount (since the second round lens has the same current in the opposite direction). Thus, at the center of the corrector the net rotation by the first two round lenses is zero. The same happens in the second step, and the net rotation at the corrector image plane is zero. To calculate aberration coefficients, we perform ray trace for several initial slopes (2, 4, 6, 8, 1 mrad) and azimuth angles (18 angles with step of ). The trajectories are shown in Figure 2.1(b). We show the current dependence of the spherical aberration coefficient 𝐶s2x3 SYLC;sim. of the 2x3-SYLC doublet for three values of inner radius, namely 𝑅 =,.5, mm, in Figure We include sim. in the superscript to differentiate it from the simplified analytical result of Equation (2.11). The colored circles are results of numerical calculations, while the dotted lines are fitted curve. We find that 𝐶s2x3 SYLC;sim. has negative sign, and its absolute value increases as a quadratic function of current 𝐼, as is expected form Equation (2.11). o Corrector Object plane (𝑧) (𝑧) 𝑟(𝑧) Corrector image plane Corrector Object plane Corrector image plane 𝑧 𝑧 (b) (a) Figure 2.1. Results of ray trace: (a) Result of ray trace for initial conditions 𝒙𝐨 = 𝟎 and non-zero slope 𝒙 𝐨, (b) ray trace for several initial slopes and azimuth angles. In the simulation model, the four ring coils have total spherical aberration coefficient 𝐶sRL 4 Obj. mm. If the coefficient of the objective lens at the corrector image plane 𝐶s 4 mm (which is a reasonably large value for typical objective lenses), we see that the required 2x3-SYLC current for correcting the total aberration of ring coils and objective lens is less than 1 AT for 𝑅.5 mm. In the following, we take 𝑅 = mm, for which 𝐶s2x3 SYLC;sim. + 𝐶sRL = for 𝐼 = AT, and 22

30 C s 2x3 SYLC;sim. + C s RL + C s Obj. = for I.4 AT. This sensitivity is quite feasible for a commercial corrector. C s 2x3 SYLC;sim. (mm) Ring coils C s RL =4 mm Objective lens C s Obj. 4 mm x3-SYLC current (AT) Figure Inner radius dependence of 2x3-SYLC sensitivity. We show the beam cross section image (henceforth called aberration diagrams) at the corrector image plane in Figure Diagrams denoted by (i) is the result of ray trace, i.e., diagram including aberrations of all order. Diagrams (ii) through (v) are aberration diagrams due to 2nd, 3rd, 4th and 5th order aberrations, respectively, and are extracted from (i) by means of Fourier analysis (Appendix A.3). In Figure 2.12(a), we show the aberration diagrams when 2x3-SYLC is turned off. The beam size is determined by 3rd order spherical aberration of the ring coils. There is also small 5th order aberration, which is 5th order spherical aberration (coefficient C 5 ) of the ring coils. When the current of 2x3-SYLC is adjusted so that C s RL + C s 2x3 SYLC = (current I = AT), the beam size is determined mainly by two 5th order aberrations, namely, 5th order spherical aberration and 5th order astigmatism (coefficient A 5 ). The values of these aberration coefficients in the cases of Figure 2.12(a) and (b) are given in Table 2.2. Table 2.2. Simulation results for aberration coefficients of the 2x3-SYLC doublet model Aberration coefficients 2x3-SYLC off 2x3-SYLC on (Current I= AT) 3rd order (C s ).42 m.19 um 5th order (C 5 ) m 25.4 m 5th order (A 5 ) 7.1 m 23

31 (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order (a) Initial slope r o : 1 mrad 8 mrad 6 mrad 4 mrad 2 mrad (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order (b) Initial slope r o : 1 mrad 8 mrad 6 mrad 4 mrad 2 mrad Figure Aberration diagrams at the corrector image plane: (a) 2x3-SYLC off, (b) 2x3-SYLC on and 3rd order spherical aberration of ring coils corrected ESTIMATION OF MAXIMUM APERTURE ANGLE FOR DESIRED BEAM SIZE In the aberration analysis of the last section, we have mainly considered only the corrector. In an electron microscope, there will be an objective lens after the corrector as shown in the schematic of Figure The magnification of the objective lens is denoted here by M obj. α is the aperture angle on the sample, which is also the image plane of the objective lens. By simple geometry, α = r o and M obj Δr Geo.Obj. = M obj Δr Geo. where, r o is the initial slope of the outermost trajectories of the electron beam, Δr Geo. is the beam spread due to geometrical aberration on the corrector image plane and Δr Geo.Obj. is the image of Δr Geo. by the objective lens, i.e., the beam spread at the image plane of the objective lens. The beam or probe size is on the sample is determined by (i) source size, (ii) diffraction effect, (iii) geometrical aberrations and (iv) chromatic aberrations. Source size depends on the type of electron guns and typical sizes can be found in the literature of electron optics [73]. Diffraction effect results in the Airy radius r A for electron beam which is given by Equation (2.14). Beam spread due to chromatic aberration is determined by Equation (2.15). where, λ is the wavelength of electron, E is electron energy, ΔE is electron energy dispersion, C c is chromatic aberration coefficients. λ can be shown to be r A =.61 λ α. (2.14) Δr c = C c ΔE E α. (2.15) 24

32 λ = 1.5 Φ (nm), (2.16) where, Φ is the electron energy expresses in volts: E = eφ. Diffraction effect can be made small by increasing the aperture angle. From Equation (2.14) and (2.16), we find that for beam energy 5 ev, beam spread r A due to diffraction is less than 1 nm for α mrad, less than.5 nm for α 4 mrad. However, beam spread due to geometrical and chromatic aberrations increases as aperture angle increases. We assume that chromatic aberration is taken care of by some other measure, e.g. by a monochromator to decrease ΔE [74-77]. Here, we estimate the limit on the aperture angle imposed by the geometrical aberrations. Now, if we require that beam spread due to geometrical aberration on the sample Δr Geo.Obj..1 nm, then the beam size at the corrector image plane Δr Geo..1/M obj. Typically, M obj.1, hence, Δr Geo. 1. nm. From Figure 2.12, we see that after C s correction, Δr Geo. is determined by 5th order aberrations, since there is no significant difference between the aberration diagram of 5th order alone (Figure 2.12(b) (v)) and that of all order together (Figure 2.12(b)(i)). Again, as explained above Figure 2.11, the current for correcting the C s of a typical objective lens will be less than.42 AT. Now, for 2x3-SYLC current I =.4 AT, the aberration diagram due to 5th order aberration alone is as shown is Figure It is clear that, for Δr Geo. 1. nm, or equivalently Δr Geo.Obj..1 nm, initial slope r o 6 mrad, i.e., aperture angle α 6 mrad. Beam spread due to diffraction is r A =. 5 nm for α = 6 mrad. Hence, if we take α = 6 mrad, the combined effect of geometrical aberration and diffraction will be a beam spread of about.37 nm. The final resolution will be determined by adding to it the effect of the source size and chromatic aberration; however, it is reasonable to expect final probe size less than 1 nm on the sample. Electron trajectory Initial slope r o Aperture angle α = r o M obj r i = r o z Corrector object plane Corrector Corrector image plane Objective Lens (Magnification M obj ) Sample/ Image plane of the objective lens Figure Schematic of a system of corrector and objective lens. 25

