Advances in Electronics and Electron Physics, Volume 78 (Advances in Imaging and Electron Physics)

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 78

EDITOR-IN-CHIEF

PETER W. HAWKES Laboratoire d’Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

ASSOCIATE EDITOR

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

Advances in

Electronics and Electron Physics EDITED BY PETER W. HAWKES Laboratoire d’Optique Electronique du Centre National de la Recherche ScientGque Toulouse, France

VOLUME 78

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CONTENTS ................................ CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE

vii ix

Theory of the Gaseous Detector Device in the ESEM G . D . DANILATOS I. I1. III . IV . V. VI . VII . VIII . IX . X.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discharge Characteristics . . . . . . . . . . . . . . . . . . . . . . Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scintillation G D D . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 16 36 42 60 80 84 91 97 99 99

Carrier Transport in Bulk Silicon and in Weak Silicon Inversion Layers S. C . JAIN. K . H . WINTERS. AND R . V A N OVERSTRAETEN List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 I1. Drift Velocity in Bulk Silicon . . . . . . . . . . . . . . . . . . . . 109 Ill . Effect of Tangential Field on Mobility and Saturation Velocity of Electrons in Inversion Layers . . . . . . . . . . . . . 112 IV . Carrier Transport and Mobility in the Weak-Inversion Region of a MOSFET. Theories Based on Macroscopic Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 V . Theories Based on Short-Range or Microscopic 128 Inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Comparison of Low-Temperature Experimental-Edge Model with the Mobility-Edge Model . . . . . . . . . . . . . . . . . . . 134 VII . Arnold’s Experiments and Macroscopic Inhomogeneity Model 138 VIII . Hall Effect and Electron-Liquid Model . . . . . . . . . . . . . . 140 v

CONTENTS

V1

IX . X. XI . XI1 . XIII .

Evidence of Deviation from Random Distribution . . . . . . . . Peaks in the Variation of pwlc with Einv . . . . . . . . . . . . . . Room- and High-Temperature Measurements . . . . . . . . . . Limitations of Theories . . . . . . . . . . . . . . . . . . . . . . . Summary of Work on Transport in Inversion Layers in the Weak-Inversion Region, and Conclusions . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 144 146 151 151 152 153

Emission-Imaging Electron-Optical System Design V. P . IL’IN. V. A . KATESHOV. Yu . V. KULIKOV.AND M . A . MONASTYRSKY I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 I1. Aberration Models of Cathode Lenses . . . . . . . . . . . . . . . . 158 111. The Variational Analysis of Cathode-Lens Optimization and 178 Synthesis Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Implementation of the Numerical Computational Methods and System Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 224 V . Automation Principles in Designing Electron-Optical Systems . . 246 IV . Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . 257 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 INDEX .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authors' contributions begin. G. D. DANILATOS (l), ESEM Research Laboratory, 98 Brighton Boulevarde, North Bondi (Sydney), NSW 2-26, Australia V. P. IL'IN( 1 5 9 , Computing Center, 630090 Novosibirsk, Lavrentiva Street 6, USSR

S. C. JAIN(103), Theoretical Physics Division, Harwell Laboratory, Didcot, Oxon OX1 1 ORA, United Kingdom V. A. KATESHOV (159, Computing Center, 630090 Novosibirsk, Lavrentiva Street 6, USSR Yu. V. KULIKOV (155), Computing Center, 630090 Novosibirsk, Lavrentiva Street 6, USSR

M. A. MONASTRYSKY (1 55), Computing Center, 630090 Novosibirsk, Lavrentiva Street 6, USSR R. VAN OVERSTRAETEN (103), IMEC, Kapeldreef 75, B-3030 Leuven, Belgium K . H. WINTERS (103), Theoretical Physics Division, Harwell Laboratory, Didcot, Oxon OX1 1 ORA, United Kingdom

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PREFACE

The first two chapters of this volume are concerned with the behavior of two very different types of devices. We open with a full account by G. D. Danilatos of a detector that has been developed for use in conjunction with the environmental scanning electron microscope, already described by the same author in Volume 71 of this series. A full analysis of the theory of this gaseous detector has not hitherto been available and this careful study is therefore all the more welcome. In the second chapter, we meet a much more familiar device, the MOS transistor. In order to understand the behavior of such transistors a very detailed knowledge of carrier transport is required. S. C . Jain, K. H. Winters and R. van Overstraeten first analyze bulk silicon and then turn to the weakinversion region of a MOSFET. This meticulous and up-to-date survey will surely be of great interest for future design studies of these transistors. The final chapter is also devoted to a device, but of a rather different kind: the cathode lens. Together with electron mirrors, these lenses have long been regarded as particularly difficult to analyze for good reason: the approximations on which electron lens studies traditionally lean break down for these lenses, in which the particles used to form the image emerge from the emissive specimen with very low energy. Their optical properties have been thoroughly investigated during the past few years by a number of Russian scientists and, although most of this work is available in English translation, it is probably less well known than it deserves. In 1987, four of the principal contributors to this theoretical development brought their work together in a book published by the Siberian Branch of the “Nauka” publishing house in Novosibirsk, and this chapter is essentially a translation of that work, which might otherwise have remained virtually unknown outside the Soviet Union. I am delighted that this highly original work is now available in a convenient form in these Advances and am most grateful to the authors for making this English translation available. It only remains for me to thank all the authors for the trouble they have taken with their contributions. As usual, I conclude with a list of forthcoming chapters, and I take this opportunity to recall that I plan to increase the number of chapters in the broad field of digital image processing. Offers of reviews on this subject or, indeed, on any of those traditionally covered in this series, are always welcome. Peter W. Hawkes ix

PREFACE

X

FORTHCOMING CHAPTERS Parallel Image Processing Methodologies Image Processing with Signal-Dependent Noise Pattern Recognition and Line Drawings Bod0 von Borries, Pioneer of Electron Microscopy Magnetic Reconnection Sampling Theory Finite Algebraic Systems and Trellis Codes Electrons in a Periodic Lattice Potential The Artificial Visual System Concept Speech Coding Corrected Lenses for Charged Particles The Development of Electron Microscopy in Italy The Study of Dynamic Phenomena in Solids Using Field Emission Pattern Invariance and Lie Representations Amorphous Semiconductors Median Filters Bayesian Image Analysis Phosphor Materials for CRTs

Number Theoretic Transforms Tomography of Solid Surfaces Modified by Fast Ion Bombardment The Scanning Tunnelling Microscope Applications of Speech Recognition Technology Spin-Polarized SEM Analysis of Potentials and Trajectories by the Integral Equation Method The Rectangular Patch Microstrip Radiator Electronic Tools in Parapsychology

J. K. Aggarwal H. H. Arsenault H. Bley H. von Borries

A. Bratenahl and P. J. Baum J. L. Brown H. J. Chizeck and M. Trott J. M. Churchill and F. E. Holmstrom J. M. Coggins V. Cuperman R. L. Dalglish C. Donelli M. Drechsler M. Ferraro W. Fuhs N. C. Gallagher and E. Coyle S. and D. Ceman T. Hase, T. Kano, E. Nakazawa and H. Yamamoto G. A. Jullien S. B. Karmohapatro and D. Ghose H. Van Kempen et al. H. R. Kirby K. Koike G. Martinez and M. Sancho H. Matzner and E. Levine R. L. Morris

PREFACE

Image Formation in STEM Information Energy and Its Applications Low-Voltage SEM Z-Contrast in Materials Science Languages for Vector Computers Electron Scattering and Nuclear Structure Electrostatic Lenses Energy-Filtered Electron Microscopy CAD in Electromagnetics Scientific Work of Reinhold Riidenberg Metaplectic Methods and Image Processing X-Ray Microscopy Accelerator Mass Spectroscopy Applications of Mathematical Morphology Optimized Ion Microprobes Focus-Deflection Systems and Their Applications Electron Gun Optics Thin-Film Cathodoluminescent Phosphors Electron Microscopy and Helmut Ruska

Xi

C. Mory and C. Colliex L. Pardo and 1. J. Taneja J. Pawley S. J. Pennycook R. H. Perrott G. A. Peterson F. H. Read and I. W. Drummond L. Reimer K. R. Richter and 0. Biro H. G. Rudenberg W. Schempp G. Schmahl J. P. F. Sellschop J. Serra Z. Shao T. Soma et a / . Y. Uchikawa A. M. Wittenberg C. Wolpers

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ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS . VOL . 78

Theory of the Gaseous Detector Device in the Environmental Scanning Electron Microscope G . D. DANILATOS ESEM Research Laboratory Sydney. Australia and Electroscan Corporation Wilmington. Massachusetts

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I1. Principles . . . . . . . . . . . . . . . . . . . . . . . . . . A . Imaging Parameters . . . . . . . . . . . . . . . . . . . . . B . Theory of Induced Signal . . . . . . . . . . . . . . . . . . . C. Induced Signals in the ESEM . . . . . . . . . . . . . . . . . . I11 . Physical Parameters . . . . . . . . . . . . . . . . . . . . . . A . Electron and Ion Temperature . . . . . . . . . . . . . . . . . B . Electron and Ion Mobilities . . . . . . . . . . . . . . . . . . C. Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . D . Recombination . . . . . . . . . . . . . . . . . . . . . . . E . Electron Attachment . . . . . . . . . . . . . . . . . . . . . F . Effective Ionization Energy . . . . . . . . . . . . . . . . . . . I V. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . V . Discharge Characteristics . . . . . . . . . . . . . . . . . . . . A . Outline of the Discharge . . . . . . . . . . . . . . . . . . . B . Amplification, Parallel Plates . . . . . . . . . . . . . . . . . . C . The First Townsend Coefficient . . . . . . . . . . . . . . . . . D . ThePaschenLaw . . . . . . . . . . . . . . . . . . . . . . E . Secondary Processes . . . . . . . . . . . . . . . . . . . . . VI . Amplification . . . . . . . . . . . . . . . . . . . . . . . . . A.Limits. . . . . . . . . . . . . . . . . . . . . . . . . . B . Geometry and Time Response . . . . . . . . . . . . . . . . . . VII . Scintillation G D D . . . . . . . . . . . . . . . . . . . . . . . VIII . Signal Spectroscopy . . . . . . . . . . . . . . . . . . . . . . A . Spectroscopy, Statistics. and Energy Resolution. . . . . . . . . . . . B . Environmental Scanning Transmission Electron Microscopy . . . . . . . C.ESEM., . . . . . . . . . . . . . . . . . . . . . . . . 1X.Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . A . On the Geiger-Muller Counters . . . . . . . . . . . . . . . . . B. Materials and Construction Details . . . . . . . . . . . . . . . . C . Future Prospects . . . . . . . . . . . . . . . . . . . . . . X . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

2

4 4 6 11 16 16 20 25 29 31 34 36 42 44 49 52 54 56 60 60 64 80 84 85 89 90 91 91 94

96 97 99 99

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Copynght R. 1990 by Acddemlc Press InG All rights of reproduction in any form reserved l S B h O-IZ-01467X9

G. D. DANILATOS

I. INTRODUCTION The environmental scanning electron microscope (ESEM), described in detail in several publications (Danilatos 1981, 1985, 1988; Danilatos and Postle 1982, 1983), is now a commercially available new instrument with the potential of becoming an established tool for research and development in many fields. This microscope allows the examination of specimens in the presence of a gaseous environment. It has created new possibilities, such as the examination of insulators, and wet and liquid specimens without pretreatment and modification. In general, solid-liquid-gas phases, their interactions, and other processes can now be studied under dynamic or static conditions. The ESEM has adapted several detection modes of the conventional scanning electron microscope (SEM) to operate in the presence of gas. Besides, it has ushered the development of completely new devices in electron microscopy, namely, the use of particular forms of gaseous devices related to those developed in other fields of science. The basic idea of using the gas in the ESEM as a detection and amplification medium was first suggested and demonstrated in 1983 (Danilatos 1983a, 1983b). Originally, the principle of the gaseous detector device (GDD) was based on the collection of current produced as a result of the ionizing action by various signals. Later, it was shown that the scintillation produced by various signals can also be used for making images, and a generalized G D D was proposed (Danilatos 1986b); according to this, the detection of products of any reaction between a particular signal and gas could be used for imaging or analysis in the ESEM. The detection of electrical charges and photons are just two particular cases of the generalized GDD. In essence, the GDD is based on the principle of classical gaseous particle detectors (ionization, proportional, and Geiger-Muller chambers of nuclear physics) adapted to the specific requirements of the ESEM. The unification of these detectors with the ESEM constitutes a novel detection practice in electron microscopy. The nearest related case is the use of proportional x-ray detectors in the SEM, but these detectors have been simply transferred to the field of microscopy. No such simple transfer may be assumed in the case of G D D without regard to complications arising from the interaction between the detector and the microscope, where the conditioning gas of the specimen chamber is to be used as the detector medium. The G D D corresponds to the open-flow type of counter, whereby the radiation source is inside the detector; here, radiation source is any ionizing (or interacting) signal generated from the electron beam-specimen interaction. Due to the multiplicity of radiations, nature of radiations, and special requirements of the ESEM (e.g., specimen positioning), it was not obvious in the beginning whether and how the G D D would perform.

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

3

The previous publications on GDD have demonstrated that the gas can be used as a medium for imaging in general. Our early understanding and image interpretation has been empirical. Most work and developments have been done with the ionization GDD. Initially, wire electrodes at low bias were used. The low bias (up to 10-20 volts) was enough to collect the signal current and the ionization current produced by energetic electrons. It has been shown how both the secondary electrons (SE) and the backscattered electrons (BSE) can be detected in the presence of gas by varying the pressure and the electrode position (Danilatos 1983b, 1986a, 1986b, 1988). In a later development based on the advice of the present author, Electroscan Corporation demonstrated that the ionization GDD can operate at high electrode bias, as a result of which the signals can be further amplified in the gas in a manner analogous to that of gaseous proportional amplifiers (unpublished results). This showed that the SE signal, in particular, can be given a preamplification having a highly beneficial effect on image acquisition. The high electrode bias has been further investigated by this author, and the experimental results are scheduled for later reports. The beam-specimen-signal-gas system is highly complex, and an understanding of the properties and efficiency of the G D D necessitates the study of the fields of ionized gases, particle impact and detection phenomena, and the testing of devices specifically suited for the conditions of the ESEM. For example, the separation of the BSE and SE, and their most efficient detection in the presence of gas, constitute one of the immediate objectives of current work. Results of this work will be reported as they become available in selfcontained parts. The principal purpose of this work is to describe the fundamental mechanisms of operation of the GDD. Whereas limited experimental work alone led our perceptions in the past, the present survey will provide the basic theory of the system and will prompt new experimental tasks. Particular aims to be achieved here are to determine the capabilities of the G D D in relation to the physical limits of amplification, frequency response, resolution, and radiation spectroscopy. This work is based on a survey of previous works in related fields coupled with current experience in the ESEM. It is essential first to understand the fundamental processes occurring in the ESEM and to determine the physical limitations and principal directions that will shape our progress. This subject is multifaceted and cannot be presented complete with experimental evidence on each or most of its facets before several years of additional work. Inclusion of experimental results currently available for parts only of this work would render it lopsided and would distract from other important issues. However, there is at present an urgent need to clarify and understand many “unusual” phenomena observed in the ESEM. Therefore, it has been decided to exclude experimental results and to present only a general theoretical guide for present and future work.

4

G. D. DANILATOS

11. PRINCIPLES It is necessary to outline some of the physical principles upon which much of the following analysis is based. For example, the mechanism of pulse induction by a moving charge among electrodes is many times ignored or forgotten, and thus important phenomena in the microscope become elusive and sometimes puzzling. A grasp of the physical magnitudes of some parameters in the microscope will also be very helpful. A . lmaging Parameters

In order to determine the capabilities of the GDD, it is necessary to establish the requirements of the microscope for satisfactory imaging. Constraints on imaging are, first of all, imposed by statistical considerations, and we should determine the magnitude of various parameters for a typical image with an acceptable noise level. Both the electron beam probe and the signals arising from the beamspecimen interaction are characterized by intrinsic noise; this sets the minimum current that can be used in the beam for a given specimen. For operation under vacuum conditions, a relationship between image parameters has been presented by Wells (1 974). In the simple case where the incident beam current I , yields a signal of strength 61,, the signal-to-noise ratio (SNR) K is related to the current as follows: I,

=

K2hr12e/4z6,

(1)

where M is the number of gray levels, e the electron charge, and z the pixel dwell time. In the above derivation, the gray levels have been allocated in such a way that the SNR is constant from the darkest to the brightest part of the image (constant reliability condition). The time constant T depends on the scanning speed (ie., on frame time) used and the number of lines per frame. We wish to determine the magnitudes of these parameters for a reference case specimen, so that these magnitudes can be used in the subsequent analysis of the GDD. It is said that the human eye can distinguish about 16 gray levels, but let us assume that we are satisfied with the round figure of M = 10 gray levels. A good image is usually made with 1000 lines over 50 seconds per frame, and therefore, we can take z = 50ps. An uncoated carbonaceous specimen (not an easy one to image) may produce a signal of about 6 = 0.1. Under these conditions, the amount of current required depends on the level of K that we are prepared to accept. If we compromise with K = 5, then the required current is 20 PA. From these values, we find that the maximum number of beam electrons striking each pixel is PIXIN = 6250, and the number of electrons

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

5

coming out of each pixel, as signal, is PIXOUT = 625. The number of electrons from each pixel at lower gray levels is less; for quick reference, these are calculated from N = 6.25 M 2 and given in pairs of numbers as ( M , PIXOUT):

(1,613

(2,251,

(3,561,

(4, loo), (5,1571,

(6,225), (7,306), (8,400), (9,506), (10,625). It is also helpful to inquire about the average time interval between the electrons striking the specimen and between the electrons emerging from it: In the beam we get Tb = e / l = 8 ns and in the signal T, = 80 ns. The average spacing (distance) between electrons in the beam depends on the accelerating voltage used: For 10 keV, we get S , = uTb = 468 mm, where u is the velocity of the electrons given by

':=c[1

-(

1 1

>']'.'

( E in keV),

+ E/511

and c is the velocity of light. The emerging electrons have a wide range of energy and spatial distribution. The average BSE energy is about half the incident energy (see, e.g., Reimer, 19851, whereas the peak of SE distribution is around 2 eV. The travel distance of a BSE by the time a new BSE emerges from the specimen would be S,,, = 3331 mm, while for a SE this distance would be S,, = 67 mm. In fact, the electrons in the signal would be apart by distances, on the average, greater than these values, as they travel in various directions. Therefore, we have only one electron (on the average) present in the ESEM chamber, as the specimen-walls distances are less than these values. This remark applies for very low pressures, where the mean free paths of the signal electrons are greater than the specimen-walls distances. For operation in the presence of gas, the theory of SNR has been presented by Danilatos (1988). In the simple case considered here, the current used in vacuum must be increased to a new value 16 in accordance to:

where q is the fraction of beam electrons surviving unscattered by the gas molecules. If the beam travels one mean free path to reach the specimen, q = 0.37, and the current should be 23.8 times higher to produce the same SNR in the image as in vacuum. In other words, the number of useful (probe) electrons reaching and leaving the specimen is the same as in vacuum, since the increased initial beam intensity compensates for the electrons scattered out of the beam. The signal parameters will be modified by the presence of gas, and this fact will be dealt with in later sections.

6

G. D. DANILATOS TABLE I PARAMETERS DEFINING THE TYPICAL SPECIMEN, IMAGE. ANII CONDITIONS IN THE ESEM K=5

M

=

7 =

6

=

10

50 p 0.1

I = 20pA V = 10kV PIXIN = 6250 PIXOUT = 625

S, = 0.47 m S,,, = 3.3 m S,, = 0.07 m

Tb = 8 ns & = 80 ns q = 0.37 1; = 480 pA

The above parameters may vary widely in ESEM. For example, the accelerating voltages usually applied are between a few hundred volts and 50 kV, the current varies from a few pA to hundreds of nA, the scanning speed is between 1/25 s (TV rate) and up to 1000 s, the number of lines between 250 and 2000, and the signal yield 6 for different specimens ranges from 0.01 up to usually about 0.5, but it may reach or exceed 1. In addition, the effects of background noise levels (not considered in the example) can be very significant, and for a detailed treatment the reader is referred elsewhere (Danilatos 1988).The aim, here, is to tabulate the rounded values determined above for an “average” case, to which we will be referring (arbitrarily) as the typical (or reference) specimen producing a typical image under typical conditions. The figures are given in Table I for quick reference. The values of the parameters determined above for the reference case would be the best output from an ideal, noise-free, detector with 100% efficiency. These can be used as a basis in our subsequent calculations to evaluate the performance of the GDD. B. Theory of Induced Signal

During the early days of imaging with the GDD, the precise mechanism of current collection was not clear. No theoretical investigation, experimental measurement, or calculations were attempted to establish the ionization state A) and beam voltage of the gas. Initially, relatively high beam current (15 kV) was used (Danilatos 1983b), and it was tacitly assumed that the gas was behaving close to a plasma state. With such a view, a biased electrode inserted in the gas would be capable of collecting charges within a certain distance around it (the Debye length). The electrode field would have little influence at longer distance. In such a situation, the plasma region outside the influence of the field would act as a cathode or an anode providing the negative or positive charges to the electrode. Thus, a reversal of the electrode bias would result in an inversion of the image contrast. The existence of a plasma situation seemed to be supported by some preliminary observations

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

7

with crude probes, whereby it was noted that the current on two probes would increase when the probes were moved several millimeters apart. Also, the current-versus-bias measurements showed a “saturation” current level reached with a few volts. However, these indications did not necessarily imply the existence of plasma, and in a later work (Danilatos 1988), the evidence available was still inconclusive. The decrease of current collection as the electrodes move close to each other and the “saturation” current can be attributed to increased loss of charge carriers by a geometry effect, to diffusion, and to other causes as discussed in detail later; the reversal of contrast when inverting the bias can be explained by the induction mechanism. Present understanding does not favour the existence of plasma conditions, at least in the vicinity of the electrodes used. Hence, it is necessary to resort to the basic theory of electricity dealing with the movement of charges in a charge-free electrical field between conductors. In the general case, we may have two conductors of any shape connected to a battery as shown in Fig. 1(a). If a charge ( + ) e moves a distance d s in the electric field E, the work done on the electron is eEds. This work is supplied by the battery in transferring a charge d q from one electrode to the other across the potential VThis transfer is necessary to maintain a zero electrical field inside the conductors, a physical law that would be violated otherwise, as the charge moves with its own field. The induced charge is calculated by the equation

-

V d q = eE ds.

(4)

Since the movement takes place in time d t and the current i is d q l d t , we obtain the general relation i = eE.v/Ci

(5)

The above two equations are equivalent and yield the intensity of current and the total charge transferred by induction. They are applicable to any geometry of electrodes. Furthermore, they are applicable, and the same charge is induced, when there is no battery and the charge moves due to its own momentum. This is so, because the ratio E / V is a constant characteristic only of the geometry of the electrodes ( E is proportional to V on account of the principle of superposition). Also, the same charge will be induced even if the electrodes are not connected, or connected through a very large resistor. This can be seen in two simple cases where the geometrical factors can be easily calculated, as is shown in Fig. l(b) for parallel plates and in Fig. l(c) for cylindrical electrodes. The electrodes are interconnected through a resistor R , which can take any value, and are shunted by an unavoidable capacitor C due and equal to the distributed capacitance of the system. The time constant RC determines the mode of charge flow through the system and the output signal V, across the resistor.

8

G . D. DANILATOS

0

R

VS 0

I'

+ -

-

+

t

FIG. 1 The motion of a charge e induces a charge on the neighbouring conductors: (a) two general-shape conductors with a nonuniform field, (b) parallel plates with uniform field, and (c) cylindrical geometry. A voltage V, appears on the circuit resistor R shunted by the distributed capacitance C. V = applied bias, E = electric field, u = charge velocity, D = plate separation, r l = wire radius, r2 = cylinder radius, r = radial position of particle, x = particle displacement from electrode, and i = current.

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

9

For the parallel plates, if the transit time T of the charge is much longer than RC, then there is no accumulation of charge on the electrodes, and the current is determined by Eq. (5), which for the parallel plates becomes .

I = -

because E/V

=

ev D'

1/D. The signal i s simply ( R C << T )

(7)

and remains constant during the transit, but becomes zero when the charge reaches the electrode. If the transit time is much shorter than R C , the induced charge q accumulates with time on the electrodes, producing a signal voltage q / C , which is given by

K=

ex ~

CD

=

~

evt CD

( R C >> T ) ,

For cylindrical electrodes, we find equivalent equations, differing by a geometrical factor, which can significantly alter the signal output. The electrical field at a distance r from the a wire of radius r t with linear charge density A, is given by (see, e.g., Purcell 1985) A

E=2mOr where so = 8.854 x

(9)

Cb/Vm. The electric potential function is

A u = --2ZE0 In r + constant.

(10)

The same equations hold when the wire is surrounded by the outer cylinder of diameter r 2 . The absolute value of E depends on the choice of bias on the electrodes. By choosing zero potential for the cylinder and V for the wire, we deduce I/

Using this relation in Eq. (9,we get

This shows that the current is not constant but decreases abruptly as the charge moves away from the wire. The voltage across the resistor depends

10

G. D. DANILATOS

again on the relationship between transit time and RC. For short R C , we simply get V, = Ri, but for long RC, we get =

1

lo '

i dt,

which yields

It should be noted that the greater fraction of the signal induced takes place during the time the charge moves within a few anode diameters. In general, the time dependence of the signal depends on the time dependence of the charge location r(t), which in turn depends on the time dependence of the velocity v(t). The latter may depend on the charge energy and the gas pressure. For a fast BSE travelling a short distance, the velocity may be assumed constant, but the velocity of a SE will depend on the field intensity and pressure as the particle drifts towards one of the electrodes. In the general case, the signal at the output depends on the time constant of the external circuit and the transit time of the system, the output being neither integrating nor differentiating; basic circuit theory can be applied to derive the voltage signal across the resistor for any magnitude of T and RC. Wilkinson (1950) presents detailed derivations of the pulse shapes for parallel plates and cylindrical geometry for both electrons and ions drifting across the field. The main conclusion here is that the amount of charge induced is proportional to the fraction of potential difference traversed by the particle. If no potential is actually applied, then the same charge would be created as if a potential existed. The above analysis is the correct way for calculating the output signal due to a particular carrier. The careless method of counting the number of particles arriving at a particular electrode can lead to erroneous results; only on certain occasions can we get the correct answer, as when we want to know the total current (due to all carriers, positive and negative) in a steady-state situation. It is not only safer to resort to the above procedures, but it is sometimes the only way to solve a problem. One immediate application is the case when one electron appears suddenly, somewhere, between two electrodes, as for example by an x-ray interacting with a gas molecule. The charge induced on each electrode during the flight time of the electron is not one --e but a fraction of - e equal to the fraction of the total potential traversed. Together with the appearance of the electron, the law of conservation of charge dictates that an equal and opposite charge is created in the form of a positive ion. This ion will travel in the opposite direction and will induce on the electrodes an additional fraction of

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

11

negative charge so that the total charge will be equal to - e . This is not a trivial conclusion, because, since the velocities of the electron and the ion differ by three orders of magnitude, the implication is that only the signal due to the electron will be recorded by an appropriate choice of RC and scanning speed, and this signal will be less than that due to the total induced charge.

C. Induced Signals in the E S E M We wish to examine how the preceding theory applies to the ESEM. The microscope parts relevant to this question are incorporated in the circuit diagram of Fig. 2 (top). The main body of the microscope is maintained at earth potential, whereas the electron gun G is biased at a high negative accelerating voltage V,. A collector electrode GDD is placed between the specimen S and the earthed top entry of the beam at point A. The electron beam passes through a hole in GDD at B and strikes the specimen at C. For illustration purposes, let the distance between specimen and detector be 2 mm, whereas that between detector and top wall (at A) be 1 mm. Let us assume for simplicity that the specimen is earthed, without any loss of generality for the conclusions we are aiming at. Many electrons will thus close the circuit at D by flowing through the bulk or the surface of the specimen; let us call these electrons class (a). A second class of electrons, class (b), will close the circuit by flowing directly to the earthed walls of the microscope at E. A third class of electrons, class (c) (say loo,/,for a typical specimen), will end up at the detector at F. We can easily draw the signal shapes versus time, if we assume a uniform field between the detector and the top wall, and between the detector and the specimen. This has been done, for the three types of electrons, in Fig. 2 (middle) for the case of very large R C , and (bottom) for the case of very small RC. For our typical imaging conditions, the pulses will be on the average 8 ns apart and discrete, because the electrode spacings are shorter than the average (spatial) separation between incident electrons. If the integrating time RC = 1000 ns, then the effect of classes (a) and (b) is null, whereas the effects of class (c) will be the accumulation of small amplitude pulses. There will be l000/80 = 12.5 pulses accumulating in this time interval, and if C = 1 pF, the volts. This is the class of total voltage developed will be 12.5e,’C = 2 x electrons responsible for the contrast as the electron beam scans from pixel to pixel. In our typical image, we wait 50 times longer on each pixel than the hereby assumed R C , and the effect of this delay will be seen as an integration on the photographic film or cathode ray tube (CRT) screen. In the hypothetical case where the RC is very short, the aoerage current from classes (a) and (b) is null, whereas class (c) gives a net current corresponding to the movement of the collected fraction moving just between

12

Earth

(RC LARGE)

I

class a

class b

class c

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

13

A and B (or C and F). These pulses produce an average signal during the dwell time on each pixel, the signal that is responsible for the contrast at that pixel. The average intensity of current over 50 ps, if all the 625 electrons of our typical image were collected, is 625e/50 x lop6= 2 PA, which is very small. For this case, the resistance can be found from RC << 8 ns to be R << 8000 ohms volts. If the output current and the resultant signal voltage V << 1.6 x were measured with a hypothetical rms meter, the useful (contrast) signal would again be very small, and in addition, it would be superimposed on a relatively high dc background signal from all other pulses. In practice, charge or current preamplifiers may be used to produce a voltage output. Great improvement can be achieved when the signal is first amplified in the gas through an avalanche formation by an external field. Gaseous gains of a factor of 100 x can be achieved relatively easily, and much higher gains are also possible with special conditions and detector design. This is one of the topics analyzed in later sections. Following the above analysis, there is a case of detector design worth special consideration. It is possible to cover the detector with a thin film of insulating material. Then, an electron of class (c) will just miss the electrode after it travels the path CF. However, it will still produce the voltage pulse ABCF of about the same magnitude. Does this really mean that we can work with an insulated electrode and still produce an image? The answer is affirmative, and it has been shown likewise experimentally in the ESEM. The explanation is as follows: The presence of the insulator will not significantly alter the electric field, especially in the conditions of ESEM. The charges lodging themselves on the insulator will somehow find their way to earth through the gas, or to the conductor itself through the bulk or the surface of the insulator. The important condition is that this movement must be either relatively slow, i.e., it must occur over times longer than the frame time, or relatively fast, i.e., it must occur over times shorter than, or equal to, the pixel time; otherwise, the image will appear smeared, or zoned with horizontal stripes, if the film is thick, or the charge goes to earth. Images have been observed using an all-around insulated wire biased with 10 V in the presence of a few mbar air. In the beginning, the image was sharp and clear, but after some time, the relative contrast decreased due to decrease of the field by the accumulated charge. However, the charging of the insulator must have resulted in less than 10 V, as imaging continued to be possible. The equivalent circuit of this mode of imaging is shown in Fig. 3. The conductor electrodes B and D create the field between them as previously, whereas the insulator J acts both as a dielectric and as a charge collector dissipating the charges through an R'C' time constant; when the time constant is R'C' >> RC, it does not matter where the charge is dissipated, i.e., to actual earth or to the high-bias (and signal) electrode as shown.

14

G. D. DANILATOS

R‘

R”

FIG.3 The insulated electrode method. B and D are conductors creating a field through the insulator J (and the specimen S). Signal is induced in the conductors by moving charges, which are collected by J.

Likewise, in the most general case, the specimen is connected to earth, through the time constant R”C”, for which we want again either with R”C” >> RC or R”C” I z. An induced signal on either electrode appears whenever a charge moves in the field of the electrodes between the specimen S and the collector J. If the charge accumulates, it will result in a field opposing the original field. The opposing bias can be calculated as follows: Let us consider that the signal 61, is amplified by the gaseous amplification factor G, which will be discussed in later sections. Then the voltage developed is V = q/C = G 61,R‘C‘IC’ = G 61,R’. With I , = 20 PA, 6 = 0.1, G = 100, and R’ = 10” R, we get only V = 2 volts. With C’ = 10 pF, we have R‘C’ = 0.1 s, which can be longer than the frame time. Thus, it is possible, in principle, to operate this design of detector with proper choice of R’ and C ’ . Also, it is possible to have a low specimen bias (i.e., charging) with a similar calculation. The small specimen bias developed will have no noticeable effect on the BSE signal, but a few volts locally are enough to modify significantly the SE signal, and therefore charging should be kept sufficiently low for SE imaging. The precise mechanism of charge dissipation in the presence of gas is not clear at this stage, but the suppression of charging artifacts that would be observed on insulators in vacuum is a definite experimental fact in the ESEM. Thus insulators may be treated as “resistive” materials in the ESEM, i.e., as materials pertaining to both insulators and conductors. Since these materials allow both the transmission of signals through them and simultaneously the conduction of electrical current, the biasing mode and the time constants can be controlled for optimum operation. The use of resistive materials in a

THEORY OF T H E GASEOUS DETECTOR DEVICE I N T H E ESEM

15

controlled manner is known in nuclear research (Battistoni et al., 1978), and similar use of them in the design and operation of the G D D could revolutionize the entire ESEM technology. In the present light, we can clarify a misconception or misunderstanding that many electron microscopists have had in relation to the specimen-current mode of imaging. This mode is often also called ubsorbed current. From the above analysis, it should now be clear that the contrast does not arise by the absorption per se of the beam electrons. It is during the flight of all electrons when pulses are induced, prior to absorption by the specimen. This is not trivial. It implies that “specimen”-current imaging is also possible with insulators in vacuum, provided that the specimen does not charge up significantly and a conductor is located nearby to pick up the signal induced by the flying electrons. Such a situation is possible to achieve in vacuum for each insulator by finding the appropriate accelerating voltage for which the beam current equals the sum of BSE + SE currents (no charging condition). Therefore, it is not the lack of conductivity that has prevented imaging of insulators with the current mode in conventional microscopy; it is rather the detrimental charging artifacts that have prevented the electron microscopists in general to think in terms of signal induction. In other words, the real mechanism of the so-called absorbed current mode of imaging has eluded many electron microscopists because of their inability to image insulators as a routine practice in SEM. Imaging of insulators is a normal practice in the ESEM, and we now clearly understand the various contrast mechanisms. Thus, the contrast on the images reported by Shah and Beckett (1979)by use of the “specimen current” cannot be simply attributed to the increased conductivity of the wet biological specimens and cannot be simply explained by the absorbed specimen current; wetness and moist environment were not the prerequisites for the operation of an ESEM as those authors thought, whereas the presence of an ionized environment and signal induction was totally ignored. The above mode of imaging with insulated wires was first accidentally noted by the present author in the ESEM, and it was later confirmed by controlled experiments. In nuclear technology, it was first Maze (1946), who developed a Geiger-Muller counter based on the same principle. This was made by painting the cathode of the counter on the outside glass surface with carbon paint. The advantage of this method is that it allows the application of high potential without an early discharge and spurious counts. The method outlined here warrants further investigation and defines one of our future tasks: for the insulated electrode method, to investigate the suitability of materials on account of radiation effects, electrical properties, efficient geometries, and configurations, as these qualities are determined by the needs of ESEM.

