Chromatic Aberration Coefficients Research Articles

Design and optimization of a conical electrostatic objective lens of a low-voltage scanning electron microscope for surface imaging and analysis in ultra-high-vacuum environment

Low-voltage scanning electron microscopy (LV-SEM) with landing energies below 5 keV has been widely used due to its advantages in mitigating the damage and charging effects to a specimen and enhancing surface information due to small interaction volume of electrons inside a specimen. Additionally, for elemental analysis of the surfaces of bulk specimens with Auger electron spectroscopy (AES) or electron energy loss spectroscopy (EELS), ultra-high-vacuum (UHV) environment is essential to maintain clean surfaces without the absorption of gas molecules during the electron beam irradiation for the acquisition of spectral data. In this study, we propose the optimal design and condition of a conical Electrostatic Objective Lens (EOL) for a UHV LV-SEM to achieve the high spatial resolution and secondary electron (SE) detection efficiency. The EOL is composed of only the three electrodes (retarding, focusing and booster electrodes) and the insulators, which is suitable for maintaining a UHV environment with less out-gassing. The cone angle of the EOL is determined as 60° to integrate a spectrometer in the UHV LV-SEM and in a large size and a higher tilt angle of the sample. Through the optimization with the simulations, the EOL achieves the minimized spherical and chromatic aberration coefficients of 0.05 and 0.03 mm at the sample side, respectively, at the landing energy of 50 eV and the shortest working distance (WD) of 1 mm for high-resolution imaging. In addition, the probe diameter of the optimized EOL is 2.3 nm at 1 keV and 5.7 nm at 50 eV with a WD of 1 mm and a probe current of 10 pA, which are comparable to previously studied compound objective lenses with magnetic and electrostatic lenses. Using a longer WD of 4 mm for analysis, the probe diameter was 5.4 nm at 1 keV and the SE detection efficiency was 83.3 % owing to the separated scintillator detector structure from the booster electrode.These results imply that the optimized EOL has the potential to be applied to a high-performance UHV LV-SEM for the surface imaging and analysis with a simple system configuration.

First- and third-order chromatic aberrations in Glaser magnetic lens for object magnetic immersion

In this paper, the Gaussian optical properties and the first- and third-order isotropic and anisotropic chromatic aberrations of the Glaser magnetic lens have been derived analytically and then calculated numerically based on two methods: the differential-algebraic DA and the electron optical aberration integrals. These coefficients have been recalculated for object magnetic immersion OMI. The expressions of chromatic aberration coefficients with the inclusion of the OMI effect of round electron lenses were not published in the literature until the present work. The numerical results of high-order chromatic aberration coefficients of Glaser magnetic lenses calculated using the DA method are shown to be in excellent agreement with those calculated using the integral aberration method and with the minor relative errors of order (10−7- 10−8), indicating that all the formulas for the chromatic aberration coefficients are entirely correct. For this purpose, COSY INFINITY 10 and Mathematica 11 were used, and both proved to be excellent computer programs for this work.

A differential algebra method for aberration analysis of variable-axis lens system

A variable-axis lens (VAL) is an effective method for achieving a large field of view with high resolution because it can preserve small aberrations by shifting the optical axis according to the deflection of the electron beam, even when the beam is deflected to regions far from the central straight optical axis. However, for systems consisting of such a VAL, conventional methods for calculating aberration coefficients that refer to the central straight optical axis could be meaningless. Here, we propose a novel approach, based on the differential algebraic(DA) method, for calculating high-order aberration coefficients for VAL systems. Instead of using the straight optical axis as a reference, we trace only one electron trajectory in such a system. Then, the linear optical properties and the geometric aberration and chromatic aberration coefficients up to 5th rank are calculated (with high accuracy) and analyzed with reference to the reference trajectory. In comparison with conventional methods, the proposed DA approach shows overwhelming advantages in having analyzed the dispersion, focus error and 2nd rank aberration coefficients. A new software package was developed, and an example was shown to demonstrate the proposed approach.

A study of Glaser's Ball – Shaped Magnetic Lens under the Effect of Current Density

A computational investigation has been carried out on the design and properties of Glaser's lens. The limiting factor in magnetic lens design is the current density that can be supported by the coil. The current density was used as parameters of the magnetic lens design. These parameters were studied for finding the minimum spherical and chromatic aberration coefficients. The computed aberration has been normalized in term of focal length under zero magnification condition.