33 Initial slope r o : 8 mrad 7 mrad 6 mrad 5 mrad Figure th order Aberration diagram at the corrector image plane for I=.42 AT. For larger aperture angles, the effect of the 5th order aberrations can no longer be ignored. In that case, they also need to be corrected, which is the topic of the next chapter CONSIDERING FINITE SIZE CROSS-SECTION OF CONDUCTING WIRES In the previous section, we have ignored the cross-section of the wires of toroidal coils comprising N-SYLC to simplify the analysis and computation. In this section, we consider of effect of non-zero cross-section of real conducting wires, e.g., Cu wires. To see the effect of different shapes of crosssection, we consider circular and half-circular cross-sections with radius d and currents I cir. and I half cir., respectively, as shown in Figure To simplify calculations, we will approximate them by finite number of line currents. We show this in Figure 2.16, where the wires are at distance R from the z-axis. In Figure 2.16(a), we show a wire with zero cross-section, i.e., an ideal line current I ideal. In (b), we approximate a circular cross-section with 9 line currents with zero cross-section, each of which is I, thus the total current I cir. = 9I. In (c), we approximate half-circular cross-section with 6 line currents, each of which is I, thus the total current I half cir. = 6I. We consider three versions of the 2x3-SYLC doublet model of Figure 2.9, where each toroidal coil is any one of the three cases of Figure 2.16, and compare the magnetic flux density distributions, aberration correction sensitivity and aberration diagrams of each case. We take d =.15 mm since common Cu wires have diameter.3 mm. All other parameters are as given in Table 2.1, and as before, computer simulation is performed by the LANTERN lens software. I cir. d z I half cir. d z (a) (b) Figure Wires with non-zero cross-section: (a) circular cross-section, (b) half-circular cross-section. 26

34 R = mm R = mm R = mm 4 4 (a) d =.15 mm (b) d =.15 mm (c) Figure Three models: (a) ideal line current with zero cross-section, (b) approximation of wire with circular cross-section, (c) approximation of wire with half-circular cross-section. Magnetic flux density distribution: We calculate magnetic flux density distribution of the sextupole component on the optic axis with the same total currents I ideal = I cir. = I half cir. = I =. 95 AT. The result is shown in Figure 2.17 for a structure with the same orientation as that in Figure 2.3(b). Here we show the coefficient of 2 in the magnetic flux density for sextupole field on the -axis, which is the coefficient of r 3 sin θ in Equation (2.7). Results for line current and circular cross-section are the same, while for half-circular cross-section it is slightly rotated, since it shows non-zero -component unlike the other two. This shows that circular cross-section is equivalent to a single line current with the same total current at the center of the cross-section. The rotation on the - plane depends on the shape of the crosssection and the orientation of 2x3-SYLC. It should be noted that the absolute value or r-components are the same for all three cases as shown in Figure 2.17(c). Figure Coefficient of sextupole magnetic flux density distributions: (a) x component, (b) y component, (c) r component. 27

35 Aberration correction sensitivity: 2x3 SYLC;sim. We calculate the 3rd order spherical aberration coefficient C s for each case, and show the results in Figure We find that the correction sensitivity is almost the same, as is expected from the similarity of the magnetic flux density distributions in Figure 2.17(c)..6 (m) C s 2x3 SYLC;sim Zero cross-section Circular cross-section Half-circular cross-section I ideal, I cir. or I half cir. (AT) Figure Comparison of sensitivity of 2x3-SYLC comprising of ideal line currents with zero cross-section, wires with circular cross-section, and wires with half-circular cross-section. Aberration diagram: We show the aberration diagrams for each case when the 3rd order spherical aberration of the ring coils is corrected in Figure Here for simplicity, diagrams only for initial slope r o = 1 mrad are plotted. We see that the diagrams are the same for line current and circular cross-section, while for half-circular cross section the diagram has the same shape and size, but is slightly rotated. This rotation is expected form the rotation of the magnetic field density lines shown in Figure Zero crosssection Circular crosssection Half-circular cross-section Figure Aberration diagrams at the corrector image plane. 28

36 From the above results, we conclude that wires with finite cross-section is equivalent to wires with zero cross-section ideal line currents, if all the wires in the 2x3-SYLC have the same cross-section. However, if there is non-uniformity in cross-sections, some parasitic aberrations might be generated, which is equivalent to the effect of mechanical errors in positions and/or orientations of line currents. We will discuss how to correct such effects in Chapter SUMMARY We have shown that N-SYLC can be realized by toroidal coils. We have proposed a 2x3-SYLC doublet model to correct 3rd order spherical aberration of rotationally symmetric lens. The required excitation current of 2x3-SYLC is less than.4 AT for beam energy 5 kev. Beam size at the corrector image plane due to the residual geometrical aberrations is less than 1 nm for initial slope less than 6 mrad. In case of an objective lens with magnification.1, this corresponds to beam spread of.1 nm due to geometrical aberrations at the image plane of the objective lens for aperture angle less than 6 mrad. Also considering the effect of diffraction at 5 kev, the beam size is about.37 nm, thereby yielding sub-nanometer resolution. 29

37 3

38 CORRECTION [78] HIGHER ORDER ABERRATION In the previous chapter we have proposed a model for correcting C s. In this chapter, we consider the correction of the higher order residual aberrations, which will allow for larger aperture angles. This will have the advantages of lower diffraction effect for low beam energy and higher probe current for better S/N (signal to noise ratio) imaging ANALYSIS OF HIGHER ORDER ABERRATIONS IN 3-SYLC DOUBLET In this section we analyze the higher order aberrations of the 2x3-SYLC doublet model of Chapter 2. We have already mentioned in the previous chapter without elaborating that after correcting C s, the most dominant higher order aberrations are of 5th order. We can calculate the 5th order aberrations analytically under the no-overlap and ideal N-SYLC conditions. The result at the corrector image plane is as follows (see Appendix A.2 for derivation). ( X 5 ) = A Y 5 r 5 cos(5θ + 6χ) o ( 5 sin(5θ + 6χ) ) + C 5r 5 o ( cosθ ). (3.1) sinθ Here, r o is the initial radial slope dr dz at the corrector object plane, and χ is the rotation of electron trajectories due to the first-round lens (RL 1 in Figure 2.7). Capital letters X, Y in quantities such as X 5, Y 5 indicate they are obtained in the rotating Cartesian coordinate system (X, Y), or in polar coordinates (r, Θ): X = r cosθ, Y = r sinθ. The first term on the right-hand side of Equation (3.1) is sixfold astigmatism (coefficient A 5 ), and it is generated as follows (also see Equation (A.16)). First, as mentioned in Section 2.2.2, the lowest order aberration due to sextupole field by 2x3-SYLC is 2nd order threefold astigmatism (coefficient A 2 ). When A 2 trajectory is deflected by the sextupole field again, a 4th order aberration is generated, which is deflected again by the sextupole field to produce the sixfold astigmatism. The second term on the right-hand side Equation (3.1) is 5th order spherical aberration (coefficient C 5 ). It is a combination of contributions by both the round lenses and 2x3-SYLC. Round lenses generate a small negative C 5, and 2x3-SYLC generates larger positive C 5 to make the overall sign of the C 5 positive. C 5 by 2x3-SYLC is a sum of two terms, which are given by Equation (A.17) and (A.18) in the Appendix A.2. The first term is generated as follows. 3rd order C s trajectory is first deflected by the sextupole field to generate a 4th order aberration, which is deflected again by the sextupole field to generate the C 5 term of Equation (A.17). In a different mechanism, combination effect of the sextupole field with the C s and A 2 trajectories generate the C 5 term of Equation (A.18). 31