16

G. D. DANILATOS

111. PHYSICAL PARAMETERS

In this section, the various physical parameters that are involved in the performance of the G D D are briefly examined. The theories pertaining to these parameters have been dealt with in numerous works for more than a century. To mention just a few: the classic works by Townsend (1947), Loeb (1955), Cobine (1941), Healey and Reed (1941), Meek and Craggs (1953), and Engel (1965), but information and other developments can also be found practically in any book on ionized gases, such as that by Nasser (1971) with detailed references to individual topics. It is beyond the purposes of this work to even outline these theories, and the researcher can refer to these works for a better understanding of the physical mechanisms involved. For the same reason, it is not always practical to refer directly to original authors of well-established theories and formulas.

A . Electron and Ion Temperature

When ions or electrons wander around inside a gas in the absence of an electric field, they come into thermal equilibrium with the gas, but, when a field is applied, they acquire additional energy, which generally results in an increase of their agitation (thermal) energy above the 3 kT/2 particle thermal energy of the host gas. This is usually expressed as a factor E :

with u being the rms agitation (thermal) velocity of the ion or electron. The distribution of electron energies is sometimes assumed as Maxwellian, but more often the Druyvesteyn one is applied. The distribution parameters depend strongly on the nature of the gas. Detailed presentations on this question can be found in the books by Healey and Reed (1941) and by Loeb (1955). The factor E plays an important role in the calculation of many practical parameters. Usually this factor appears together with another dimensionless factor A as a product AE, where A is expressed in terms of various means (averages) of the electron velocity u: A=

3(u) 2(t.2)(v-1)

The value of A depends on the actual velocity distribution. For approximately E / p < 0.5, the distribution is closely Maxwellian and A = 1. For

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

17

5 < E / p < 25, thedistribution is best represented by the Druyvesteyn distribution and A = 1.14 (Huxley and Zaazou, 1949). In the present work, we quote as values of E its product with A for a Druyvesteyn distribution, as is commonly done, and in various formulas the factor A has likewise been omitted. Many parameters are expressed as a function of the ratio E / p ; the meaning of E / p is that it is proportional to the energy acquired by the charged particle between two successive collisions. This is because the mean free path L , is inversely proportional to pressure and hence E / p is proportional to E L , . For ions (positive or negative) and for low enough E L , (i.e., E / p ) , practically all this energy is transferred to the gas molecules during each collision, and the coupling is said to be strong. The coupling between ions and the gas is very effective for about E / p < 20 V/Pam, and their temperature is practically equal to that of the neutral gas. However, the electron temperature is generally higher than that of the gas, even for low values of E / p , because of weak coupling. This is seen as an increased factor E , values of which are given in Tables II(a-g) for various gases at different E / p values and for gas temperature T = 288°K. In the same table are also given the agitation velocity u of the electrons; the other parameters included are discussed in following sections, and all have been converted to SI units.

TABLE IIa TOWNSENI) FACTORS I N AIR(HUXLEY AND ZAAZOU, 1949) ElP 0.375 0.75 1.5 2.25 3.00 4.5 6.0 1.5 11.25 15 18.75 22.5 37.5 15 112.5 1 50 187.5

E

10-5

vd

x

10-3

9.43 11.0 3.58 13.2 20.5 4.88 18.8 37.5 6.62 23.5 52.5 7.78 27.4 62.0 8.5 35.8 75.0 9.35 43.8 10.1 87.0 51.4 10.7 98.0 68.5 12.0 124.0 84.0 13.0 146.0 98.0 13.9 166.0 (From Townsend and Tizdrd, 1913) 115

169 268 356 435 500

Le x 102

6.20 5.93 5.60 5.33 5.20 5.07 5.00 4.93 4.93 4.93 4.93

18

G. D. DANILATOS TABLE IIb TOWNSEND FACTORS I N OXYGEN (HEALEY ANII KIRKPATRICK, 1939) x 10-5

t:

5.5 9.5 19.0 32.0 43.0 52.0 57.5 62.0 7 1.o 95.0

0.188 0.375 0.75 1.5 3.75 7.5 11.25 15 22.50 37.5

L'd

2.7 3.55 5.01 6.5 7.54 8.29 8.71 9.05 9.7 11.2

x

L, x lo2

13.3 16.1 16.3 16.9 27.3 49.5 67.0 81.5 99.5 159.0

h x

13.33 10.53 7.55 5.07 3.81 3.81 3.61 3.43 2.97 3.29

10.4 1.25 2.2

5.2 16.7 17.2 7.0 0.0 0.0

TABLE IIc TOWNSEND FACTORS I N NITROGEN AND BAILEY, 1921) (TOWNSEND

ElP 0. I88 0.375 0.75 1 .so 2.25 3.75 7.50 15.00 30.00 45.00 37.5 45.0 52.5 60 67.5 75 112.5

I 50 225 300 375

E

7.5 13.0 21.5 30.5 35.5 41.3 48.5 59.5 89.0 126.0

u x 10-5

ud x 1 0 - 3

3.15 5.15 4.14 6.2 5.35 8.7 6.35 13.1 6.85 17.8 7.4 27.0 8.0 48.5 8.85 86.0 10.8 146.0 12.9 193.0 (From Gill and Engel, 1949) 232 289 330 369 402 435 488 560 590 667 740

lo5

L: x loz 6.00 4.73 4.27 3.84 3.76 3.69 3.59 3.55 3.67 3.85

TABLE IId TOWNSEND FACTORS I V HYDROGEN AND BAILEY, 1921) (TOWNSEND ElP 0.188 0.375 0.75 1.50 3.75 7.50 15.00 30.00 37.50

4.85 4.33 3.81 3.19 2.85 2.73 3.33 4.89 5.6

6.5 9.0 11.9 16.0 25.5 38.0 70.0 160.0 217.0

2.02 2.62 3.5 4.3 5.9 7.62 10.15 13.1 14.0

3.1 5.4 9.3 15.0 26.4 44.0 78.0 130.0 148.0

TABLE IIe TOWNSENDFACTORS IN WATER VAPOR AND DUNCANSON, 1930) (BAILEY E P

E

9.00 10.51 12.00 15.00 18.00 24.00

3.78 5.67 8.64 18.9 37.0 48.9

x 10-5

cd

2.21 2.72 3.37 4.98 7.00 8.04

x 10-3

L: x 102

h x lo5

0.49 0.64 0.84 1.43 2.20 2.25

0.6 1.9 13.0 30.0 45.0 50.0

30 35 42 62 81 96

TABLE ]If TOWNSEND FACTORS IN HELIUM AND BAILEY, 1923) (TOWNSEND ElP

e

0.0 10 0.015 0.038 0.075 0.15 0.375 0.75 1.50 2.25 3.00 3.75

1.77 2.12 3.68 6.2 11.3 27.0 53.0 105.0 137.0 152.0 172.0

x

10-5

1.53 1.68 2.12 2.87 3.87 5.96 8.4 11.8 13.5 14.2 15.1

ud

x 10-3

1.11 1.33 2.14 2.96 3.93 5.74 8.25 12.7 17.5 23.5 30.2

L: x loz 12.19 10.40 8.80 7.93 7.07 6.40 6.47 7.00 7.33 7.80 8.53

20

G . D. DANILATOS TABLE IIg

TOWNSEND FACTORS I N ARGON (TOWNSEND A N D BAILEY, 1922)

ElP

E

0.094 0.146 0.266 0.394 0.533 0.7 13 0.938 3.75 7.50 11.25

100 120 160 200 240 280 320 310 324 324

x 10-5

ud

11.5 12.6 14.5 16.3 17.8 19.3 20.6 20.2 20.7 20.7

x 10-3

3.1 3.25 3.6 4.15 4.85 6.0 1.7 40.0 65.0 82.0

L: x 102 26.67 19.60 13.73 12.00 11.33 11.33 11.87 15.07 12.53 10.53

The general rule is that coupling is weaker with monatomic than with diatomic gases. The coupling becomes very strong with polyatomic gases. The degree of coupling affects other parameters such as mobility and diffusion. B. Electron and Ion Mobilities In the presence of an electric field, the electrons and ions, apart from their agitation velocity, drift towards one of the electrodes with a constant velocity cd that is a function of the field, pressure, and nature of the gas. The absolute and relative values of drift velocity of ions and electrons moving under the forces of an electric field in a gas is of paramount importance in determining the frequency response of the GDD. More precisely, it is these velocities together with electrode geometry and accompanying form of electric field that determine the time response of the system. It depends on this geometry whether we will use the ions or the electrons for signal detection, and therefore, we need to examine here the behaviour of both types of charge carrier. 1. Ions

The attainment of the constant drift velocity is actually achieved in the steady-state (equilibrium) condition after a number of initial collisions (Engel, 1965). The following defining equation relates the drift velocity ud and E: ud

=

KE,

(17)

where K is the ion mobility (e.g., Engel, 1965). Because K is inversely proportional to pressure, the constant K ' , referred to as reduced ion mobility

THEORY OF T H E GASEOUS DETECTOR DEVICE IN THE ESEM

21

(Loeb, 1955), is sometimes also used in an equivalent equation: ud

= K'(E/P),

(18)

and care should be taken as to which parameter is used each time ( K ' is the mobility at (not per) unit pressure that is (here) one pascal in SI units). For low and moderate fields ( E / p < 20 V/Pam), Langevin has derived the formula (Cobine, 1941):

where mi is the molecular mass of the ion and m that of the gas. L,, is the mean free path of the ions in the gas: Ll, =

1 nn(r, + r)*'

with II being the gas concentration, and ri and r the molecular radii of the ion and the gas, respectively. By converting Eq. (19) as a function of pressure p , molecular weight M and viscosity y, we obtain in SI units: K=

8 x lo7 y

PM

The above formula yields values of about three to five times the measured ones. Elaborate theories and detailed discussions can be found in the literature, but it would be futile here to seek the best formula, for the reason that the mobility depends strongly on the impurities in the gas; it is not expected to use pure gases on a regular basis in the ESEM. For example, the mobility of highly purified helium at one atmosphere, O'C, is 17 x m2/Vs. The cause of m2/Vs, whereas for ordinary helium is 5 x change of mobility with impurities is the formation of clusters around the ions, which are thus becoming heavier. The mobility of negative ions is only little different from the positive ones. The constant K ' = K p for ordinary gases is usually between (Cobine, 1941):

8 < K ' < 80 m2Pa/Vs.

(22)

Measured values of K ' for some positive ions in their own gas are quoted in Table 111. It is noted how the values reported differ for the reason mentioned, and therefore, we cannot expect to predict the precise mobility in the ESEM. In Table I11 the range of E / p , in which the mobilities quoted are valid, is given for two cases. As mentioned before, the temperature of the ions is raised above the temperature of the gas, and the mobility ceases to be constant as the field is increased beyond some characteristic value. After passing a complex

22

G D DANILATOS TABLE I11 K FOR POSITIVE 10% I \ m'Pa V\ (ADAPTED FROM COBI\LE 1941, A N D EVCEL 1965)

REDuCtD M o H i L i r Y

I3 6 18.7

12.7 26.7

13.1 13.3 18.62(-1on)

59 133.3

50.9 106.6

99 44 for Elp < 8

18 16 for E,p

i40

variation, the drift velocity eventually varies in proportion to (E/p)'". For ions in their own gas, the following formula has been adapted from Engel (1965):

(y2,

cd = 7.428 x lo4 v1127'1/4M-3'2

(23)

which yields a close order-of-magnitude result. Experimental results on mobilities of ions (or drift velocities) over a wide range of E i p for various gases can be found in the literature; for convenience, three examples have been adapted from Loeb's book and presented in Figs. 4, 5 , and 6 for nitrogen, oxygen, argon, and helium. 2. Electrons

With electrons, the situation is different. First, because of its relatively small mass, we expect its mobility to be at least three orders of magnitude

40

c

30 -

2 E

M-

x

15-

7s 3

100-

6-

5-

4'

'

20

"

'"":

40 60 80 100

4

200

400 600 1000

E/p, V/Pam FIG.4 Drift velocity of nitrogen ions in nitrogen. (Adapted from Varney, 1953.)

THEORY OF THE GASEOUS DETECTOR DEVICE I N T H E ESEM

v1

E

m

23

6 -

43-

'8 2 X

=2" 1.0 0.6

--

E1 0.2

,

20

, , , , , ,,, 40 60 100

, 200

, , , , , , ,, 400 600 1000

E / p , V/Parn FIG.5 Drift velocity of oxygen ions in oxygen. (Adapted from Varney, 1953.)

100

-

N

4 -

'

;;8 l O

20 ' 4'0

QO'lOO ;00'4bO' 1 'k O

E/p, V/Pam FIG.6 Drift velocity of helium and argon ions in their respective gases. (Adapted from Hornbeck, 1951.)

greater than that of ions. Then, they quickly attain an increasing energy even in weak fields, and we cannot assign a constant mobility. The electron drift velocity is inversely proportional to the thermal velocity of the electrons and, assuming a Maxwellian distribution of velocities, is given by the equation (Healey and Reed, 1941):

2eLL E 3um, p '

u*=--

24

G. D. DANILATOS

where m, is the electron mass and LL the mean free path at (not per) unit pressure (i.e., at 1 Pa):

(25)

L: = L,p.

This may be referred to as the reduced mean free path, and values of it are given in Tables II(a-g). In the same tables are also given the corresponding drift velocities. By using the values of u and Lk from these tables in Eq. (24),we note that the latter yields a close result to that given in the tables. Theoretical predictions of the electron mobility in the extended range of E / p are either nonexistent or are quite complicated. In addition, the strong dependence of mobility on gas constitution makes such predictions of little practical value in the ESEM. Generally, the variation of zjd versus E / p takes a near-parabolic form (square-root curve) initially and then exhibits an inflection upwards. In the latter range, the electron starts losing energy due to inelastic collisions, so its thermal (agitation) movement is not an obstacle to achieving a higher drift velocity. The drift velocities for several gases are listed in Tables II(a-g), with nitrogen, in particular, over an extended range. The drift velocity for helium, also over an extended range, is given in graphical form in Fig. 7 (Phelps et al., 1960). Impurities affect the electron mobility in different ways. An electronegative impurity may capture and release the electrons during its transit, thus effectively reducing the mobility, or the fraction of free electrons. Mixtures of different gases have mobilities in accordance with their constitution. For

/

6 4 ul

\

/

2 -

E

lo5-

6 4 -

Elp, VlParn FIG. 7 Electron drift velocity in helium. (Adapted from Phelps et al., 1960.)

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

25

example, argon/nitrogen mixtures show higher drift velocity than pure argon. Addition of polyatomic impurities can increase the mobility considerably. Detailed theories and mobility listings can be found in the books by Loeb (1955) and Tyndall (1938). More recently, Christophorou et al. (1979) have presented detailed data on electron drift velocities for argon/hydrocarbon mixtures. Gas mixtures have been studied for the purposes of radiation counters, but most of these may not be suitable for ESEM work. Therefore, a new research task is defined for finding appropriate gas mixtures with high electron mobilities suitable for ESEM applications. We need to know the mobilities for predicting the transit time of charge carriers between two electrodes (the transit time of a given carrier should not be confused with the propagation time of a spark). The mobility is one parameter in the determination of the time response, but additional parameters can jointly affect this determination. For example, the field variation for a given electrode shape can have effects of the same magnitude as that determined by the mobility alone.

Diffusion can influence the performance of the G D D significantly, and this phenomenon must be understood and quantified accordingly. Particles suddenly liberated at a given point in a gas diffuse spherically. Standard diffusion theory predicts that the mean displacement rD of the particles from the “centre of gravity” at a time t after their release is given by: rD = (6Dt)”2,

where D is the diffusion coefficient. We need to inquire what the relative importance of this displacement is in comparison to the interelectrode distance and electrode dimensions. Figure 8 shows an electrode configuration that establishes a uniform field between electrodes. The top electrode is surrounded by a concentric annular plate held at the same potential, but only the central electrode is used for signal detection. The outer electrode is placed only as a guard electrode to help define a uniform field. In such a configuration, if r,, is greater than the size of the collecting electrode rE, electrons will be lost to the guard ring; these electrons will produce pulses of variable magnitude in the central electrode during their transit, but the net result is null. The ratio j’ = r,/d is a measure of the electron losses, and we can calculate it as in Wilkinson (1950): We have d = u d t and f’ = ( 6 D / ~ , d ) ” ~Further, . theory gives the following relationship between mobility and diffusion:

26

G. D. DANILATOS

l------

G

G

+v

+v

\

\

\

I I

\ \

‘

\ I

‘\1 s

A

FIG.8 Electrons are released at point S of earthed plate electrode A and drift in a uniform field towards electrode E surrounded by guard G . Diffusion displaces the electrons at a distance rD away from the axis of E.

and we get v,/D

= eE/EkT, E = f

V/d, and finally, =

(ET)1’2

This shows that losses, for this particular geometry, depend on the applied voltage and the value of E. For ions, with V=7.5 volts across 2 mm at lo00 Pa (i.e, E / p = 3.75) and c = 1, we get f = 0.14, and the losses become significant for rE < 0.28 mm. For electrons in nitrogen, E = 41.3, and under the same conditions, f = 0.91, and the losses are significant for rE < 1.8 mm. In argon, the losses can be very pronounced for small electrodes (rE < 5 mm). With nitrogen again, applying V = 65 volts (i.e., E / p = 45), we find an improvement o f f = 0.54, for which rE < 1 mm. Unfortunately, for most gases we have no figures for E at much higher E / p (e.g., E / p = 150) that can be applied in the microscope when gaseous amplification is desired, but it appears that the electrons follow the lines of force and the losses are small, according to a discussion by Wilkinson (1950). However, the degree of diffusion effects may be pronounced in the ESEM, where the electrode can be of the order of the pressure-limiting aperture (PLA 1) diameter. An alternative, more precise, way to examine the effects of diffusion is by calculating the fraction of electrons arriving at the electrode E in Fig. 8. If the total number of electrons originating from S is N o , and the number arriving at E is N,, the fraction R = N,/N,, can be calculated from the diffusion equation, which was derived by Huxley and Zaazou (1949) and is adapted

THEORY OF T H E GASEOUS D E T E C T O R D E V I C E I N T H E E S E M

here as: R

=

1 - (1

+ $)-1’2exp{&

[l

-

(1

+

27

5)1’2]}.

(29)

This equation was derived for slow electrons drifting without multiplication. However, according to Townsend and Tizard (1913), in the case of avalanche multiplication, the fraction of electrons collected by E relative to the total number of electrons arriving at the top after multiplication is the same as R. The numerical examples considered previously in conjunction with Eq. (28) can now be repeated with Eq. (29) to find the precise fraction of electrons collected. Thus, for r E = 1.8 mm, d = 2 mm, V = 7.5 V, p = 1000 Pa, and E = 41.3, we get R = 0.78, i.e., when we equate the radius of the collecting electrode to the average displacement rD of the charges, 78% of them are collected. Another case of interest is to apply Eq. (29) when biasing the anode (collecting and guard electrodes) with 400 V, for d = 2 mm and p = 1000 Pa in nitrogen. From Table IIc we find by extrapolation E = 580. If we wish to collect, for example, 90% of the electrons, we find that we must use a collecting electrode with radius rE = 1.17 mm; for 78”/,,we need rE = 0.93 mm. For practical purposes, it is useful to note that when we increase the specimen chamber pressure, we normally decrease the specimen distance from PLA 1 in inverse proportion. Thus, if the detector is at the aperture plane, E / p will remain constant with constant bias, and hence the value of E is also constant. From this, we find that, by fixing rE = 1 mm and V = 400 V, we get R = 0.07 at p = 200 Pa and d = 10 mm; R = 0.82 at p = 1000 Pa and d = 2 mm; R = 0.95 at p = 1333 and d = 1.5 mm; R = 0.997 at p = 2000 Pa and d = 1 mm. To maintain a constant R, one would have to vary rE in proportion to d, so that the ratio r,/d remains constant. To fix R = 0.9, we should maintain a ratio r E / d = 0.585 under the conditions of the previous example, i.e., with pd = 2000 Pam. In the ESEM, the emerging electrons from the beam-specimen interaction have varying initial energies. The BSE will follow their own trajectories. The SE have their energies distributed mainly around 2 eV, which is comparable with the thermal electron energy gained from a moderate field. For example, at E / p = 0.75, the electrons in argon average 10.6 eV, whereas in nitrogen they average 0.8 eV (since 3kT/2 = 0.037 eV at room temperature, or dividing E by 27). This means that the SE lose their “memory” after only a few collisions. The case of a fine wire biased at a high potential is more complicated to analyze. In this case, the field is strongest only a few diameters around the wire, and if the electrons start at a considerable distance from it, they may become lost by diffusion (e.g., to a nearby wall). Whereas this might appear to suggest

28

C . D. DANILATOS

that plate electrodes are to be preferred, such a choice should be deferred until other factors are also examined. There are both advantages and disadvantages between plates and wires, with sometimes opposing effects. From the above analysis, it is apparent how useful the collection of data included in Tables II(a-g) is for technical design. However, no complete data have been found, especially for the conditions of ESEM. Townsend and Tizard (1913) have reported measurements of E and ud for air with E / p in the range up to 180 V/Pam. These measurements have been criticized by Huxley and Zaazou (1949), whose values of ud differ mainly in the low range of E / p . The data of the early authors are included in Table IIa for the higher range, in view of lack of information from other sources. For gaseous mixtures, the situation is more complex and data more sparse. Townsend and Tizard (1913) have reported that moisture in air reduces the diffusion of electrons for low values of field. This is due to electron attachment to water molecules, the negative ions thus having a temperature close or equal to that of air. For example, with E up to 800 V/m and p = 3.6 mbar air in their apparatus, this phenomenon was pronounced, but at higher fields, the electrons moved freely. Bortner et al. (1957) quote drift velocities for Ar, N,, CH,, CO,, C,H,, cyclopropane, and mixtures of these gases for E / p up to a few V/Pam. Den Boggende and Schrijver (1984) present data on electron cloud sizes and drift velocities of a number of gases and their mixtures, including N, and CO,, which can be introduced in the ESEM. The data are for fields up to lo5 V/m at atmospheric pressure. Binnie (1985) reports on drift velocities and diffusion (both longitudinal and transverse) for argon/CO, mixtures. Other recent measurements have been reported especially for gas mixtures used in nuclear instruments; Piuz (1983) gives data on drift velocities and longitudinal diffusion for Ar-CH,, Ar-CH,CH,, Ar-CO,. The diffusion coefficient of ions can be estimated from Eqs. (27) and (21) as (30) Experimental values are given in Table IV. In either way, the actual values are again affected by the purity of the gas. TABLE IV EXPERIMENTAL VALUES OF DIPFUSIO~U COEFFICIENT €OK SOME IONS IN THElK OWNGASI N rnZ/s, AT p = 1 Pa (ADAPTED FROM ENCEL.1965) Air 0.29( +)

N* 0.31(+)

0 2

H2

0.43(- ) 0.28( + )

1.3(+)

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

29

D. Recombination

Recombination results in signal loss due to the neutralization of charge by way of electron capture by a positive ion or charge transfer from negative to positive ions. Our approach in tackling the problem of recombination is to find first rough estimates of it, and if the magnitudes are not important, we need not pursue unnecessarily the complexities of detailed derivations and calculations. In the ESEM, there are two different regions of concern. One is the region inside and close to the primary beam travelling through the gas. The effects of ionization and recombination will appear as a background-level signal not carrying information from the specimen. This problem was only outlined previously (Danilatos, 1988) and will not be analyzed here either, because it requires special attention. Its effects may be important. The other region is the one away from the beam, as it is influenced by the signals emerging from the specimen. This region is relatively extensive. It is the effects associated with the signals in this region that we consider here. In nuclear physics, there is a distinction between preferential, columnar, and volume recombination. There is an excellent discussion by Wilkinson (1950) on the three types of recombination, and his reasoning can be profitably transferred to the ESEM conditions. The following conclusions are based on his work. Preferential recombination is that which takes place between a negative ion, or electron, and the positive ion from which the electron was originally separated. This happens when two oppositely charged particles find themselves at such a short distance that the electrostatic forces can overcome the thermal energy tending to diffuse them away. This type of recombination becomes of any practical significance only at pressures above 100 atm. Up to one atmosphere, where we expect the ESEM to operate, the probability of such recombination is extremely small. Columnar recombination is the type occurring between ion pairs formed along the track of an ionizing particle. This phenomenon occurs mainly with heavy particles in nuclear physics travelling along almost straight tracks, but with the electrons in the energy range of the microscope, the tracks are zigzag. Perhaps, the primary beam as a whole could be treated as in the theories of columnar recombination, but this will not be considered here. If there is any recombination, this can be classified under the third type: Volume recombination occurs when two oppositely charged particles from different tracks are neutralized. Wilkinson has concluded that this type is negligible for pulse counter devices, since for such devices to be effective, the pulses are separate in time and space. He considers the case when there is a steady background ionization source as a possible cause for recombination. Such a possibility may be applicable in the case of ESEM. The transit time, for

30

G. D. DANILATOS

example, between two plates 2 mm apart with 40 volts across, in nitrogen, at 1000 Pa ( E / p = 20), is about 20 ns for electrons and about 4000 ns for ions. That means that for our typical case of imaging conditions, the electrons are being collected discretely one at a time, while the positive ions lagging behind create an accumulated positive ion density ever present in the gas. If this density is n', the fraction f' of electrons captured is gived by (31) where R is the recombination coefficient. Wilkinson (1950) discusses the integration of the above equation and presents the result

f=

1O18Rit t +

-

where i is the steady current due to the background ionization occurring uniformly throughout the volume x, and t + and t - are the transit times of positive and negative ions, respectively. The coefficient R depends on the pressure; this, starting from a low value at low pressures, reaches a maximum of the order of m3/s at one atmosphere and decreases again at higher pressures. This is the maximum value reached in a situation of positive and negative ion recombination; in free-electron gases, the coefficient can be four orders of magnitude smaller. For i, we can take the order of magnitude of the primary beam, say i = lo-'' A, and in the worst case of dealing with ions only, ti = t - = 4 x s at one atmosphere; if, arbitrarily, we take x = m3, we find f' = 1.6 x which is exceedingly small, and it should be even smaller in the conditions of ESEM. Above, we assumed x = lop6 m3 as the active volume of the signals. Recombination rates may be significant if we want to consider the volume of the electron beam only and the recombination effects thereby associated with the electron beam alone: If the beam has a diameter of 200 angstroms, the m3. This small volume volume over 2 mm distance is roughly x = would tend to produce a high f ; however, we cannot find f without knowing t+ and r - , which must now be taken as the lifetimes of the charges within the beam volume (they must be much shorter than the transit time between electrodes). The general conclusion is, therefore, that recombination effects inside the signal volume of gas are unlikely to be a cause for signal loss. The above discussion has not considered a special case that is a strong candidate for recombination; namely, recombination during the avalanche formation. In the particular case of high gaseous amplification, the head of the avalanche achieves a high density charge. Space-charge effects can modify the

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

31

whole amplification process. Space charge may result in pulse peak shift in proportional counters according to Hendricks (1969), but the peak shifts have also been attributed to volume recombination, and there is a controversy on this matter (Mahesh, 1978). It is early to determine the role of recombination in the avalanches of ESEM at this stage. E . Electron Attachment

A proportion of the signal can be lost due to electron capture in electronegative gases. When an electron is captured to form a negative ion, the negative ion is more than 1000 times slower. If during the dwell time per pixel the ions move only through a small fraction of the potential difference between electrodes, these ions will contribute only a small fraction to the signal, and the electrons captured should be considered lost. The loss of signal is not simply in proportion to the number of electrons attached; this loss also depends on the location of capture in the field: If an electron is lost after it traverses most of the potential difference, the signal loss is minimal, otherwise it can be significant. The configuration of field is important in this respect, and the electron-capture effects will be different in the field between plates from the field in cylindrical geometry. The theory of electron attachment can be found in many works (see, e.g., Massey, 1969), but a concise derivation of the signal loss in particle counters can be found in a review survey by Franzen and Cochran (1962). The same procedure is adapted here. If L , is the electron capture mean free path, then the electron capture mean free path L , in the direction of' the applied jield is simply

L,

= (Cd/U)L,,

(33)

where, as usual, cd and u are the drift and thermal electron velocities. If No electrons are liberated in a uniform field at xo distance from the terminal (collecting)electrode, then after travelling a distance x from xo, there will be N electrons still free, given by N = N o exp( - u/L,).

(34)

The total signal voltage V, induced by the free electrons after time T, when all the electrons have been collected, is given by the integration ( R C << T ) :

and

32

G. D. DANILATOS

The signal without electron capture can be found either from the above equation by letting L, become very large, or directly from Eq. (8) as

Thus the transmission factor is

as opposed to the transmission factor N / N Oof the number of particles alone (without induction effects). To evaluate the magnitude of this effect, we only need to know the electron-capture mean free path or an equivalent parameter. This may not be easy under all conditions. Usually in the literature, we find the electron-capture (or attachment) cross section o c , or the probability of attachment h when a collision occurs. These quantities relate to the total collision cross section G and the mean free path L, of the electron as follows:

and from Eq. (33) we get

since L , = l / n o and L, = L;/p. Already, it has been shown in the past that the number of electrons attached can be significant (Danilatos, 1988). Now, we can estimate the transmission factor in Eq. (37) by a numerical example for the actual SE signal in the ESEM typical image. The number of electrons leaving each pixel is 625, and a fraction of them will be captured in an atmosphere of oxygen. Let us choose values that are readily available for the various parameters. Let us put V = 7.5 volts, d = 0.002 m, p = 1000 Pa, so that for E / p = 3.75, we get (from Table IIb) for oxygen, u = 7.54 x los m/s, L’,, = 27.3 x lo3 m/s, L:, = 3.81 x m, and h = 5.2 x lo-’; from these values, we get L, = 0.027 m. If all the SE start at xo = 0.002 m, we get f = -0.96 (the minus sign arises from the negative charge of the electron producing a negative pulse); therefore, the losses are mild. We can increase the distance xO by placing the electrodes further apart and by increasing the bias accordingly so that E / p is constant; then L , remains the same, and we immediately find f = 0.83 for yo = 0.01 m, which is significant. It is further noted that the transmission factor .f of Eq. (37), based on the induction principle, predicts a much better signal than simply counting the transmitted fraction of electrons N / N O ( e g , only one third is transmitted at xO = L,).

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

33

There are some peculiarities with oxygen worth special attention. The attachment cross section is an irregular function of energy (see data and reviews by Wilkinson, 1950; Engel, 1965; and Franzen and Cochran, 1962). It shows a sharp peak at 0.3 eV, a lower and broader peak around 2 eV, and a very pronounced peak around 4-8 eV, with other irregularities elsewhere. Between peaks and troughs, the variation is by up to a factor of 10. This can be significant. The example quoted above was for conditions of E and p corresponding to 1.6 eV. By increasing the bias so that the electron thermal energy is 2 eV, the attachment cross section increases and may result in a significant loss of signal. By further increasing the bias, we will operate away from the peak, and the lossess will be minimized again, until the next peak. The end result on the transmission of electrons depends on how several parameters interplay. A prediction of the transmission is complicated by the fact that the actual electron energies may be broadly distributed about the average energy considered here. Added to this difficulty is the lack of agreement on the values of attachment cross sections (or probability of attachment factors) in the literature. Nonetheless, we may use the data of Table IIb to get an idea of the variation off versus E l p . The result is shown in Fig. 9, where we note that the transmission is at a minimum between 5 < E / p < 10; this corresponds to an electron energy of about 1.7 eV.

Oxygen p 2000 Pa D = 5mm

O'*

t

0"

0

I

10

"

'

20

'

'

30

.

I

40

E/p, V/Pam FIG.9 Variation of transmission factor (induced signal) versus E / p due to a variation of electron attachment probability in oxygen (pressure p D = 5 mm).

=

2000 Pa, electrode separation

34

Ci D. DANILATOS

The “peculiar” variation of the attachment cross section of oxygen (and possibly of other gases such as water vapor) may have additional significance: The majority of SE in the microscope have an energy around 2 eV, and by purposeful control of the various parameters, we could eliminate or facilitate the passage and detection of SE. That is, oxygen and other suitable gases and mixtures of gases can be used as “natural” filters. In the example above, only oxygen was considered. but the presence of oxygen in another gas can have similar effects. The same equations above can be used by simply using the partial pressure of oxygen, but the electron temperature and drift velocities pertaining t o the mixture or the predominant gas should be taken. For example, 0.1% of 0, in argon has been found to cause an appreciable attachment loss; this is due to the high thermal velocity of electrons in argon corresponding to a peak of the electron attachment coefficient. Addition to this mixture of 2% CO, diminishes the losses again because the new mixture results in lower electron temperature corresponding to a low attachment cross section (Franzen and Cochran, 1962). Obviously, much more study and carefully designed experiments are needed to further explore these new possibilities. That is, work should be devoted to find suitable gases and mixtures of gases in conjunction with suitable electrode configurations for controlling and filtering the types of signals collected in the ESEM. In the above analysis, we assumed throughout that the scanning speed is fast and the negative ions do not have much time to contribute to the signal. In our typical image, we took 50 p s as the waiting time for each pixel. The drift velocity of negative oxygen ions at E / p = 3.75 is 14.9 m/s [from Eq. ( 2 3 ) ] , and the transit time for a distance of 0.002 m is 130 ps. Therefore, there will be a loss of signal, unless we, at least, double the exposure time. The slow arrival of ions (if the time constants are comparable) can produce a smearing of the image, a phenomenon already observed and referred to as zoning in the first publication (Danilatos, 1983b). Hence, with appropriate choice of scanning speed and circuit time constants, we can eliminate this effect, but at the possible loss of signal intensity. The application of the highest possible bias, to produce additional amplification, may not result in the avoidance of electron capture, but this case must be carefully studied. The literature on Geiger- Miiller and proportional counters (which operate at high bias) suggests that electronegative gases have always been a problem, but this understanding should be reexamined in the specific conditions of ESEM by use of special geometries.