Computational Design of a Magnetic Mirror

A computer-aided design (CAD) has been carried out to investigate the properties of the magnetic electron mirror design. The work has been focused on suggesting a mathematical formula to represent the radial displacement. The function that has been taken into consideration was suggested to give rise to the mirror action. A numerical solution is carried out for solving the Paraxial-ray equation for determining the optical properties such as the focal length, the spherical and chromatic aberration coefficients and the excitation of the mirror. The pole shape of the mirror has been determined in two dimensions. In the present work, the profile of the mirror determined from the suggested trajectory is the single-pole types. The coefficients of the chromatic and spherical aberrations of the magnetic mirror are determined and normalized in terms of the focal length. The operational requirements are determining the choice of the mirror.

100 keV vacuum sealed field emission gun for high resolution electron microscopy

A standalone 100 kV field emission gun (FEG) has been developed that can be installed and operated on a standard transmission electron microscopy electron optical column or custom designed high voltage electron optical columns. The FEG comprises a thermally assisted field emission cathode and an asymmetric electrostatic lens that can operate from 20 to 100 kV in an ultrahigh vacuum (UHV) chamber. In its current configuration, the FEG has spherical and chromatic aberration coefficients (Cs and Cc, respectively) in the range of Cs = 607–670 mm and Cc = 60–87 mm at 100 keV over a range of working distances of 50–206 mm from the exit plane of the FEG unit. A dedicated high voltage supply unit with voltage ripples of less than 1 ppm at 100 kV has also been developed. The FEG is transported under UHV and does not require the use of SF6 gas during operation, as is customary in high voltage FEG TEMs. Preliminary results of operating the FEG on a Philips Tecnai 12 and a JEOL JEM-1400HR TEM show the resolution of gold (111) crystal planes at 0.235 nm and (200) planes at 0.202 nm.

Simple program for computing objective optical properties of magnetic lenses

The paper describes the basic features of a new program denoted Magnetic Electron Lens Optical Properties (MELOP), primarily intended for providing free and simple way to calculate the objective focal properties of rotationally-symmetric electron magnetic lenses with the presence of its axial magnetic field distribution. The calculation is done by solving the paraxial ray equation using the fourth-order Runge-Kutta formula. For specific beam voltage, the program computes the excitation parameter, the object or image plane, the objective principal plane, the objective focal length, the objective magnification, the spherical aberration coefficient, the chromatic aberration coefficient and the magnetic flux density at the object or image plane. These parameters are solved for zero, low, high or infinite magnification condition. The program can handle instantaneously plotting the variations of the calculated parameters relatively. In addition to that, the data can be transferred to xlsx or txt file format.

High order aberrations calculations of Wien filters using differential algebra methods

In this paper, a novel scheme based on the differential algebraic (DA) method is proposed to analyze the electron optics properties of the Wien filters. A new software package is then developed, to compute the geometrical and chromatic aberrations up to the fifth order of the Wien filters. For examples, the aberrations of single filter and the double filters are calculated. The calculated geometrical aberration coefficients are comparable with the counterpart calculated by Tang's theory. However, both theories can only match with each other in the calculation of the chromatic aberration coefficient when the dispersion ray is taken into consideration in Tang's theory.

Measuring chromatic aberration in LEEM/PEEM

Measurement of chromatic aberration in a Low Energy Electron Microscope (LEEM) or Photo Electron Emission Microscope (PEEM) is necessary for quantitative image interpretation, and for accurate correction of chromatic aberration in an aberration-corrected instrument. While methods have been developed for measuring the spherical aberration coefficient, C3, measuring the chromatic aberration coefficient, Cc, remains a more difficult task. Here a novel method is introduced to simplify such measurements. The viability and accuracy is demonstrated using detailed electron-optical ray-tracing calculations. Experimental results show that the method is easily reduced to practice.

Third-rank chromatic aberrations of electron lenses

In this paper the third-rank chromatic aberration coefficients of round electron lenses are analytically derived and numerically calculated by Mathematica. Furthermore, the numerical results are cross-checked by the differential algebraic (DA) method, which verifies that all the formulas for the third-rank chromatic aberration coefficients are completely correct. It is hoped that this work would be helpful for further chromatic aberration correction in electron microscopy.

Aberration measurement of the probe-forming system of an electron microscope using two-dimensional materials

The geometric and chromatic aberration coefficients of the probe-forming system in an aberration corrected transmission electron microscope have been measured using a Ronchigram recorded from monolayer graphene. The geometric deformations within individual local angular sub-regions of the Ronchigram were analysed using an auto-correlation function and the aberration coefficients for the probe forming lens were calculated. This approach only requires the acquisition of a single Ronchigram allowing rapid measurement of the aberration coefficients. Moreover, the measurement precision for defocus and two-fold astigmatism is improved over that which can be achieved from analysis of Ronchigrams recorded from amorphous films. This technique can also be applied to aberration corrected STEM imaging of any hexagonal two-dimensional material.