39 A 5 term X 5 Y 5 = A 5 r o 5 cos5θ sin5θ C 5 term X 5 Y 5 = C 5 r o 5 cosθ sinθ A 5 term + C 5 term A5 C5 = 1 A5 C5 = 1 1 A5 C5 = 1 Figure th order Aberration diagrams at the corrector image plane for different ratios A 5 C 5. The total effect of A 5 and C 5 is a sixfold symmetric aberration diagram as shown in Figure 3.1. Aberration diagrams due to A 5 term and C 5 term are both round-shaped. However, one rotation of r o corresponds to five rotations of the A 5 term in the opposite direction, whereas a single rotation in the same direction of the C 5 term. The final beam shape depends on relative values of A 5 and C 5 as shown in Figure 3.1. These diagrams are obtained by simply plotting the right-hand side of Equation (3.1), where we have set χ = for simplicity. The case A 5 C 5 = 1: matches well in shape with Figure 2.12(b)(v), where A 5 C 5 = 1:.5 as can be calculated from Table 2.2. In this chapter we will develop methods for correcting A 5 and C 5, which has also been treated by several authors. Haider et al. have optimized the parameters of the system to make A 5 small [36], but 32

40 do not correct it. As for C 5 correction, the focus length of transfer lenses is changed to separate the coma-free plane of the objective lens from the center of the sextupoles, which produces a negative 5th order spherical aberration to correct the residual C 5. However, this changes the paraxial rays, which affects the final scanning area on the sample, i.e., the observation magnification in SEM or STEM. This shift of magnification changes as the observation conditions (pattern thickness, beam energy and so on) change. We are especially interested in SEM for the measurement of semiconductor devices, for which, it is extremely important to maintain the correct observation magnification. This will make controlling an electron optical system with the conventional C 5 corrector described above extremely complicated, and risks of error in observation magnification increases. Hence, we will be interested in correcting all the aberrations with fixed paraxial rays. Sawada et al. have proposed a method to correct A 5 with dodecapole triplet, using sextupole fields [79]. However, for C 5 correction, they reply on the method of Haider et al. described above. Pohner proposed a 6-step system for C s and C 5 correction [8]. However, the correction of A 5 is not addressed, and also the large number of required steps makes the system complicated. Our aim is to correct both A 5 and C 5 simultaneously and independently, without changing the paraxial rays, and with as few steps of N -SYLC as possible STRATEGY FOR HIGHER ORDER ABERRATION CORRECTION To devise methods for correcting A 5 and C 5, we first derive the general form of the aberrations by N-SYLC with any value for N. We will demonstrate here that the A 5 term can be cancelled by 6- SYLC, whereas C 5 by 4-SYLC. If several types of N-SYLC, i.e., N-SYLC with different values for N are present simultaneously, there will also be combination aberrations between different N-SYLCs, which we will consider in later sections. We consider a singlet consisting of two round lenses and an N-SYLC in between them, as shown in Figure 3.2. The round lenses have opposite signs of excitation currents, and they focus electron trajectories with initial slope r o so than they are parallel to the z-axis or the optic axis between the round lenses. We calculate the aberration due to the N-SYLC (i.e., 2N-fold field) at the image plane. The lowest order aberration is an (N 1)-th order aberration called N-fold astigmatism (the coefficient of which is denoted henceforth by A N 1 ), whose integral form at the image plane is as follows. ( X N 1 ) (G H Y N Idz) r N 1 cos((n 1)Θ + Nχ) o ( ), (3.2) N 1 sin((n 1)Θ + Nχ) where, ( X N 1, Y N 1 ) is the (N 1)-th order aberration, G and H are paraxial trajectories given by Equation (2.1), and I is the N-SYLC current. The aberration coefficient A N 1 is then A N 1 (G H N Idz). (3.3) Since it is proportional to the excitation current I, we can manipulate its sign simply by changing the sign or direction of the current. 33

41 Object plane Image plane f f f r o RL 1 RL 2 z o z i z N-SYLC Figure 3.2. Schematic of N-SYLC singlet with two round lenses. The next higher order aberration by N-SYLC is a combination aberration of the above A N 1 term with the 2N-fold field. One way of understanding it is as follows. When the A N 1 ray passes through the 2N-fold field, it is deflected further, which results in the higher order combination aberration. It is a spherical aberration of order 2N 3, and at the image plane is as follows. ( X 2N 3 ) {G H Y N 1 (G H N Idz 2N 3 (3.4) H GH N 1 Idz) Idz} r 2N 3 cos Θ o ( sin Θ ). The aberration coefficient is as follows. C 2N 3 {G H N 1 (G H N Idz H GH N 1 Idz) Idz}. (3.5) It is proportional to the square of the excitation current I, hence its sign is independent of the sign of I. It can in fact be shown that C 2N 3 has minus sign. Correcting A 5 with 6-SYLC: From Equation (3.3) we see that, by superposing 6-SYLC (N=6) onto 2x3-SYLC, and adjusting the excitation current of 6-SYLC, A 5 of Equation (3.1) can be cancelled. Correcting C 5 with 4-SYLC: As for C 5, by superposing 4-SYLC onto 2x3-SYLC, we can generate negative 5th order spherical aberration (since N =4 makes N = 5 in Equation (3.5)), which will cancel the positive C 5 of Equation (3.1). However, according to Equation (3.2), it will also generate a 3rd order fourfold astigmatism (coefficient A 3 ), which is unwanted and hence need to be cancelled. We will address this issue Section

42 In conclusion, we can correct the residual aberrations of 2x3-SYLC doublet by superposing two more types of N-SYLC to it: 6-SYLC (sextupole field) to correct A 5, and 4-SYLC (octupole field) to correct C 5. Next, we will consider aberration correction with such a structure X(3+4+6)-SYLC FOR HIGHER ORDER ABERRATION CORRECTION STRUCTURE OF 2X(3+4+6)-SYLC We need to superpose 6-SYLC and 4-SYLC onto 2x3-doublet to correct A 5 and C 5, respectively. As with the case with 2x3-SYLC for C s correction, we will consider 2x6-SYLC and 2x4-SYLC since their higher order components are negligible and the sensitivity higher. We denote the superposition of 2x3, 2x4 and 2x6-SYLC as 2x(3+4+6)-SYLC. A schematic of it is shown in Figure 3.3, where for clarity, 2x3 and 2x6-SYLC have the same radius, while 2x4-SYLC has a different radius. I N i stands for excitation current of 2xN-SYLC in the i-th step (N=3, 4, 6). In Figure 3.2 there is only one step; however, as we will show in the subsequent sections, several steps are required for desired results. 2x4-SYLC 6 2x(3+6)-SYLC R 1 4 I 4 i I 4 i I i R 2 I 3 i + I i I 3 i + I i 6 Figure 3.3. Schematic of 2x(3+4+6)-SYLC seen on x-y plane. z-axis is the optic axis. Here, for simplicity and clarity, the radii of 2x3-SYLC and 2x6-SYLC are the same, while 2x4-SYLC has a different radius. However, they all can have the same or different radii. As already mentioned in the last section, 2x4-SYLC generates A 3 at the lowest order, and negative C 5 as higher order aberration. We need to cancel this A 3, just as A 2 is cancelled in the 2x3-SYLC doublet for C s correction. Moreover, since there are three types of N-SYLC present simultaneously in Figure 3.3, other unwanted combination aberrations will be generated, which also need to be corrected. We determine all the aberrations of 2x(3+4+6)-SYLC in the following section to investigate how to cancel all the unwanted aberrations CONDITIONS FOR CANCELLING UNWANTED ABERRATIONS We consider a multiplet of 2x(3+4+6)-SYLC which has m steps, with two adjacent 2x(3+4+6)-SYLC being connected by a pair of rotationally symmetric transfer lenses. We show a schematic of this in 35