F. Effective Ionization Energy An electron passing through a gas loses energy not only in the form of ionization, but also in the form of other processes, such as excitation,

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

35

0 1 2 3 4 5 6 7 8 9 1OOO1112 13 1L1500 Energy in Volts. FIG. 10 Total number of ion pairs produced versus initial electron energy in nitrogen. (From Cobin,? 1941.)

dissociation, and so on. Figure 10 shows the total number of ionizations caused by an electron in nitrogen for various initial electron energies (Cobine, 1941). The average, or effective energy W spent between successive ion-pair formations, is not constant in this range (0-1500 5V). For energies in the range 2-5 keV, Cohn and Caledonia (1970) have reduced a value W = 34.6 eV. For many gases the effective ionization energy is between 30 and 40 eV. Knowledge of W is needed in the calculation of the maximum possible number of ion pairs a SE and a BSE can form, and hence the maximum intrinsic (initial) amplification we can expect from the gas-signal interaction. In the case of gas mixtures, we can calculate the effective ionization energy of the mixture qj from the values and Wj of the component gases i and j by the following equation (Franzen and Cochran, 1962):

pi

where Zij = Pi

,with pi the partial pressure of the gas i and the constant

+ aijpj

& and Wj for various mixtures. From the above data, we can estimate that a 5 keV BSE will generate about 140 electrons in nitrogen, if allowed to travel its full range. In practice, this is rather difficult to achieve, because the BSE will strike some wall well before it exhausts its energy. This is particularly the case in the very short working space usually demanded by ESEM. Only the low takeoff angle BSE, especially at near atmospheric pressure, may approach this limit of initial amplification. This wall effect, i.e., the interception of a signal electron by a boundary, should aij is given in Table V together with the

36

G . D. DANILATOS TABLE V

EFFECTIVE IONIZATION ENEKGIES AND CONSTANTS OF BINAKY GASMIXTURES (FRANZEN AND COCHKAN, 1962)

N* N, N, He He He He H, H2

H2 A 0 2

A H, N* CH, A CH4

36.3 36.3 36.3 30.1 29.7 29.7 30.3 37.0 37.0

37.0 26.4 32.2 26.4 37.0 36.3 29.4 26.4 29.4

0.28 0.53 I .06 0.75 3.55 8.47 0.68 1.78 4.03

be addressed simultaneously with the electrode geometry and in conjunction with the geometry around the PLAl (main pressure-limiting aperture), the gas pressure inside and above the aperture and, generally, the gas dynamics requirements of the ESEM. 1V. TERMINOLOGY

Until now, we have used the conventional terminology of SEM. However, the advent of ESEM has created the need for new terms, the introduction of

which will facilitate the description of more complex phenomena. In particular, the use of gas as a detection medium is a novel development in electron microscopy, and an appropriate new scheme of terms need to be introduced, as will become evident from the following discussion and from the developments presented through this work. An effort is made for the new terms to be consistent with those used in related fields outside the field of electron microscopy. The number of new terms is kept small but sufficient to serve the new developments. The introduction of terminology is made on a tentative basis, first to be used in the present work, and later to be discussed and amended by the scientific community. This will be determined largely by the techniques that will prevail and establish themselves in ESEM. The terms gaseous device and gaseous detectors have been used quite extensively for a long time in the literature of ionized gases and of nuclear methods and instruments. Hence the term gaseous detector device (GDD) has been used by the present author as a collective name for all devices that use the gas as a detector in the ESEM. There is a wide variety of ways to use the gas as

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

37

a detection medium. Accordingly, the term G D D requires further modification to denote its special operation and function in each case. As mentioned earlier, when we use the ionization products, we refer to it as ionization GDD, and when we use the gaseous scintillation, we refer to it as scintillation GDD. In electron microscopy, we are mostly interested in displaying and differentiating the various types of signal emanating from the specimen, such as the secondary (SE) or backscattered (BSE) electrons, and the detector name should also contain this information. As there are many possible combinations of methods and types of achieving equivalent results, it is left for the future to determine which ones will prevail and then to determine appropriate abbreviations. At present, there is a more urgent need to refer to the various signals generated in the ESEM with the most concise, descriptive, and accurate terms. In vacuum electron microscopy, there is a multiplicity of signals; this multiplicity becomes much more diverse and complicated in ESEM. As the electron microscopy field developed, a new terminology was introduced to describe the needs of this field. For example, the term secondary electrons has been arbitrarily assigned to those electrons coming out of the specimen with energy less than 50 eV. The rationale of this definition was that these electrons carry information about the top layer of the specimen (a few tens of angstroms). The electrons with energy higher than 50 eV are generally referred to as backscattered electrons. However, the general physicist still refers to the secondary electrons as being those produced by the primary electrons (e.g., an electron beam) regardless of the energy of the former. Similarly, we may have production of tertiary electrons and so on. In SEM, the tertiary low-energy electrons generated from the pole piece of the final lens by the BSE electrons from the specimen have been used to produce BSE images, and the detection system has been referred to as converted backscattered-electron detector or CBSE (Moll et al., 1979).The same electron signal has also been referred to as SE-111, whereas the SE-I is the SE produced from the surface of the specimen by the primary electron beam, the SE-I1 is the SE produced from the surface of the specimen by the exiting BSE, and the SE-IV is the SE from the electron optics column produced by the primary electron beam (Peters, 1982). In the ESEM, the number of possible interactions has increased greatly, as the number of combinations between beam, specimen, gas, signals, and walls is very large (Danilatos, 1988). Thus we have electrons produced in the gas, which may be of low or high energy (8-rays), also other signals such as scintillation, etc., all of which can be produced by a large number of causes. Obviously, the numbering scheme such as SE-I, etc., would be very difficult to memorize, whereas the introduction of a large number of further arbitrary terms could become cumbersome and, if not universally accepted, very confusing.

38

G. D. DANILATOS

The attempted scheme proposed here has been thought to take into account the above-mentioned difficulties and also to bring in line the field of electron microscopy with established terms in other physical sciences, instead of a tendency to create a self-contained discipline oblivious of the advances and practices elsewhere. The objective here is to economize in the number of terms and, at the same time, to be as consistent as possible with established terms both in electron microscopy and elsewhere. A minimum number of key words and symbols are selected, which, when combined, yield a term or acronym with no ambiguity. There are three broad classes of words: The first pertains to the various sources from which various signals are produced, the second pertains to the various types of signals, whilst the third pertains to a further differentiation of these signals according to energy. The various sources of signals are the electron gun, the instrument walls, the specimen, and the gas. The electron gun generates an electron beam or probe that reaches the specimen chamber, and thus it is convenient to think of the probe itself as the source. The word probe is selected to denote the source of any primary irradiation and refers to an electron beam in the ESEM, but it may also refer to an ion beam, x-ray beam, etc., in the most general case for other instruments. The letter P refers to a probe and is read as “from probe.” The specimen, sample, or object under examination is denoted by the word object and symbolized by the letter 0 that is read as “from object.” The gas or mixture of gases introduced in the ESEM is denoted by gas and symbolized by G read as “from gas.” The various walls in the instrument, such as the walls of the specimen chamber or the electron optics column, are denoted by walls and symbolized by W read as “from walls.” Whereas the probe is the primary source of signals, all others (i.e., walls, object, and gas) are secondary, tertiary, quaternary, etc., sources. It will become helpful and economizing if we separate the probe into two fractions: the fraction that reaches the object totally unimpeded and the fraction that interacts with the instrument walls (electron optics column, apertures, etc.) and/or the gas. The first fraction is identified here with the probe that is associated with useful information from the object and usefully contributes to image contrast: It is exactly this fraction that is encoded with the letter P. The second fraction is associated with no, or generally little and limited, information from the specimen and generally contributes to the background noise of the image; it is responsible for the creation of the stray fast electrons, the SE-IV, the electron skirt in the gas, the secondary electrons, and other signals in the gas, all of which are formed outside the useful probe. This second fraction is encoded by P‘ ( P primed) and may be read with a convenient expression such as “scattering probe.” “scattered probe,” or “stray beam.” To avoid any ambiguities, P‘ strictly means any electron (or part) of the probe that has already undergone or is

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

39

due to undergo a scattering event prior to striking the specimen; it may or may not reach the specimen after the scattering event. There are three basic types (or carriers) of signals: The electrons, E, the gaseous ions, I, and the rays, R. By rays we denote all types of photons in the visible, invisible, and x-ray regions. It is further useful to differentiate the various signals according to their energy. The main division is that between low energy and high energy. The low energy is denoted by the word slow, S. The high energy is denoted by fast, F. Thus, in electron microscopy, the electrons with energy less than 50 eV are the low energy (slow)electrons and can be symbolized as SE (slow electrons), while those with higher energy are symbolized as FE (fast electrons). The low-energy photons usually produced in the microscope in the infrared, visible, and ultraviolet regions are denoted by the word light, L. The high-energy photons are usually the x-rays, which are symbolized by X. Thus, using the above symbolism, we can encode many signals simply and precisely. For example, SEO reads as “slow electrons from object” and is simply the conventional secondary electrons issuing from the specimen surface. FEO reads as “fast electrons from object” and is simply the conventional backscattered electrons issuing from the specimen. LRW reads as “light rays from walls” and is the scintillation produced from the walls. The order of the wording is: [energy level, i.e., high or low] [type of carrier, or signal] [source]. The above scheme is yet incomplete because it fails to differentiate between same signals having different origins. For example, the SEG can be slow electrons from gas produced by the scattering probe (P’)or by the FEO (fast electrons from the specimen), or by other irradiations. It is important in electron microscopy to specify the cause of a particular signal. We should, therefore, encode additional information. This can be done simply by adding the preceding generations of signals as far back as is necessary to describe the signal of interest unambiguously. For clarity, the various generations are separated by a hyphen- which is read as “directly caused by” and signifies that the signal written on the left side of the hyphen has been caused or generated in the medium (body) of the left side directly by the signal shown on the right side of it. By direct we mean that any processes between cause and effect are unobservable or of a small magnitude and significance. Thus, more specifically, a probe electron that undergoes only a few scattering events in the specimen and loses practically no or little energy is considered to produce directly a FEO (i.e., a low-loss electron), and this relationship is denoted as FEO-EB (fast electrons from object directly caused by electrons from probe). Thus, we can now distinguish the SEG-EP’ (slow electrons from gas directly caused by the electrons from scattering probe) from the SEG-FEO (slow electrons from gas directly caused by fast electrons from object). In the latter

40

G, D. DANILATOS

example, there would be no ambiguity of the code SEG-FEO, if the FEO are produced only by one cause or are well identified and understood; if there are more causes than one, we must keep adding the preceding causal generations separated by hyphens, until there is no ambiguity. To simplify unnecessary complexities, or long codes, arising from the above scheme, we can introduce an additional type of linking between generations of signals. On many occasions, we may skip the intervening processes (signals) by connecting the final signal only to one of the preceding signals, if we are not concerned about what happens in-between. To show this type of connection, we introduce the symbol which is read as “indirectly casued by” and signifies that the signal written on the left side of the symbol has been caused by or was generated in the medium (body) of the left side indirectly from the signal shown on the right side of it. Thus, the notation FEO EP signifies that the FEO have been produced indirectly by the probe electrons via multiple electron scattering in the specimen, or via multiple scattering combined with x-ray generation in the specimen. To maximize the flexibility of the system, the combination of symbols = is read as “caused by” and signifies the linkage of signals both directly and indirectly. Thus, SEW = FEO N EP (slow electrons from walls caused by fast electrons from object caused by electrons from probe) are all the conventional secondary electrons produced from the walls by all the conventional backscattered electrons. SEO ZT EP (slow electrons from object caused by electrons from probe) are all the conventional secondary electrons from the specimen. Finally, a term is introduced to describe the special case of avalanche or cascade formation when the slow electrons drift in a gas in the presence of an electric field with sufficient energy to excite and/or ionize the gas. The drifting electrons under the influence of usual electric fields may acquire an energy of up to several decades of eV. Although they can be classified as slow electrons, they deserve a term of their own, because they constitute a special group of electrons. In the same way, a drifting electron can generate a sequence, cascade, or avalanche of photons. These groups of electrons or photons are associated with one initial electron, or electron/ion pair of specified origin. The word introduced for these carriers is cascade, denoting cascade, or avalanche, or sequence, and is encoded by C. Thus we may write CEG (cascade electrons from gas), but because the cascades considered here are always in the gas, the word gas is superfluous, and we can simplify by writing only CE, i.e., simply “cascade electrons.” Similarly, we have CLR for “cascade light rays.” Table VI summarizes the above definitions. The terminology introduced here can describe all the conventional signals and, in addition, all the new ones whenever they arise. The most usual

-

-

41

THEORY OF T H E GASEOUS DETECTOR DEVICE I N T H E ESEM TABLE VI AND ENERGY KEYWORDSFOR SOURCES,CARRIERS,

Source (from)

Carrier

Energy

Link

Probe P Scattering Probe P’ Object 0 Gas G Walls w

Electrons E Ions I Rays R

slows Fast F Light L x-rays X Cascade C

DirectlyIndirectly Directly and indirectly 2

-

acronyms for the conventional signals and those of ESEM are given in the proposed system below.

Concentional signals in vacuum: Electron probe, now EP. Secondary electrons (SE), now SE. High-resolution secondary electrons (SE-I), now SEO- EP. Low-resolution secondary electrons (SE-II), now SEO EP. Converted backscattered electrons (SE-III), now SEW = FEO Column secondary electrons (SE-IV), now SEW 2: EP’. Backscattered electrons (BSE), now FEO. Low-loss BSE, now FEO-EP. Auger electrons, now FEO-XRO. Cathodoluminescence (CL), now LRO. x-rays, now XR.

-

1

EP.

Note that P’ in vacuum stands for the stray electrons from the probe striking the walls of the electron optics column, whereas P’ in the ESEM is primarily the electron skirt made of the scattered electrons by the gas. In addition to the above signals, new signals appear in the ESEM, or the above signals must be further modified. E SE M signals: Scattering probe (electron skirt in most cases) EP’. Total incident beam EP&EP’. Slow electrons from object SEO N (EP&EP’). Useful (information) slow electrons from object SEO N EP. Background slow electrons from object SEO N EP‘. Fast electrons from object FEO [i.e., FEO N (EP&EP’)]. All useful fast electrons from object FEO N EP. Background fast electrons from object FEO N El”. Useful slow electrons from gas SEG 1 EP.

42

G. D. DANILATOS

Background slow electrons from gas SEG = EP‘. Direct ionization electrons from gas SEG-SEO 2 EP. Direct ionization electrons from gas SEG-FEO 5 EP. Direct ionization electrons from gas SEG-XRO 2 EP. Direct ions from gas SIG-FEO etc. Cascade electrons CE 2: FEO 2 EP. Cascade electrons CE 2: SEO 2 EP. Cascade electrons (no information) CE N EP’. Cascade ions CI N FEO N EP. Cascade ions CI 21 SEO N EP. Cascade ions (no information) CI 2 EP’. Gaseous cathodoluminescence LRG- EP’. Gaseous scintillation LRG 1EP. Direct scintillation due to slow electrons LRG-SEO N EP. Direct scintillation due to fast electrons LRG-FEO = EP. Cascade scintillation due to slow electrons CLR 2: SEO N EP. Note that the symbol & (and) was used above without any prior stipulation. Similarly, one may use , (comma) and other convenient punctuation, as long as the notation is clear. The above list contains only part of the many more possible signals to be encountered in the ESEM. This list demonstrates (a) that the number of new signals is indeed large, hence the need for new terminology, and (b) that the phrase descriptions given in the list are incomplete and ambiguous, unless long explanatory sentences were incorporated; however, the accompanying new acronyms are short, descriptive, and precise. The proposed scheme has been devised to accommodate those phenomena occurring most frequently and with the highest probability. One may think out some very complicated possibilities, in which case the system could appear cumbersome and perhaps not sufficient; however, such cases occur with very low probability, and an attempt to complicate the system in order to accommodate those possibilities would only be a pedantic exercise. The system is sufficient at least for the purposes of the present work.

V. DISCHARGE CHARACTERISTICS

We can distinguish two phases (or stages) in the operation of the GDD. The first phase is the direct interaction of the various signals with the gas. The second phase pertains to all subsequent processes including the collection, or better, the detection of the products of the first phase. Thus, we first consider the ionization products and then the ensuing discharge in the gas between the electrodes. We need to understand and evaluate the properties of this dis-

THEORY OF T H E GASEOUS DETECTOR DEVICE I N THE ESEM

43

charge in order to determine the limits of frequency response and amplification of the GDD. Following the newly introduced terminology, there are three basic ionizing signals from the specimen in the ESEM, namely, the SEO, the FEO, and the XRO. Their range of action during the first phase is in the same order: The SEO would diffuse from their point of origin, and it depends on the applied field how far and in which direction they will go; the F EO will practically instantaneously penetrate much further, depending on their energy, and the XRO will penetrate still further. The approximately 10-50 eV SEO will cause some ionization, but the majority of them, having around 2 eV, will not. The FEO will ionize the gas quite readily throughout their tracks, and in addition, they will generate secondary SEW and FEW from the walls they strike. The X R O will also ionize the gas, but the relative efficiency of ionization should be taken into account. In addition to all these, the primary beam itself will cause ionization in a well-predetermined fashion, independently from the signal carrying information from the specimen. Thus, the various signals and the beam can be considered as sources of ionization in the bulk and the boundaries of the gas during the first phase. Any complete analysis of the discharge during the second phase should take into account the spatial distribution of ionization during the first phase. This problem should be considered as a separate task; that is to find theoretically and experimentally the spatial distribution of ionization in the ESEM, for a given geometrical configuration of specimen, chamber walls, and electrodes. To be able to proceed with the analysis below, we consider the configuration of Fig. 11 and make some simplifying assumptions: The electron beam passes through an annular electrode (A), which is surrounded by larger concentric electrodes (B) and (C), all of which are held at the same potential,

Beam

1

I

I

FIG. 1 I Separation of detection volumes by concentric annular electrodes and uniform field Volume (a) detects mainly the S E O signal, while (b) and (c) detect various types of F E O signal.

44

G . D. DANILATOS

but are all independently connected to separate amplifiers. The specimen is located at the other plate electrode (E) at earth potential, and the whole configuration is one producing a uniform electric field. By virtue of the electrode independence at the top, the space between the electrodes is separated into different detecting volumes, each giving different information about the signals. Volume (a) contains mainly the SEO, and we assume that only small ionization is produced by the FEO, because of the small lateral dimension traversed by them. In this, the source of initial electrons is at the specimen with initial agitation energy of around 2 eV. Volume (c) is predominantly ionized by the low takeoff angle FEO, while volume (b) is ionized by a mixture of different kinds of FEO. In the latter volumes, the source of initial ionization (ion pairs) is in the bulk of the gas. The two different types of initial ionization source, namely, one point source at the electrode (E) and the other in the bulk of the gas, result in quantitatively different discharges during the second phase. These types of discharge have been analyzed in the literature, the results of which will be considered below. Knowledge of particle counter techniques in nuclear physics is very valuable for the development of the G D D in the ESEM. However, the conditions and objectives in the two disciplines are quite distinct; in nuclear physics, we have mainly high energy, or heavy particles, and we are interested in simple counting or in radiation spectra. In the ESEM, we mainly have (relatively) low-energy beta radiation at very high counting rates, or x-rays, in conditions of relatively low pressure, all mixed-up in a constrained space, generally without a well-specified gas composition, and indeed with types of gases not usually recommended for efficient counter designs. The aims are also quite distinct, namely, apart from radiation spectroscopy and perhaps particle counting, we wish to construct images and mappings as fast as possible with satisfactory SNR, contrast, and resolution. Therefore, to properly understand and develop the GDD, we need to resort to the basic physical concepts on which the nuclear devices have been based, so that we can rebuild on new grounds a device to most efficiently serve the requirements of the ESEM. A brief survey of the principles of the gaseous discharge will indicate the parameters that we should manipulate in future experimentation.

A . Outline q j the Dischurye

A n electric discharge is usually described by a current-versus-voltage characteristic (ILV curve). There are three major regions in this curve: the Townsend discharge, the glow discharge, and finally the arc discharge, with inbetween transition regions of special significance. Detailed descriptions and further references are given in any book on ionized gas and gaseous discharges

THEORY OF T H E GASEOUS DETECTOR DEVICE IN THE ESEM

45

I

0 J 0

FIG. 12 Schematic variation of current versus bias for two different initial intensity signals for a Townsend discharge. There are five characteristic regions.

(Townsend, 1947; Cobine, 1941; Loeb 1955, Engel, 1965; and others). We are mainly interested in the Townsend discharge, since there, the output depends on the input, and the discharge can be controlled. Two typical curves of Townsend discharge are shown in Fig. 12. These curves are characterized by five regions. Region (I) corresponds to the continuous increase of current with increase of bias until practically all of the initially created charge that can be collected is collected. At very low bias, charge is lost mainly due to direct escape, diffusion, back-diffusion, and electron capture (attachment). If the collection time is long, electron capture does not result in charge loss. In the ESEM conditions, static I-V measurements appear to reach a saturation level with only a few volts bias (Danilatos, 1988), but this apparent saturation will be reexamined separately in a following discussion. Region (11) corresponds to the situation where the current increase over the saturation level I, is very small. The electrons on the average do not have enough energy to ionize the gas, but because the energy distribution is broad enough, there are always a number of electrons with energy above the average energy sufficient to ionize the gas. By further increase of the bias, the electrons drifting in the field acquire enough energy to ionize the gas, a process that develops into an avalanche. As a rule of thumb, the threshold field for ionization to occur is about 10 V/Pam. Once the average energy of the electrons reaches and exceeds a certain level (usually the gas ionization energy), the measured total (positive and negative carrier) current I increases sharply over I , . In the classical theory of ionized gases, it is shown that the collected current I is amplified exponentially in

46

G . D. DANILATOS

accordance with the derivation I,’l,, = exp(ctD)

(CE&CI) 2 SEO,

(41)

where z is the first Townsend coefficient. For this derivation, it is assumed that all of the (initial) current I , is produced very close to or at the cathode, as in the case of SEO. The signal code following the equation signifies that it can be applied when the detected signal is caused by both electron and ions in the avalanche triggered by the electrons from the object. This equation describes most of region (111), in which the amplified output is proportional to the input. With still further increase of bias, the density of charge may become so high as to alter the externally applied field. The space charge increases either due to a single avalanche or to accumulated positive ions from different avalanches. Space-charge effects become obvious when the space-charge density is of the same order as the surface-charge density of the anode. Under these conditions, the output current ceases to be proportional to the input, and region (IV) is referred to as a region of limited proportionality. Gradually, at first, secondary ionization processes appear, which, with further increase of bias, become so pronounced that they alone control the discharge (here the primary process is the initial avalanche). These mechanisms, or y-processes, result in the release of additional electrons from the cathode. This “external” infusion of electrons is caused by different processes, such as (a)the positive ions striking the cathode, (b) photons emitted by excited gas, and (c) the metastables arriving at the cathode. These processes have the common characteristic that, when they produce enough new electrons, they can trigger a self-sustained discharge running independent of the microscope signal (e.g., the SEO); in other words, variations in the intensity of the signal will not be detected, and the sensing electrode will be flooded with a high discharge current. The derivation of the basic equation governing the discharge, when this secondary electron emission is present, can be found in most books on ionized gases. For this derivation, the y-coefficient is introduced, which is defined as the average number of electrons emitted from the cathode for each positive ion arriving at the cathode. The original signal current together with the additional cathode current produced by the y-processes will be amplified in the gas by the a-process. The governing equation in the steady state is easily found to be (e.g., Cobine, 1941): 1 1,

--

exP(aD) 1 - y[exp(aD) -

11

(CE&CI) 1 SEO

In the absence of secondary processes, y = 0, and the above equation reverts to Eq. (41). From this equation, it is found that the current becomes infinite

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

47

(breakdown) when the denominator becomes zero, i.e., when y exp(clD) = 1 (since usually, exp(ctD) >> 1). In that case, the output is independent from the input, and the system is in region (V), or the Geiger-Muller region. Both, regions of limited proportionality and Geiger-Muller regions require special attention in the design of G D D in order to understand the precise physical processes involved that ultimately determine the limits of amplification. This understanding will help in the possible manipulation of certain parameters and the achievement of optimum operation of the GDD. It should be noted that SEO in Eq. (42) includes all the SEO of Eq. (41)plus the SEO caused by y-processes. The difference could be described by some long expressions and there is no need to introduce additional codes for this case. The difference is simply seen by the presence or absence of the coefficient ;‘. Back-difluusion and Saturation Current, It is often thought that the saturation current level I, is reached in region (11) of the curve in Fig. 12 when it appears that this curve has leveled-off. However, such an approach can be misleading. Loeb (1955), has presented a critical review on this matter. There are four factors that hinder the actual achievement of saturation current. These are: (a) geometrical factors affecting the total collection of charge carriers, (b) the random diffusive motions of carriers, (c) space charge fields, and (d) recombination. We can ignore the last two factors for the present purposes. It has been shown that geometrical factors can result in severe loss of charge carriers, difficult to recover even at very high electrode bias (Loeb, 1955). Work has been done by releasing photoelectrons uniformly from the surface of the cathode of a parallel plate system in vacuum. The photoelectrons had an average energy below I eV, and yet the current curve appeared to level off well below the expected saturation value. Even at very high electrode bias, which would have led to strong ionization in the presence of gas, the losses can be significant. The amount of losses depends on the dimensions and separation of the electrodes, on the energy of the electrons released, and on the applied voltage. The equations of this problem have been solved, but these do not have a general application to the case of ESEM. Specifically for the SEO, they are released practically from “a point” at the cathode, whereas the anode can be relatively extended above this point. With this geometry, we expect the losses to be minimal. However, there is a wide range of geometries of G D D to be employed for specific purposes, and each case should be examined separately. Thus, attention should be directed to the possibility of electron losses due to geometry alone, as experience in the literature shows. Diffusion is very relevant to the GDD. Already, we have examined the possible losses due to diffusion in the bulk of the gas in Section 1II.C. Here,

48

G. D. DANILATOS

special attention is given to the fact that electrons are being “reflected,” or diffused back to the surface that they came from. This possibility can result in significant modification of the value of the saturation current I , and the calculation of the actual gaseous amplification in the discharge. This effect is particularly significant for the SEO signal. Thomson (1928) developed the theory of electron back-diffusion, which was later modified by other workers to better account for the experimental observations (Loeb, 1955). The current fraction I / 1 , that is measured in region I for parallel plates can be derived from the following equation:

_I -I,,

4KE u,,+4KE’

(43)

where uo is the average escape velocity of the electrons from the cathode, K is the electron mobility, and E is the applied field. This derivation is correct when the gas, the pressure, and the field are such that the escape velocity is not influenced by the applied field and the gas molecules, and that the initial energy is dissipated over a short distance from the cathode relative to the interelectrode distance. These conditions can be fulfilled beyond a certain pressure, with the other parameters being constant. It is interesting to note here the dependence of electron loss on the initial energy. If two processes are producing electrons with equal rates at the cathode, the process producing the lower electron energy will play a dominant role in the subsequent discharge. Theobald (1953) has shown in detailed experimental studies the strong presence of such effects in various gases. He further showed that the back-diffusion depends on E / p and not on p alone. The theory of back-diffusion is complicated in the case of electronegative gases, which increase the losses, but the difference between electronegative and electropositive gases is small at high E / p and low current densities. In addition, Eq. (43) must be modified for nonuniform fields. In practice, these complications can be accounted for by the general relation developed by Rice ( 1946):

where A , B, A’, and B‘ are constants characteristic of the particular system. The form of this equation is the same as Eq. (43). It has an asymptotic form approaching the saturation value I, = A . This equation can be used for the calculation of the saturation current, since in most cases this saturation current cannot be directly measured because of the early onset of ionization in the discharge. To calibrate the ionization current, we solve the above equation for V = A ( V / I ) - B and plot the values of V versus V/I experimentally

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

49

obtained. The graph should normally yield an initial straight line portion, from which the constants A and Bare found. Use of theseconstants in Eq. (44) can predict the value of I , that escapes back-diffusion; it is this I , that is amplified in the subsequent discharge and contributes to the useful signal. One implication of back-diffusion is that the SEO signal can be significantly modified through the interplay of various parameters. Most of the SEO have an energy around 2 eV and they can undergo back-diffusion, whereas the minority of SEO at the two extremes of the distribution will get through without much loss. The electrons towards the 50 eV range do not qualify for back-diffusion, because they can ionize the gas and can shoot through a relatively long initial distance from the specimen surface. Electron back-diffusion should be studied in conjunction with the filtering effects of electron capture.

B. AmpliLfication, Parallel Plates In this section, the amplification factors are calculated under the condition that the y-processes are absent. Referring to Fig. 11, the saturation current I , in the detection volume (a) may be taken, for simplicity of presentation of the derivations here, to be equal to the SEO current I , , which is the charge emitted during the pixel time:

The actual value of I , can be lower than I, due to back-diffusion and should be taken into account in the laboratory. For simplicity, we may assume that the SEO are at the cathode. In volume (b) or (c), the saturation current is proportional to the FEO current I,,, emitted from the specimen and consists of a fractionfl,,, directly impinging on the electrode and a fraction resulting from the ionization of the gas,

I,

dN dr

= eDA-

+ fZFEo

(SEG&SIG)-FEO&FEO,

where A is the area of the electrodes, D is the distance between them, and dNldt is the uniform rate of ion-pair production per unit volume throughout the volume. The FEO traversing the gas in all directions are assumed as radiation source uniformly distributed in the bulk of the gas, since this will help us deduce first-order solutions. The actual spatial ionization density in the gas constitutes a separate topic for special study. If the ionization is not uniform,

50

G. D. DANILATOS

an integration must be performed over the volume I0 =

Y

+ fIF,,

-dv

e

P,

(SEG&SIG)-FEO&FEO.

(47)

The above derivations are correct for the steady-state situation, where both the negative and positive carriers (total) are collected. If we discard the slow positive carriers by use of short pixel time and appropriate circuitry, we must integrate the contribution to the pulse by all the electrons alone throughout the volume. In the case of uniform ionization, this integration yields for the dynamic (electron) current

I

1 dN - -eDA‘-2 dt

+ fIF,,

(SEG-FEO)&FEO,

(48)

which is less than the static (total) current by the fraction 1/2. A different fraction would result with nonuniform ionization and nonuniform field. Generally, the direct component f I , can be much less than the ionization component and may be discarded in the above equations. The “saturation” region (11) persists throughout a voltage range, depending on the electrode geometry, pressure, and nature of gas. If the initial electrons are produced uniformly throughout the volume between the electrodes, then the collected current from avalanche amplification is given by (Engel, 1965): I

_ -10

exp(aD) - 1 CYD

(CE&CI) ‘v FEO.

(49)

This is a continuation from Eq. (46) with f = 0. Whereas the ratio I / I o in Eq. (41) appears greater than in Eq. (49), the numerical ualue may be greater in the second case, because of the initial amplification occurring within the I , by the ionizing FEO energy; the extent to which this initial amplification (within I,) occurs depends on the overall configuration of specimen electrodes and their relative positioning with respect to the surrounding walls, for given electron beam and gas conditions. For the dynamic current, i.e., the component due to the electrons alone, we have to calculate the signal induced by them. Each SEO drifting with vd produces an avalanche of charge exp(uu,t); by integrating the induced charge as the avalanche develops and adding all the avalanches for all electrons of each pixel, we find I

--

I0

exp(aD) - 1 MD

which is identical to Eq. (49).

CE

N

SEO,

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

51

The equation of amplification for the case of uniform initial ionization, as with FEO in volumes (b) and (c), can be found by a double integration. The charge per unit time in the volume Adx is e(dN/dt)Adx, which, travelling a distance x to the collecting electrode, induces a current to be (after the first integration):

dN A -(expax dt aD

e-

- 1)dx.

By integrating the contribution from all the volume elements in the distance from 0 to D, we find

Taking into account Eq. (48) with f = 0, we finally find for the amplification factor for BSE, due to drifting electrons alone,

CE

N

FEO.

The above derivations lead to an important conclusion. The amplification factor is reduced aD times in both cases of SEO and FEO, when electrons alone are used. This reduction can be very significant. For example, if aD = 7, the total possible amplification from Eq. (41) is 1097 x, but with electrons alone, the amplification is reduced to 157 x. Therefore, if the transit time of positive ions is not compatible with the scanning requirements of the image, we cannot expect very high amplification. This may be the case with parallel plate geometry (depending on the electrode separation, the gas, and the bias). We will see, however, that in cylindrical geometry, the ions can more readily contribute to the signal, since they need only travel a few anode radii near the anode surface. It would be a simple matter to calculate the gaseous gain of signals in the ESEM, if it were not for three major problems. The first problem is that the ionization coefficient tl is not a constant: It depends on the nature of gas, the pressure and the electric field. This dependence is quite complex, and it has not been possible to find an analytical expression in the literature yet. The second and more serious problem is that the experimentally measured values of x show a strong dependence on the purity of the gas, and high purity is a condition not usually found in the ESEM. The third and most severe problem is that the equations of amplification presented above are not valid when the secondary ionization (y-processes) becomes pronounced and an unstable condition, leading to breakdown, results. These problems will be analyzed below.

52

G. D. DANILATOS

C . The First Townsend Coeficient The a-coefficient is defined as the number of ion pairs per unit length in the direction of the applied j e l d produced by an electron as it drifts in the (opposite) direction of the field. Its value varies nearly, but not exactly the same, as the ionization eficiency coeficient s, (ion pairs per unit length along the actual electron path) versus electron energy. The a is in some cases twice the value of s, since the actual electron path in an electric field is increased by scattering collisions (Engel, 1965). The coefficient s is related to the ionization cross section cri and to the ionization mean free path Liby the following equations:

where n is the density of gas particles, p the pressure, N A the Avogadro number, R is the universal gas constant, and T the absolute gas temperature. There is an abundance of data on ionization cross sections and ionization efficiency coefficients in the literature, especially in the energy range up to a few hundred eV. Data on the ionization coefficient is usually presented in the form r / p = f ( E / p ) , both for convenience and because 01 is proportional to pressure, whereas E l p is proportional to E L , , i.e., the energy acquired between ionizing collisions. The ionization coefficient starts from a vanishingly low value at the ionization energy of the gas and increases steadily up to a maximum, followed by a continuous decrease with increase of electron energy. As the ionization coefficient enters into calculations of other quantities, it would be helpful to have an analytical expression for it in order to avoid cumbersome numerical methods for each individual calculation. Attempts to provide an analytical expression in the complete energy range have failed due to the multiplicity and complexity of intervening causes. However, there is an expression representing this coefficient fairly accurately in a limited electron energy range for pure gases. This expression is (Engel, 1965): tl = A ex.(

P

-

&),

(54)

where the constants A and B are given in Table VII for various gases. The range of validity of Eq. (54) corresponds to the ascending part of the curve up to its maximum value and has an S-shape, as is shown in Fig. 13. The gain of the gaseous amplifier as derived in the previous section is given as a function of interelectrode distance for each fixed value of a. There are two ways to fix this coefficient: (a) by fixing p and E, which means we should increase the voltage in proportion to the increase of distance, and (b) by

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

53

TABLE VII VALUES OF

Gas

THE

CONSTANTS A AND B

OF

EQ. (54) (ENGEL,1965) E J p range of validity (V mPa)

A 1,mPa

B V/mPa

9.00 4.05 11.25 15 9.75 9 2.25 15

256.5 104.3 273.8 349.5 217.5 135.0 25.5 (18.8) 277.5

75-450 15-75 75-600 375-750 112.5-750 75-450 15-1125 150-450

(23-75)

7t

ElP F IG . 13 The variation of the first ionization coefficient a l p versus E / p for nitrogen. The tangent from the origin defines an optimum operation point on the curve.

varying E and p simultaneously in accordance to

VE=-= D

BP ln(Ap/a)’

(55)

as can be derived from Eq. (54). It is seen that it is possible to gain no amplification at a fixed D by simply increasing the bias and the pressure simultaneously in the manner indicated by the above equation. For the general case, we note that the amplification formulae contain the product r D

54

G . D. DANILATOS

(the average number of ion pairs produced in the gap), which can be found from Eq. ( 5 5 ) to be aD = A p D exp( - B p D / V ) .

(56)

This shows that there is an equivalence between p and D,since they enter together as a product. Thus for a given p D , we can find the amplification ( [ / I o ) versus applied voltage, in the range of validity of Eq. (54). An alternative way of looking at these phenomena is to consider the equivalent coefficient q expressing the average number of ion pairs per Volt of potential difference between the electrodes. This is

The last member of these equations implies that the maximum value of this coefficient is attained when the values of cc/p and E / p yield a maximum ratio. This ratio is maximum where the straight line from the origin is tangent to the curve in Fig. 13. For each gas, there is a uniquely defined value of E / p for which the ionization efficiency is maximum. This optimum condition can be derived analytically (Cobine, 1941) as: (E/p)opt,mum

=

B.