A novel electron mirror pulse compressor

An electron mirror-based pulse compressor design has been developed for improving the temporal resolution of dynamic/ultrafast transmission electron microscopes and ultrafast electron diffraction cameras. The improvement will enable electron microscopes and diffraction cameras to better resolve the dynamics of reactions in the areas of solid state physics, chemistry, and biology. The design utilizes a combination of mirror optics and a magnetic beam separator, which exploits the symmetry inherent in reversing the electron trajectory in the mirror in order to compress the pulse. This system can also simultaneously correct the spherical and chromatic aberration of the objective lens for improved spatial resolution. For pulsed experiments with a practical bunch charge, the correction of the chromatic aberration coefficient counters the spread in the electron energies induced by the space charge of the pulse to make possible the probing of the sample with high spatial resolution. The pulse compressor can accommodate pulses with a range of electron densities and energy spreads. Furthermore, it is designed to fit into both ultrafast electron diffraction cameras and dynamic/ultrafast transmission electron microscopes. Consequently, this instrument is suitable for enhancing the study of the structure, composition, and bonding states of new materials at ultrafast time scales.

Low-energy scanning electron microscope using a monochromator with double-offset cylindrical lenses

The nanoelectronics industry demands continuous improvement in the performance of scanning electron microscopes (SEMs). Extremely low energies of less than 1 keV are required for SEM observations to allow the subsurface and nanoscale information of target specimens to be measured with minimized charge-up and beam damage depths because of the reduced interaction volumes. In this article, the authors propose a new monochromator (MC) structure and investigate its applicability to SEMs operating at such extra-low energies. The proposed MC, which uses double-offset cylindrical lenses, can perform energy filtering in its midsection and form a stigmatic and nonenergy dispersive image at the exit. The energy resolution is expected to be better than 10 meV for a pass energy of 4 keV. The MC has the additional advantage of a simple but robust structure, which is essential for industrial applications. Assuming the use of ideal and high-performance SEM optics, for which the spherical aberration coefficient and the chromatic aberration coefficient are both 1 mm, beam diameters measured in terms of the full width that contains 50% of the beam current (FW50) are calculated at the specimen position. Use of the MC improves the beam diameter dramatically to 4.4 nm, as compared to the diameter of 19.7 nm for the SEM without the MC, at a landing energy of 100 eV. The chromatic aberration contribution also becomes negligible because of the MC. The beneficial effects of the MC with regard to the beam diameter become increasingly prominent at lower landing energies ranging down to 10 eV. A SEM using this MC can generate highly monochromatic (10 meV) electron probe beams with small size (5 nm) and low energy (100 eV), which indicates the additional possibility of a new surface electron microscope that uses phonon signals. Based on this theoretical investigation, the authors conclude that this MC can effectively improve the SEM's performance capabilities in the extra-low-energy region.

An adjustable electron achromat for cathode lens microscopy

Chromatic aberration correction in light optics began with the invention of a two-color-corrected achromatic crown/flint lens doublet by Chester Moore Hall in 1730. Such color correction is necessary because any single glass shows dispersion (i.e. its index of refraction changes with wavelength), which can be counteracted by combining different glasses with different dispersions. In cathode lens microscopes (such as Photo Electron Emission Microscopy – PEEM) we encounter a similar situation, where the chromatic aberration coefficient of the cathode lens shows strong dispersion, i.e. depends (non-linearly) on the energy with which the electrons leave the sample. Here I show how a cathode lens in combination with an electron mirror can be configured as an adjustable electron achromat. The lens/mirror combination can be corrected at two electron energies by balancing the settings of the electron mirror against the settings of the cathode lens. The achromat can be adjusted to deliver optimum performance, depending on the requirements of a specific experiment. Going beyond the achromat, an apochromat would improve resolution and transmission by a very significant margin. I discuss the requirements and outlook for such a system, which for now remains a wish waiting for fulfilment.