43 Figure 3.4. The round lenses are denoted by RL n, with n = 1,,, and the 2x(3+4+6)-SYLC are represented by rectangles. Electron trajectories at each step are as shown in Figure 3.2, which are of the form r(z) = r o H(z). Assuming no-overlap and ideal N-SYLC conditions, we analytically calculate all the on-axis aberrations up to 5th order. We list these aberrations, with their dependence on the N-SYLC excitation currents in Table 3.1. These are derived in Appendix A.2. Table 3.1. List of axial geometrical aberrations up to 5th order at the image plane of 2x(3+4+6)-SYLC multiplet. Here, n=order (total power of initial angles) of aberration. means that the corresponding aberration is not generated. Dependence of the aberrations on N-SYLC currents is also shown. Sources of aberrations n 2x3-SYLC (Sextupole field) 2x4-SYLC (Octupole field) Combination effects of 2x3-SYLC and 2x4-SYLC 2x6-SYLC (Dodecapole field) 2 Threefold astigmatism A 2 (I 3 1 I I 3 3 ) 3 C s {(I 3 1 ) 2 + (I 3 2 ) 2 + (I 3 3 ) 2 + } Fourfold astigmatism A 3 (I I I ) 4 Aber 4,3-SYLC a{(i 3 1 I I 3 3 ) 3 } + b{(i 3 1 ) 3 (I 3 2 ) 3 + (I 3 3 ) 3 (I 3 4 ) 3 + }, a, b constants Aber 4,3-SYLC 4-SYLC ( I 3 1 I I 3 2 I 4 2 I 3 3 I ) 5 Sixfold astigmatism (A 5 ), Fifth order positive spherical aberration (C 5 ) C 5 {(I 4 1 ) 2 + (I 4 2 ) 2 + (I 4 3 ) 2 + (I 4 4 ) 2 + } Aber 5,3-SYLC 4-SYLC (five aberrations) A 5 (I I I ) As is already shown in Chapter 2, the lowest order aberration from 2x3-SYLC is 2nd order threefold astigmatism (coefficient A 2 ), which generates negative C s as higher order combination aberration. Further higher order aberrations are 4th order aberration (Appendix A.2, Equation (A.11) (A.12)) and two 5th order aberrations (coefficients A 5 and C 5 ) as discussed in Section

44 Corrector object plane As for 2x4-SYLC, the lowest aberration is 3rd order fourfold astigmatism (coefficient A 3, Appendix A.2, Equation (A.1)), which generates 5th order negative spherical aberration as higher order combination aberration (Appendix A.2, Equation (A.19)). There are no other aberrations solely from 2x4-SYLC up to 5th order. The combination effect of 2x3-SYLC and 2x4-SYLC generates unwanted 4th order (Appendix A.2, Equation (A.13) (A.14)) and 5th order (Appendix A.2, Equation (A.2) to (A.24) aberrations. Finally, 2x6-SYLC produces A 5 at the lowest order (Appendix A.2, Equation (A.15)), with no other aberrations up to 5th order. f f f f f f f f f f f f 4 z o RL 1 RL 2 RL 3 RL 4 RL 5 RL 6 R Electron trajectory z L 2x(3+4+6)-SYLC 2x(3+4+6)-SYLC 2x(3+4+6)-SYLC Step 1 Step 2 Step 3 Total m steps H trajectory G trajectory Figure 3.4. Schematic of 2x(3+4+6)-SYLC multiplet with total step number m. Now, we can achieve a 2x(3+4+6)-SYLC multiplet system with all the on-axis geometrical aberrations up to 5th order corrected as follows. First, the current of 2x3-SYLC is adjusted in each step so that the negative C s generated by this cancels the positive C s of round lenses. Next, the currents of 2x4-SYLC is adjusted so the negative C 5 from it corrects the positive C 5 by 2x3-SYLC. Then, 2x6-SYLC current is adjusted to correct the A 5 by 2x3-SYLC. It is to be noted that corrections of C 5 and A 5 are independent of each other, so they can be performed in any order. Now the unwanted aberrations, which are shown in red in Table 3.1, are left to correct. We first require that A 2, A 3, Aber 4,3-SYLC 4-SYLC and Aber 4,3-SYLC must vanish, the requirements for which are as follow. Condition for A 2 = : I 1 3 I I 3 3 = (3.6) Condition for A 3 = : I I I = (3.7) Condition for Aber 4,3-SYLC 4-SYLC = : I 1 3 I I 2 3 I 2 4 I 3 3 I = (3.8) Condition for Aber 4,3-SYLC = : (I 1 3 ) 3 (I 2 3 ) 3 + (I 3 3 ) 3 (I 4 3 ) 3 + = (3.9) 37

45 We first derive the conditions for Equation (3.6) to (3.9), and then determine the additional conditions for the remaining unwanted aberrations Aber 5,3-SYLC 4-SYLC to vanish. The reason for doing so is that, the former four types of aberrations have simple dependence on the excitation currents of N-SYLC and hence are easier to analyze. It is immediately clear that we need at least three steps (m 3 in Figure 3.4) to satisfy Equation (3.6) to (3.9). Some possible combinations satisfying these conditions with number of steps m =, 4 are shown in Table 3.2. By calculating Aber 5,3-SYLC 4-SYLC for all these cases, we find that at least four steps (quadruplet) are required so that Aber 5,3-SYLC 4-SYLC=. Three such possible combinations are given in Table 3.2. These are the desired models in which all the axial geometrical aberrations up to 5th order are corrected. Table 3.2. Combinations of currents for several triplet and quadruplet systems satisfying A 2 = A 3 =Aber4,3- SYLC 4-SYLC =Aber4,3-SYLC =. Aber5,3-SYLC 4-SYLC vanish only for certain quadruplets (No. 3, 4, and 5). No. Number of steps, m N-SYLC type 1st Step N-SYLC currents 2nd Step 3rd Step 4th Step Aber 5,3-SYLC 4-SYLC 1 3 (Triplet) 2 3 (Triplet) 2x3-SYLC I 3 I 3 2x4-SYLC I 4 I 4 I 4 2x3-SYLC I 3 I 3 2x4-SYLC I N4 I 4 I 4 Non-zero Non-zero 3 4(Quadruplet) 2x3-SYLC I 3 I 3 I 3 I 3 2x4-SYLC I 4 I 4 I 4 I 4 Zero 4 4(Quadruplet) 2x3-SYLC I 3 I 3 I 3 I 3 2x4-SYLC I 4 I 4 I 4 I 4 Zero 5 4(Quadruplet) 2x3-SYLC I 3 I 3 I 3 I 3 2x4-SYLC I 4 I 4 I 4 I 4 Zero 6 4(Quadruplet) - SYLC 4 - SYLC I 3 I 3 I 3 I 3 I 4 I 4 I 4 I 4 Non-zero 38