(58)

This implies that for a given gas, bias, and electrode distance, there is an optimum pressure yielding maximum amplification; these optimum parameters are known as the Stoletow constants. With the preceding investigation, we can predict the amplification gain of parallel plates for a given set of parameters in the range of validity of A and B, before breakdown occurs. D. The Puschen Law

The presence of y-processes represents the main limitation on the maximum gain of the GDD. A more detailed investigation of the mechanisms responsible is necessary, and reference to the various theories in the literature would be most helpful. The value of the y-coefficient depends on the nature of gas, electrodes, and electrode configuration. In this section, we wish to consider the general equation governing the sparking potential versus pressure and electrode separation. There is an abundance of experimental measurements in this respect, but a good approximation can be predicted for plate electrodes by obeying Eq. (54). Based on this, it can be found that the sparking (breakdown or starting) potential V, is (Engel, 1965):

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

55

1000

> 6

m

800

700600-

>”c Y U

2

:

o

o

500-

400-

300-

l 0 0

1

2

3 4 5 6 7 8 91011 PO, Pam

FIG. 14 The breakdown potential versus p D in air for three different values of y-coefficient.

where

The main features of this equation are that the pressure and distance enter again as a product and that it predicts a minimum value for the breakdown potential at some characteristic value of p D . This equation, known as Paschen’s law, was discovered by Paschen in 1889. It agrees satisfactorily with experimental measurements, provided we choose the correct value of the coefficient y. An example is given for air with y = 0.01 in Fig. 14. The two other curves have been plotted for the hypothetical cases with y = 0.001 and 7 = 0.0001. As expected, the breakdown values are increased, but not too much. Data on ionization coefficients, Paschen’s curves, and other technical information can be found in the books by Weston (1968) and Espe (1968). It can be shown that the minimum breakdown potential Vminoccurs at (PD),~, given by the equations (Engel, 1965): A

Although Paschen’s law dictates the upper limits of amplification, it allows us considerable flexibility for manipulation. The breakdown processes can be controlled to a certain extent. By postponing the breakdown to a higher bias, the amplification is controlled (caused only) by the cr-processes together with

56

G. D. DANILATOS

their associated equations derived in Section V.B. The practical implication of this is that we can achieve simultaneously both higher amplification and stability of the discharge with proper controls.

E. Secondarjl Processes Paschen’s law is always obeyed, since it determines the breakdown potential for a given set of conditions. By changing these conditions, we can modify the shape and level of the Paschen curve. This encourages us to undertake research for the extension of the limits of breakdown as far as possible in the conditions of the ESEM. The secondary or y-processes are different mechanisms operating singly or simultaneously. They all result in the injection of new electrons from the cathode, over and above the electrons produced by the primary process of ionization caused by the microscope electrons (electron beam and useful signal) plus (i.e., over and above) the electrons produced by secondary processes in the gas from the primary processes connected with the microscope electrons. All primary and all secondary electrons augment the detected signal and hence may contribute towards amplification, but it is only the excess electrons from the cathode which threaten a breakdown. These excess electrons from the cathode result in a discharge independent from the microscope electrons and, hence, in a white image at the breakdown point. When we operate below the breakdown bias, which electrons may contribute to the image depends on the scanning speed of the image contrast and the clipping time of the electronic circuit. The following discussion is a brief exposition of the theories on secondary mechanisms and is sufficient for the immediate purposes of this work. Detailed descriptions of this topic (streamer formation, etc.) can be found in most standard textbooks on ionization. There are three major mechanisms producing additional electrons at the cathode. One very common process takes place via the photons emitted from excited species in the bulk of the gas. These photons travel in all directions; some of these photons are absorbed by the gas, but some can reach the cathode and eject photoelectrons. This process is very fast (lops s). We can code these electrons as SEO-CLR = (SEO&FEO) 2 EP when they carry useful information. Another also very common mechanism involves the positive ions striking the cathode. There are different theories to explain this mechanism. The prevailing theory is that, when a positive ion approaches the cathode at a distance of a few atomic radii, coulombic forces detach an electron from the cathode and neutralize the ion; the neutral atom is in an excited state and emits

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

57

a photon, which, in turn, ejects a photoelectron from the cathode. For this to happen, the ionization energy of the gas (parent of the ion) must not be less than twice the work function of the cathode. This two-stage liberation of a new electron is very fast (less than s), but the whole mechanism involving, initially, the transit of the causal ion through the gas is slow. Therefore. this mechanism may or may not contribute to the image contrast. An alternative theory proposes the mechanism that the electrons are ejected by the momentum energy of the ion striking the cathode. We can code these electrons as SEO-CI N (SEO&FEO) N EP when they carry useful information. A third mechanism involves the metastables. Many gases and mixtures of gases produce significant number of metastables, which are generally longlived species emitting a photon. The metastables emit photons as they wander around or when they strike the cathode, and these can liberate a new electron. This electron is far removed, in time, from the causal signal and will, generally, contribute towards background noise in the image. We can code these secondary electrons in the same way as the photoelectrons. In a very recent publication, more light is shed on the mechanisms of breakdown from experiments with a parallel plate chamber inside a magnetic field (Gruhn et al., 1986). It has been found that at a given gain, the sparking rate actually increases with increase of the magnetic-field intensity. This is attributed to a reduction of the avalanche size in the presence of magnetic field. The increase of charge density is correlated to the increase of sparking rate. In general, the sparking probability increases with gain, with magnetic field, and with primary (initial) ionization. The gas composition affects the sparking probability as follows: For a fixed gain, the gas producing the greatest diffusion has the lowest sparking probability. From these findings, we can conclude that we should shape the electric fields via a suitable electrode configuration in the ESEM, so that the avalanche(s) spreads as much as possible as it approaches the final anode. When operating the GDD in the ionization mode, we should not seek amplification with the aid of y-processes. One reason is that some of these processes are slow, but the main reason is that they occur intensely over a short range of bias prior to breakdown, and hence they create an unstable condition. This can be seen in Fig. 15, where Eq. (42) has been plotted for nitrogen for different values of 7 ; Eq. (56) has been used, and the curves are shown in the range of validity of A and B. It is noted that the amplification deviates abruptly from the curve corresponding to the complete absence of secondary processes. There are several ways to restore stability and hence extend the useful amplification as discussed below. In proportional and Geiger-Muller counters, quenching gases are introduced to suppress the photomechanism. As a general rule, polyatomic gases are introduced together with the main filler gas. These agents have the

58

G. D. DANILATOS

0-001,

0

/o.o

100m300w)o500600700800

Bias, V FIG. 15 The logarithm of gain Log(l/l,) versus bias in nitrogen using Eqs. (42) and ( 5 6 )for different values of 1'-coefficient shown.

property of absorbing photons in the bulk of the gas by producing products of dissociation rather than free electrons. Furthermore, the ions of the primary gas readily exchange their charge with an electron from the additive gas. By the time the positive ions reach the cathode, they are mainly composed of large ions, which are relatively slower; when these ions strike the cathode, they dissociate in preference to ejecting a new electron from the cathode. In addition, when a metastable emits a photon, this photon is usually absorbed by the agent. All in all, these gases quench the secondary processes, and we can apply much higher bias to achieve a more stable and higher amplification. The use of quenching gases can be applied to ESEM, but with restrictions. The main quenchers in nuclear devices are organic gases that can cause severe contamination in the microscope from the dissociation products. Experience in the ESEM, when using acetylene, has shown that the gun filament was burned out after a half or one hour of operation. Observation of the burned gun filament under the optical microscope showed carbonaceous filaments present. It can be inferred that a similar problem could result through the introduction of the commonly used CH, as quencher, but it should be tried. Liquid acetone has 21 mbar saturation vapor pressure at room temperature and has good quenching properties (Hempel et al., 1975). Perhaps, by suitably improving the evacuation design of the ESEM, such gases could be used, except that the dissociation products may contaminate the specimen under

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

59

examination. Some of these gases, such as alcohol, can also cause excessive electron beam scattering. Very importantly, these agents should be used with proper care, because they can be hazardous. One probable use of these gases could be for the sake of study of the dissociation processes themselves. However, they should not be a priori discarded for possible use in the ESEM before appropriate experiments, since certain mixtures could prove just right. Fortunately, there are other gases used as quenchers, such as halogens (Liebson and Friedman, 1948), inorganic mixtures and CO,, which may hold promise for possible general use in the ESEM. Dwurazny et al. (1983) have experimented with various inorganic mixtures and found the best amplification with Ar + 10% Xe + 8% N,, Ar + 3.5’1” K r + 8% H,, and Ar + 107; Kr + So/:, N,, but the optimum proportions may differ at the different pressures used in the ESEM. A gas mixture of 65% He 35% CF, has been investigated by Kopp et al. (1982). In proportional counters, mixtures of 97.5% Ar + 2.5% CO, and 90% Ar + 10% CH, (known as P10) are used, and Fuzesy et al. (1972) have found that the addition of Xe extends the proportional region to about five times higher gain than without it. Hendricks ( I 972) has studied the gas amplification in counters filled with Xe-CH, and Xe-CO, mixtures. Koori et al. (1984) have reported gaseous gains in excess of lo8 with “magic mixtures” such as argon-hydrocarbon-freon mixtures. Koori et al. (1986) have further found that Ar-C0,-Freon mixtures behave also as magic gas; this could be of interest in the ESEM, since the gas does not contain hydrocarbons and is incombustible. These extremely high gains were obtained with a single wire counter operating at atmospheric pressure with a few-kV bias. However, again, a prediction of all possible effects is not easy to make at present, and experimentation is the easiest way to find new gases, agents, mixtures, etc., best suited for ESEM applications. Special consideration should be given to the Penning mixtures. These mixtures have a very high ionization efficiency and are usually associated with low breakdown potentials. This implies that high gain can be achieved at a relatively low bias (prior to the instability region), which is advantageous from the point of view of design requirements. Since the Penning principle is based on metastables, these mixtures should be studied to determine their time response. High gain, low bias, and fast response are qualities sought for good G D D performance. Apart from the choice of gas composition, there are other methods to suppress the secondary mechanisms. Choice of the cathode material may help. The obvious rule is to choose materials with high work function. For example, copper and brass are good materials for cathode, whereas zinc, alkalis, aluminum, and soft solder should be avoided (Korff, 1946). Choice of the cathode geometry can also be beneficial. The cathode is wherever the positive ions end up. This can be the specimen chamber walls, a

+

60

G. D. DANILATOS

selected wire, or the specimen itself. The cathode geometry determines the electric field around it and hence controls the momentum with which the ions will strike the cathode during their last few mean free paths. Sharp points should be avoided. The cathode geometry can influence the discharge also by minimizing the number of photons striking it. Thus, a perforated cathode can contribute towards this objective (Korff, 1946).Similarly, the cathode can be made from a number of wires, and generally it can be hidden from the photon-producing discharge volume. This is a well-known rule employed in electron multiplier vacuum devices. In channeltrons and microchannel plate technology, the channels are curved, or tilted (like venetian blinds), to stop photons reaching the end cathode and thus producing an early breakdown. With this notion, it would be very worthwhile to research into the possibility of making channels, or microchannels from special materials, which will allow the gaseous discharge to develop unhindered to high gains. Essentially, the whole of Section V.E defined a broad topic for further research.

VI. AMPLIFICATION Modern electronic amplifiers can produce a high gain with good frequency response and low noise, and a gaseous gain of around 1OOx can be used without deterioration of the overall SNR for the signals encountered in the microscope. However, any additional gaseous gain that could be achieved would greatly improve the design of electronics, cost, and overall performance of the ESEM. In this part, we will concentrate on better understanding the mechanisms of a high gaseous gain. A . Limits

Based on the previous analysis, we are in a position to see how various parameters interplay in establishing the amplification of the GDD and, from this, to find what the limits of amplification are. We will consider the case of parallel plates and use the equations and constants derived in the previous sections. This will give us a good idea about how to handle this complex question and in which direction to experiment in order to determine the actual amplification and to derive optimum results during the operation of the GDD. We shall use the equations presented in Section V.B. A small modification of these equations is needed before we use them to find the expected gain. Ionization in the gas by the drifting electrons is not possible below a limiting electrode bias. For a given pressure, the bias must have at least a minimum level V,, so that the electrons can acquire sufficient energy between collisions

THEORY OF T H E GASEOUS DETECTOR DEVICE I N THE ESEM

61

to ionize the gas. Likewise, the distance between the electrodes must exceed a minimum value 0, in order for the electrons to be able to ionize the gas. These limiting values of bias and distance are determined by the conditions of each system and relate by V, = ED,. Therefore, the gain equations are not applicable below these limiting values. These equations must be modified by replacing the parameter D with D - 0, and V with V - V,: D

-

Dm for D

and

V

-

V, for I/.

(61)

The minimum bias is about the ionization potential of the gas or the effective ionization potential of a mixture of gases, and to this corresponds an absolute minimum distance between the electrodes. Since pressure and distance enter in the equations of amplification as a product, it is a common practice to use this product as an independent variable to find the amplification, with all other parameters being fixed (see Engel, 1965). Thus Eq. (42) has been plotted versus p D in Fig. 16 for different fixed values of y , at V = 300 volt bias, and with (arbitrarily) V, = 15.5 volts. The parts of the curves between the two vertical broken lines correspond to the range of validity of the constants A and B, but for illustrative purposes, the curves have been plotted also at very low p D using the same A and B. The curves are discontinued where breakdown occurs. We note that the curves deviate quickly from the case of y = 0. The case with y = 0.01, or less, is close to a real case and does not differ much from the case of an absence of

1

I J

3

4

I

0

1

2 P O , Pam

FK,. 16 The Log(l;f,) bersus p D for different ;-coefficients, using Eq. (42) with V = 300 V.

62

G. D. DANILATOS

secondary processes. The important fact that these curves show is that there is an optimum p D . This can also be found analytically by equating the derivative of the amplification function to zero. The result is identical to the optimum Stoletow parameters given by Eq. (58). It is fortunate that the optimum p D is in the range of the ESEM conditions. For example, at D = 1 mm and p = 1500 Pa, we operate at near-optimum amplification with nitrogen and parallel plates at 400 V. It is important to note that this is the maximum possible amplification produced both by electrons and ions (total gain). That is the case if the time scale of imaging is such as to allow the use of positive ions. In the above numerical example, we have E / p = 267, and from Fig. 4 we find zjd = 1820 m/s; then the transit time is 0.5 p s . Fig. 17 shows the limits of amplification with total current for different fixed biases, for nitrogen and y = 0, in the range of validity of the constants A and B. The general conclusion is that the amplification can vary by orders of magnitude, even when using a high bias (e.g., 700 V), by simply varying the distance and pressure. The other conclusion is that the same amplification can be achieved at two different sets of p and D,and of course, there is an optimum operation condition. The maxima lie on a straight line. We are also interested in faster scanning rates in the ESEM, and hence, we should also consider what happens to these curves at those rates. Figure 18 shows the amplification of the SEO signal generated only by the fast

4-

3-

--0

2

A

2-

1-

pD, Pam FIG 17 The Log([ I , ) versus p D for different bias in nitrogen using Eq (42), with FIG maxima lie on d strdight line

, - 0: the

.I

~

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

63

pD, Pam FIG. 18 The Log(l;Z,) versus p D for different bias in nitrogen using Eq. (50)

ionization electrons, using Eq. (50) at the different fixed electrode voltages. As expected, the amplification is reduced by about one order of magnitude. Similarly, Fig. 19 shows the amplification limits for FEO (uniform bulk ionization) signal generated only by the fast ionization electrons using Eq. (52).

“I m a 0

2-

1-

0

0

1

2

3

4 5 PO, Pam

6

7

8

9

FIG. 19 The Log(l!f,) versus P D for different bias in nitrogen using Eq. (52).

64

G . D. DANILATOS

Again, the amplification is further reduced, but the overall FEO gain can be a lot higher on account of the initial ionization multiplication by the FEO energy (incorporated in Io). B. Geometry und Time Response

The preceding analysis was done on the basis of parallel plate geometry. We can expect a similar behaviour for other geometries except for quantitative differences that may result in qualitative advantages (or disadvantages) of the GDD. The geometrical configuration is important in many respects, such as the type of signal detected, the frequency response, and the amplification. 1 . Detection and Amplificution Volume

Some concepts borrowed from nuclear instruments may serve to design an efficient GDD. In the well-known gridded counters, a fine-mesh grid can be inserted between two electrodes, with a simple way being as shown in Fig. 20 (Frisch, 1944). Let the distance of the grid from the earthed electrode be much greater than that from the biased electrode, and also let the bias of the grid be only a little higher than the earthed electrode, so that the voltage difference V between the grid and the top electrode I/ = V, - V, is used either for gaseous amplification or simply for drifting the charges passing through the grid. The volume between the grid and earth is known as the detection (or conversion) volume (or gap), whereas the volume between the grid and the high bias electrode is the amplification (drift) volume (or gap). By such a scheme, we can increase the gain of the FEO signal for a given voltage if room can be found in the design of ESEM to accommodate the electrodes profitably. The FEO traversing the detection volume will ionize this volume, and all the ionization electrons generated there will contribute equally to the amplification

= FIG.20 Definition and separation of detection and amplification volume by a grid electrode

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

65

when they all cross the grid and all induce equally the resulting signal at the top electrode. The slow ions in the detection volume are screened out. Thus Eq. (50) can also be applied to the FEO, resulting in a higher gain than with Eq. (52), provided the amplification volume is the same as that used with Eq. (50). A similar phenomenon can be observed with SEO: A higher gain can also be observed for the SEO by simply passing them through a grid. The hitherto presented theories on signal induction and gaseous gain can help us see and understand phenomena that might appear “strange.” Such a case might be when the bias on the grids in Fig. 20 is such that the volume between earth and grid is both detection (conversion) and amplification volume, whereas that between grid and top electrode is only a drift volume, without amplification. Let us initially assume that the grid is lOOo/, transparent to electrons and that i t perfectly shields and separates the two volumes, i.e., the field lines of the two volumes do not mix. Let the total gain (CE&CI) be G. If we employ a fast detector and the ions are excluded, the gain due to electrons alone is G / a D . When all the electrons from the first volume traverse the full length of the second volume, they induce a charge equal to the number of electrons, i.e., equal to G. Therefore, the signal collected by the top electrode is crD times higher than that induced on the grid, despite the fact that there is no gaseous amplification in the second stage! The paradox is resolved if we remember, that the ions that are left behind will slowly drift and induce the “missing” charge on the grid in duecourse. In reality, of course, the induced signal on the top electrode is not as much higher, because the grid is not IOO‘Z transparent and because the field lines “leak” through the grid. These effects will show up as a deterioration factor f , so that the net gain i s fotD, which hopefully i s greater than unity. The whole question of separating various detection and various amplification volumes reduces to the art of shaping the electrostatic field in a purposeful manner. This art is a separate topic in itself. The grid mesh size, spacing, and shape can be calculated to produce a desired field direction and intensity. O n this specific topic, references can be found in a review survey of nuclear instruments by Franzen and Cochran (1962). Information on how to make strong grids is given by Priestley (1971). There are more elaborate electrode configurations in nuclear physics devices that can be adapted for detection purposes in the ESEM. These techniques relate to coincidence and anticoincidence detection. Coincidence is when two or more detectors produce a count within a predetermined time interval (resolution time), whereas in anticoincidence, a pulse appears at one detector without a pulse at a second detector within a short time interval. Of particular usefulness can be the “equal compensating chamber,” or “differential chamber”; this has the collecting electrode (e.g., a mesh) in the

66

G. D. DANILATOS

middle of the field, so that if a particle traverses both sides, little or no signal is detected, but, if the particle stops in one side, a pulse is fired (Korff, 1946; Fenyves and Haiman, 1969). The needle chamber consists of needle arrays, with each needle behaving as a separate proportional amplifier (Grunberg and Le Devehat, 1974; Fujita et al., 1975; Ranzetta and Scott, 1967). This type of electrode configuration is characterized by high amplification and counting rates. With proper adjustments, it could prove extremely valuable in the design of the GDD. Independent multiwire anodes in the same detection volume can also have a specific significance both for imaging and for spectroscopy. This method, or a wire with resistance, can be used for position measurements and detection. Dense grids as parallel electrodes, and combinations of parallel grids with multiwire systems have also been used t o advantage in nuclear physics (Hilke, 1983; Hendrix and Lentfer, 1986; Charpak and Sauli, 1978). The main disadvantage of these systems is that they require sufficient space for assembly and manipulation, and it is not immediately obvious how they could be adapted to the ESEM environment in a space of only a few mm, or even a fraction of a mm. However, the many possibilities of electrode shapes and configurations open a new research task to establish their best exploitation for the needs of the ESEM. A first comprehensive design in this direction is described below. 2. A Basic Configuration of GDD

Already, the geometrical configuration of electrodes in Fig. 11 serves as one way to separate various signals. This way of separation is further justified by the quantitative conclusions reached in the previous section. Let us consider a numerical example. From Fig. 18, we find an optimum gain of 28 for the SEO at 400 V. From Fig. 19, the corresponding gain for FEO in the same detection volume is 11 multiplied by the number of ionization electrons per FEO produced in this volume. This number of ionization electrons per FEO can become a small fraction of unity by making the diameter of the detection volume not much greater than the diffusion displacement of the electrons. In Fig. 21, an electrode configuration together with the pressure-limiting aperture (PLA1) is shown. For PLAl = .5 mm, the electrode (El) in the SEO volume should be reduced to only a thin circular ring slightly larger than the PLAl diameter and placed as close as possible to the PLA1, but separately from it. The PLAl should be either at earth potential or at some other appropriate potential so that the SEO cannot escape through the hole. The fine-wire ring electrode should be surrounded by one or more annular flat and concentric electrodes (E2 and E3) at the same potential as the ring electrode to ensure uniformity of field and definition of the detection volumes. A further

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

Y

PLAl

E

67

4 EO

El E2 E3

ICONDUCTOR)

FIG.21 A practical multielectrode configuration of GDD for the definition of detection volumes and signal separation in the microscope. PLAl = pressure-limiting aperture as controlling and detecting electrode; E l , E2, E3, and E4 are separate detection and/or guard electrodes.

ring or annular electrode (E4) can be placed above the PLAl to detect the ionization caused by the FEO escaping through the hole. The electrode E l will collect practically most of the SEO together with an amplification, while all others will collect the CE _N FEO. The outermost electrode will collect the ionization due to low-angle FEO showing topography, while E4 above the PLAl will collect ionization due to the high-angle FEO showing atomic number contrast. A very small fraction of CE ‘v FEO may be mixed with the CE = SEO, and if this is visible at all, it can be subtracted electronically with the aid of the signals from the other electrodes. All signals from the various electrodes can be manipulated first by adjusting the electrode bias and then by electronic means (mixing and processing). This system can be made more versatile by splitting each electrode E2 and E3 into two equal segments, preferably at normal directions with each other. By electronically subtracting the output of one half from the output of the other half, topographic shading with Z-contrast suppression can be effected in the usual way as is done with FEO detectors (see Reimer, 1985). In the above, the detection and the amplification volume coincide. By changing the bias of the various electrodes, it is possible to allow the SEO, or better the CE 2 SEO, to pass through the PLAl and be detected by electrode E4. This is possible because the electron avalanches triggered by

68

G. D DANILATOS

SEO are all located directly under the PLAl and, at high magnifications, close to the axis of the aperture. In addition, the size of the avalanches when they arrive at the aperture plane are of the order of the PLA1, as calculations have shown in Section 1II.C. Furthermore, if the magnetic field lines of the objective lens traverse the PLA1, the CE N SEO would tend to cluster closer together, thus facilitating their passage through the aperture. It remains to properly shape and bias the electrodes, especially the electrodes EO and E4, in order to maximize the electron passage. This approach may achieve exceptionally high gains for the SEO for various reasons. First, the aperture would tend to reduce the number of photons from the head of the avalanche above the aperture to reach the cathode below the aperture; this will suppress one of the most important ?-processes responsible for breakdown. Secondly, many of the field lines starting from the anode would finish on the aperture, and since the ions have a totally different mobility and spatial distribution from the electrons, they would be captured by the aperture grid; in particular, as most of the ions cluster and start around the anode, they are quite spread, and only a small fraction would escape back through the aperture. Thus, another y-process would be greatly reduced. Third, we can also expect the metastables to be hindered from returning to the cathode as they diffuse randomly, thus reducing the third 7-process. Fourth, the ions are blown upstream by the supersonic speeds attained above the aperture. Fifth, the aperture grid screens the two regions below and above it, thus all the electrons from the first step (from below) will induce a full electron pulse on the collecting electrode in the second step (above); as with conventional grids, allowance should be made for the loss of electrons that do not pass through the aperture, and also for the “leak” of field lines between the two regions. A sixth reason why the configuration in Fig. 21 could achieve a very high gain is proposed here as a hypothesis: As was pointed out in Section V.E, it is advantageous to spread the avalanche head as much as possible; this is exactly what we expect to happen with the present electrode configuration. A single SEO triggers one avalanche that reaches the PLAl plane. If the electrode E4, or a system of electrodes, is such that electrons from different locations at the aperture disk are made to follow different trajectories to E4 far apart, then a family of separate avalanches would start above the aperture. Thus, parullel amplijcations would take place, the sum total of which should produce a higher amplification than would be achieved if all the produced charges had remained in a single avalanche with a high concentration of charge over a smaller volume. To test this hypothesis, special care is needed to position and shape the electrodes EO and E4. Comparatively, the contribution to the SEO contrast by the (CE&CI) 2 FEO generated above the aperture is expected to be small, because the

THEORY OF THE GASEOUS DETECTOR DEVlCE IN T H E ESEM

69

SEG-FEO would not amplify by as large a factor as the SEO. For the same reason and because the probe energy is highest (smallest ionization cross section), the unwanted (CE&CI)-PE‘ should be minimal. The proposed electrode configuration may not be the best possible, but it is considered a basic one for general experimentation and studies of many phenomena and properties of the GDD at present. 3. Cylindrical Geometry Our inquiry here is to see if there are any benefits for the ESEM from employing the techniques of cylindrical counters. How is the detection volume determined in the ESEM conditions, and is there a better amplification with such a geometry? Can we apply the conclusions from nuclear applications to the GDD? The following survey will help our understanding of this configuration. In the majority of particle counters, the cylindrical geometry is used. In this, a thin wire (anode) is located at the axis of a cylinder (cathode). There are some immediate advantages for doing this. One is that almost all the volume of the cylinder is effectively a detection volume, and only that small volume contained within a few wire radii is effectively the amplification volume. This can be seen immediately from the way the electric field and the potential are distributed between the electrodes [see Eqs. (10) and (1 l)]. For an earthed cathode and an anode with potential V , (or field Ea), we can briefly rewrite those equations as

r

E = L E r

a?

(63)

yielding the potential and field intensity at radius r. Half of the potential drop occurs within a radius rli 2,

Thus for r2/r1 = 500, half of the voltage drop occurs within 4% of the total volume. The fast voltage drop around the anode creates a grid effect without an actual grid. For better and more effective and controlled separation of the detection and amplification volume, an actual grid can be sometimes inserted around the anode, like a spiral grid (Campion, 1971). Let us assume that space charge effects, y-processes, and electron attachment are absent in the following considerations. Then the amplification

70

G. D. DANILATOS

G in a nonuniform field can be derived from the integral

where r , is the radius from which the electrons can (start to) ionize the gas as they continue their drift towards the anode. If the first Townsend coefficient is a known function of position (or field), the above integration can be carried out and the gain can be predicted. For example, the Townsend equation (54) can be used in its range of applicability. However, there is a fundamental limitation in the validity of the above integration. As Morton (1946) has pointed out, the above equation cannot be applied in the cases where the variation of field is significant over one ionization mean free path in the direction of field. He has found experimentally that when the variation of field over one ionization mean free path is more than about 2.5%, integration of the above equation will yield an erroneous result. This can be the case, for example, at such a low pressure, for which the electron has to travel a considerable distance between ionizing events. Clearly, the ESEM operating between a few P a up to one atmosphere can be affected by this limitation over a particular range. Morton’s work was extended to higher pressures by Johnson (1948).Based on these works, Loeb (1955) points out that the inapplicability of the Townsend equation is not caused by the gradient of the field per se. It is rather caused by the lack of equilibrium. Even if we knew the function of a versus field (usually a l p versus E / p ) , we cannot apply it for integration, because the Townsend coefficient refers to the steady-state (equilibrium) drifting of the electron; for each particular value of E / p , the electron must undergo a set number of collisions, at constant E / p , to reach the equilibrium value of r / p given by the Townsend function (or any other equilibrium equation, for that matter). In such conditions, the best course is therefore the experimental determination of gain. The establishment of the range of conditions (pressure and voltage) for which a prediction of gain can be based on an equilibrium equation is also difficult for the same reasons. However, we may get only an idea of this range if we could tentatively assume that Eq. (54) can be used only to derive an approximate ionization mean free path L:; then we can find the field at r and r + L: and, hence, the rate of its variation over this distance. Thus, we get L: = l/r, and from Eq. (63) we find E ( r + Li)/E(r) = r/(r + Li). In Fig. 22, the latter fraction is plotted for different combinations of bias and pressure (V,p ) for a cylinder with rI = 0.1 mm, rz = 5 mm, filled with nitrogen. We note that only at the high V = 10000 volts and p = 10000 Pa do we have a small field variation close to the Morton criterion over a particular range of radial

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

71

:\\

~2000,10000) (400,1500)

0

I

0

1

2

3

4

5

Radial distance r, mm Fir:

77 F r i r t i n n a l variatinn nf plprtrir firld n v e r n n r innirrntinn mean free n x t h in r v h -

drical geometry with various combmations ot (bias, pressure) glven with each curve. Nitrogen, r, = 0.1 mm, r z = 5 mm.

distances. For the case of 400 volts and 1500 P a frequently encountered in ESEM applications, the Morton criterion is clearly violated. The most relevant work to the ESEM conditions is that of Morton and Johnson. Their results are presented below first. There are also several formulae giving the gain of counters, but these formulae are applicable for conditions of special gas fillings at high pressure and bias, because most counters are designed to operate at near or above atmospheric pressures in the kV range of anode bias. Historically, the ESEM has operated mostly in the lower pressure regime, namely at, or below, the saturation water vapor pressure at room temperature, and for these conditions, the derivations for counters are not applicable to GDD. However, the ESEM has been shown to operate up to atmospheric pressures (Danilatos, 1985),and a brief review of the theories of counter amplification is in place here to facilitate future research of the GDD. The various formulae presented below correspond to different dependence on the ionization coefficient, in different environments. The form of the curve depends on the electron energy distribution, the electron mobility, and the probability of ionization (Charles, 1972). These parameters vary widely with the nature of the gas or gas mixture, and there is no predictable form available. Some gases show one preferred behaviour, whereas others show another,

72

G. D. DANILATOS

and therefore, one formula gives a better prediction for one set of circumstances than another (Zastawny, 1966). The main derivations are given below; modifications and further references are given in the works quoted. Morton and Johnson. For an understanding of the behavior and an estimation of the expected gain of the GDD at low pressures (around 2000 Pa) and bias of several hundred volts, we can resort to some experimental measurements presented by Morton (1946) for hydrogen and by Johnson (1948) for nitrogen and hydrogen. Morton used two different diameter cathodes, namely 3.2 and 11.1 mm, inside a 88.9 mm diameter anode, with hydrogen gas at pressures below 1370 Pa. By use of the principle of similitude (Cobine, 1941), we can deduce the expected behavior for smaller cylinders as might be encountered in the ESEM. This can be done by dividing the geometry by a given factor and by multiplying the pressure by the same factor. The curves in Fig. 23(a) and 23(b) have been reproduced (in SI units) from measured values given by Morton for three different voltages. By dividing the geometry and by multiplying the pressure by a factor of around 50, we can obtain the gain for systems of dimensions and pressures likely to be encountered in the ESEM. We note the maximum gain at some optimum pressure for the chosen geometry parameters. A t very low pressure, the gain is very low because of the small number of collisions. At very high pressure, the gain is again small because the mean free path is small and not enough energy is gained to cause ionization. The maximum gain occurs when, or close to when, the ionization fills the entire volume. Campion (1971) estimates that the ionization fills the entire volume at a pressure twice that corresponding to the maximum, but he bases this on a double differentiation of Eq. (75), to be given below. (Note, in Morton’s paper, the numbers of 150 and 180 volts are transposed on the corresponding curves; this is believed to be a printing error, since no comment on this irregularity is made either by Morton himself or by Loeb (1955), who later reviewed that work.) Morton develops a differential-difference equation for the electron current as a function of the electron energy and distance from the cathode. By numerical methods, he establishes good agreement between his measurements and theory for the range in which he could obtain adequate data. He finds the location of ionization and the energy distribution of electrons reaching the anode; his results are radically different from those predicted by use of the Townsend equation in the nonuniform field. Johnson confirmed and further extended the work of Morton in a higher pressure range (between 1.33 and 101300 Pa) and bias both for hydrogen and nitrogen. He used two different diameter wires as cathode (2.37, 5.55 mm) and three different cylinders as anode (15.1, 28.6, 43.6 mm). The electrode

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

73

180

7

OO

2

1

3

Log P, Pa FIG.23(a) The gain ( I , I o ) versus pressure at different bias in cylindrical geometry with hjdrogen: wire cathode diameter = 11.1 mm, anode diameter = 88.9 mm. (Adapted from Morton. 1946.)