The microbeam system at Tohoku University

A microbeam system called MB-I was developed at the Dynamitron laboratory at Tohoku University in 2002 for use in biological applications. Although the system was designed to achieve a submicron beam spot size, parasitic field contamination from tungsten carbide slit chips and an annular Si surface barrier detector have limited the beam spot size to [Formula: see text]. By replacing these components, parasitic field contamination of the system was reduced and the performance of the microbeam system was remarkably improved. A measured beam spot size of [Formula: see text] at a beam current of several tens of pA has been achieved. MB-I has been used for simultaneous in-air/in-vacuum particle-induced X-ray emission (PIXE), Rutherford backscattering spectroscopy (RBS), secondary electron (SE), scanning transmission ion microscopy (STIM) analyses, and three-dimensional PIXE micron computed tomography (PIXEμCT), with applications in various fields. To obtain a higher spatial resolution of several hundred nm and a higher beam current with a resolution of several μm, a triplet lens system was designed and installed in MB-I. The triplet system has a larger demagnification than the existing system; however, it also has larger chromatic and spherical aberration coefficients. Therefore, stricter requirements are imposed on the accelerator performance, particularly the beam brightness and energy stability. In addition to the microbeam, the Dynamitron accelerator was also upgraded to obtain a higher beam brightness. The beam brightness is [Formula: see text], with a half-divergence of 0.07 mrad. The energy resolution of the accelerator was improved by developing a terminal voltage stabilization system (TVSS), to achieve an energy resolution of [Formula: see text]. Thus, the effects of the increased chromatic and spherical aberration were mitigated by restricting the divergence angle, without reducing the beam current. A beam spot size of [Formula: see text] was obtained with a beam current of 150 pA. While the analysis system of MB-I can be used for simultaneous in-air/vacuum PIXE, RBS and STIM analyses, as well as 3D PIXEμCT, without changing the target chamber, changes are required in the experimental setup for these techniques, which is time-consuming, Thus, a new microbeam system, MB-II, was developed. The MB-II is a doublet system and is not equipped with a high-resolution energy analysis system. It is connected to a switching magnet. The beam brightness is [Formula: see text] at a half-divergence of 0.1 mrad and this does not decrease as the beam divergence increases. This property is matched to obtain a higher beam current using the doublet system. With a current of 300 pA, the MB-II provides a higher beam current for a [Formula: see text] beam spot size than is achievable with the MB-I. At present, both microbeam systems are in routine operation at the Dynamitron laboratory.

Electrostatic afocal-zoom lens design using computer optimization technique

Electron optics is the key to the successful operation of electron collision experiments where well designed electrostatic lenses are needed to drive electron beam before and after the collision. In this work, the imaging properties and aberration analysis of an electrostatic afocal-zoom lens design were investigated using a computer optimization technique. We have found a whole new range of voltage combinations that has gone unnoticed until now. A full range of voltage ratios and spherical and chromatic aberration coefficients were systematically analyzed with a range of magnifications between 0.3 and 3.2. The grid-shadow evaluation was also employed to show the effect of spherical aberration. The technique is found to be useful for searching the optimal configuration in a multi-element lens system.

Space-Charge Effect on the Theoretical Design of Electron Gun Operated under Zero Magnification Condition

A computational investigation is carried out in the field of charged –particle optics with the aid of numerical analysis method using the personal computer. The work is concerned with the design of electron gun with space-charge effect. The Finite element method (FEM) used in the solution of Poison's equation for determine the axial potential distribution of the two-electrode immersion lens operated under zero magnification condition , and from the solution of the paraxial ray equation the optical properties such as the focal length , spherical and chromatic aberration coefficients are determined, also a calculation of the brightness and perveance for the lens. The electrodes geometry was determined in two and three dimension with the aid of new drawing software called SIMION 7 and the electron beam trajectory under zero magnification conditions has been determine for these electrodes.

Space-Charge Effect on the Theoretical Design of Electron Gun Operated under Zero Magnification Condition

A computational investigation is carried out in the field of charged –particle optics with the aid of numerical analysis method using the personal computer. The work is concerned with the design of electron gun with space-charge effect. The Finite element method (FEM) used in the solution of Poison's equation for determine the axial potential distribution of the two-electrode immersion lens operated under zero magnification condition , and from the solution of the paraxial ray equation the optical properties such as the focal length , spherical and chromatic aberration coefficients are determined, also a calculation of the brightness and perveance for the lens. The electrodes geometry was determined in two and three dimension with the aid of new drawing software called SIMION 7 and the electron beam trajectory under zero magnification conditions has been determine for these electrodes.

Aberrations of a volume holographic optical element obtained by means of a cylindrical object wave and a spherical reference wave

This paper presents expressions that make it possible to calculate the third- and fifth-order aberrations of a volume holographic optical element (HOE) recorded by means of a cylindrical reference wave and a spherical object wave. It is shown that the aberration coefficients of this volume element can be represented as a sum of the monochromatic and chromatic aberration coefficients, as well as the aberration coefficients that result from the photoinduced, thermally induced, and strain variations of its refractive index and dimensions. It is pointed out that the selectivity of a volume HOE reduces its aberrations by comparison with the corresponding aberrations of a thin HOE.

Spherochromatic aberration of a single electrostatic lens of simplified design

Parameters of a single lens composed of two coaxial cylinders have been numerically calculated for the case of energy spread in a charged-particle beam. The spherical and chromatic aberration coefficients of this lens have been determined. It is shown that the focus-spot size calculated from the classical formulas taking into account the coefficients found is in good agreement with the results of numerical calculations.