46 It is to be noted that, to correct A 5, theoretically it is not essential to add 2x6-SYLC to all steps; only a single step is enough. Thus, after correcting C s with 2x3-SYLC doublet as described in Chapter 2, if we intend to correct only the residual A 5 but not C 5, a simple 2x(3+6)-SYLC doublet is enough; we do not need a 4-step corrector. Similar doublet model to correct C s and A 5 with conventional sextupole and dodecapole is not practical, since the dodecapole component lacks the required sensitivity due to magnetic saturation of the pole pieces. On the other hand, N -SYLC is free of magnetic saturation, and as shown later in Section 3.4, 2x6-SYLC has high enough sensitivity CORRECTING SENSITIVITY The sensitivities of 2x4-SYLC and 2x6-SYLC in 2x(3+4+6)-SYLC for C 5 and A 5 correction, respectively, depend on the system parameters as follow. C 5 correction sensitivity: I 4,C R 4 4 Φ 1 2f 3 L 3 2, (3.1) A 5 correction sensitivity: I,C R Φ 1 2f L 1, (3.11) where, I 4,C : The absolute value of 2x4-SYLC current required for C 5 correction, I,C: The absolute value of 2x6-SYLC current required for A 5 correction, R 4 : 2x4-SYLC inner radius, R : 2x6-SYLC inner radius, eφ: Electron energy, L: Length of N-SYLC COMPUTER SIMULATION The arguments of the previous sections are based on analytical calculations where, for simplicity, we have assumed the no-overlap and ideal N-SYLC conditions of Section 2.2.2, and also discarded the combination aberrations of N -SYLC and round lenses. In this section, in order to evaluate the aberration correction effects with a realistic 2x(3+4+6)-SYLC, we perform numerical calculation using the LANTERN software, in which these simplifications are not imposed. We consider a quadruplet or four-step 2x(3+4+6)-SYLC as shown in Figure 3.5. In the numerical calculation, as with the case of 2x3-SYLC doublet model, ring coils are used as round lenses and toroidal coils as N-SYLC. Two adjacent ring-shaped coils have opposite signs of excitation currents. We consider electron trajectories of the form r(z) = r o H(z), and the various parameters in the numerical calculation are given in Table

47 z o =-2 Corrector object plane R R L L Toroidal coils as 2x(3+4+6)-SYLC (here only three coils are shown for simplicity) Ring coils as rotationally symmetric lens R L R L 175 z i =2 Corrector image plane z Figure 3.5. Schematic of 2x(3+4+6)-SYLC quadruplet model for numerical calculation. The ring coil current required to focus the paraxial trajectories is 5 AT. Here we present the results of aberration calculations for three quadruplet models, namely, No.3, 4 and 5 of Table MODEL 1 First, we consider the model in which each step of 2x3-SYLC has the same value of excitation current, with the signs in the 1st, 2nd, 3rd and 4th step being +,+,+,+, respectively. Each step of 2x4- SYLC has the same absolute value of excitation current, with the signs in the 1st, 2nd, 3rd and 4th step being +,+,-,-, respectively (No.3 in Table 3.2). In Figure 3.6, we show the aberration diagrams at the corrector image plane z i of Figure 3.5. Here, diagrams denoted by (i) is the result of ray trace, i.e., diagram including aberrations of all orders. Diagrams (ii) through (v) are aberration diagrams due to 2nd, 3rd, 4th and 5th order aberrations, respectively, and are extracted from (i) by Fourier analysis as described in Appendix A.3. 4

48 Table 3.3. Values of various parameters of the 2x(3+4+6)-SYLC quadruplet system. Parameter Coil length of each 2xN-SYLC Inner radius of each 2x4-SYLC Inner radius of each 2x3-SYLC and 2x6-SYLC Outer radius of each 2xN-SYLC Radius of each RL i Focal length of each RL i: f Length of the system (z o -z i distance): 16f Beam energy: eφ Value 35 mm 3 mm 2 mm 65 mm 1 mm 25 mm 4 mm 5 kev Initial position: r o = ( o) 2 + ( o) 2 Initial slope: r o = ( o ) 2 + ( o ) 2 2, 4, 6, 8, 1 mrad Azimuth angles at the corrector object plane, 2, 4,, 358 degrees: 18 angles for each r o With all the N-SYLC turned off, C s at z i is about 845 mm, which is the most dominant aberration, and the beam diameter is about 169 nm (Figure 3.6(a)). Then we turn on the 2x3-SYLC, and adjust the coil currents to correct C s. We find that C s is corrected for I 3 1 = I 3 2 = I 3 3 = I 3 4 = I 3 =. 144 AT, and the dominant residual aberrations are of 5th order, which results in an aberration diagram with sixfold symmetry (Figure 3.6(b)). Next, by turning on the 2x4-SYLC, and setting I 4 1 = I 4 2 = I 4 3 = I 4 4 = I 4 = AT, C 5 is corrected, and the residual aberration is mainly due to A 5 (Figure 3.6(c)). To correct it, we turn on the 2x6-SYLC and adjust the coil currents. We found that we need to adjust the orientation of the 2x6- SYLC as well. By rotating 2x6-SYLC by 6.45 in the anticlockwise direction with respect to 2x3-SYLC and 2x4-SYLC (which have the same orientation) at each step, and by setting I 1 = I 2 = I 3 = I 4 = I =. 9 AT, the residual A 5 is also corrected (Figure 3.6(d)). As a result, we obtain a beam size of less than 1 nm for initial slope r o 1 mrad. The changes of the values of aberration coefficients C s, C 5 and A 5 at the corrector image plane z i as different types of N-SYLC are turned on are given in Table 3.4. The residual aberrations after C 5 and A 5 correction is of 6th order and higher. 41

49 (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order (a) (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order (b) (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order (c) (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order (d) Initial slope r o 1 mrad 8 mrad 6 mrad Figure 3.6. Beam cross section of Model 1 at the corrector image plane obtained by numerical calculations for initial slopes r o = 6, 8, 1 mrad. 42

50 Table 3.4. Values of aberration coefficients computed by the LANTERN Lens software. All N-SYLC turned off -SYLC turned on: C s correction 4-SYLC turned on: C 5 correction 6-SYLC turned on: A 5 correction Aberration coefficients ( I 3 =. 144 AT, I 4 = AT and I = AT) ( I 3 =. 144 AT, I 4 = AT and I = AT) ( I 3 =. 144 AT, I 4 = AT and I =. 9 AT) (Figure 3.6(a)) (Figure 3.6(b)) (Figure 3.6(c)) (Figure 3.6(d)) C s.84 m.4 μm.4 μm.4 μm C m 55.5 m 11.4 mm 11.4 mm A 5 m 16. m 16. m 6. mm It can be seen from Figure 3.6(d) that the residual aberrations higher than 5th order have a deflection effect, since the center of the diagram is away from the z-axis. Some adjustment of the N- SYLC currents can cancel this deflection. For example, as shown in Figure 3.7, by setting I 4 1 = I 4 4 = AT and I 4 2 = I 4 3 =.6 AT, there is a residual 4th order deflection (Appendix A.2, Equation (A.14)) which cancels the deflection effect of Figure 3.6(d). As a result, the beam crosssection can be brought to z-axis, with the total beam size being about.8 nm for initial slope 1 mrad (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order Initial slope r o 1 mrad 8 mrad 6 mrad Figure 3.7. Result of adjustment of 2x4-SYLC currents in Model 1 (initial slopes r o = 6, 8, 1 mrad). The deflection effect of 4th order term balances the deflection effect of higher than 5th order aberration MODEL 2 Next, we consider the model in which 2x3-SYLC currents have signs +,+,+,+, and 2x4-SYLC +,-,-,+ (No.4 in Table 3.2). For 2x3-SYLC currents I 3 1 = I 3 2 = I 3 3 = I 3 4 =. 144 AT, 2x4-SYLC currents I 4 1 = I 4 2 = I 4 3 = I 4 4 = AT and 2x6-SYLC currents I 1 = I 2 = I 3 = I 4 =. 9 AT, and with the same 2x6-SYLC rotation as Model 1, axial geometrical aberrations up to 5th order are almost cancelled at the corrector image plane z i. The resultant beam cross section at z i is shown in Figure 3.8. The final beam size at the corrector image plane is less than.4 nm for initial slope r o 1 mrad. 43