24

E

iao

bias used was up to several thousand volts. The primary electron current source was by photoelectric emission from the cathode by illumination with ultraviolet light. He took care to ensure the purity of the gases used. He also accounted for back-diffusion by use of the Rice Equation (44) when this effect became significant at pressures above 700 Pa. Therefore, his data should be

74

G. D. DANILATOS

N

x

0

0

0

1

3

2

4

5

Log p, FIG.24 The gain I i I , versus pressure in nitrogen at different (wire) cathode fields. Dotted parts of curves together with solid curves represent the gain when inverting polarity. (Adapted from Johnson, 1948.)

taken as a reliable source for reference in our future measurements of gain. Figure 24 is a reproduction from his paper, converted to SI units. Each curve corresponds to a fixed field strength E , at the wire surface. The curves are discontinued where breakdown occurs. Johnson found that the gain curves were the same when he inverted the electrode bias (a case of interest for the GDD). Interestingly, his curves show an extended working range (higher breakdown bias) when using the wire as anode. He further found that the gain differs when inverting the polarity only for the low bias (200 and 300 V) at low pressures (at the peaks and below) in the case of hydrogen. Hence, we cannot assume that we can use Morton’s curves with reverse polarity. Most interestingly, he found that his results could be plotted with a single reduced (universal) curve for each gas. By calculating the quantity log[( l;rlp) ln(l/Zo)] as ordinate and E , / p (maximum reduced field at wire surface) as abscissa, he produced the universal plots shown in Fig. 25 (adapted). This type of information has general engineering application. We can choose any cylindrical geometry with given r l , r 2 , p , and V, find E , / p , and from the corresponding ordinate, we can finally deduce the gain l / I o . The only limitation for this is that Johnson’s data may not cover the complete possible range. For an arbitrary set of geometry, pressure, and bias, there are two possibilities if they fall outside the range of these curves: either they correspond to a breakdown regime, or Johnson’s data require completion. Whereas the total gain is not practically affected by changing the polarity in the indicated range, the ionization density distribution in the gas is radically

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

75

FIG.25 Reduced gain characteristics for hydrogen and nitrogen. (Adapted from Johnson. 1948).

different. For a detailed discussion of the physics of such systems, the reader is referred to Loeb’s book. Rose and Korff. Rose and Korff (1941) have derived the following expression for the gas gain G = I l l o :

where n is the gas concentration, I/ the applied bias, and cp(0)relates to the energy distribution of electrons (unknown); for a great variety of distribution functions, this parameter varies between 1/2 and 2, but for the great majority of cases, it is close to unity. The constant V, is the bias at which the ionization starts and is best determined experimentally for each counter by extrapolating the straight portion of the gain curve plotted on log G- versus V-axes. The constant a is characteristic of the nature of gas and is the constant of proportionality between ionization cross section and electron energy V, (in volts): Di

= av,

(67)

The constants a for different gases are given in Table VIII. For a mixture of i gases, a = aipj,where pj is the partial pressure of the component i. I

The above derivation, as many others, suffers from not knowing the energy distribution of electrons, the assumptions for which led to putting the ratio r / p

76

G. D. DANILATOS TABLE VlII RATEOF INCREASEOF IONIZATIONCROSSSECTION WITH ELWTKON ENERGY (ROSEAND KORFF,1941) IN x lo2’ m2/volt. A 1.81

Ne

He

H,

0 2

0.14

0.1 1

0.46

0.66

as being proportional to the square root of

N, 0.7

CH, 1.24

Elp, namely:

cp(O)N, E

a=[R7p]

’

where NA is the Avogadro number; this dependence is at variance from the Townsend equation, but it is well applicable to the particular conditions of many counters. Rose and Korff, and later others, have verified the gain formula for several gas mixtures, such as Ar-CH,, at pressures above 13000 Pa. Wilkinson. Wilkinson (1950) presents a derivation for gain based on the general expression for a l p as

-I

B P WP)“ ’ where K , B, and m are constants characteristic of the gas. The gain is then given by G( = Kexp[

G = exp[Cr(l/m,D)],

(70)

where

and

r(x,y ) =

:’j

t X -

1

exp( - t )dt.

The voltage V, at which we can expect a little gas amplification (G with 6 << 1) is given approximately by

=

1+6

Wilkinson cites the following numerical values in the esu system: For argon, K = 12.6, B = 28.2, and m = 0.575; for hydrogen, K = 73, B = 18.6, and m = 0.3. He emphasizes that the above expressions are guides to probable

THEORY OF THE GASEOUS DETECTOR DEVICE IN THE ESEM

77

behavior, and the constants are for pure gases. At high pressures and small r l , as is used in nuclear physics, the following approximation can also be used: V , = 7.5pr, ln(rz/rl).

Diethorn. as follows:

(72)

Diethorn (1956) has derived an alternative formula for the gain

where AV and S, are characteristic gas constants. The physical significance of AVis the potential difference through which an electron moves between two consecutive ionizing collisions, and S, is the critical value of E/p above which ionization occurs. For the derivation of this equation, it is assumed that a linear relationship expresses the ionization constant: @ / P = (In 2/AV)(E/P).

(74)

Measurements of the constants for several gas mixtures used in proportional counters are given by Wolf (1974). Some of these mixtures could be used in the ESEM.

Williams and Sara. By simply assuming the Townsend equation (54), we get for the gain, through integration of Eq. (65),

(75) where r, is the critical radius at which ionization occurs. For high enough pressures, the second exponential term in the brackets can be neglected, leaving what is known as the Williams and Sara (1962) equation for gain. At low pressures, both terms should be used, yielding a maximum of gain when plotted versus pressure. Nasser (1971) presents the equivalent result by assuming that the critical radius is at the point where the electron has fallen from the cathode through a potential difference equal to the ionization potential of the given gas. In this case, the gain is given by G

= exp

exp( - X ) - exp( - Y ) exp(B/Ar,E,) - 1

where X

=

[

Apr, exp

___

( A r 3 - l1

78

G. D. DANILATOS

and

with El being the field at r t . By reversing the polarity in cylindrical geometry, we obtain a result that differs from Eq. (76) only in the denominator, which becomes 1 - exp( - B/ArlE,). This result indicates the asymmetry of this configuration and is significant at low pressures. Zastawny. Yet a further derivation has been provided by Zastawny (1966). This is an improvement of the Diethorn derivation. It assumes again a linear dependence for the ionization coefficient as

(77)

a l p = C(ElP - SO)>

with C and So being characteristic constants of the gas. The gain is then given by

IL

G = e x p E,rl K

+C

(

In-+-oP:

2 111 1

,

with K being a constant of integration characteristic of the gas. Aoyamu. Aoyama (1 985) has derived a generalized gas-gain formula fitting very well the experimental data of proportional counters:

where n is the gas (particle) concentration and L , m, and M are constants experimentally determined for each gas mixture.

Kowalski. Another recent derivation is reported by Kowalski (1985, 1986), with which he has found also a very good agreement for the majority of gases used in proportional counter fillings:

G

= exp{raEa[La - 1

(5y-1+ p

R ] } , d # 1,

where the constants A , d , and R are found experimentally. He tabulates values of these for a variety of Ar- and Xe-based gas mixtures.

4. Discussion of Gain Bambynek (1973) and Charles (1972) have reviewed the topic of gain. However, those reviews, as well as the original authors’ derivations of gain presented above, make no reference to the works by Morton (1946) and

T H E O R Y OF T H E GASEOUS D E T E C T O R DEVICE 1i3 T H E ESEM

79

Johnson (1948). Charles has criticized the validity of previous works. He has pointed out that many experimental measurements on gas amplification were in error through amplifier-pulse-shaping effects. In particular, the pulse size recorded depends on the shaping of time constants, the anode radius, and the gas pressure o n which the ion mobility depends. He considers the Diethorn and Zastawny formulae to give the best approximations, and he proposes a new equation to fit the correct experimental data. Gold and Bennett (1966) suggest that the Diethorn formula should provide an adequate approximation in the region E l p < 750 V/Pam, whereas Specht and Armbruster (1965) indicate that the Rose and Korff formula is better in the region E / p > 750 V/Pam. Kowalski (1983) has reviewed the various formulae and found the best fit by the Zastawny expression in the range E i p > 70 V/Pam, and by the Diethorn one in the range E / p > 90 V/Pam; the Sara and Williams formula failed, whereas the one by Rose and Korff fitted only in narrow intervals of E / p . The best approach in the ESEM is, therefore, to experimentally determine the gaseous gain, with the previous information providing a good guide in this respect. Fujii et al. (1986) have presented a device for measuring gain and drift velocities in various gases. It should be emphasized that the amplification formulae given refer to the total signal induced by the electrons and the positive ions. The relative strength of the two components depends very much on the parameters of geometry, pressure, gas, and bias. Generally, the component due to electrons is a small fraction of that due to the positive ions. This is because the electron avalanche develops very fast only as it approaches the anode wire, but by then the electrons traverse only a small potential difference, according to Eq. (14). On the contrary, the majority of positive ions are formed near the anode and traverse most of the total potential difference. An estimate of the very small fraction (less than of signal due to electrons is given by Wilkinson (1950). 5. Discussion of Time Response The transit time for electrons is much shorter than for ions, but in cylindrical geometry, the ions produce most of the total signal within only a short distance from the anode, and hence the effective transit time can be very short also for the ions. Indeed, if the ions are the signal carriers in this case, their transit time will determine the frequency response of the GDD. The transit time for an electron or ion depends on the function of drift velocity versus position. By assuming the simple relationship ud = K ' ( E / p )for ions, we obtain by integration

80

G. D. DANILATOS

With more complex situations, this time can be found numerically, if the drift velocity is given graphically, but the difference from the above simple derivation is not great (Wilkinson, 1950). The pulse shape in cylindrical geometry has a very sharp rising part. By sacrificing some of the peak height, we can increase the time response electronically by a proper choice of clipping time, when we are interested in the individual pulses (as in spectroscopy). Otherwise, the pixel signal is the accumulation of all the amplified pulses during the integration time. From the analysis of cylindrical geometry presented here, we obtain a general understanding of the behavior of the G D D with similar geometry. It could help design controlled and meaningful experiments in the actual conditions of the ESEM, and it provides directions for a further task, namely, to determine experimentally the gain and frequency response of a fine-wire anode contained in various shapes and sizes of cathodes, with various gases and gas mixtures.

VII. SCINTILLATION GDD As was mentioned in the introduction, the GDD can also operate in the scintillation mode. This is based on the production in the gas of photons in the ultraviolet, visible, and infrared region by various microscope signals. The production of photons can take place either directly from the exciting source, e.g., LRG-(FEO, SEO, XRO), or indirectly by drifting electrons in a field; these electrons are, for example, SEG-(FEO, SEO, XRO), or CE z (FEO, SEO, XRO). The direct production of photons (primary photons) corresponds to the first phase of signal detection, whereas the indirect production of photons by drifting electrons corresponds to the second phase of signal detection. Thus, what in fact we seek to exploit here is what was considered a nuisance in the previous sections, namely, the gaseous scintillation. Both the line spectra and the continua are usually present, and most of the radiation is restricted to the ultraviolet region. One component in gas mixtures can act as wavelength shifter to convert the UV radiation to visible light. Alternatively, certain plastic materials and coatings can act as wavelength shifters. Whereas the above ideas are new to electron microscopy, they are well established in gaseous and nuclear physics research. The detection of the primary photons in the gas is the principle of the gaseous scintillation detectors. The phenomenon of light amplification in the presence of an electric field has been studied by several workers (e.g., Legler, 1963; Szymanski and Herman, 1963). The detection of the secondary photons produced by the

THEORY OF THE GASEOUS DETECTOR DEVICE IN T H E ESEM

81

drifting and multiplying electrons in the applied electric field is the basis of “gas proportional scintillation (GPS) counters” (Conde and Policarpo 1967; Policarpo et al., 1967). The early experiments were done with alpha particles in argon, in a radial electric field. They measured the time response and found that the primary scintillation gave a rise time of lo-’ s, the secondary scintillation gave a rise time of 2 ps, and a pulse tail extended for several decades of ps. The pulse amplitude increased with anode voltage. In a similar way, the above principles could be applied to the ESEM, but not without study and experimentation to determine the specifics of operation and the limits of applicability. For example, at low pressures (around 2000 to 3000 Pa), where much work is done in ESEM, the nuclear instruments are usually suitable for detection of heavy particles and fission fragments (Mutterer et al., 1977), whereas in the ESEM, we have soft x-ray and beta radiation. Also, the time response requires close attention. The feasibility of the scintillation G D D has already been established (Danilatos, 1986b), even though the experiments conducted were far from the optimum conditions. Thus, the early observations contained a large proportion of primary photons. The efficiency of primary photon production in gases is relatively low in comparison with standard scintillators. However, the secondary light yield can be several orders of magnitude greater than for NaI(Ti) crystals (Policarpo et al., 1972). Furthermore, the acrylic material (perspex) used by the present author in early experiments as light pipe transmitted wavelengths above 390 nm. Using nitrogen, the main wavelengths emitted were around 390 and 340 nm, and therefore, there were considerable light losses in the light pipe. Yet, despite the poor experimental setup then at hand. it was possible to record clear LRG-(FEO&SEO) images. Therefore. it can be safely inferred that the imaging possibilities can be immensely improved by proper choice of materials, bias, and light transmission and detection systems. In the same way as Eq. (65) for electron amplification, we can write an equation for light amplification with a corresponding scintillation coefficient 6. Engel (1965)presents some data on the dependence of this coefficient on the electric field and pressure. The data in Fig. 26 are taken from Engel’s book and give the curves for nitrogen and hydrogen. The two curves for nitrogen correspond to the two different spectral lines. For comparison, the ionization coefficient of nitrogen is also plotted. It is noted that the light production starts at much lower E j p than for ionization. Updated data on S/p for nitrogen and hydrogen can be found in a paper by Legler (1963). The data in Fig. 27 are taken from that paper; they show the relative variation of light production in the ratio 6 j a versus E / p . It is noted that there is a dependency on pressure in addition to the dependency on Ejp. This is due to quenching, which increases with increase of pressure.

82

G . D. DANILATOS

-a UJ

s- "O

-

a v)

c

c 0

2 a

0.5 -

,

N2(340nm)

0

I

0

10

I

20

30

40

50

60

70

E/p, V/Pam FIG.26 Coefficient of photon production 6,'p versus E i p for nitrogen at two wavelengths (390 nm and 340 nm) and hydrogen; dashed curve is the ionization coefficient of nitrogen for comparison. (Adapted from Engel, 1965.)

I

20

30

40

50

60

E/p, V l P a m FIG.27 Relative production of photons over ions d/a versus E:p in nitrogen at three different pressures. (Adapted from Legler, 1963.)

Data on the scintillation properties of gases have been reported by numerous workers. Szymanski and Herman (1963) give information on rare and other gases in the presence of an electric field. Mutterer (1982) has studied the luminescence properties of Ar-N, mixtures at various pressures and the effect of low electric fields on light yields (for E / p < 1 V/Pam). Policarpo et al.

THEORY OF THE GASEOUS DETECTOR DEVICE I N T H E ESEM

83

( 1967)have shown that the secondary scintillation amplitude increases quickly with nitrogen concentration in N,-Ar mixtures, reaching a maximum at 2.5”, N,; this was found at near-atmospheric pressure, and it may differ at lower pressures, because the wavelengths emitted depend on pressure. Dondes et al. ( 1 966) have undertaken a detailed study of the spectroscopic properties of various gases with and without the presence of field, for currents not exceeding the saturation level. The obvious direction of action is then to design a proportional scintillation G D D . We need to enhance the photon production while inhibiting these photons from triggering a breakdown. This can be done by “hiding” the cathode, wherever it is. This measure, if possible, will suppress one of the ?-processes, namely, the liberation of electrons from the cathode by the useful photons. A breakdown is eventually unavoidable on account of the other ;+processes. However, we may not have to resort to high fields at all. Policarpo et al. (1972) have established that the light production is so intense that it is not necessary to work in the region where ionization amplification is present. Above a threshold bias, the secondary light output increases linearly with applied voltage up to the point where ionization commences. There are some immediate advantages in using the scintillation GDD. One is that it is practically free of microphonic noise generated by mechanical vibration. The output is not produced by the induction mechanism, but by the amount of light reaching the photosensor. Also, electromagnetic noise, which could be detected by the electrodes, will be filtered out by the photon mode. Another great advantage is the time response. Many excited states usually have a very short lifetime (nanoseconds). The development of a cascade can be very fast, and, since we don’t have to wait for the ions to traverse back a substantial potential difference, the light production from the multiplying electrons is very fast. Metastables, if present, have a relatively long lifetime, but they can be excluded by proper choice of wavelength filters and proper external electronic circuits. Therefore, the scintillation GDD can have a very fast time response suitable for real TV scanning rate imaging. Campion (1968) has conducted simultaneous observations on both the electrical pulse and the photon production as a discharge develops. He observed the main and satellite light pulses using a 10% methane and 900/0 argon mixture. He measured the gas amplification, the electron transit time, and the optical radiation response. He found that the mean life time of the excited states is of the order of 3 ns, and the development of a single Townsend avalanche is between 20 to 50 ns (with pure methane, less than 10 ns). Spurious pulses are produced with a variety in the shapes of time distributions of these pulses; this is indicative of the presence of several mechanisms by which the pulses are produced, although not all of these mechanisms are necessarily

84

G. D. DANILATOS

operative for any given counter. By employing a grid at ground potential in front of the cathode at + 25 V, the light pulses are suppressed (Campion, 1973). Noble gas scintillators with internal electrodes have been used for highaccuracy position-sensitive detectors (Charpak et al., 1975). Policarpo (1982) has reported on the coupling of proportional or primary scintillation devices with multianode or multiwire proportional counters for the detection and localization of the incoming radiation. Similar methods may open novel possibilities in the ESEM. The scintillation G D D has many possibilities of application, and developments can extend well beyond the brief outline above. For example, it is possible to continuously monitor any impurities in a gas by observing selected impurity lines with a spectroscopy system (Thiess and Miley, 1974). Various products can appear either from the beam-specimen interaction or from the signal -gas interaction. Dissociative reactions can readily occur in the gas at great rates. Engel (1965) cites the following numerical examples showing the relative intensity of three reactions of an electron swarm in hydrogen: At E / p = 40, we have 70 dissociations, 25 excitations, and 5 ionizations; at E / p = 100, we have 60 dissociations, 20 excitations, and 20 ionizations. Further, Corrigan and Engel (1958) have shown that for electrons of low energy, the metastable atoms far exceed the emission of quanta in the far UV, but this is reversed at higher electron energies. It is too early to fully assess the implications. This poses a further broad research task ahead, in order to determine the efficiency and limitations of the scintillation GDD. Further reference material can be found in the extensive reviews by Birks (1964) and Platzman (1961); Policarpo et al. (1972); Alegria and Policarpo (1983); detailed studies of and data on such systems with various gas mixtures have been reported by Thiess and Miley (1974).

VIII. SIGNAL SPECTROSCOPY

The ESEM is effectively a radiation source immersed inside a gas, and therefore, we should be able to apply the methods of nuclear and particle physics for spectroscopic analysis. No experimental work has been reported yet in this area, and the following presentation is a theoretical approach to establish the broad guidelines for future work. Whereas the idea of using the gas in the ESEM specimen chamber as the medium for spectroscopy seems reasonable, it is necessary to reexamine the general principles of spectroscopy in the specific conditions of the microscope. The principle of using the gas of the ESEM as a detection medium for imaging has already been established both experimentally and theoretically

T H E O R Y O F T H E GASEOUS DETECTOR DEVICE IN THE ESEM

85

(Danilatos 1983a, 1983b, 1988). In this section, we wish to see to what extent it would be feasible to use the gas for characterizing the signals emitted from the beam-specimen interaction in terms of energy or wavelength distribution. In other words, we seek to examine the signal spectroscopy. In the vacuum operation of electron microscopes, signal spectroscopy has been widely practiced. This comes under the two general categories of x-ray microanalysis and electron energy loss spectroscopy. These areas are well known and established in the electron microscopy field, and there is no need for particular references. Photon and electron detectors of various kinds are used. For x-ray detection, both proportional counters and solid-state detectors are in general use (Reed, 1975). For electron loss spectroscopy, the spectrometers used are of the vacuum electromagnetic field type for separation of electrons of different energies (electron-optical systems, see, e.g., the review by Reimer, 1985). Thus, all signal spectroscopy methods employed in vacuum operation of electron microscopes have the common characteristic of using devices detached from the signal source and separated from the specimen by an envelope of vacuum space. In the ESEM, the GDD is in intimate contact with the specimen. This, together with the fact that the specimen is at, or near, its natural state, opens some unique new possibilities. However, the advantages may be offset by certain compromises that accompany the GDD operation. The precise nature and extent of the advantages and disadvantages is not easy to predetermine theoretically, except in the most general terms. Adaptation of various existing detectors, from operating in vacuum conditions to the conditions of the ESEM, is an important approach. Such adaptation may range from a simple geometry change (as was done with the scintillating BSE detectors, or as can be done with an existing x-ray detector) to more complex changes of the electron-optical elements of spectrometers to achieve a result in conjunction with the spectroscopical properties of the gas.

A . Spectroscopy, Statistics, and Energy Resolution

A description of some of the principles of spectroscopy can be found in numerous textbooks of the field and in manuals on radiation counters (see, e.g., Knoll 1979). We usually distinguish the wavelength-dispersive and the energydispersive analysis. In wavelength-dispersive analysis, we detect and measure the various wavelength photons produced. In vacuum microscopy, the photons (x-rays, etc.) originate directly from the specimen about which they carry information. In the ESEM, photons are also produced in the gas by the signal-gas interactions. It may be that the latter interactions overshadow the

86

G D. DANILATOS

former. In any case, we are concerned in the present work with the gaseous reactions and their relationship to the specimen information. The observation of light spectra coming from the gas and their relationship to various radiations is common practice in nuclear physics. The gas-proportional counters referred to previously operate in this fashion. Hence, we may expect a similar operation of the G D D in the ESEM. The energy-dispersive mode is very popular on account of its simplicity. This is based on the principle that the amount of ionization by a given energy signal (electron or photon) produced in the detection medium (gas or solidstate) is proportional to the energy of the signal. Thus, each electron or photon will generate an initial number of ion pairs (charge) in the gas of a proportional (or ionization) counter, the pulse height of which will be proportional to the initial charge. By feeding this output into a multichannel analyzer,we obtain the energy distribution of the ionizing signals. The simplest and easiest form of spectrum is the integral pulse height giving the total number of pulses within a broad energy range, usually greater or smaller than a set value. By increasing the number of channels, we obtain a dijjerential pulse height distribution. The resolution of a detector producing a spectral peak caused by monoenergetic radiation is usually defined as the full width at half maximum (FWHM) divided by the location of the peak centroid. The resolution is ultimately limited by statistical fluctuations in the detection medium. It is well known that the solid-state (or crystal) detectors have much better energy resolution than the gaseous counterparts (see reviews by Knoll, 1979, and Reimer, 1985), especially in the higher energy range. We can make an estimate of the statistical noise of a gaseous detector. If No are the average number of ion pairs formed by a particle, we can assume a Poisson distribution with N;’’ as the standard deviation. If the number No is large, the peak shape is Gaussian, and the relationship between FWHM and standard deviation is FWHM = 2.35 NA’’. The pulse amplitude together with its FWHM at the output of a proportional amplifier are in the same proportion amplified, and the resolution limit is simply 2.35/N;I2. Thus the resolution limit is expected to deteriorate when No is small. Actual measurements have shown that the resolution limit is better by a factor F called the Fano factor. The Fano factor is defined as the ratio of the observed variance to the Poisson variance, and the resolution limit R is given by: I/’

R

=

2.35(&)

The physical significance of the Fano factor can be understood as follows: When the ionizing collisions No are only a small fraction of all collisions occurring (e.g., by an energetic electron), they take place randomly, and the

THEORY OF THE GASEOUS DETECTOR DEVICF I N T H E ESEM

87

standard deviation is simply N:l2 (Poisson statistics). When all the particle energy E is spent for ionization only, the number of ions is strictly determined by the ratio E/W = No,and the standard deviation is zero ( W is the average energy spent per ion pair). In reality, we are between those two extremes as measured by F. The Fano factor is substantially less than unity in gasproportional (also in semiconductor-diode) detectors as opposed to being unity in scintillation detectors (Knoll, 1979). Let a, be the standard deviation of No (the initial number of ion pairs), N the average number of electrons developed after multiplication having a standard deviation 0,corresponding to an average amplification G for all the avalanches, with standard deviation oG . Then, from the error propagation law, we get

Let A be the average multiplication factor for the individual avalanches caused by single electrons. Then A = G. If the variance of a single-electron multiplication is o;, we have, again from the error propagation law,

From the definition of the Fano factor, we also have

(z) ’

F

=N,’

and Eq. (83) becomes

The second term of the above equation represents the fluctuations due to the single-electron avalanches, and by putting ( o * / A ) ~= f for the relative variance, the resolution is expressed by the Frish- Fano equation

+

R = 2 . . 3F 5 (f ’~ l” )

The factor f ’ can be found theoretically or experimentally (Alkhazov, 1969, 1970; Genz, 1973). This is zero at A = 0, increases monotonically with amplification, and gradually approaches a constant level, with its numerical value depending on the nature of the gas. Equations (86) and (87) are sometimes found in terms of another factor, as follows. A theoretical prediction for the distribution P ( x ) in the number of

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G. D. DANILATOS

electrons x produced in an avalanche with amplification A is (the Furry distribution)

If A is greater than about 50, then the above is reduced to P(x) = A

exp( - x/A).

(89)

The last expression yields ( o * / A ) ~= 1, which is confirmed by experiments at low electric field. At high electric $field, Byrne (1962) has proposed the Polya distribution

* + 0)

-x(l

+

x(1 0) [-T]heXP[

1.

where the parameter 0 is 0 < H < I and relates to the fraction of electrons acquiring an energy higher than the ionization energy of the gas. Later, Byrne (1969) derived an equation for this parameter: (91) where 2 < c < 3, V, is the ionization potential of the gas, a is the first Townsend coefficient, and E is the electrid field. For the Polya distribution, the relative variance is

The factor h represents the fraction of electrons with energies above the ionization energy, or, more precisely, with energies two or three times the ionization energy (the number of electrons being large). Finally, the overall statistical limit given by Eq. (86) and the resolution by Eq. (87) become (for large A ) :

' R = 2.35

F+b

(93)

+ (r) F

b

"2

(94)

The Fano factor varies typically between 0.05 and 0.2, while the Polya parameter h varies between 0.4 and 0.7, and thus the amplitude fluctuation is mainly dominated by the fluctuation in the avalanche size (Knoll, 1979). The factors F, f , and b depend on the gas mixture, and data on these have been published by various authors (Alkhazov et al., 1967; Marzec and Pawlowski, 1982).

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

89

The above-quoted energy resolution is the theoretical ultimate limit that is irreducible. In practice, the resolution can be worse, because it is affected also by other factors (reducible), such as the external electronics, the wire diameter uniformity, diffusion, gas purity (electron attachment), and others. Corrections for counting losses, background, inefficiency of detection, and, in general, the theory of pulse evaluation and spectroscopy, can be found in most books on radiation detection and measurements (e.g., Wilkinson, 1950; Fenyves and Haiman, 1969; Korff, 1946).

B. Environmental Scanning Transmission Electron Microscopy The type of electron microscope with which to practice signal spectroscopy with the least complications would be the scanning transmission electron microscope (STEM) modified with differential pumping chambers to accept a gaseous environment in its specimen chamber. A thin specimen can be placed very close to the main pressure-limiting aperture (PLAl), so that a pressure level comparable to that used in conventional proportional chambers is achieved. The electron beam strikes the specimen, and a multiplicity of signals emerges from the other side of the specimen. The space on this side is free to be fitted with electrodes comprising one, several, or a system of proportional counters, together with multichannel analyzers. The configuration is equivalent to the use of a 2n windowless flow-type counter. Such a system should in principle be capable of performing signal spectroscopy, as in nuclear instruments, of the radiation source being confined at a small area of the specimen. There can be certain difficulties with such a system, not usually encountered in nuclear physics. These are mainly the following. The counting rate required can be extremely high, and the system may be limited by the speed response of the external circuit and the transit time of the counters. There is a wide spectrum of electrons, mainly characterized by a peak around 2-3 eV (most SEO). This broad spectrum will constitute an intense background; it is not clear at this stage, if this background will allow the resolution of the superposed characteristic peaks due to x-rays and Auger electrons. Some general measures can be taken to reduce these difficulties. The anode(s) can be shielded by grids with appropriate bias to screen the unwanted signals. For example, the SEO can be subtracted by a suitable negative bias, thus eliminating the exponential background peak at the low energies of the spectrum [given by Eq. (89)l. Plane or cylindrical grids can be used to define the detection and amplification volumes. The techniques of coincidence, anticoincidence, and related methods can be used in the usual manner. Furthermore, a “sink” can be introduced to eliminate the unscattered or verylow-angle scattered probe electrons; this can be done, for example, by a very

90

G. D. DANILATOS

sharp conical aperture subtending a small angle below the specimen. Special electrode configurations may be introduced. Modern nuclear methods have been developed to handle very fast counting rates. Hammarstrom et al. (1980)have developed multitube detectors matched with fast electronics to reduce dead time. By connecting inductance between the inputs of the differential preamplifiers, counting rates in the range of MHz/wire can be handled. Fast, low-noise electronics have also been reported by Benson (1946) and Fischer et al. (1985, 1986). In x-ray astronomy, events have to be identified in an environment with a high intensity of charged particles, i.e., electrons and protons; for this, a positive identification discrimination is required. Andressen et al. (1976) describe a Xe-filled gas scintillation proportional counter with which they studied the pulse trains of signal output; they suggest that these are associated with particles producing extended ionization tracks and that they could be used for discrimination of x-rays and direct digital energy measurements. Mathieson and Sanford (1963) have reduced the cosmic background in an x-ray proportional counter through rise-time discrimination. Pulse-shape discrimination techniques have also been reported by Sudar et al. (1973), Harris and Mathieson (1971), and Isozumi and Isozumi (1971). It is not expected that all the above techniques can be transferred to the ESEM field without further development, but it is hoped that at least some similar methods could benefit our aims.

C. ESEM Signal spectroscopy in the ESEM will have the same difficulties as in the environmental STEM (ESTEM), with the addition of the limited space available above the specimen. This limited space together with low pressures usually employed can result in pronounced wall effects. The specimen has to be placed relatively close to the PLAI, and the shape of this aperture cannot be extremely sharp and conical (with small apex angle), since it is limited by electron optics requirements. The gas dynamics requirements are additional constraints for an optimum geometry of the aperture. Thus, the space available for detection and amplification in the gas is severely limited, contrary to the case of environmental STEM. The space available has two areas. One is defined “above” the aperture, i.e., downstream of the aperture. In general, the utilization of this space for detection has been shown in practice by the present author in several articles, but its suitability for signal spectroscopy remains to be seen in practice. For this, a better understanding of the gas dynamics (gas-flow properties) is required in the meantime (a paper in this area is in preparation). The other space available for detection is “below” the aperture, i.e., upstream in the area between the specimen, the aperture

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91

grid, and the specimen chamber walls. The use of this space to fit our detector electrodes can be optimized by proper design of the final lens system of the microscope. The wall effect is minimal for the low-angle FEO especially at high pressure. The wall effect is also less for the low-energy electrons (and x-rays), but for these, the energy resolution is also low. An improvement could be achieved by introducing a magnetic field parallel to the wire anode; this will tend to bend the trajectories of the electrons away from the walls around the wire, increasing at the same time the number of ionizing collisions (multiplication) (Franzen and Cochran, 1962).However, the feasibility of such schemes in the ESEM remains to be seen. A n additional difficulty for practicing spectroscopy i n the ESEM (and ESTEM) is the possible variability of the gas composition both between different specimens and during the observation of a single specimen. The calibration of such systems may be difficult. An immediate application of signal spectroscopy, other than material characterization, can be the improvement of image resolution. The FEO have a characteristic peak at energies close to and equal to the primary beam energy (low-loss signal; Wells, 1974).The energy resolution of the dispersive mode of G D D is, for example, as follows: If we use argon, the Fano factor F = 0.17, and the Polya factor h = 0.5 (Knoll, 1979); for a 10 keV beam, the low-loss electrons have the same energy, and because W = 26.2 eV/ion pair, we would have No = 10000/26.2 = 382 ion pairs. Using these values in Eq. (94), we find an energy resolution R = 9.8%; this is good but not adequate for a low-loss image. However, there can be a lot of image resolution improvement if electrons in this top 10% range of energy are used. The resolution could even be better if we used Eq. (87) with f = 0, meaning that we have no gaseous gain; then the resolution would be R = 5 % , but this would also require that the electronics will not introduce additional noise. At this stage, experimental data are scarce in this area. On the evidence given in this section, we can conclude that a new frontier is clearly open and a whole new task for future work is defined.

TX. REMARKS A . On the Geiger-Muller Counters

Although the Geiger-Muller (GM) counters may not be directly applicable to the GDD, study of their literature may at least yield valuable data for our purposes. The G M device operates in the region of electrode bias in which all the curves I-V in Fig. 12 merge in a single line. The output pulse in

92

G. D. DANILATOS

this region is independent of the input signal; it carries no specific information about the nature of incident signal, and hence it is used only as a counter. The development of the discharge is characterized by a phase of electron multiplication and transport, lasting about 1 ps, followed by a phase of sweeping away the ions, lasting between 100 and 1000 p s . The main advantage of this device is its very high amplification, the output of which is usually in the range of volts and can be directly recorded. The main disadvantage of existing devices is their slow counting rate that makes them impractical for imaging purposes. A hypothetical G M could be used for imaging, if it could count the particles of a particular kind coming out of each pixel and if it could produce an output voltage proportional to the counting rate for each pixel. Obviously, known G M cannot be used under the fast counting and scanning rates encountered during the formation of a FEO or a SEO image. A G M is made like a proportional counter except that it is biased at a potential above a “starting voltage” where all pulses are of the same height regardless of the type of particle. The G M regime is reached gradually as we increase the electrode bias in the region of limited proportionality, where the individual avalanches interact with each other appreciably. In that region, space-charge effects start becoming increasingly important. This happens when the space-charge density is comparable with that required to charge the device to the working potential. For parallel plates, these effects are unimportant, if the current 1is much smaller than a limiting value given by (Wilkinson, 1950): 2.65 x 10~’’,4KV2 1 << D3 (95) 9

where A is the plate area, K the ion mobility, I/ the applied voltage, and D the plate separation. In nitrogen, for A = mz, K = 0.01 mz/Vm, I/ = 400 V, and D = 0.001 m, the above equation yields I << 4.2 x lo-’ A. However, for D = 0.005 m, we get I << 3 x lo-’ A; if the amplification is 1000, then the primary beam current should be 1 << 3 x lo-’’ A, and therefore, it is possible to reach this limit with the beam currents used in the ESEM. As a rule of thumb, space-charge effects appear when the number of electrons in the avalanche exceeds about 3 x lo6 (Richer, 1959). Further studies on charge effects have been reported by Hendricks (1969) and Mathieson (1986). The G M are usually made with cylindrical geometry using a very fine wire. The discharge spreads along the wire length (it “burns” the wire) very fast (about l o 5 m/s). The space charged formed close to the wire alters the external electric field to an extent that the initiated breakdown is discontinued. The positive ions formed by the discharge are swept away by the field in a finite time interval, in which the G M is not operative. Dead time is defined as the time from the start of a pulse during which no particles can be counted. Recovery

THEORY OF THE GASEOUS DETECTOR DEVICE I N THE ESEM

93

time is the time from the start of a pulse where pulses are recorded, but with decreased height. The discharge breakdown can be quenched either externally by the electronic circuit or internally (self-quenching). There are three types of selfquenching(Korff, 1946). These are electrostatic quenching of the avalanche by the space charge of the positive ions, quenching of the photons in the initial avalanche, and quenching of secondary emission when the positive ions strike the cathode. Special gas mixtures achieve these properties. Most diatomic gases produce a breakdown after a short proportional amplification region of seldom more than a factor of 100 x . Such gases can be used with non-self-quenching (external circuits) for (relatively) fast counting. A special technique was developed by Simpson (1944); with a special electronic circuit, he reversed the electrode bias once the pulse started to rise, and collected the ions on the wire. He managed to reduce both the dead time and the recovery time down to 20 ps, and there was no indication that this was the limit. There are low-bias-operating G M using the Penning eflect (Fenyves and Haiman, 1969), which can be of interest in the ESEM. These are the halogen-quenched counters operating at 250- 500 volts with a plateau length of 100-300 volts and slope 3-10% per 100 volts. They can be used for the measurement of large radiation intensities by determining the electric current instead of reading the pulse rate. The maximum measurable radiation intensity can be significantly increased, both in the mode of pulse counting and in the mode of current measurement, by biasing the anode in a pulsed way (Fenyves and Haiman, 1969). This is done in either of two ways: (a) by applying a steady voltage lower than the threshold of the G M and adding a pulsed component to reach in the plateau, (b) by applying a steady voltage in the plateau and feeding negative pulses to it, thereby decreasing the effective voltage of the counter below the threshold. The dead time depends on the counting rate and decreases as the counting rate increases, but the pulse height also decreases. The pulse shape and the counting rate are constant in the plateau of the GM, but the pulse height and charge release increase with voltage (Fenyves and Haiman, 1969). The dead time can be further shortened by operating at higher overvoltage, low pressure, small geometry, operating several small diameter counters in parallel, or glass beads sealed on the wire (Wilkinson, 1950). Despite all these improvements, Wilkinson does not recommend the G M for fast counting, since the proportional counters produce superior results. Yet, this question is not closed for the new situation of the ESEM, where the irradiation source is finely controlled over a very small area, and minute “hybrid,” GM-like, detection configurations still remain to be tested.

94

G . D. DANILATOS

B. Materiuls und Construction Details The object of this work is to help in the design of meaningful experiments and ultimately to construct a multipurpose GDD for expanding the scope of the ESEM. To avoid misleading and valueless observations, the experiments must be systematic under controlled conditions. Now, the basic factors for this design can in the main be recognized. Some practical suggestions have been made in the preceding analysis; below, some further practical tips compiled from the literature can be helpful and time saving for future work. The detecting electrode can be either biased or at ground potential, and the results can be (not always) equivalent. When it is at ground potential, the cathode is biased at a negative or the anode at a positive voltage. This configuration has the great advantage that very simple electronic circuits can be used. Another advantage is the low input capacitance, since the anode can be only a short wire connected directly to the input of the preamplifier built inside the specimen stub. Also, some forms of breakdown are avoided. This author has used this method to bias the anode up to 400 volts quite successfully with the simple preamplifer-amplifier circuit used in previous work at low electrode bias (between - 10 and 10 volts). With certain geometries, the advantage of biasing the collector electrode is that all field lines terminate on this, and the loss of electrons is minimized. Thus, the latter mode may be more versatile in the ESEM applications, but it requires special electronic circuit design to float the electrode at high bias and also suffers by higher input capacitance; the latter can be minimized by placing a high resistance close to the collecting electrode. These are some basic considerations, but modern electronics can alleviate many of the difficulties. Breakdown can take place in many places other than in the detection and amplification electric field, such as in the condensers, the high-tension cable, and across insulators. To avoid coronas and breakdowns over the greater part of the system, opposite-sign bias can be used; by this, both cathode and anode are biased by opposite-sign potential relative to earth, so that a maximum field is established between the electrodes but not between each electrode (or wiring) and the walls of the system; thus, we will be limited only by the expected breakdown between the electrodes and not elsewhere. Of course, a satisfactory result can be achieved by a well-designed singly biased system using guards and screens. The detection and amplification volumes must be well defined or known. This can be achieved by a system of electrodes and grids. A guard “ring” should be used at the potential of the collector electrode in order to prevent effects due to leaks and fluctuations arising from the polarization of the dielectrics at high electric field. When the guard ring is not operated at ground potential, it should be connected to ground through a large capacitor, so that

THEORY OF T H E GASEOUS DETECTOR DEVICE IN T H E ESEM

95

its potential will stay as constant as possible. The guard also shields from room mains and other source interference. No part of the whole system that is connected to the head amplifier should be allowed to “see” the outer world, and complete screening is recommended. All screened parts should be connected to a single earth without loops, preferably at the head amplifier when this is possible. The insulators placed in high fields may produce spurious effects due to polarization, distortion of field, and local discharge across the surface. Equilibrium fields are established across insulators, and they stabilize after a few minutes or hours. The frequency spectrum of discharge and breakdown across insulators is generally low. Fused quartz is an excellent insulator, but it should make good contacts with metal parts. The properties of the various insulators intended for use should be studied in advance. When surface insulation is a problem, a layer of natural ceresin wax, or a silicon varnish is very effective (Wilkinson, 1950). Plastic insulated cables from the sensor electrode to the amplifier should stay undisturbed during operation to avoid a sort of piezoelectric effect. Plastics may also accumulate charge and thus become polarized and disturb the field; the magnitude of this effect depends on the pressure used and the state of ionization of the surrounding gas, but this is not generally a problem in the ESEM. Insulators inside the active volume allow the field through, but they create a dead space, thus reducing this volume. I t should also be remembered that the resistance of plastics is greatly affected by ionizing radiations. Cavities with low electric field in the neighbourhood of the active volumes may harbour ions for relatively long times. These ions may cause spurious background noise (avalanches) during imaging. Microphonic noise can be a serious problem, but it can be eliminated if identified. This can be reduced by increasing the capacity or decreasing the voltage, but these measures reduce the time response and the amplification. The best solution is to make a rigid construction to eliminate vibrations. The guard rings can also help in this by restricting the active field away from vibrating parts. After all the above precautions, we can concentrate on the active, detection, and amplification area. The shapes and materials of the electrodes must be thoughtfully chosen. Tungsten, stainless steel, platinum, or nichrome is recommended for the anode. The cathode should be made from high work function materials such as graphite and stainless steel. Corona discharge can be avoided by careful construction and polishing of the inner surfaces. Dust particles should be prevented from lodging on the electrodes. Censer and Walczak (1987) provide a latest account of the influence of gas composition, cathode material, and geometrical parameters on the operation of a thin multiwire chamber working at high amplification.

96

G . D. DANILATOS

According to Wilkinson (1950), all building materials may have some radioactivity, which should be avoided. Brass is bad, aluminum worse, and steel better. Data on water and water vapor, a key constituent in many ESEM applications, are scarce. A recent report by Aoyama et al. (1987) examines the influence of water and ethanol vapor on the operation of air-proportional counters. They found an anomalous current caused by positive ions of water and ethanol following their adsorption on the anode wire. The proportional counter region shifted to the higher voltage side by water vapor, and to the lower voltage side by the ethanol vapor. The detection efficiency was decreased due to electron attachment to the vapors. Spurious pulses were generated by leakage current through the surface of an insulator, facilitated by the deposition of water on the surface; water-repellent material (silicone resin) was effective to reduce this. Many of the above problems can be effectively eliminated by suitable electronic circuit design. The frequency response of the circuit can be properly adjusted both to eliminate, or reduce, these problems and to be compatible with the scanning and recording requirements of the microscope. These designs depend on the mode of operation of the GDD. For signal spectroscopy, the requirements are strictest.

C. Future Prospects

Throughout this work, frequent mention was made of the creation of new research topics. Some of these topics, or research tasks, will have already undergone testing by the time this work is sent to the press, and the results seem to confirm the theory presented here. Several more topics are also under way for experimental implementation. The theoretical together with the practical results obtained to date clearly show that the GDD will play a most fundamental role in electron microscopy. In particular, the GDD offers the possibility of directly detecting two very useful signals, namely, the routinely used “backscattered” and “secondary” electron signals. Both of these signals can be detected and amplified directly and immediately after their generation from the specimen. All stages, from the specimen to the display of an image, provide a monotonically ascending amplification sequence, and hence the noise bottleneck is at the specimen beam-to-signal conversion level. This is not generally the case with the conventional solid scintillating detectors, where the transmission of light quanta in the light pipes can show significant losses. Such losses can be severe particularly in the ESEM where the shapes of the detectors have special constraints. It has been shown that particular shapes of light pipes result in the

THEORY OF THE GASEOUS DETECTOR DEVICE I N T H E FSEM

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noise bottleneck being in the light pipe. In addition, significant losses can result from the fact that in many solid scintillating detectors, a significant portion of the signal electrons never reach the detector. All these difficulties are alleviated in the case of the GDD, since all electrons immediately and directly enter the detection medium, i t., the gas. By obtaining the maximum possible gain at the very first amplification stage inside the gas, followed by ascending electronic gain afterwards, an optimum detector results. The possibility of detecting the secondary electrons from a true specimen surface, in particular from surfaces of insulators and wet or moist specimens, has added a new dimension in materials and surface physics research. These electrons originate from the top molecular layers of a specimen and can reflect the variation of the surface properties. Clearly, the vacuum SEM is limited only to dry specimens and to some low accelerating voltages with resulting contrast mechanisms not yet fully evaluated. The ESEM is free from these limitations, and the use of secondary electrons with plastics has revealed completely new contrast mechanisms. The gas at various pressures results in various contrasts not previously seen with conventional detection methods. Therefore, the G D D has ushered in a new era in microscopy with very promising future prospects.

X. CONCLUSION The preceding analysis has shown that the G D D is not just another detector for a particular signal detection. It has evolved to be rather a whole new method of detection unique to the conditions of ESEM. It is a method in its own right for the detection of different signals created by the uncoated natural surfaces of specimens. It is a multipurpose detection system both for imaging and signal spectroscopy. The complexity of the system is both a deterrent and a blessing. I t has deterred us from developing the system speedily and from proving its full potential convincingly in the short term. However, its complexity allows a lot of flexibility in the way the system can operate and the things it can achieve. Experimental work to date gives strong support to some daring speculations about its future potential. It is the large number of parameters characterizing the G D D that, through their interplay, make it both complex and versatile. The type and pressure of gas are two broad and general variables that determine a host of subvariables such as the type of charge carrier, mobility, and time response. The geometry or electrode configuration is another general variable that is interwoven with the gas properties through the electric field potential that they generate in the

98

G . D. DANILATOS

gas. The interaction of the external field and gas is expressed in terms of electron and ion temperature, and through the resulting gas ionization, excitation, dissociation, and other reactions. These reactions serve in the detection and amplification of various signals emerging from the beamspecimen interaction. The main limitation of the system arises from the inevitable breakdown described by the Paschen law, This breakdown takes place because of the ?-processes. These are the mechanisms by which new electrons are liberated from the cathode, followed by instability and breakdown. Therefore, the key to improving the gain of the G D D and the stability of its operation is to learn more about these mechanisms and how to control them. This can be done both by proper choice of gas composition and by electrode materials and configuration. The photoemission of the gas excited by the electrons drifting and multiplying in the electric field can be used for the detection of signals. Photons produced in the gas, if they reach the cathode, can produce photoelectrons, and this, being a 7-process, will contribute effectively to breakdown. However, in the region prior to the breakdown point and even prior to avalanche formation, nuclear devices successfully operate in the gasscintillation mode with high gains. This is a clear precedent for the G D D to operate on the same basis; the advantages of this mode are simplicity of electronics, much reduced noise,and relatively high gain, and it can be used with a different menu of gases for different applications. It appears that the ionization and scintillation modes of GDD have a complementary role to play. The limits of amplification may vary widely depending on what compromises can be made in each application of ESEM. If the gas composition can be selected freely, then there are gas mixtures with quenching properties that would allow high gains, namely above 1OOOx. If a particular application restricts the gas composition, then we can expect a lower gain. The gain will depend also on the electrode configuration and materials, and the physical limits of these parameters are yet to be found. The frequency response of the G D D also varies widely. The physics of the gaseous discharge does not pose an obvious practical upper limit to it. This limit is rather determined by our technological limitation for making and fitting into the microscope electrodes of appropriate size and shape, in conjunction with the appropriate gas mixture for a particular application. Present-day electronics technology is a vast improvement over that used with early proportional and G M counters, and this advantage allows us more choices of experimentation. The preceding analysis has shown that a time response around 1 ps is usual, but this time can be easily varied by one or two orders of magnitude with a little extra care. Therefore, the G D D can operate in the full frequency range normally encountered in the SEM.

THEORY OF THE GASEOUS DETECTOR DEVICE l h T H E ESEM

99

The suggestion has been made to investigate the possibility and the limits of using the GDD as analyzer. Based on experience in other fields, it is expected to be able to perform at least some basic signal spectroscopy. Both wavelength (from the gas) and energy dispersive spectroscopy seem good possibilities for future developments. However, the practical usefulness of these remains to be seen. It would be extremely useful if the low-loss peak could be isolated, because this would produce excellent high-resolution imaging. Because of lack of sufficient experimental work at the present, further speculation on this question should be made with caution. However, this caution should not deter us from exploring what appears to be an extremely promising field.

ACKNOWLEDGMENTS ElectroScan Corporation is gratefully acknowledged for supporting this work. I thank

Mr. N. Baumgarten for reading and checking the manuscript.

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.

AIlVAhCLS I > ELtCTKONICS A N D ELECrROh PHYSICS \ O L 7'8

Carrier Transport in Bulk Silicon and in Weak Silicon Inversion Layers S. C . JAIN and K . H . WINTERS Theoretical Physics Diuision Harwell Lahoratory

Didcor. United Kingdom

R . VAN OVERSTRAETEN IMEC Leucen . Belyruni

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11. Drift Velocity in Bulk Silicon . . . . . . . . . . . . . . . . . . . 111. Effect of Tangential Field on Mobility and Saturation Velocity of Electrons in Inversion Layers . . . . . . . . . . . . . . . . . . . . . . . . A . Effect of Field Inhomogeneity . . . . . . . . . . . . . . . . . . B. Work of Cooper and Nelson . . . . . . . . . . . . . . . . I\'. Carrier Transport and Mobility in the Weak-Inversion Region of a MOSFET . Theories Based on Macroscopic Inhomogeneity . . . . . . . . . . . . . A . General Features of the Mobility and the Inhomogeneity . . . . . . . . B. Density of States and Percolation in Inversion Layers at Low Temperatures . . C . Effective-Medium Theory . . . . . . . . . . . . . . . . . . . D . Resistor-Network Model . . . . . . . . . . . . . . . . E . One-Dimensional Model . . . . . . . . . . . . . . . . . . V . Theories Based on Short-Range or Microscopic Inhomogeneities . . . . . . . A . Brews's Small-Fluctuation Theory . . . . . . . . . . . . . . . . B . One Particle Mobility-Edge Model Applicable to Low-Temperature Work . . C Electron-Liquid Model . . . . . . . . . . . . . . . . . . . Vl . Comparison of Low-Temperature Experimental-Edge Model with the MobllitFEdge Model . . . . . . . . . . . . . . . . . . . . . . . . . V I I . Arnold's Experiments and Macroscopic Inhomogeneity Model . . . . . . V I I I . Hall Etfecl and Electron-Liquid Model . . . . . . . . . . . . . . . 1X . Evidence of Deviation from Random Distribution . . . . . . . . . . . X . Peaks in the Variation of pWlcwith n,,, . . . . . . . . . . . . . . XI Room- and High-Temperature Measurements . . . . . . . . . . . . . XI1. Limitations of Theorles . . . . . . . . . . . . . . . . . . . . . XI11 . Summary of Work on Transport in Inversion Layers in the Weak Inversion Region . m d Conclusions . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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S. C. JAIN, K. H. W I N T E R S A N D R. VAN O V E R S T R A E T E N

List of’ Symbols

velocity of sound in silicon oxide capacitance per unit area semiconductor capacitance per unit area inversion layer capacitance per unit area depletion layer capacitance per unit area density of states in the presence of inhomogeneity density of states in a uniform channel. In a two-dimensional system, it is constant independent of E electric field condensation energy kinetic energy electric field in the direction of current flow electric field perpendicular to the interface Fermi energy local band edge or inhomogeneity parameter mobility edge percolation edge optical phonon energy source drain field fraction of area occupied by insulating regions Fermi distribution function conductivity of pseudometallic region conductivity of pseudoinsulating region conductivity conductivity in electron liquid model value of 9 at 1/T = 0 Fig. 18 and 19 and text referring to figures minimum metallic conductivity tunnelling conductivity magnetic field drain current with inhomogeneity in the channel drain current in homogeneous channel current density per unit width Boltzmann’s constant Kinetic energy in the extended and the localized states respectively Fermi wave vector channel length mean effective distance between collisions

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

105

mean free path between collisions with acoustic phonons mean free path between collisions with optical phonons electron effective mass free electron mass density of acceptors average density of carriers in the channel carrier density on top of the barrier carrier density in the well carrier density in the inversion layer at a distance y from the source value of ninvat E , = E,, Hall carrier concentration concentration of carriers localized in the band tail probability of local potential energy at V within dV fraction of volume allowed magnitude of electronic charge same as ninv density of fixed oxide charge at the interface per unit area average values of Q,, f Q,, density of fast surface states per unit area Hall coefficient value of ith resistor network oxide thickness absolute temperature effective thermal velocity saturation velocity drift velocity local potential energy in the channcl average randomly directed electron velocity in thermal equilibrium gate voltage effective gate voltage drain voltage first band voltage average band edge substrate bias voltage exchange energy threshold voltage distance from the interface activation energy distance from the source along the channel

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

number of electrons in a group that moves as a whole for Eq. (61) and number of charges q at each site in Eq. (18) permittivity static dielectric constant of silicon static dielectric constant of oxide mobility at tangential field E , mobility at E , -+ 0 and without inhomogeneities mobility in weak inversion obtained by dividing the observed channel conductivity (at small values of V,) by the average carrier density in the channel Hall mobility true microscopic mobility variance surface potential mean effective time between collisions standard deviation of surface potential average distance of mobile carriers from interface screening parameter used by Arnold standard deviation

I. INTRODUCTION

At high-field strengths, the carriers in a crystal gain more energy from an applied electric field than the energy they can lose to the lattice, and they become hot. Since the time between collisions is a function of energy, the velocity of hot carriers does not increase in proportion with the field. The drift velocity saturates, the mobility decreases, and the electrical behavior becomes nonohmic. The early work on drift velocity in bulk silicon was done by Ryder (1953), Duh and Moll (1967) and Norris and Gibbons (1967). The energy that the carriers gain from the applied field is lost to the lattice by collisions with acoustic phonons at low fields and with optical phonons at high fields. The intervalley phonons also play a significant role in the scattering process. As a result, the drift velocity saturates at around 1 x lo7cm/s at room temperature when the applied field exceeds 2 x lo4 V/cm (Canali et al., 1975; Schwarz and Russek, 1983a). The carriers in the inversion layer of a metal-oxide-semiconductor (MOS) transistor are confined to a thin two-dimensional layer. They suffer additional scattering by the roughness of the interface, Coulomb scattering by fixedoxide charges and charges in the interface states, by surface phonons and

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

107

other defects at the interface. Since the vertical field due to the gate voltage determines the number of carriers and their average distance from the interface, the mobility is a very sensitive function of this vertical field. In weak inversion regions, overall mobility pWIcdefined as the ratio of observed conductivity to average carrier density in the channel is also degraded seriously by interfacial potential fluctuations. The fluctuations occur due to a nonuniform or inhomogeneous distribution of fixed-oxide and interface-state charges. Thus the mobility of carriers is determined by the quality of the interface and by the vertical field, and also by the tangential field if the drain voltage V, is not sufficiently small. The early theoretical and experimental studies of carrier scattering in the inversion layer were made by Schrieffer (1955), Green et al. (1960), Stern and Howard (1967), and others. These studies formed the basis of much of the work that followed. In particular, the papers of Fang and Fowler (1968), Chen and Muller (1974), and Guzev et al. (1972) are important. These papers established that observed macroscopic mobility in the weak-inversion region is strongly degraded due to potential fluctuations caused by inhomogeneities in the interface charges. The mobility increases with temperature as exp( - q A / k T ) (where A is a constant) in this region. As the gate voltage is increased, the carrier density in the inversion layer also increases. The effect of inhomogeneity becomes small as the strong-inversion region is approached. The mobility attains a peak and then starts decreasing again as the gate voltage is further increased (see Fig. 1). At low temperatures, more than one peak has been observed (Fang and Fowler, 1968).

ImV)

Fic,. 1. Experimental mobility behaviour versus surface potential for p-channel ( 1 1 I ) MOSFET No. 4, Q,. = 5.0 x 10" cm-'. (After P. A. Muls et al., 1978.)

108

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

Considerable further work was stimulated immediately afterwards. Brews (1975) gave a three-dimensional perturbation theory of conduction in the channel when there are small potential fluctuations at the interface. Muls, De Clerck, and Van Overstraeten (1978) have written a review on the subject covering the transport and mobility in weak-inversion regions at room temperature. In the strong-inversion region, the results reported by Sabnis and Clemens (1979) are significant. They showed that the plot of mobility as a function of effective vertical field is a universal curve at room temperature. Plotted in this manner, it does not depend on oxide thickness or substrate doping. Extensive theoretical and experimental work was done in the years that followed. Ferry (1978) and Ferry et al. (1981) gave a comprehensive theoretical model of the transport in inversion layers. Sun and Plummer (1 980) extended and confirmed the main results of Sabnis and Clemens. These authors and Schwarz and Russek (198315) gave empirical expressions for the inversion carrier mobilities that are useful for circuit design and simulation. Arora and Gildenblat (1 987) extended these expressions to include mobilities at low temperatures. In veiw of the considerable interest in low-temperature operation of complementary MOS (CMOS) circuits (Sun et al., 1987), the behavior of the mobility in this regime has been studied extensively; several papers have also appeared on the effect of very thin oxides on the mobility. Until recently, the saturation velocity of electrons in inversion layers was thought to be about 6 x lo6cmjs as compared to 1 x lo7cmjs in bulk silicon. Muller and Eisele’s work (1980) showed that this is not the case. At large drain voltages, necessary to observe the saturation in velocity, the tangential electric field in the channel is not uniform. If a correction is made to account for this fact, saturation velocities close to the bulk value are observed. Cooper and Nelson (1983) have measured the high-field drift velocity by using the improved time-of-flight technique (Cooper and Nelson, 1981; Nelson and Cooper, 1982)and have confirmed that the saturation velocity in the inversion layer is indeed close to 1 x l o 7 cmjs. Referring to Fig. 1, we can divide the mobility curve into the following two regions: 1. The weak-inversion region, where the carrier density in the inversion layer is less than loL2cm-2; 2. The region of strong inversion, at carrier densities greater than 10l2 cm-2, where the mobility decreases monotonically with carrier density (or with increasing gate voltage).

We first review briefly the work on drift velocity in bulk silicon in the next section (Section 11). The change of drift velocity with tangential field in inversion layers is discussed in Section 111. Theoretical models of the effect of inhomogeneity on the conductivity g and mobility pwlc are discussed in

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

109

Sections IV and V; Section VI discusses the interpretation of experiments. Region 2 (strong-inversion region) is not discussed in this chapter. The effect of irradiation on the behaviour of a MOS field-effect transistor (MOSFET) operating in weak inversion is of great practical interest in developing both sensitive stable dosimeters and radiation-resistant transistors. Radiation damage produces a large density of oxide-fixed charges and interface states. The inhomogeneity and potential fluctuations increase, and both the conductivity g and mobility pWlCare considerably modified. It is hoped (region 1) given in this chapter will that the review of the behavior of pWlC help in interpreting experiments on irradiated transistors. 11. DRIFT VELOCITY IN BULK SILICOK High-field transport in bulk silicon is of great practical and academic interest. It has been extensively studied and debated since the pioneering experimental and theoretical work of Shockley (1951) and Ryder (1953). Early theoretical treatment of the subject is given in the book by Conwell (1967). The dependence of drift velocity on electric field can be divided into four regions: (i) In the low-field region, E < lo3 V/cm, the velocity is determined by acoustic and intervalley phonon scattering; (ii) in the intermediate-field region, lo4 V/cm > E > lo3 V/cm, scattering by optical phonons becomes increasingly important as the field strength increases; (iii) when E > lo4 V/cm, optical phonon scattering becomes the dominant process, and the drift velocity saturates; (iv) at very high field strengths, E > lo5 V/cm, carriers become very hot, and breakdown phenomena occur.

Simple expressions for ud and p ( E ) can be obtained for regions (i) and (ii). Region (i) The conservation of momentum and energy is expressed by the following two equations (Schwarz and Russek, 1983a) in region (i):

Equation (2) ignores scattering by optical phonons and is not valid at high is the mean length between collisions with acoustic phonons field strengths. lac and is assumed to be independent of the field E . z and 1 are the mean effective

110

S. C. JAIN, K. H. WINTERS A N D R. VAN OVERSTRAETEN

time and distance between collisions, u is the effective thermal velocity and c is the velocity of sound in silicon and has a value of 9.1 x lo5cm/s (Schwarz and Russek, 1983a). The effect of electron heating, if any, is included in these quantities. Equations (1) and (2) can be solved simultaneously for ud and yield an expression similar to that derived by Shockley’s (1951) quantum-mechanical analysis. Taylor expansion of this expression yields (Schwarz and Russek, 1983a)

The low-field region is well described by Eq. (4). However, the value of obtained from Eq. (4) exceeds the measured value by more than one order of magnitude. The discrepancy arises because, as the subsequent work showed (Jacoboni et al., 1977), intervalley phonon scattering is a major source of energy loss, and this has not been taken into account in the above treatment. Region ( i i i ) At high fields, when optical phonon scattering is the dominant process for energy loss, Eq. ( 2 ) is replaced by

where E,, is the optical phonon energy and has a value of 0.063 eV for silicon, and I,, is the mean length between collisions with optical phonons. Solving ( I ) and ( 5 ) for u d , we now obtain

Experimental data for electron diffusion in high fields suggests that the electron distribution with energy is much cooler than predicted by Eq. (5). Again, intervalley scattering and nonparabolicity of the bands may be responsible for this discrepancy. The hot-electron phenomenon is still not well understood. To get over these difficulties, Schwarz and Russek (t983a) assumed that the average hot-electron energy is the sum of the zero-field thermal energy and the drift energy i m * v j , which yields

C A R R I E R T R A N S P O R T IN W E A K SILICON INVERSION LAYERS

1 11

where z' is the effective mean velocity of the electron for high fields as defined earlier. Schwarz and Russek obtained the following expression for ud :

The plot of Eq. (8) is compared with experimental data (Jacoboni et al., 1977) in Fig. 2. The straight lines (a) are low-field ( I ' = Idc) and high-field ( I ' = lop) approximations. Curve (b) is the plot of Eq. (8) and agrees well with the experimental data of Jacoboni et al. (1977) shown by open circles. The dashed curve is the plot of an empirical equation given earlier by Scharfetter and Gummel ( 1 969). The value of c, calculated in the high-field approximation agrees well with the experimental data of Jacoboni et al. (1977) at different temperatures. It has a value of about 1.3 x l o 7 cm/s at 4K, 1 x l o 7 cm/s at room temperature and about 0.9 x lo7 cm/s between 500 and 600K. Schwarz and Russek have pointed out that although Eq. (7) underestimates the electron temperature for high fields, it takes into account correctly the phonon scattering in this limit. Since Eq. (8) is based on crude physical assumptions, it should be regarded as semi-empirical. Also, it should

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lo5

1 V Icm) Fib. 2 , Velocity-field profiles for Si (Ill) at 3 0 K . (a) low- and high-field approximations of the model; (b) Eq. (8) of the present model; (c) empirical Scharfetter-Gummel equation. Data points a r e taken from Fig. 17 of Jacoboni et al. (1977). (After S. A. Schwarz and S. E. Russek. Electric Field

1983a.l i 1983 IEEE.

I12

S. C. JAIN. K. H. WINTERS AND R. VAN OVERSTRAETEN

be noted that a good fit of the data is only obtained by using m* = 0.19m0, and comparison with experiment is confined to data in Si in the (1 11) direction where anisotropic effects are minimal. Although such detailed studies for hole transport have not been made, Schwarz and Russek (1983a) have shown that their model is in rough agreement with the experimental data of Jacoboni et al. (1977) for hole transport also.

FIELDON MOBILITY AND SATURATION 111. EFFECTOF TANGENTIAL OF ELECTRONS IN INVERSION LAYERS VELOCITY A . Effect of' Field lnhomoyeneity

When a lpm channel transistor is operated at V, = 2.5 V, the tangential electric field exceeds lo4 V cm-', mobility does not remain constant, and hotelectron effects become important. For longer transistors, the effect will occur at higher values of V,. The early measurements of Fang and Fowler (1970) yield a value of 6 x lo6 cm/s for the saturation velocity of electrons in high fields in the inversion layer. The saturation velocity of electrons in the inversion layer has been investigated theoretically by Basu (1978) and by Ferry (1978) and Ferry et al. (1981). Since for strong fields of lo4 V/cm many subbands are populated, the electron transport becomes three-dimensional, and theoretically the saturation velocity should be close to its bulk value. Indeed, theoretical values of L', are close to the bulk value of 1 x l o 7 cm/s (Basu, 1978). Recent experimental work (Muller and Eisele, 1980; Cooper and Nelson, 1983) discussed below revealed that the earlier experimental values are in error. These low values were obtained because the effect of inhomogeneity in the channel field was not taken into account while interpreting the experiments. The current density j in the channel is given by (see for example Muller and Eisele, 1980), j

= ~~~,nv(Y)U(y),

( 9)

where y is the distance from the source in the direction parallel to the interface. Since the total voltage drop in the channel is V,, the average carrier density Einv is obtained by assuming

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

113

and the average drift velocity is given by

at an average electric field

E = v,,L .

(13)

The average quantities given by Eqs. ( 1 1)-(13) assume that V ( y )and E(J) decrease linearly with y. The validity of this assumption is not obvious. The above treatment cannot predict the actual variation of velocity with field. It is possible to calculate V ( y ) and E ( y ) along the channel approximately (Muller and Eisele, 1980). For this purpose, we use the following semianalytical equation (Muller and Eisele, 1980; Cooper and Nelson, 1983),

where (x varies between 1 and 2. Note that since E, is a function of y, then so is 1. Equation (14) with a = 2 fits the experimental data of Muller and Eisele (1 980) and the theoretical results of Ferry (1978) except at very high fields where smaller values of x are required. The experimental data of Cooper and Nelson (2983) are fitted by Eq. (14) with IY = 1.92. between Eqs. (9), (lo),and (14) and solve We can eliminate t;(y)and fiinV(y) the resultant equation for E ( y ) . The results obtained by Muller and Eisele (using c, = lo7 cm/s and M = 2) in this manner are shown in Fig. 3. The channel length is 0.5 pm, the oxide thickness to, is 50 nm and V, = 20 V. The results of Fig. 3 show that as the drain voltage increases to larger values, the electric field near the drain end increases sharply, whereas it is pinned at a constant value - 5 x lo4 V/cm in the neighbourhood of the source. If similar calculations are made for longer transistors, the value at which the field near the source saturates is smaller. This result can be understood from general physical considerations. Since carrier density given by Eq. (10) is smaller at the drain, a larger field is always required near the drain to keep the total current constant. As the velocity near the drain begins to saturate and mobility starts to decrease, in order to keep the current constant, a much larger drop of voltage near the drain is required to produce a very high electric field. Very soon, the effect of any further increase AV, of VD occurs totally near the drain end, and the electric field near the source remains practically constant. Since the electric field near the source saturates earlier, the electron velocity at the source should also “saturate” at lower values, whereas the velocity at the drain end should keep increasing and saturate at larger values. The velocities at the two ends calculated from the

114

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN 7 . -

,-O5r---Vicrr

O

/

0

-

01

02

O L prr 0 5

03

Distance From Source

y

FIG 3 Calculated electric-field profiles for different drain voltages near current saturation The transistor is the same as in Fig 4 with V , = 20 V (After W Muller and I Eisele, 1980) ( 1980 Pergamon Press PLC

measured I, V, characteristics and from Eqs. (9) and (10) are shown in Fig. 4. The velocity at the source “saturates” at about 6.7 x lo6 cm/s, whereas at the drain it increases up to 8.5 x lo6 cm/sec. If a longer transistor is used, the final “saturated” values of velocity at the source end are much smaller. It should be emphasized that this velocity saturation at the source is due to the fact that the field there has stopped increasing with increasing drain voltage V, and is not related to the saturated velocity in the usual sense. The value of v, obtained by using Eq. (12) will therefore be smaller than the true saturation velocity. This explains the lower values obtained by Fang and Fowler (1970) and others. The velocity-field relation near the drain should be used to obtain the true saturation velocity. Muller and Eisele fabricated eight identical transistors with channel lengths ranging from 0.5 to 2 pm and measured their I,, V, characteristics. The “saturated” velocities at the source calculated from the observed I , , VD characteristics, as described above, monotonically decreased with channel length from a value of about 6.8 x lo6 cm/s at L = 0.5 pm to about 5 x lo6 cm/s at L = 2 pm. The highest velocity at the drain was constant (independent of channel length L ) at a value of 8.5 x lo6cmjs. The maximum drain voltage that could be used was limited by the onset of avalanche multiplication.

C A R R I E R TRANSPORT I N WEAK SILICON INVERSION LAYERS

crnlsec

f 15

t

q

c c

G

1

0

lo3

5

L

.

I

, l , . . l

VD

-

Vlcm

10 IL

10

Frc;. 4. Drift velocity at source (+,-) and drain (0,---) versus V,/L for two different gate voltages. Upper curve, V, = 20 V; lower curve, V, = 30 V. Transistor parameters: L = 0.5 pm, I,>,= 50 nm, V, = 0 V. (After W. Muller and I. Eisele, 1980.) 1980 Pergamon Press PLC.

B. Work of Cooper and Nelson Reliable measurements of the velocity-field relation have been made recently by Cooper and Nelson (1983). To avoid the complications created by inhomogeneous field distribution in the channel, they used a different technique to study high-field transport of inverted carriers in the channel. In their method, a packet of discrete charge introduced by a pulsed laser drifts in a uniform applied tangential field. The time of flight of the charge packet is observed, and electron velocity is thus determined. The measured drift velocity of electrons in the (01 1) direction for a (100) silicon surface at different tangential and normal field strengths is shown in Fig. 5. The maximum tangential field is limited by the breakdown voltage, which was 550 V for the devices used by Cooper and Nelson. The observed drift velocity is 8.9 x lo6 cm/s for a tangential field of 4 x lo4 V/cm and a normal field of 9 x lo4 V/cm. The fit of the data by Eq. (14) (with a = 1.92) and assuming

-

116

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN 10

v-7-w L 16 3 12 2 3L 156

101

0 728

0 520 0 36L 0 260 Tangenh 0

I

Field (Viprn 1 I

i

10 15 Effective Normal Field (Vlprn)

5

0

t

b

0

FIG. 5. Measured velocity of electrons on (100) silicon at room temperature. Charge transport is along the (011) direction. The solid curves are the best fit to the data using Eq. (16) as described in the text. (After J. A. Cooper, Jr. and D. F. Nelson, 1983.)

( E , and e are the adjustable parameters, see Table I) gives an extrapolated value of 9.2 x lo6 cm/s for the saturation velocity u s , independent of the normal field En.The values of other parameters needed for fitting the data are given in Table I along with the value of us. It can be seen from Fig. 5 that Eqs. (14) and (15) fit the data extremely well. Cooper and Nelson have discussed extensively the factors that determine the accuracy of their results and conclude that the velocity-field data obtained by them is accurate to within +5%.

TABLE 1

PARAMETERS OF THE MATHEMATICAL MODELUSEDI N FITTING THE DATABY EQS. (14) AND (15) (AFTERJ. A. COOPER,Jr. A N D D. F. NELSON,1983) ~~~

Parameter

Value

PO

1 105 cm*/Vs 30.5 V/pm 0 657 9 23 x 10'cm s 1.92

Ec

e 1.

1

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

1 17

Before concluding this section, we would like to discuss briefly the fieldvelocity relations measured by Coen and Muller (1980). These authors used specially designed and constructed MOS transistors having resistive gates. With these devices, they were able to obtain uniform tangential and normal fields in the channel by appropriately biasing the gate. Unfortunately, experiments could not be made for tangential fields higher than 1.3 x lo4 V,icm, and they obtained the saturation velocity only by extrapolation, finding a value of (5.5 to 4.0) x lo6 cm/s. Since a large extrapolation from low field values was involved, this saturation velocity could be in error.