51 (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order Initial slope r o 1 mrad 8 mrad 6 mrad Figure 3.8. Beam cross section of Model 2 at the corrector image plane obtained by numerical calculations for initial slopes r o = 6, 8, 1 mrad MODEL 3 In this model 2x3-SYLC currents have signs +,+,-,-, and 2x4-SYLC +,-,+,- (No.5 in Table 3.2). For 2x3- SYLC currents I 3 1 = I 3 2 = I 3 3 = I 3 4 =. 144 AT, 2x4-SYLC currents I 4 1 = I 4 2 = I 4 3 = I 4 4 = AT and 2x6-SYLC currents I 1 = I 2 = I 3 = I 4 =. 9 AT, and with the same 2x6-SYLC rotation as Model 1, axial geometrical aberrations up to 5th order is almost cancelled at the image plane z i. The resultant beam cross sections at the corrector image plane z i are shown in Figure 3.9. The final beam size is less than.5 nm for initial slope r o 1 mrad (i) Ray trace (ii) 2nd order (iii) 3rd order (iv) 4th order (v) 5th order Initial slope r o 1 mrad 8 mrad 6 mrad Figure 3.9. Beam cross section of Model 3 at the corrector image plane obtained by numerical calculations for initial slopes r o = 6, 8, 1 mrad. It is to be noted that the absolute values of -SYLC, 4-SYLC and 6-SYLC currents required in Model 1, 2, and 3 are the same, hence all three models have the same sensitivity. We see from Figure 3.7, 3.8 and 3.9 that aberrations of 6th and higher order are negligible in models 2 and 3, hence these two are preferable to model 1. However, in a real system, the most suitable model should be determined also by taking into account the effect of parasitic aberrations that might occur due to imperfections in sizes and shapes of the N-SYLC and the overall performance including an objective lens. It is also to be noted that we can switch from one model to another just by changing the signs of N-SYLC currents, with no structural or hardware change. 44

52 For simplicity, here we have considered only the 1:1 imaging of the object plane z o to the image plane z i in Figure 3.5. When there is an objective lens after z i plane, the excitation current of 3-SYLC should be increased compared to the value in Table 3.4 to correct the C s of the objective lens. Consequently, the effect of C 5 and A 5 will increase, and the currents of 4-SYLC and 6-SYLC should also be modified to correct them. However, the principle of correction remains the same as describe above DISCREPANCIES BETWEEN ANALYTICAL AND SIMULATION RESULTS There are few discrepancies between the analytical results and computer simulation results. First, the adjustment of the orientation of 2x6-SYLC, namely its rotation with respect to 2x3-SYLC and 2x4- SYLC does not follow from the analytical calculations of Section 3.3. One of the reasons for this rotation is the overlap of ring coil field and N-SYLC fields. In analytical calculations, we assume that they do not overlap (the no-overlap and ideal N-SYLC conditions of Section 2.2.2). However, the fields of the ring coils in computer simulation enter the regions of the N-SYLC, which makes the electron trajectories to rotate inside the N-SYLC, an effect not taken care of in analytical calculations. The effect of this rotation is different for A 5 originating from 2x6-SYLC as the lowest order aberration, and the one originating from 2x3-SYLC as a higher order combination aberration. The adjustment of rotation is smaller when there is less overlap of the fields of N-SYLC and those of the ring coils. However, it should be possible to realize this rotation electrically with few supplementary line currents if necessary. Second, the non-vanishing 3rd order aberration (A 3 ) in Figure 3.8(iii) and 2nd order aberration (A 2 ) in Figure 3.9(ii) are due to deviation from Equation (3.7) and Equation (3.6), respectively. This happens because ring coil field in each step, which determine the paraxial trajectories, does not have the desired symmetry due to their wide field distribution. However, these are not fundamental problems; the beam size is already smaller than 1 nm at the corrector image plane, and using round lenses with narrower field distributions in real corrector should decrease this effect and yield an even smaller beam size. Narrower field distribution can be achieved by optimizing the winding of the ring coils or using round lenses with proper shielding SUMMARY In this chapter we have proposed a novel 2x(3+4+6) quadruplet model for correcting axial geometrical aberrations up to 5th order. The required N-SYLC current is less than 1 AT for beam energy 5 kev. After correction, the beam size at the corrector image plane is less than 1 nm for initial slope less than 1 mrad, which is nearly twice the initial slope allowed for similar beam size in the 2x3- SYLC model of Chapter 2. 45

53 46

54 Corrector object plane Corrector image plane Corrector object plane Corrector image plane ABERRATION CORRECTION WITH IN-LENS N-SYLC [81] The magnetic material free structure of N-SYLC can make the design of electron optical system more flexible, simpler and versatile, and also make the system more compact. In this chapter, with such advantages in mind, we consider immersing N-SYLC into round lens fields, which we denote as in-lens N-SYLC model. This would be impossible with conventional multipoles since the pole pieces would absorb any surrounding magnetic fields and produce unwanted and uncontrollable effects. We describe here the in-lens N-SYLC structure and consider 3rd order spherical aberration correction with in-lens 3-SYLC doublet IN-LENS N-SYLC STRUCTURE We show a schematic of in-lens N-SYLC doublet model in Figure 4.1(a). For comparison, we also show a schematic of the 3-SYLC doublet of Chapter 2 in Figure 4.1(b), which we henceforth denote as the Rose-type model for convenience. We show the outer-most electron trajectories with initial slope r o. The trajectories are fixed by the rotationally symmetric lenses, which are denoted by RL with a subscript to enumerate. In Figure 4.1(a), RL 1 and RL 2 have opposite signs of excitation. In Figure 4.1(b), RL 1 and RL 3 have positive excitation, while RL 2 and RL 4 have negative excitation. Threefold symmetric line currents (3-SYLC or 2x3-SYLC) are represented by a pair of rectangles as before, and overlap the rotationally symmetric lenses in Figure 4.1(a). d d d d f f f f f f f f r o Electron trajectory z r o RL 1 RL 2 RL 3 RL 4 z z o RL 1 RL 2 zi z o z i 3-SYLC/ 2x3-SYLC (a) 3-SYLC/ 2x3-SYLC 3-SYLC/ 2x3-SYLC (b) 3-SYLC/ 2x3-SYLC Figure 4.1. Schematics of (a) in-lens 3-SYLC (or, 2x3-SYLC) doublet and (b) the Rose-type model. 47