AND MOBILITY IN THE WEAK-INVERSION REGION IV. CARRIER TRANSPORT OF A MOSFET. THEORIES BASEDON MACROSCOPIC INHOMOGENEITY

A . General Features of the Mobility and the Inhomogeneity

The very low carrier mobility in the weak-inversion region and the maximum in the plot of mobility versus 4s(Fig. 1)were observed by Fang and Fowler (1968)and confirmed by Murphy et al. (1969),Guzev et al. (1972),Chen and Muller (1974), Muls et al. (1978), and more recently by Wikstrom and Viswanathan (1986). These investigators found a large decrease in the observed conductance mobility in the weak-inversion region as shown in Fig. 1. The mobility increases sharply as the strong inversion region is approached, reaches a maximum, and decreases again as the gate voltage further increases. Several early attempts were made to explain theoretically the large suppression of mobility in the weak-inversion region, based on the assumption that Coulomb scattering by interface charges is mainly responsible for the reduction of the observed mobility. These attempts were not successful in interpreting the experiments satisfactorily (see Brews, 1975,for discussion of this point). Guzev et al. (1972), Chen and Muller (1974), Mott et al. (1975), Brews (1975) and Stern (1974) invoked the channel inhomogeneities to interpret the experimental mobility data. As pointed out by Brews (1975), these nonuniformities reduce the channel conductance without altering the microscopic mobility. Since these nonuniformities or inhomogeneities introduce regions of lower surface potentials (regions of potential barriers) where carrier density is small, the carriers have either to surmount the barriers or to take longer paths around them while going from source to drain. This process results in a drastic reduction of channel conductivity, and the mobility, determined by using the relation

118

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

becomes very small. To distinguish the mobility determined in this manner from the microscopic mobility, we designate it as MOSFET weak inversion conductance mobility pWlc. It is necessary to model the potential and carrier distribution in the inhomogeneous channel before we can discuss the transport of carriers in the channel. Two approaches have been used to describe the potential distribution. In one model, the inhomogeneity consists of a Poisson or some other distribution of Q (including surface-state charges) at the silicon-silicon dioxide interface and is essentially microscopic. A microscopic model of potential fluctuation has been used by Mott and coworkers (1975), by Brews (1975), and more recently by Adkins (1978a). Mott treats the conduction in the channel according to the mobility edge and variable-range hopping theory, whereas Brews has given a perturbation theory in which fluctuations are considered to be small. Adkins (1978a, 1978b) (see also Tkach, 1986) has given a theory, the electron-liquid theory, based on strong correlation effects. In the other model, known as the quasi-uniform model, the inversion layer is divided into small areas or cells, and each area is assumed to have a uniform local potential corresponding to the value of Q for that cell. The potentials of these cells are then assumed to vary in a random manner. The quasi-uniform model permits the modelling of macroscopic long-range inhomogeneities (Arnold, 1982). The interaction between the neighbouring cells is ignored in this model, however. In the presence of macroscopic inhomogeneities, the conduction model must be at least two-dimensional, since a one-dimensional model does not permit the carriers to go around the barriers. In the presence of macroscopic inhomogeneities, the inversion layer consists of local regions of high and low conductivities. The two-dimensional theories of conduction in binary mixtures can be used in such cases, These theories are discussed in the remainder of this section. The theories of conduction in a channel with microscopic inhomogeneities are discussed in Section V.

B. Density of States and Percolation in Inversion Layers at Low Temperatures Arnold (1976) treated the inversion layer at low temperatures as a twodimensional medium with macroscopic inhomogeneities. According to the model developed by Arnold, the probability P ( V ) d V of finding the local electronic potential energy within dV around V is assumed to be Gaussian, P( V ) = (2n0;)-"~ exp( - V2/2a3,

(17)

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

with standard deviation a , given by Arnold (1974, 1976), CTA 2 = 27CQ(Z~2j-~)2/A2(Es

+

119

(18)

where Q is the total density of charged centres at the surface, each carrying a is the average distance of the minority charge zq,,?. is the screening constant, i. carriers from the interface, and E, and E,, are dielectric constants of the semiconductor and the silicon dioxide, respectively. The screening constant i., is given by ,? = ~ ~ / 2 n q ~ ~ ( E , ) . (19) We now calculate the effective density of states in the inhomogeneous inversion layer. The density of state Do in a homogeneous two-dimensional system in the lowest subband is constant, Do

m* 7rh

= 2.

The effective density of state is given by (Arnold, 1976)

where m* is the appropriate effective mass of the carrier. With the random distribution of cells, D(0) = $Do = 1 x 1014cm-2 e V ' . D ( E ) increases linearly with E near E = 0 and becomes Do for E >> 0. According to Eq. (17), the number of cells that have a local band edge r/; above V = 0 is the same as of those below V = 0, such that the average band edge V,, = 0 (see Fig. 6). Note that the rectangular shape of the band edge implies that the interaction between the different cells is ignored. a,b,c Fermi level positions at 4'K

k-

Intrinsic level

4 Y-

FIG.6. Conduction band edge in inversion layer with macroscopic inhomogeneity. The lines a. b. and c denote three positions of Fermi level corresponding to three values of gate voltage.

120

S. C. JAIN, K. H. WINTERS A N D R. VAN OVERSTRAETEN

The number of carriers in any given cell depends on the position of the Fermi level determined by the effective gate voltage V G .Consider the Nchannel MOSFET with a very small positive effective gate voltage such that the Fermi level is at position a (Fig. 6). The effective density of states is determined only by the area of cell 4; this is the only cell with allowed (filled) density of states. The other cells are empty. As the gate voltage V &increases and E , = 0, the effective density of states is D , / 2 . These half-cells have finite (nonzero) electron density. Thus half the space is now allowed, the other half is prohibited. As E , moves further up to position c, most of the space becomes allowed. The inversion layer is now a two-dimensional composite or inhomogeneous medium with two kinds of inclusions: (1) the allowed cells that contain electrons and are metallic in nature, and (2) the prohibited cells that have no electrons and arc insulators at T = 0. At T > 0, the conductivity of the prohibited cells is not zero. It is small and thermally activated, The model is illustrated in Fig. 7. At E , < E,, the allowed cells or allowed regions “form isolated lakes in a continent of prohibited space,” as shown in the lower figure. If the Fermi energy is high, however, most of the space is allowed. The prohibited regions can be regarded as “islands in an allowed sea.” According to percolation theory, if the total area of the cells for which E , > (where r/T is the local potential energy) is 3,the allowed regions for electrons will form contiguous regions (Arnold, 1976), i.e., they will form an infinitely

A

Y

X

FIG.7. Local potential barriers in inversion layer. (After E. Arnold, 1976.)

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

121

extended region. The threshold energy E,, for percolation is given by Kirkpatrick (1971),

which gives E,,

= 0.

(23)

Thus, as the gate voltage increases, E , rises, and when it crosses Ecp,metallic conduction by percolation becomes possible. A first approximation to the conductivity gives

where y o is the conductivity of the large allowed region, p ( E )is the fraction of the volume allowed; P ( p ) is the fraction of p ( E ) that forms the extended channels and can be easily calculated. Attempts have also been made to make yo a function of p ( E ) to allow for the change of mobility due to scattering from the boundaries between the ordered and the disordered regions. The dashed curve in Fig. 8 shows a plot of y/go = P ( p ( E ) ) .Just above the percolation edge p = i, percolation theory gives higher values of g as

0 pc

P

~ I G X. . Conductivity of a two-dimensional square net of conductances with 25 x 30 nodes. binary disorder. Values of the conductivities are y, = 1 (with probability 1 ) and q z = 0.5,O.Z. and 10- (with probability 1 p). The data points are based on resistor-network theory and sohd lmes o n effective-medium theory. The dashed line corresponds to P ( p ) ,i.e., the conductivity based on percolation theory, and p , corresponds to E,. = E,, = 0. (After S. Kirkpatrick, 1971.) -

122

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

Drain

Source

(a)

Drain

Source

(b)

Drain

Source

(C)

FIG.9. Schematic illustration to show that P( p ) overestimates the conductivity.

compared to the results obtained either by the resistor network or by the effective-medium theories discussed later in this section. Intuitively, one can see that the percolation theory should overestimate the conductivity. We illustrate this schematically in Fig. 9. We assume that the fraction P ( p ( E ) )is constant in Figs. 9(a), 9(b), and 9(c), but the configuration of P ( p ( E ) ) with respect to the prohibited medium is different. It can be seen that it is only in Fig. 9(a) that g will be proportional to P, g will be zero in Fig. 9(b) and will have an intermediate value in Fig. 9(c), which is a realistic distribution. The real medium corresponds to Fig. 9(c), and percolation theory overestimates the current. C. Effective-Medium Theory

In this theory (Bruggeman, 1935; Landauer, 1952), the average effect of random g i i s can be expressed by giving all of them a single value gm. The conductance g is chosen in such a manner that the average effect of changing any one of the conductances back to its true value is zero. The method is similar to that used for constructing coherent potentials (Soven, 1967; Velicky et al., 1968) in the theory of electrons in alloys. In a binary distribution, gm is given by (Kirkpatrick, 1971)

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

where z = 6 for a simple cubic lattice in 3D, and z in 2D. In 2D, Eq. (25) reduces to (Arnold, 1976) gm = $ ( 2 ~

~ ) ( Y I- Y Z )

+ Cb(2~- ~)’(YI

=4

- 92)’

123

for a square lattice

+ g1g21”~.

(26)

For 91 >> Y Z , we have Ym =

Yl(2P

-

I),

which becomes zero for p = $when EF = Ecp,as expected at the percolation threshold. The effective mobility can be determined by dividing g m by qEi,,, where Einv is the average carrier density in the inhomogeneous medium. The values of gm calculated by using effective-medium theory are also shown in Fig. 8 by continuous lines. It is seen that except in the neighbourhood of p = and except for g 2 = the agreement between the results obtained with the resistor-network model and the effective-medium theory is excellent. The discrepancy near p = +is also small. It should be noted that a discrete medium with two values of conductivities is used instead of the actual continuous medium in the effective-medium theory. Now we apply the above theory to the inversion layer, Following Arnold (1976), we designate the allowed cells as pseudometallic with average conductance yl, and the prohibited cells as pseudoinsulating with average conductance g 2 . We calculate g1 and g 2 as a function of E,, which is determined by the applied effective gate voltage V‘. Let the average conductance of the pseudometallic regions be

4

At low temperature, the contribution to conductance g from a cell with V E , is proportional to the Boltzmann factor exp[(E, - V ) / k T ] ,and the average conductance of the pseudoinsulating regions is given by

+

= (g1/2.f;)exp(a~/2k2T2 E,/kT)

erfc(EF/&.k

+ aAA/fikT),

(30)

wheref; = 1 - p (E , ) = ) e r f c ( E , / & , , ) i s the fraction of areaoccupied by the insulating regions. The conductivity g1 of the pseudometallic regions is somewhat smaller than the conductivity yo of a large crystal with the same mobility and carrier concentrations because the carrier mean free path is reduced by scattering from the boundaries of the pseudometallic and the pseudoinsulating regions. Arnold has shown that this reduction is negligible if the correlation length of

124

S. C. JAIN. K. H. WINTERS AND R. VAN OVERSTRAETEN

the potential fluctuations is more than 100 A. Using Eqs. (26), (28), and (29),g, can be calculated for an inversion layer. D . Resistor-Network Model In this two-dimensional model, the inhomogeneous medium is replaced by a square network of conductances gi (or resistances R i ) . The value of the conductances yi form a random distribution. Applying Kirchhoff’s current law to the node i, we obtain (Fig. lo), Cgij(V i

V,) = 0,

(31)

where gij is the conductance of the link between the adjacent nodes i and j . Such a network is a discrete model of the medium whose conductivity g varies continuously with position. The voltages V;. at each node of the network (and from them the total current flow for a fixed external voltage) are obtained by a relaxation procedure (Kirkpatrick, 1971).A specific case of binary distribution of gijwith probabilities p and 1 - p has been treated by Kirkpatrick (1971). A value of g1 equal to 1, and three values of g 2 equal to 0.5,0.2, and were used. The results of this calculation are also shown in Fig. 8. We have already compared these results with those obtained by the effective-medium theory earlier. Kirkpatrick used a network of conductances, whereas Arnold (1982) and Wikstrom and Viswanathan (1986) have used resistors instead of con-

FIG.10. Resistor network for simulation of two-dimensional current flow. (After J . A. Wikstrom and C. R. Viswanathan, 1986.) 1986 IEEE.

:c

C A R R I E R T R A N S P O R T I N W E A K S I L I C O N I N V E R S I O N LAYERS

125

ductances in two-dimensional networks to simulate the flow of current in an inhomogeneous inversion layer. The effective-medium theory leading to Eq. (26) treats the inversion layer as a binary mixture of two types of regions. The average properties of the two regions are taken into account. The composition of the “mixture,” i.e., the relative abundance of the regions, is changed by changing the Fermi level through variation of the gate voltage. The theory does not take into account the fact that both pseudometallic and pseudoinsulating regions have positiondependent, continuously varying conductivity values. Moreover, it is not easy to include any anisotropy in the inhomogeneity in this model. The resistornetwork model discussed above is somewhat superior in these respects, and anisotropy can be included by giving more weight to resistors in one direction than in the other. In applying the theory to inversion layers, the resistor values can be chosen by using the expression for a two-dimensional electron gas (Stern and Howard, 1967), R, = {eDpkT[l

+ exp(E,

- E,)/kT]}-’,

(32)

where D is the density of states in the lowest subband, p is the true microscopic mobility, and Ei is the local band edge. Wikstrom and Viswanathan (1986) used an alternative approach. They used the following expression for normalized R i ,

The distribution f(V,,) V,, was determined experimentally, and V,Bi was determined from the equation

I-.. VFSi

f’( VFB)d V,, = a random number between 0 and 1.

(34)

Here x

(Qi(V’,)>

=

f(f/pB)Qi(V;j -

FV)”FB=

(35)

~- r*

is the charge density over the whole device. The inhomogeneity is introduced through V,,, and the Fermi level through the gate voltage V ; . E. One-Dimensionul Model As we emphasized earlier, the two-dimensional models of conduction are necessary since the one-dimensional model does not permit the carriers to go

126

S. C . JAIN, K . H. WINTERS AND R. VAN OVERSTRAETEN

around the barriers. Chen and Muller (1974), however, developed a onedimensional model that showed good agreement with experiment. We discuss this model here. The schematic distribution of potential in the channel and its simplified form used by Chen and Muller are shown in Fig. 11. They assumed that the barrier height A in the simplified model of Fig. 1 l(b) is given by A

vs

= 2aAkT/q.

(36)

I X -

Ib) FIG. 11. (a) Solid line: probable surface potential q5$ (Gaussian variation). Dashed h e : approximation proposed on Nicollian and Goetzberger near constant u, in regions of dimension (a)"*; (b) binomial distribution for us with equivalent mean and deviation to the models of Fig. 6(a). (After J. T. C . Chen and R. S. Muller, 1974.)

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

127

This is a quasi-uniform model, and a , is given by (Chen and Muller, 1974)

Chen and Muller showed that the mean and the standard deviation of the actual statistical potential distribution (Nicollian and Brews, 1982)is the same as that of the simplified model. Here Coxand C,, are the oxide and the semiconductor capacitances, Q is the average interface charge density, and a is the size of the characteristic area. The relation between the carrier density nb on top of the potential barrier and n, in the potential well is given by nb = n, exp( - qA/kT).

(38)

It is assumed that, on average, each region occupies half the area. The average carrier density is given by

Assuming that the microscopic mobility is the same in both regions, and the current density is the same throughout, the potential drop across the barrier is (nw/nb) times the drop across the well, (Chen and Muller, 1974), we obtain

where Jd is the drain current and L is the channel length. The conductance becomes mobility pWIC PWJC =

4Y

2

+ exp( -qA/kT) + exp(qA/kT)

(41) ’

If qA/ kT >> 1, Eq. (41) reduces to pwIC= 4p exp( - qA/kTh

-1 --_1 p

Here

Pb

Pb

+-.1

(42) (43)

ps

is the bulk mobility, and

is the surface mobility without inhomogeneities and at low fields, C is a constant and En is the surface field perpendicular to the interface.

128

S. C . JAIN. K. H. WINTERS AND R. VAN OVERSTRAETEN

V. THEORIES BASEDON SHORT-RANGE OR MICROSCOPIC INHOMOGENEITIES A . Brews’s Small-Fluctuation Theory Brews (1975) adapted the theory of nonuniformities in bulk semiconductors (Herring, 1960) to describe conduction in nonuniform channels of MOSFETs. In this theory, the distribution of charges in the silicon-silicon dioxide interface is assumed to be random and given by a Poisson distribution. Brews assumes that, in the presence of inhomogeneity, the deviations of the potential and carrier density from their values in a uniform channel are small, so that a perturbation theory can be used to solve the Poisson and transport equations for an inhomogeneous channel. By using the perturbation theory and the Green function method and for very small fluctuations in the potential and carrier density, Brews obtains

where ID,is the current in the absence of fluctuations and AIDis the change in the drain current caused by fluctuations; ID =

IDO(l -

PWlC = A 1 -

)<‘z>)>

(46)

3)>

(47)

where (0,’)is given by Muls et al. (1978, p. 244), and the value of p is given later in Eq. (50); (of) =

[

4

kT(8,

y$$ln[l

+ cox)

+(

ci = ( E , + &,,)In.

cox

+ cs

y],

(48)

(49)

Here C, = CD + Cinv,C, is the depletion layer capacitance, Cinv is the inversion layer capacitance, Qox is the average interface charge density, and 2 is the average distance of the minority carriers from the interface. Minority carriers do not see the fluctuations in interface potential that have wavelengths smaller than i. The three-dimensional perturbation theory of Brews discussed above is elegant. However, it should be restated that the theory is applicable only for very small potential fluctuations. Moreover, experiments of Castagne and Vapille and of Baccarani (Nicollian and Brews, 1982, p. 245), who obtained C-V curves for grossly nonuniform MOS capacitors cannot be interpreted on the basis of a simple Poisson distribution of point charges at the interface. To compare Eq. (47) with experiments, a value of p is required. This value is the microscopic mobility and can be taken from the data in the strong-

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

129

z

-c

0

400-

h

=200A

3001 20F

1

I

1

to9

108

I

I

10’0

1

1

I

10”

10’2

inversion layer carrier density I crn3 FIG.12. Mobility behaviour for a fixed-charge density near the level where the mobility peak disappears. The carrier density at the nominal 2pFthreshold is indicated. (After J. R. Brews, 1975.)

inversion region where the effects of fluctuations have decreased to a negligible value. Brews used the empirical formula

with 7=2

to

3,

f i - 1.5, and p , = 5 0 0 c m 2 V ~ ’ s ~ ’ .

(51)

The mobility calculated using Eq. (47) is shown in Fig. 12. B. One-Particle Mobility-Edge Model Applicable to Low-Temperature Work I t is known from the work of Anderson (1958) that if there are large fluctuations in the depths of a regular array of potential wells, all states in the band become localised. Mott (1966,1967) pointed out that if the fluctuation in the potential is not large enough to localise all states, some localised states in the wing of the band will still be produced. These states will be separated from the extended states towards the centre of the band by a mobility edge E,, shown in Fig. 13. Far away from the mobility edge E,,, band theory applies to both the extended and the localised states. Just above the mobility edge, the mean free path is of the order of lattice spacing. Just below the edge. though the envelope of the wave function decays as C a rCI,is not given

130

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

Ec rn

Energy FIG.13. The smeared band edge and density of states in a disordered two-dimensional inversion layer. The mobility E,, separates the localized and the extended states.

by the normal band theory and is not equal to the tunnelling coefficient valid for the localised states deep in the tail. For the deep states, c1 is given by E,,

-

E = h2a2/2m*,

(52)

whereas near the edge. a cc ( E c m

-

ElS,

(53)

where s varies from 0.75 to 1 or becomes even larger (Adkins, 1978a). It is clear that the conductivity of such a system should change from thermally activated when E, < E,, to metallic when E , > E,,. In two dimensions, the minimum metallic conductivity gmm comes out to be a constant and is given by qmm 0.12e2/h = 3 x S. (54) The smeared-out band and the mobility edge E,, in an inversion layer with disorder were shown schematically in Fig. 13. When E , = E,,, the conductivity y should equal the minimum metallic conductivity g,,. The conductivity increases monotonically as EF increases to larger values. For EF < E,,, g is given by g

=

g,,exp(-

W / k T ) for

k T 5 E,, - E,,

(55)

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

131

where W = Ecm- E,. At very low temperature and for E , still less than E,,, thermally activated g conductivity given by Eq. ( 5 5 ) becomes small, and transport by variable-range hopping (thermally activated tunnelling) becomes more important. The conductivity is now given by

i,r=-,[r2(E,) Yo

J

1’3

n N ( EF)k T

’

where go is a constant (Adkins, 1978a). The values of conductivity expected from Eqs. ( 5 5 ) and (56) are shown schematically in Fig. 14. C. Electron-Liquid Model

The correlation effects in the electron fluid in inversion layers have been discussed by Russian authors (see Tkach, 1986) and by Adkins (1978a, 1978b). We give below a simple treatment due to Adkins.

I

t

1/T FIG. 14. The conductivity (schematic) according to the one-particle mobility-edge model The curvature at low temperatures is due to variable-range hopping.

132

S. C. JAIN, K. H. WINTERS A N D R . VAN OVERSTRAETEN

The typical binding energies of carriers in the localised states are of the order of 10 meV. The mutual potential energy of two electrons in an inversion layer is approximately given by V ( r )= q2/47COr= 5.8(n/10" cm-2)1'2.

(57)

Here Fo = 7.8 is the mean permittivity of the silicon and the silicon dioxide. For carrier densities 10" cm-2 to 10l2 cm-2, V(r)is of the order of 5-20 meV. These values are comparable to the binding energies deep in the localised states. When the carrier densities are large, Coulomb correlation energies increase, and binding energies due to inhomogeneity decrease. This shows that correlation must be important close to an Anderson transition. In the metallic state, correlation can cause localisation by Wigner condensation. Wigner condensation occurs when the reduction in potential energy is more than the increase in kinetic energy due to localisation. The energy gain E , due to Wigner condensation is given by

=

-

1.69(n/10" cm-2) + 4.4(n/10" cm-2)1/2meV,

(59)

for a (100)silicon surface. Here RE,,, and RElocare themean kinetic energy per carrier in the extended and the localised states, respectively, and I/exch is the exchange energy and is equal to the change of potential energy on localisation. The variation of condensation energy E , with electron density in the inversion layer is shown in Fig. 15. The values are so large that correlation must be important at low temperatures. Tkach (1986) solved the Schrodinger equation for electrons in the N-channel inversion layer, taking into account the correlation effects. His results also show that such effects in the inversion layer are important. Correlation effects lead to condensation of the electron gas at T = 0. There is n o long-range order in two-dimensional systems (Landau and Lifshitz, 1980);the frozen solid is amorphous. At finite temperatures, the behaviour of the inversion layer cannot be described by an excited state consisting of normal modes or phonons (Meissner et al., 1976; Bonsai1 and Maradudin, 1977). Adkins has suggested that at finite temperatures, thermal motion will provide the energy for collective motion of the "electron liquid leading to the flow of current under the action of applied field. In this model, designated as the electron liquid model, all particles participate in the conduction, and mobility is treated as thermally activated. The energy of activation is determined by potential fluctuations as well as by correlation effects.

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

133

>

-E c

0

L c.

V

-0

a2 L

Q,

Q

x Ep Q,

C

a2 C

.-0 4-

$ C 0

U

C 0

V

FIG.15. Estimated condensation energy for Wigner localization of electrons at a (100) silicon surface. (After C. J. Adkins, 1978b.)

In the model of Tkach, two weakly interacting subsystems of electrons exist at finite temperatures. There are localised electrons in one subsystem and delocalised electrons in the other. The energy separation between the two subsystems and their relative population depends on the total density of electrons as well as on the temperature. At a critical temperature T,, all electrons become delocalised. According to Adkins’s electron liquid model, the conductivity gelq is given by

where yo = ze2J87ch = z x lO-’S,

(61)

z is the number of electrons in the group that moves as a whole due to correlation effects, and W is the activation energy of mobility. The dependence of getqon n comes through z. It is difficult to derive an expression for z in terms of n and potential fluctuations. However, the theory can be used to interpret the experiments qualitatively. The region where the electron-liquid model is applicable is shown in Fig. 16.

134

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

10 n I I 0'1 c rn-2

FIG.16. A tentative phase diagram showing the region in which the electron-liquid model should provide a good description of inversion-layer behaviour. The independent-particle model should always become valid in the limit of low carrier concentration. (After C. J. Adkins. 1978b.)

VI. COMPARISON OF LOW-TEMPERATURE EXPERIMENTAL-EDGE MODEL WITH THE MOBILITY-EDGE MODEL Adkins (1978a, 1978b) has reviewed the early work done at Cambridge. Most of this work was carried out on (100) silicon surfaces and with Q,, 5 2 x 10" cm-2. Typical results (Adkins et a]., 1976) of conductivity for a sample with Q,, = 2.5 x 10'l cm-2 are shown in Fig. 17. The carrier density ninvvaries from very low to quite high values, and the temperature range considered is from less than 1.4K to more than 10K. Most of the predictions of the mobility edge model (Section V.B) agree with these results. The straightline portions converge to gmm= 2 x lo-' s, which is close to the theoretical value of 3 x lO-'S. The activation energy decreases as ninvincreases as expected. The density of states calculated from these results agrees with the theoretical value to within 30%. Figure 17 shows that there are deviations from linear behaviour both at low temperatures and at high temperatures. The low temperature plots seem to agree with the l/T''3 law expected from variable-range hopping. The deviation at high temperatures is not well understood. It could be due to an increase in mobility, because at high temperatures, carriers may be excited to levels much higher than Ecm.We shall discuss the high and low temperature regions again in the next section. The Hall effect measurements made by the Cambridge group also appear

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

135

T (K) 10

4.0

2.0

1.4

I

I

I

I

-L

-9-

0

I

I

I

I

I

0.2

0.4

0.6

0.8

-

T - l ( K-' FIG.17. Behaviour of the conductiwty near threshold as observed in a sample w ~ t ha low number of localised states (about 2.5 x 10" cm-*) (After C. J. Adkins, 1978a.)

to be consistent with the mobility-edge model (Pollit et al., 1976). However,

there are several other experimental results that cannot be interpreted on the basis of this model (Adkins, 1978a). Against the prediction of the mobilityedge model, it is found that go(l/T = 0) is not constant independent of Einv even when sin" < ncp. In many cases, it increases as Einv increases. Adkins has established a correlation between go(1/T = 0) and Einv (see Fig. 18), go(l/T = 0)

=

(%l")1.2>

(62)

based on the experimental results of a large number of authors. Similarly, for n,,, = ncm and E , = E,, = 0, Adkins finds (see Fig. 19),

g(E,

=

0 )Q;:.~*,'

and this is usually more than gmmpredicted by the mobility-edge model.

(63)

136

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN I

I

I

x 10-5

100 .

-

0

m

... ./-= .

10 k

%

'r /

"

1 I

L-

0.1

10

1

100x10"

n (cm-2) FIG. 18. A plot of yo against carrier concentration n. The gradient of the line is approximately 1 .2. (After C. J. Adkins, 1978a.)

x1~-5' loo -

I

I

I

I

I

I

I

I -

-

-

-

-= v)

- 10 -

-

E

-

-

0

1

1

I

I

1

I

I

I

I

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

137

Finally, Allen et al. (1975) found it necessary to assume that E,, was not a constant but a function of Einvin order to interpret their experiments. It is clear that the mobility-edge model in the present form cannot be used generally to describe the behaviour of the inversion layer. Several difficulties had similarly arisen in applying the mobility-edge model to experiments on amorphous silicon. These difficulties have been removed by suitable improvements in the theory (Mott, 1987a, 1987b). Attempts have been made to modify the model; it has been suggested that as the electron concentration increases, the mobility edge moves up due to Wigner localisation (Pepper, 1985). Another difficulty has arisen with the mobility-edge model. Abrahams et al. (see Mott, 1987b) pointed out that in a truly two-dimensional system, all states are localised and there is no real metallic conduction. In the weak disorder limit, they found (see also Gorkov et al. and Vollhardt and Wolfe both quoted in Mott, 1987b) that the conductivity is given by

where yB is the Boltzmann conductivity, I is the mean free path, and Lis the inelastic diffusion length, with the size of the specimen being much larger than L . The value of iis given by

where k , is the wave vector at the Fermi surface. The inelastic diffusion length varies as 1/T, which means that y decreases with decreasing temperatures. At very low temperatures, when L > L o , where Lo is the localisation length of the exponentially localised wave function, the conductivity is given by 9 = constant e-LILn.

is

Mott and coworkers assumed, on the other hand, that the wave function not exponentially localised but that for large r, it behaves as

where s < 1 for k,l>> 1, i.e., for higher energy states, but s -+ rx; as E , + E,. According to this model, a mobility edge does exist, and it separates the quasimetallic and hopping conductivity regimes. Qualitatively, the behaviour of the experimental data can be made consistent with any of these models. A more detailed discussion of this topic is beyond the scope of this chapter, and it can be found in the excellent review of Pepper (1985).

138

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

VII. ARNOLD’SEXPERIMENTS AND MACROSCOPIC INHOMOGENEITY MODEL

Arnold (1974, 1976, 1982) measured conductivity and the Hall effect in p-channel MOSFETs at low temperatures. The Hall mobility

was plotted as a function of T in the range 1.5 to 25K (Arnold, 1974) and 1.5 to 10K (Arnold, 1976). The latter results are shown in Fig. 20. Both Arnold (1974, 1976) and Thompson (1978) found that Hall carrier concentration was constant and independent of temperature. It is clear from Eq. (64) that the mobility peff = pH values plotted in Fig. 20 can be converted to y values by multiplying them with the corresponding nH values, which are independent of temperature. Arnold’s results, therefore, can be compared with In g, 1/T plots, such as those shown in Fig. 17.

T(K) n,.

10

2

4

1.5

\”

\

\6.7x10”

\

4.8~10~~

I

1

FIG.20. Experimental points: effective mobility in a p-channel MOS sample, relative to the maximum value p,, versus reciprocal temperature.-The value of po is 400 an2 V - ’ s-‘, at a carrier concentration n 2 1.6 x 10l2 cm-’. Solid curves: calculated from effective-medium theory. (After E. Arnold, 1976.)

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

139

The number of carriers in the inversion layer is given by

.=Iz EF

D(E){l

+ exp[(E

-

E,)/kT])-'dE,

(654

and

n = n, when E , = Ecp=O, (65b) where E,, is the percolation edge. For E,,, 2 n,, conduction becomes metallic. If n, is known from the experiments, a, can be evaluated by using Eqs. (21) and (65). For lower values of n, E , can now be calculated by using Eq. (65a), since a, is already known. In his earlier paper, Arnold found that, against the expectation of the theory, W is always smaller than E , as shown in Table 11. It is interesting to estimate the variance a,. Assuming i, to be constant and equal to 25 A, Q,, = 6 x 10" cm-', and Do = 2 x loL4cmP2eV-', the values of 0 , can be obtained by using Eqs. (21) and (65). The calculated value of 0, is about 19 meV at E , = 0 for i. = &. Thus we see that the observed activation energy is much smaller than a, and is considerably smaller than E,, - E,. The results shown in Fig. 18 and in the above discussion suggest that metallic conduction makes an appreciable contribution in the n, T regions from which values of W are derived. The experiments of Arnold (1974, 1976) and Arnold's theory based on macroscopic inhomogeneity remove the restrictions on go( 1/T = 0) and gmrn(Er= E,,), which are placed by the mobility-edge model and which are not supported by experiments. However, it is not clear whether correlations between go and Einv and between g,,, and Qoxshown in Figs. 18 and 19 can be explained on the basis of this theory. TABLE I1 HOLECONCENTRATION E,,, , FERMI LFVELE , A N D ACTIVATION ENERGY W I N p-CHANYEL INVERSION LAYER OF A MOSFET BASE RESISTIVITY 2 Q-Cm II-TYPE, 1200-A-THlCK A N D ALUMINIUM GATEWERE DRYOXIDE USED (DATATAKENFROM ARNOLD.1974) cm->

fi,,, x

4.7 6.5 8.3 1.2 16.0

E , meV

W meV

- 16

1.23 0.542 0.245 0.066 0

- 12 -9 -4 0

140

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

The solid line curves in Fig. 20 are the fit of the effective-medium theory (discussed in Section 1V.B) to the experimental data points. The values of various quantities used in these calculations are Q,, = 6 x 10" cm-2, m* = 0.5 m,, and /I= 25A. At the percolation threshold, E , = E, = 0, ncp = 4 x 10" cm12. It is seen that both experimental points and theoretical plots are curved even at n > ncp, and the apparent activation energy seems to increase at higher temperatures. The activated conduction persists above the percolation threshold. Arnold (1982) has also calculated the tunnelling conductance g1 and finds that it should become comparable t o g2 only for T < 1.7K and n 5 4 x 10" cmp2. At the higher concentrations and temperatures appropriate to Fig. 18, tunnelling currents must be negligible. The curvature indicating larger peff (or g) at low temperatures is therefore not caused by variable-range hopping. Since Einv used in the figure is larger than ncp for all the curves in Fig. 20, the metallic conduction is important and becomes dominant at lower temperatures, giving rise to the observed curvature. At higher temperatures, thermally activated conduction becomes increasingly important, even though E , > E, = 0 and we are in the metallic regime. This is understandable because the carriers in the pseudoinsulating regions will continue to be thermally activated. Arnold has also found that metallic conduction extends into band tails, i.e., it contributes to conduction even when E , < E, = 0. This result is difficult to reconcile with the theory with a sharp mobility edge.

VIII. HALLEFFECTAND ELECTRON-LIQUID MODEL

Measurements of the Hall effect have been most difficult to explain using any of the theories discussed above, except perhaps the electron-liquid model. In the metallic regime, the Hall carrier concentration is slightly smaller than the average carrier concentration determined from the value of applied gate voltage and inversion-layer capacitance. The difficulty arises when the system enters the activated regime. The Hall carrier concentration becomes equal to the total carrier concentration and is independent of temperature. Arnold (1974, 1976) has tried to explain these results on the basis of percolation theory with limited success. The expression (60) for conductivity based on the electron-liquid model can explain essential features of experimental results at low temperatures and not very high values of Einv when metallic conduction dominates. In the thermally activated regime, as n increases, W decreases due to screening, and the correlation energy V ( r )increases. This should result in an increase of the

141

C4RRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

number z in the group of electrons that are associated with a site and move collectively. This explains the observed behaviour that go zz n1.2. Since the number of pinning sites increases with No,, z decreases as No, increases, and go decreases consistent with the observed behaviour g,,srN;:''. For small values of z , z 3, go becomes equal to gmmin agreement with the results shown in Eq. (54). The transition to a metallic state occurs when the zero point energy, which varies approximately as n3I4 (Adkins, 1978a), becomes equal to the binding energy of the electron in the group pinned to the site. Adkins finds that the transition occurs at n = 8 x 10" cm-2 for a binding energy of 10 meV. Since the calculations are very crude, the results can be considered in reasonable agreement with experiments. This model does not predict the observed region of very low temperatures and low carrier densities where W becomes temperature dependent. It is possible that a process involving thermally activated tunnelling and similar to variable-range hopping becomes more important in this region and that the temperature dependence of the log of resistivity becomes weaker than l,T. Since correlation causes a collective flow of electrons, as in a viscous fluid, the Hall effect gives the total number of electrons in the inversion layer. The Hall mobility now becomes thermally activated as observed. Englert and Landwehr (see Adkins, 1978a, for details of this work) measured the resistivity of p-channels on (100) silicon at different small values of drain voltages. At about 1.25K, the resistance drops from a value 1.4 x lo6 R at a source drain field E,, of .025 V cm-' to 4.3 x lo5 R at E,, = 0.25 V cm-', and to 5 x lo4 Q at EsD = 1.25 V cm-'. Adkins (1978a, 1978b) has shown that these results are consistent with the electron-liquid model. His calculations show that the electric field causes heating of the electron plasma, its temperature rises from 1.25K at the lowest field strength to 1.53K and 2.82K at the two higher strengths, consistent with the observed reduction in resistivity.