55 In case of in-lens 3-SYLC, it is desired that RL 1 and RL 2 are also free of magnetic materials, otherwise the lenses might affect the 3-SYLC field and vice versa. Thus, the model may lose its advantages as a corrector free of magnetic materials. One way to avoid this is to use ring coils as RL 1 and RL 2, which will be discussed in more details in Section ABERRATION ANALYSIS OF IN-LENS 3-SYLC In this section, we will consider C s correction with in-lens 2x3-SYLC doublet model. To make the analytical calculation simpler, we impose thin lens approximations on the rotationally symmetric lenses, and ideal N-SYLC approximations on 2x3-SYLC, which allow us to calculate the aberrations analytically APPROXIMATIONS FOR ANALYTICAL CALCULATIONS We assume the following thin lens approximations for the rotationally symmetric lenses of Figure 4.1(a). (i) (ii) The lenses only change the slopes of electron trajectories, and do not cause any shift of the trajectories in the radial direction. The rotation χ of the trajectories due to the lenses is a step function and is constant in between the round lenses. As for the 2x3-SYLC, we will use the ideal N-SYLC condition of Section 2.2.2, which we repeat below. (Note that we cannot use the no-overlap conditions mentioned in Section since the lenses and the N-SYLC here do overlap each other physically.) (iv) (v) We use sharp cut-off fringing field (SCOFF) approximation for the N-SYLC fields. The fields are taken to be non-zero only in the region of the N-SYLC. In the region of N-SYLC, we take the fields to be that for infinitely long line currents. With these approximations, we analytically investigate the basic properties of in-lens 2x3-SYLC model PARAXIAL TRAJECTORIES To analyze aberrations, we first need to fix the paraxial trajectories. In Fig 4.2(a), we show the paraxial trajectories G(z) and H(z) in the rotating coordinate system in accordance with thin lens approximation (i) mentioned above, and, in Figure 4.2(b), the rotation due to magnetic field of the rotationally symmetric lenses in accordance with thin lens approximation (ii). G(z) and H(z) are paraxial trajectories satisfying G(z o ) = 1, G (z o ) = ; H(z o ) =, H (z o ) = 1. The focal length of each lens is f, and the total length of the corrector 4d. For the G trajectory given in Figure 4.2(a), parameters such as f, q and l in the H trajectory follow from lens equation and simple geometry. They are expressed in terms of d in Figure 4.2(a). The region of 2x3-SYLC with length L in each step is shown by the shaded grey areas in Figure

56 d d d d H trajectory G trajectory G z o = 1 G z i = 1 z = z z o Corrector object plane 2x3-SYLC length L f = d l = d 4 z i Corrector image plane q = 2 1 G z o = G z o Thin lens RL 1 Step 1 (a) Thin lens RL 2 Step 2 d d d d Rotation z o Corrector object plane 2x3-SYLC χ z = 2x3-SYLC Thin lens RL 1 Thin lens RL 2 Step 1 Step 2 (b) z i z Corrector image plane Figure 4.2 Trajectories in the in-lens 2x3-SYLC model under thin lens approximations: (a) paraxial trajectories in the rotating coordinate system and (b) the rotation angle ABERRATION CALCULATION We calculate the aberrations of the model of Figure 4.2 up to 3rd order by solving the equations of motion by means of successive approximation (for details of the mathematics involved, see Appendix A.2, or Chapter 1 of [71]). There are no combination aberrations between the 2x3-SYLC and RL i up to 3rd order. The lowest order aberration is second order astigmatism (coefficient A 2 ) due to the sextupole field of 2x3-SYLC, and is given at the corrector image plane as follows. [ X Y ] = ημ R 3 Φ r o 2 G(z i )(I 1 I 2 ) H 3 cos( Θ + χ) [ ] dz. (4.1) sin( Θ + χ) 49

57 The integration is over the region of the first 2x3-SYLC. Here, R is the inner radius of 2x3-SYLC, r o is the initial slope at the corrector object plane (see Figure 4.1), and χ is the rotation of paraxial trajectories due to RL 1. We have denoted the 2x3-SYLC current in step 1 as I 1 and in step 2 as I 2. The next higher order aberration due to 2x3-SYLC is of 3rd order. After lengthy calculations, it can be shown that at the corrector image plane, the aberration is as follows. [ X Y ] = r o 3 cos Θ {C [ sin Θ ] + C cos(θ + χ) + [ sin(θ + χ) ] + C cos(θ χ) [ ]}. (4.2) sin(θ χ) It is a combination of three spherical aberration terms with phases, χ and χ. Here, C = C 1 {(I 1 ) 2 + (I 2 ) 2 } + (C 3 + C 4 )I 1 I 2, C + = C 2 (I 1 ) 2 + ( C 3 + C 4) I 1 I 2, where, C = C 2 (I 2 ) 2 + ( C 3 + C 4) I 1 I 2, with C 1 = κ G(z o) ( 1 f 84 L7 d L d L5 d L 4 d L3 d 5 ), C 2 = κ G(z o) ( 1 f 16 L8 5 1 L7 d L d L5 d L4 d 4 1 L 3 d 5 ), C 3 = κg(z o ) ( 1 L L d 7L 5 d L 4 d 3 1L 3 d 4 + 4L 2 d 5 ), C 4 = κ G(z o) ( 1 f 8 L L7 d 14 L d L 5 d L4 d L 3 d 5 4L 2 d ), κ = 18 ( ημ R 3 Φ ) 2. We introduce a notation ΔX lmpq for the coefficient of X o l Y o m X o p Y o q (l, m, p and q are integers) in X, and similarly ΔY lmpq for Y. Then Equation (4.2) can be rewritten as follow. ΔX 3 C + (C + + C ) cos χ ΔX X: [ 21 (C ] = + C ) sin χ, (4.3) ΔX 12 C + (C + + C ) cos χ ΔX 3 [ (C + C ) sin χ ] ΔY 3 (C + C ) sin χ ΔY Y: [ 21 C ] = + (C + + C ) cos χ. (4.4) ΔY 12 (C + C ) sin χ ΔY 3 [ C + (C + + C ) cos χ] Now, if we impose the condition I 1 = I 2 = I, then A 2 of Equation (4.1) vanishes. Also, under this condition, C + = C, thus ΔX 21 = ΔX 3 = ΔY 3 = ΔY 12 =. Then Equation (4.3) and (4.4) become 5

58 [ ΔX 3 ΔX 21 ΔX 12 ΔX 3 ] = [ ΔY 3 ΔY 21 ΔY 12 C s in C s in ], (4.5) [ ] = [ ], (4.6) in ΔY 3 C s which are of the form of 3rd order spherical aberration with coefficient C in s. Here, C s in with, C s in = C + (C + + C ) cos χ C s uncoupled + C s coupled, (4.7) C uncoupled s = (C 1 + C 2 cos χ){(i 1 ) 2 + (I 2 ) 2 }, (4.8) coupled C s = (C3 + C 4 )(1 + cos χ)i 1 I 2. (4.9) Here, C s coupled is a coupled term, i.e., it is a cross term of contributions from step 1 and 2, since it includes I 1 I 2. This term is non-zero only when both I 1 and I 2 are non-zero, i.e., both steps are turned on. On the other hand, C s uncoupled is an uncoupled term, i.e., it is simple addition of the contributions from step 1 and PARAMETER DEPENDENCE OF THE SIGN AND VALUE OF C s in In this section, we investigate the characteristics of the spherical aberration coefficient C s in of inlens 2x3-SYLC based on Equation (4.7) to (4.9). We set the values of different parameters as shown in Table 4.1. Table 4.1. Parameters in analytical calculation Parameter Value 2x3-SYLC length L 4 mm 2x3-SYLC radius R 2 mm Corrector length 4d 2 mm Ring coil focus length f = d 25 mm Electron energy eφ 5 ev 2x3-SYLC current I 1 = I 1 = I 1 AT Rotation χ Variable With the values of Table 4.1, Equation (4.7), (4.8) and (4.9) can be plotted as functions of χ, the rotation of the paraxial trajectories due to the rotationally symmetric lenses. The result is shown in Figure 4.3. The value of χ in a real corrector will depend on the detailed design of RL i, for example, its geometric dimensions and shapes, focus length, and so on, and may not be always easy to control. Here we have evaluated the aberration coefficients for various values of χ. We see that the sign of C s coupled does not change depending on χ as is clear from Equation (4.9), and is always negative: C s coupled. On the other hand, C s uncoupled can be negative or positive depending on χ. The overall spherical aberration coefficient C s in is negative and its absolute value largest for χ satisfying cos χ = 1, while it is positive around χ satisfying cos χ = 1. In between, there are points where C s in =. 51

59 Aberration coefficient (m) Aberration coefficient (m) Aberration coefficient (m) I1 と I2 合計 C s uncoupled I1I2 total C s coupled Cs1 C in s = C uncoupled coupled s + C s χ (deg) Figure 4.3. χ dependence of aberration coefficients for fixed currents I 1 = I 2 = 1 AT. C s in also depends on 2x3-SYLC length L. With the values of Table 4.1 (except for L), we calculate the L dependence of the aberration coefficients for two values of χ, namely χ = 5 o and 51.5 o. The results are shown in Figure 4.4. For χ = 5 o, which is away from the point cos χ = 1 in Figure 4.3, both C s coupled and Cs uncoupled, and Cs coupled is much larger than Cs uncoupled. Hence, C s in is negative and its value is determined primarily by C s coupled, which increases as the 2x3- SYLC length L increases. For χ = 51.5 o, which is in the region near cos χ = 1 in Figure 4.3, C s coupled and C s uncoupled, and they both have comparable absolute values. Cs in is negative for smaller L while positive for larger L, and the absolute value of C s in is also smaller than in the case χ = 5 o L (mm) L (mm) uncoupled I1 Cと I2 合計 s coupled I1I2 C total s Cs1 in C s C uncoupled s C s coupled (a) (b) Figure x3-SYLC length (2L) dependence of aberration coefficients for (a) χ = 35 o and (b) χ = o, with I 1 = I 2 = 1 AT. 52

60 To function as an aberration corrector, C s in should be negative, and its absolute value should also be large for high sensitivity. Hence, case like Figure 4.4(a) is desirable for in-lens 2x3-SYLC model. Although it might not always be possible to control χ fully, it should be made sure that we have negative C s in with enough sensitivity for some optimized value of 2x3-SYLC length L COMPUTER SIMULATION The analytical results in the last section are obtained under the thin lens approximation for the rotationally symmetric lenses, and ideal N -SYLC approximation for 2x3-SYLC. This allowed us to investigate the basic properties of in-lens 2x3-SYLC model analytically. In this section, to evaluate a realistic case, we perform computer simulations without imposing the thin lens and the ideal N-SYLC approximations MULTI-WINDING RING COIL MODEL As is already mentioned at the end of Section 4.1, it is desirable that the rotationally symmetric lenses are realized without magnetic materials. We consider using ring coils as rotationally symmetric lenses for this purpose. We first need to find a realistic structure for the ring coils. For this, we will consider using not a single coil, but many windings to distribute the current, with the aim of making the current through a single coil required for focusing of electron trajectories less than 1 A. Making the radius of ring coils smaller makes the total required excitation current smaller; however, the positive 3rd order spherical aberration due to the ring coils become larger. This will in turn require larger excitation current for 2x3-SYLC to cancel that aberration and have enough negative spherical aberration to cancel the 3rd order spherical aberration of an objective lens. Hence, we need to optimize the model so that both the ring coil current and N-SYLC current are within acceptable ranges. In the simulation, we use multi-winding ring coil structure as shown in Figure 4.5. In each step, there are 51 turns in the z direction, while 1 turns in the radial direction. Thus, the total turn number is 51 1 = 51. The space between two adjacent coil currents both in z direction and radial direction is.3 mm, since real Cu wires will have similar diameter. The innermost diameter of ring coil is 7.3 mm, while the outermost diameter is 1. mm. We define the z position of the multi-winding ring coil to be the center of it (the position of the 26th ring coil in the z direction) SIMULATION MODEL We show the simulation model in Figure 4.6. Each of the multi-winding ring coil is as shown in Figure 4.5. For 2x3-SYLC we use toroidal coils. The values of various parameters are given in Table 4.2, and as the simulation tool, the LANTERN Lens software is used. 53

61 15. mm Corrector object plane R Corrector image plane.3 mm 1 Turns.3 mm 7.3 mm z z 51 Turns 1. mm 1 Turns 51 Turns (a) (b) Figure 4.5. Multi-winding ring coil: (a) Outer view, (b) cross-section schematic. 2x3-SYLC Multi-winding ring coils 2x3-SYLC z L L z = 1 mm z = 5 mm z = z = 5 mm z = 1 mm Step 1 Step 2 Figure 4.6. In-lens 2x3-SYLC doublet model with multi-winding ring coils. Large rectangles represent the toroidal coils of 2x3-SYLC. Shaded rectangles show cross-section of multi-winding ring coils. 54

62 Table 4.2. Parameters in computer simulation Parameter Value Corrector length 4d 2 mm Multi-winding ring coil position -5 mm, 5 mm 2x3-SYLC position -5 mm, 5 mm 2x3-SYLC (toroidal coil) inner radius R 2 mm 2x3-SYLC (toroidal coil) outer radius 65 mm Corrector object plane -1 mm Corrector image plane 1 mm Electron energy eφ 5 ev Sign of ring coil excitation in step 1 Positive Sign of ring coil excitation in step 2 Negative Initial slope r o of electron trajectories at the 2, 4, 6, 8, 1 mrad corrector object plane Azimuth angles at the corrector object plane, 2, 4,, 358 degrees: 18 angles for each r o SIMULATION RESULTS First, we compute the focusing condition of the ring coils, with 2x3-SYLC turned off. The result is as follows. Current through a single coil = A, Total current in each step = = AT. Spherical aberration coefficient due to the ring coils, C s RL =2.19 m Thus, the current through a single ring coil is less than 1 A. We show the trajectories calculated by the LANTERN Lens software in Figure 4.7, and the rotation angle χ in Figure 4.8. Corrector object plane component component Corrector image plane z Focusing by the multi-winding ring coil in Step 1 Focusing by the multi-winding ring coil in Step 2 Figure 4.7. Electron trajectories calculated by the LANTERN Lens software in the fixed (non-rotating) coordinate system. Here, trajectories for three initial slopes r o = 6, 4, 2 mrad, and several azimuth angles for each of them are shown. 55