-

-

Ix. EVIDENCE OF DEVIATION FROM RANDOMDISTRIBUTION In a later paper, Adkins (1979) has examined more carefully whether the effective-medium theory combined with the percolation theory can explain the Hall effect measurements. The conductivity results obtained by Adkins are similar to those obtained by Arnold and are shown in Fig. 20. Adkins has made calculations for much smaller values of ninvand finds a dominantly activated regime. In the metallic regime just above the threshold, Adkins also finds that both thermally activated and metallic conduction are important.

142

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

The results of Hall effect calculations are shown in Fig. 21. Below the threshold, the Hall carrier concentration decreases drastically. The Hall mobility becomes activated, but it becomes temperature independent again at low values of Einv, in contrast to predictions of the electron-liquid model. Adkins has rightly emphasized the need for more accurate Hall measurements

n

3

I

.

r r 2 0

c

1

0 n FIG.21. (a) The variation of the normalised Hall mobility Hall mobility pl, with carrier concentration n as predicted by application of effective-medium theory to the macroscopicinhomogeneity model of inversion layers; (b) the variation of the Hall carrier concentration l/eR,, with actual carrier concentration n as predicted by application of effective-medium theory to the macroscopic-inhomogeneity model of inversion layers. (After C. J. Adkins, 1979.)

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

143

FIG. 22. Resistor network to simulate conductivity and Hall effect in inversion layer. (After E. Arnold, 1982.)

at very low carrier concentrations. Such measurements are difficult. We should also mention that Adkins made several simplifying assumptions to obtain the results of Fig. 21. The effect of these assumptions on the calculated values shown is not known. Arnold (1982) made a further attempt to explain the observation that Hall carrier concentration is independent of temperature in the thermally activated regime. He used the resistor-network model discussed in Section 1V.D to obtain the conductivity and Hall carrier concentrations theoretically. The Hall voltage was simulated as shown in Fig. 22, where VHi =

PHV(R,)

(66)

is the local Hall voltage in the resistor i, and H is the magnetic field. The input parameters are D (density of states), p, E,, T , and inhomogeneity parameter Ei [see Eq. (32)]. A value of standard deviation 5 was chosen to adjust the onset of thermally activated behaviour. If Ei is assumed to be Poisson distributed, the conductivity results are similar to those obtained by the effectivemedium theory and shown in Fig. 21. However, Hall carrier concentration was found to be thermally activated as shown by the dashed lines in Fig. 23. Arnold used another model for Ei (designated as long-inclusion model); as distinguished from a random distribution, he assigned a given value of Ei (or Ri,see Eq. (32)) to several contiguous cells. Several groups of these cells were randomly oriented in two perpendicular directions. The remaining cells

144

S. C. JAIN, K. H. WINTERS AND R. VAN OVERSTRAETEN

6-

5-

*

*

*

04

05

\\V \

\

Experimental Random network

Long inclusions

0

01

02

03

06

C

I

1/T (K)-'

FIG.23. Computed and experimental Hall carrier concentration versus 1/7 in an inversion . E. Arnold, 1982.) layer, n = 6.3 x 1 0 ' o c m ~ 2(After

were assigned a value Ei = 0. The conductivity results were more or less the same in the two models. However, the Hall carrier concentration obtained with the long-inclusion model is very different, as shown by the continuous line in Fig. 23. These results, if found to be of general validity, are important. They suggest that distribution of inhomogeneity Ei is not random in actual inversion layers.

X. PEAKS IN

THE

VARIATION OF pWlcWITH ninv

Before concluding the discussion on low-temperature conduction in inversion layers, we discuss briefly some measurements of Fang and Fowler (1968). Their results of conductance mobility in n-channels are shown in Fig. 24. At 297K, a sharp peak in the plot of mobility versus V, is found; this peak will be discussed more fully below. As the temperature is lowered, the peak decreases in intensity and a new peak appears at a higher value of VG. At 4.2K, only this new peak remains.

145

CARRIER T R A N S P O R T IN WEAK SILICON INVERSION LAYERS

3426

1.88 OHM C M (100) 6 =IlOOA

1500

I

1250 m I

> cu

E >.

1000

c

..-

0

E

-

n

297OK

n

750

U

aJ

c

aJ

z .!?

LL

500

250

0 -10

0

10

20 30 40 Gate voltage (V)

50

0

FIG.74. Field-effect mobility for a low-resistivity silicon (100)surface. (After F. F. F a n g and A. B. Fowler, 1968.)

As pointed out by Brews (1975), this behaviour can be understood with the help of Fig. 25 taken from Stern (1972). It is seen from the figure that at 4.2K, all the carriers are in the lowest subband. In this case, the effect of fluctuations is stronger because the inversion layer is thinner, and so a larger carrier density is required to eliminate the effect of the fluctuations. At 297K, most carriers are in higher subbands, and the effect of fluctuations can be eliminated by a smaller number of mobile carriers, giving rise to a peak at lower concentrations. At intermediate temperatures, most carriers are in higher subbands at lower carrier concentrations but move to the lowest subbands at higher concentrations. This gives rise to the two peaks observed.

146

S. C. JAIN, K. H. WINTERS A N D R. VAN OVERSTRAETEN

--Fraction 0.8

-

in lowest valleys

Fraction in lowest subband

,

-

ul

1

10

I

10

10

(fiinv + ndcp) (cm-2) FIG 25. Fraction of carriers in the lowest subband for (100) n-channel device. (After F. Stern, 1972.)

Wikstrom and Viswanathan (1 986) have also measured conductance mobility, and their results in the temperature range 82K to 295K are shown in Fig. 26. At low temperatures, their results are somewhat different from those of Fang and Fowler (Fig. 24). We already see two peaks at lO5K in Fig. 24, but there is only one peak even at 82K in Fig. 26. The reason for this discrepancy is not clear at this time.

XI. ROOM-AND HIGH-TEMPERATURE MEASUREMENTS Most investigators have plotted the dependence of mobility pWlcon V , or room temperature only. We have already shown results of Muls et al. (1978) in Fig. 1 and those of Wikstrom and Viswanathan (1986) in Fig. 26. Chen and Muller (1974) made measurements on both n-channel and p-channel devices in an extended temperature range up to about 350K. Their

ninv at

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

c

147

T=82K

3200

-.2400 U

v

= 2000 >. .c

n

o 1600

E

2 8l

1200

0 .c

u

800

400

0 Average inversion charge density

(As/cm2)

FIG.26. Mobility versus inversion charge density at different temperatures. (After J. A. Wikstrom and C. R. Viswanathan, 1988.) 1988 IEEE.

results for n-channel transistors are shown in Fig. 27. It is seen from the figure that at low values of iiinv,the conductivity is thermally activated; Einv increases, the energy of activation decreases, and at about Einv= 1 x 10" cm-2 in Fig. 27(a) and at 2 x lo9 in Fig. 27(b), conduction becomes metallic. In this respect, the results are similar to the low-temperature measurements. Chen and Muller have fitted Eqs. (42) and (44) to the experimental results by plotting pWlcas a function of 1/T as shown in Fig. 28. A linear dependence is found as predicted by theory with activation energies equal to 0.1 eV for transistor 25, and 0.08 eV for transistor 17. The authors plotted pWIcaccording to Eq. (41) along with Eqs. (43) and (44) and compared it with their experimental data. The results are shown in Fig. 29. In view of the fact that the theory is one-dimensional and is very crude, the agreement is surprisingly good. Qualitatively, most of the results can be interpreted according to Brews's three-dimensional theory discussed in Section V.A. A comparison of Brews's theory with some of the early experimental results (Guzev et al., 1972)is shown in Figs. 30 and 31.

pH measured 3L3"K 323°K 0 296OK @ 279OK

-

0 x

Y)

> LOO

N '

5 I

by Fang and Fowler o n a ( I l l ) . lohrn-crn sample at 295'K

0 x

0

0 x

170"

1

0

200

0

o

0

e

m

e I

0

I

108

I

I

109

,

10'0

,No 2SL io12

iiinv ( c m-2) (a)

600

0

erne

3L6OK

0 8 B O O , @ x x 08

. x 323'K

2

.

0 296OK LOO - @ 2 7 6 ' K

x&oQ;xo@ 0 xo

xOo O

N

6

0

I

200

",o

0 X

e13

OX

oxo e

g

-0 X

XO

-g

0

@

I

0

I

I

No 17J

I

vl

I

>

VI

>

N

I

100 2.5

3.0 OS / k T

3.5

-

4.0

2.5

.

3 .O U'sI k T

I

I

3.5

L .o ___)

10) (b) FIG.28. Inversion-layer mobility versus 1000/T(K) in the low normal-field region for the two devices of Fig. 27, respectively (surface densities - 5 x 10' ern-'. (After J. T. C. Chen and R. S. Muller, 1974.)

CARRIER TRANSPORT I N WEAK SILICON INVERSION LAYERS

149

-

->

N

No 25

LOO - T = 23OC Experimental

E,

I

1-

200 -

-Theoretical

O ~ ' " " ' ' ' ~ ' ~ ' - ~ 20 2L 28 32

36

U)S/kT

(a)

-

VI

No 17

>

.

T = 23OC

N

2oo

I

Experimental

- Theoretical 20

2L

28

32

36

OS/kT

(b) FIG.29. Experimental results and theoretical predictions Lfrom Eqs. (41) and (44)J for fl for two n-channel transistors. The theoretical component owing to surface field p5 is also plotted; values for C LEq. (44)] are 1.76 x lo7 and 1.84 x lo7 in (a) and (b). respectively. (After J T. C Chen and R. S. Muller, 1974.)

The shape and carrier densities at which pwlc starts decreasing or shows a peak agree quite well with the experimental results of Guzev et al. (1972) in Fig. 30. The effect of temperature on pwlc seen in Fig. 31 agrees quite well with the behaviour observed by Chen and Muller (1974) and shown in Fig. 27. Similarly, Muls et al. (1978) have found that their measurements (see, e.g., Fig. 1) can be fitted with Brews's theory.

0’ lo9

I

I

1 olo

10”

I

1012

3

1

Inversion layer carrier density / crn2

FIG.30. Conductance mobility versus carrier density for various degrees of interface uniformity as determined by the number of interface charges per unit area, Q, effective in causing fluctuations. The inset shows data of Guzev et al. (1972) for p-channel devices that have similar behaviour. For the calculation, the parameters are the bulk doping N = 2 x 1015/cm3;the oxide thickness, 2200 A;and i, = 200 A.(After J. R. Brews, 1975.)

270° K 0

I

I

*

@F I

I

Inversion layer carrier density I crn2

FIG.31. Mobility behaviour at higher levels of fixed charge. The carrier density corresponding to the traditional strong-inversion threshold, 2pF, is indicated. The parameters for the calculation are bulk doping N , = 2 x oxide thickness, 2000 A;interface charge 2 x 10”./cmz; and i = 200 A.(After J. R. Brews, 1975.)

CARRIER TRANSPORT IN WEAK SILICON INVERSION LAYERS

15 1

XII. LIMITATIONS OF THEORIES In spite of this success, there are several shortcomings of the theories, listed below.

1. Muls et al. (1978)found that the values of A required to fit experimental results with Brews’s theory are unrealistically large. 2. If 0 , or A are small, the exponent in Eq. (41) can be expanded, and experimentally speaking, Eq. (42) of Chen and Muller (1974) becomes indistinguishable from result Eq. (47). Equation (47) is valid for a, << 1. The values of G-, used in Fig. 31 by Brews are so large that Eq. (47) based on Brews’s theory is not valid. It should be noted that the interface charges and potential fluctuations in transistors degraded by irradiation or hot electrons will be much larger, and the values of o, will be even larger. The theories are based on the assumption that the charges are Poisson distributed. Arnold’s work (1982) and some of the work on decorated interface clusters (Nicollian and Brews, 1982) show that this may not be so in many cases. XIII. SUMMARY OF WORKON TRANSPORT IN INVERSIONLAYERSI N THE WEAKINVERSION REGION,AND CONCLUSIONS 1. Between 2K and 5K, In y versus 1/T plots are linear for very low values of Einv with an activation energy of W of a few meV; W decreases as iiinv increases. For intermediate values of iiinv, the plots become curved. For sufficiently large values of Gin”, conduction becomes metallic. 2. The conduction is thermally activated for carriers localized in band tails below the mobility edge (microscopic inhomogeneity) or below the percolation edge (macroscopic inhomogeneity). When E , > E,, (mobility edge) or E , > E,, (percolation edge), conduction is metallic. The one-particle mobility-edge model is consistent only with a limited number of available experimental data. Most of the data seem to fit better with macroscopic inhomogeneity, percolation, and two-dimensional effects. The thermally activated and metallic conduction occur simultaneously when E , > Ecp. 3. At very low temperatures, Ing versus 1/T plots curve upwards. The conduction might be by variable-range hopping. 4. At temperature >5”K, the plots bend upwards, as if the activation energy is now increasing. This behaviour is not well understood. 5 . At room temperature, pWlcversus Einv plots show a peak at about fiin, = lo1’cm-*. As the temperature decreases, this peak becomes less pronounced, and a new peak starts developing at higher values of Einr. As the

152

S. C. JAIN. K. H . WINTERS AND R. VAN OVERSTRAETEN

temperature decreases further, this new peak continuously increases and the room-temperature peak decreases. Near 77K, both peaks are clearly seen. At 4.2K, the low-temperature peak becomes strong and the room-temperature peak disappears. The 4.2K peak is attributed to carriers in the lowest subband, and the room-temperature peak to the carriers in the higher subband. However, this behaviour is not observed in more recent experiments (Wikstrom and Viswanathan, 1986). 6. Fang and Fowler studied the effect of source-to-substrate bias on the observed conductance mobility. As the value of the bias is changed to increase the vertical field, for the same value of Einv, the room-temperature peak decreases in intensity and finally disappears. The low-temperature peak becomes stronger. An intense vertical field promotes population in the lower subband. The room-temperature and high-temperature behaviour is different. 7. For Qox < 3 x 10" cm-2, pwIcis constant for values of E,,, < 10" cm-' and decreases for larger values of ninv(see Fig. 30). For Q,, > 3 x 10" cm--2,the behaviour of pwlcis different. It decreases drastically and remains constant for small values of E,,, between 10' and lo9 cm-'. As Einv increases to higher values, the conductance mobility pwlc increases sharply and attains a maximum value between Einv = 10" cm-2 and 10" cm-' and then decreases again, giving a step in the mobility versus Ylinv plot. (See Figs. 1, 26, and 27). 8. The maximum in the pWIc,Einv plot moves to higher values of E,,,, and the step height increases as the value of Q,, increases. The conclusion of the above discussion is that although our understanding of transport in inversion layers in the weak-inversion region has improved considerably, a lot of work needs to be done to fill in the gaps that still exist in our knowledge. The more important areas for future work are (i) a theory of transport when inhomogeneity is large and interaction between the two neighbouring cells is important; (ii) an experimental investigation of the inhomogeneity, i.e., whether or not it is randomly distributed. In particular, E,, may be distributed, ie., it may be different in different portions of the crystal, which may explain many results that are not yet understood.

ACKNOWLEDGMENTS SCJ is grateful to Dr. A. B. Lidiard for making his visit to the Theoretical Physics Dikision possible in 1987, when some of this work was done. The work described in this report is part of the longer-term research carried out within the Underlying Programme of the United Kingdom Atomic Energy Authority.

C A R R I E R TRANSPORT I N WEAK SILICON INVERSION LAYERS

153

REFERENCES Adkins. C. J. (197%). J . Phy.5. C. 11, 851. Adkins. C. J. (197%). Phil. Mug. B. 38, 535. Adkins. C. J. (1979). J. Phps. C. 12, 3395. Adkins. C. J. Pollit, S., and Pepper, M. (1976). J . Physique 37 C4, 343. Allen. S. J., Isui, D. C., and DeRossa, F. (1975). Phys. Rev. L e f t . 35, 1359. Anderson. P. W. (1958). Phps. Rev. 109, 1492. Ando. T.. Fowler, A. B., and Stern, F. (1982). Rev. M o d . Phys. 54, 437. Arnold. E. (1974). Appl. Phps. Lett. 25, 705. Arnold. E. (1976). Surf. Sci. 58.60. Arnold. E. (1982). Surf. Sci. 113, 239. Arora. N. D.. and Gildenblat, G. Sh. (1987). I E E E Trans. Electron Devices ED34, 89. Basu. P. K. ( 1 978). S d i d Stare Commun. 27, 657. Bonsall, L., and Maradudin, A. A. (1977).Phys. Rec. B15 1959. Brews. J. R. (1975). J . Appl. Phys. 46, 2193. Bruggeman, D. A. G . (1935). Ann. Phys. (Leipzig) 24,636. Canali. C., Jacobini, C.. Nava, F., Ottaviani, G., and Alberigi-Quaranta. A. (1975). Phys. Rev. 12, 2265. Chen. J. T. C., and Muller, R. S. (1974). J . A p p l . Phys. 75, 828. C‘oen. R W., and Muller, R. S . (1980). Solid Stare Electron. 23, 35. Conwell, E. M. (1967). “High Field Transport in Semiconductors.” Academic Press, New York. Cooper, Jr., J. A., and Nelson. D. F. (1981). I E E E Electron Deuice Lett. EDL-2, 171. Cooper. Jr.. J. A., and Nelson, D. F. (1983). J . Appl. Phys. 54, 1445. Duh. C. Y., and Moll, J. L. (1967). I E E E Trans. Electron Devices 14, 46. Eversteyn, F. C., and Peek, H . L. ( 1 969). Philips Res. Rep 24, 15. F-ang. F. F.. and Fowler, A. B. (1968). Phys. Rev. 169, 619. Fang. F. F., and Fowler, A. B. (1970).J . A p p l . Phps. 41, 1825. Ferry. D. K. (1978).Solid-Stare Electron. 21, 1 IS. F u r ) . D. K.. Hess, K., and Vogl, P. (1981).“VLSI Electronics,” (N. G. Einspruch, ed.). Academic Press, New York. Green. R. F., Frankl, D. R.. and Zemel, J. (1960).Phps. Reo. 118, 967. Gurev. A. A.. Kurishev. G. L., and Sinitsa, S. P. (1972). Phys. Status Solidi 14, 41. Herring. C. (1960).J . A p p l . Phys. 31, 1939. Jacoboni, C., Canali, C., Ottaviani, G., Alberigi-Quaranta, A. (1977).Solid State Ele<,tron.20. 77. Kirkpatrick. S. (1971). Phps. Rev. Lett. 27, 1722. Landau. E. D., and Liftshitz, E. M. (1980). “Statistical Physics,” 3rd ed. Pergamon Press, Oxford. Lmdauer, R. (1952).J . Appl. Phys. 23, 779. Meissner. G., Namaizawa, H., and Voss, M. (1976). Phys. Rev. B13, 1370. Mott. N. F. (1966). Phil. Mag. 13, 689. Molt. N. F. (1967). Adi.. Phps. 16,49. Mott. N. F. (1987a).J . Phys. C: Solid State Phps. 20, 3075. Mott. N. F. (1987b). “Conduction in Non-crystalline Materials.” Clarendon Press, Oxford. Mott. N . F., Pepper, M., Pollit, S., Wallis, R. H.. and Adkins, C. J. (1975).Proc. Roy. Soc. A345, 169. Muller. W.. and Eisele, I. (1980). Solid Stare Commun. 34,447. Muls. P. A,, Declerck. G. J.. and Van Overstraeten, R. J. (1978), Adkances in Electronics and Electron Physics 47. 197. Murphy. St. J. N., Berz, F., and Flinn, J. (1969). Solid State Electron. 12. 775. Nelson. D. F., and Cooper Jr. J. A. (1982). Surf. 3C.i. 113, 267.

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Nicollian, E. H., and Brews, J. R. (1982).”MOS Physics and Technology.” John Wiley and Sons, New York. Norris, C. B., and Gibbons, J. F. (1967). I E E E Trans. Electron Devices 14, 38. Pepper, M. (1985). Contemp. Phys. 26,257. Pollit, S., Pepper, M., and Adkins, C. J. (1976). Surface Science 58, 79. Ryder. E. J. (1953).Phys. Rev. 90,766. Sabnis, A. G.,and Clemens, J. T. (1979). I E D M Tech. Dig., 18-21. Scharfetter, D. L., and Gummel, H. K. (1969). I E E E Trans. Electron Devices ED-16, 64. Schrieffer, J. R. (1955). Phys. Rev. 97. Schwarz, S . A,, and Russek, S. E. (1983a). I E E E Trans. Electron Devices ED-30, 1629. Schwarz, S. A., and Russek, S. E. (1983b). I E E E Trans. EIecton Devices ED-30, 1634. Shockley, W. (1951). Bell Sysr. Tech. J . 30. 990. Soven, P. (1967). Phys. Rev. 156, 809. Stern, F. (1972).Phys. Rev. B5,4891. Stern, F. (1974). Phys. Rev. B9, 2762. Stern, F., and Howard, W. E. (1967). Phys. Reti. 163,816. Sun, J. Y.-C., Taur, Y., Dennard, R. H., and Klepner, S. P. (1987). I E E E Trans. Electron Derices ED-34, 19. Sun. S. C. and Plummer, J. D. (1980).I E E E Trans. Electron Devices ED-27, 1497. Thompson, J. P. (1978). Phys. Lett. A 66, 65. Tkach, Yu. Ya. (1986). Sot.. Phys. Solid State 28,909. Velicky, B., Kirkpatrick, S., and Ehrenreich, H. (1968). Phys. Rev. 175, 747. Wikstrom, J. A., and Viswanathan, C. R. (1986), “Analog M O S Modeling,” Final Report, Electrical Engineering Department, UCLA. See also Wikstrom, J. A,, and Viswanathan, C. R. (1988). IEEE Trans. Electron Devices ED-35,2378.

A I I V A M E S I\

E L K rROhICS A h D F L E C T K O N PH\rSICS \ O L

11:

Emission-Imaging Electron-Optical System Design V. P. IL’IN V. A. KATESHOV YU. V. KULIKOV M. A. MONASTYRSKY Computing Center Novosibirsk USSR

1. Introduction . . . . . . . . . 11. Aberration Models of Cathode Lenses.

IV.

V.

VI.

,

. .

.

,

.

.

. . .

. . . . . . . . . . . . A. The Traditional Approach to the Aberration Theory of Cathode Lenses. . . .

111.

.

.

. . . . . .

. .

B. Differentiation-Based Development of an Aberration Model with Respect to the Parameter . . . . . . . . . . . . . . . . . . . . . . . . C. The Main Criteria of Emission System Quality . . . . . . . . . . . . The Variational Analysis of Cathode-Lens Optimization and Synthesis Problems . . A. The Formulation of Problems of Parametric Optimization of Cathode Lenses . B. Integral Equations in Variations. A General Three-Dimensional Case . . . . C. Integral Equations in Variations for Axially Symmetric Surfaces and Surfaces with Weakly Disturbed Axial Symmetry . . . . . . . . . . . . . . . . D. Variations of the Limiting Paraxial Equation Solutions . . . . . . . . . E. The Problem of Axial Synthesis of Cathode Lenses . . . . . . . . . . . Implementation of the Numerical Computational Methods and System Optimization A. The Solution of Integral Equations for the Axially Symmetric Potential of a Simple Layer . . . . . . . . . . . . . . . . , . . . . . . . B. The Computation of Axially Asymmetric Disturbances by the Bruns-Bertein Method. . . . . . . . . . . . . . . . . . . . . . . . . . C. The Algorithms for Computing the Derivatives of the Potential. . . . . . . D. Integration of Paraxial Equations for Electron Trajectories . . . . . . . . E. Numerical Solution of the Problem of Optimization of Electron-Optical Systems Automation Principles in Designing Electron-Optical Systems . . . . . . . . A. Software Requirements for CAD EOS . . . . . . . . . . . . . . . B. Automation of Algorithm Construction in the EFIR Program Package . . . . C. Basic Components and Characteristics of the APP EFIR . . . . . . . . . Numerical Experiments. . . . . . . . . . . . . . . . . . . . . . A. Calculation of the Potential for the Perturbation Function . . . . . . . . B. Calculations of the Nonaxial Symmetric Perturbations by the Bruns-Berteln Method . . . . . . . . . . . . . . . . . . . . . . . . . . C. Computation of the First-Order Parameters and the Aberration Coefficients . , D. Solution of the Practical Problems. . . . . . . . . . . . . . . . . References . . . , . . . . . . . . , . . . . . . . . . . . . .

156 158 159 167 170 178 178 181 190 198 207 224 225 233 237 240 244 246 247 250 254 257 257 260 262 265 276

155 tngltsh translatmn cop)nght 7 1990 by Academlc Press. Inc All rights of reproduction in any form re\er\ed ISBN 0-12-014678-9

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V. P. IL’IN et nl

1. INTRODUCTION

This work deals with mathematical and software support for the computer-aided design (CAD) system of emission-imaging electron-optical systems (EOS). Computer-aided design is based on mathematical simulation that combines algorithmic, programming, and hardware tools in order to provide a deeper insight into the objective properties and laws of the objects or processes under study. The computing experiment-in the broad sense-is in fact a new technique integrating all kinds of methods from different branches of science. I t includes the statement of physical and mathematical models of a particular device, the development of numerical methods and programs, computations including automated initial data processing and output visualization, the analysis of the data obtained, and the final decision based on the comparison with the results of physical experiments. The successful solution of these problems depends on mathematical and software characteristics, such as the adequacy of models, the economy and accuracy of numerical algorithms, the reliability and effectiveness of the program implementation as well as on the automatization level, and the “intellectualization” of the communication with the user. The work presented here gives an example of solving such problems in order to investigate and to design emission EOS, which is one of the specific tasks of mathematical and technical physics. All mathematical problems of imaging electron optics may be divided into two large classes: direct problems (analysis problems), whose solution would show the properties of the image in the electron-optical system of the given design and with given electrode potentials; and inverse problems (synthesis problems), whose aim it is to determine the desired configuration of electrodes and feeding conditions providing for the required electron-optical parameters. The fast development of computers, as well as of numerical methods for solving boundary-value problems of the potential theory for the domain with complex configuration, triggered the recent progress in creating program complexes designed to solve analysis problems. Numerous investigations by Soviet and authors of other nations have shown that the synthesis problem is by far more complicated. In its general form, it can be considered within the problem of optimal control of the boundary and boundary conditions in systems with distributed and concentrated parameters. Similar problems arise in many fields of applied physics. In the approach to perturbation theory (which in electron optics is usually called the aberration theory), one can single out three main tasks from the

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

157

general synthesis problem: 1. Calculating the axial distribution of the electromagnetic field in order to meet the necessary requirements of electron-optical characteristics. 2. Extending the calculated axial distribution into space and forming electromagnetic field sources, such as electrodes, magnetoconductors, etc. 3. Error estimation of the general task by direct calculation of electron-optical characteristics; optimization (or “finishing”) of these characteristics on the restricted set of varying parameters that determine the structure and the feeding conditions of the initial approximation system based on the solution of problems 1 and 2.

The present work covers a range of problems that, in our opinion, are the basis for the solution of the general synthesis problem. Here special emphasis is placed on specialized methods of analysis and optimization. In Section I1 the authors consider the ways of building the aberration model of cathode lenses and, on its basis, determine some practically important functionals of emission systems. Section 111 discusses variational analysis of parametric optimization and synthesis problems. A general formulation of cathode-lens optimization problems is provided. Careful attention is given to calculating the potential perturbations due to the distortion of the calculation domain boundary and boundary conditions; variations of some electron-optical functionals have been evaluated. The possibility of reducing axial-synthesis problems (see I ) to the problems of optimal control over dynamic systems has been studied. Section IV sets the numerical aspects of the main stages for computing and optimization of emission systems. The solutions of the first types of integral equations for the accurate evaluation of the potential and its perturbation are considered in detail; a high-performance method for integrating the paraxial equation is suggested; the formulation and solution algorithms of cathodelens parametric optimization problems are specified. Section V discusses general principles of mathematical and software support for electron-optical system computer-aided design as well as the approaches to applied program packages (APP) development. It also dwells on the technique of computer-aided generation of EFIR APP algorithms reported in the previous sections; these algorithms featuring a natural input language and advanced service provide effective calculations and the optimization of emission systems. Section VI analyzes the results of numerical experiments carried out to solve model and practical problems. The problems treated in the present work certainly do not give universal coverage of the directions along which the mathematical model of emission systems is developed.

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V. P. IL’IN et ul.

The authors are grateful to Y. A. Flegontov, whose salutary criticism has greatly improved the presentation of this paper. The work has also benefited from discussions with Y. V. Vorobyov. We are indebted to all who have promoted the publication of this chapter.

11. ABERRATION MODELSOF CATHODE LENSES

The term “emission system” will be applied to an electron-optical system made up of an electron emission source, i.e., a cathode, an electrostatic or a combined (electromagnetic) electron lens-a cathode lens; and a receiver of the electron image-a screen. Considering photoelectron emission, assume that the density of distribution of the photocurrent emitted by the photocathode surface into the vacuum volume of the emission system is proportional to the illumination produced on the cathode by the outer light source. The distribution of photoelectrons with respect to angles and energies (we take it to be prescribed) is known to be a rather important characteristic of photoemission reflecting the physical properties of a photocathode. In the general case, a cathode lens is a region of the electromagnetic field being created between cathode and screen by a certain system of electrodes and magnetoconductors. We shall confine ourselves mainly to discussing the axially symmetric electrostatic cathode lens, which can have a complex enough system of electrodes. This system is usually characterized by scale variety, the presence of vacuum, conducting or dielectric gaps, apertures, and fine-structured grids set in the way of the electron flow. Affected by nonhomogeneous focusing fields concentrated in the volume of a cathode lens, the photoelectrons are accelerated and directed onto the image receiver, creating on its surface a certain distribution of the current density called electron image. A typical image receiver used in emission systems is a luminescent screen. The physical properties of cathodes as electron sources, and of screens as electron image receivers, are treated in detail in monographs by Berkovsky et al. (1976) and Melamid and Soboleva (1974). There is also a lengthy list of books on this problem. The major feature of emission-imaging electron-optical systems, which determines their characteristics as well as the well-known computational difficulties, lies in the fact that the cathode plays a double role: being a source of photoelectrons, it is also (from the point of view of image generation) one of the most essential electrodes in a cathode lens. Similarly, the screen can also be involved in producing the focusing field that affects the image quality. The most important criteria by which one can judge how well the emission system

EMISSION-IMAGING ELECTRON-OPTICAL SYSTFM DESIGN

159

works (emission-system functionals) are the electron-optical magnification, the crossover plane position, the best focusing surface position and shape, the degree of geometric distortions, modulation transfer function, space and time resolution, and others. Depending on the constructive implementation and the field of application of specific emission systems, the requirements imposed on functionals may vary to a large extent. Most essential for emission systems operating in static conditions are the requirements of electron-optical magnification, crossover and image plane positions, as well as modulation transfer function, space resolution, and geometric distortions. Dynamic emission systems used particularly in the devices for recording and for examining ultrafast physical processes place an additional requirement on high time resolution, which in turn depends on a number of factors. The present section considers some approaches to the aberration theory of cathode lenses, and it reports on emission system functionals of most practical importance. It also discusses computational techniques and analyzes model problems. A . The Traditional Approach to the Aberration Theory of’ Cathode Lenses

Consider the motion of electrons in a source-free region of the electromagnetic field. Restricting ourselves to the case of stationary fields and ignoring the charge and beam current proper, we present Lorentz equations of motion (in nonrelativistic approximation) in the form

x=

e

--

m

Ex -

~

e . ( yBz - i B J ,

m

The initial conditions for Eq. (1) may be written as x(0) = xg,

X(0) =

160

V. P. IL’IN er al.

In Eqs. (1)-(2) the following notation is used: x, y, z are Cartesian coordinates; E x , E,, E, are components of the strength vector of the electrostatic field; B,, B y , B, are components of the magnetic displacement vector; e;” = eli2 sin d o , e ; l 2 = e1j2 cos 0, are projections of the initial velocity of the particle onto the x0y-plane and Oz-axis, respectively ( E = ep ez is the initial energy of the particle expressed in the units of the potential; 0, is the angle formed by the initial velocity direction and the Oz-axis); clo is the angle formed by the initial velocity projection onto the x0y-plane and the Ox-axis; elm is the specific charge of the particle. In the x0y-plane, we introduce the vector r that will be written as a complex number r = x + iy and, respectively,

+

According to Sherzer, if we exlude time from the equations of motion (1) using the energy integral

we shall reduce ( 1j to the trajectory equation: r“

1

+ cp

+ r J r * ’ (r’

-

cp,,

+c

icp

2 iz

~cp) +

id*

2m c p - c p o + c

(2+ rtr*t)”]iir*

(4)

= 0.

Equation (4), which is an exact consequence of the equations of motion ( 1 j, will be referred to as a general equation of trajectories of charged particles in stationary combined fields. In Eq. (4), cp, cp, are the electrostatic and scalar magnetic potential, cp, is the value at the initial point with coordinates x,, y,, z,; the asterisk stands for complex conjugation; the prime denotes differentiation with respect to the space coordinate 2. The initial conditions (2) take the form i;1/2

r(z,) = rneiPO,

r’(z,) = PcjI2 e i z o.

(5)

where Po = arg r(z,), Y, = arg rf(zo), The aberration theory problem of cathode lenses could then be formulated as solving Eq. (4) with initial conditions (5) in the form of an asymptotic expansion with respect to the set of small parameters &:I2, c ; l 2 , r o . The difficulties arising in solving this problem were pointed out in the well-known works by Recknagel (1941) and Artsimovich (1944). They are due to the fact

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

161

that the electron initial energy is small, the object (cathode) is the lens field, and the beam aperture on the cathode can reach 90". The conditions stated lead to the existence of a singular point in Eq. (4) that at E = 0 coincides with the particles takeoff point. In accordance with the terminology used, Eq. (4) belongs to the class of nonlinear singularly perturbed differential equations that are known to reject the formal perturbation theory. In numerous works on the aberration theory of cathode lenses (Artsimovich, 1944; Vorob'ev, 1956; Kas'yankov, 1966; Monastyrsky, 1978; Kulikov and Monastyrsky, 1978; Monastyrsky and Schelev, 1980; Chou Li-Wei et al., 1983; Uchikawa and Maruse, 1969; Ximen et al., 1983; etc.), various approaches were worked out that help to overcome the difficulties mentioned. Unfortunately, most of the above-mentioned works contain no data allowing objective evaluation by using the criteria given in the introduction, of the effectiveness of an aberration model as an element of CAD mathematical software. Two approaches for which the authors have such data available will be treated later. The approach presented in this section analyzes the general trajectory equation (4); the second approach, based on the construction of so-called 7-variations of equations of motion, is treated in Section 1I.B. Assuming that the axial component of the E ~ " particle velocity has an arbitrary nonzero fixed value, we construct the expansion of the solutions (4)-(5) to the Cauchy problem with respect to the set of small parameters E:", r o , and shall then examine the coefficient asymptotic functions of the resulting expansion at small c,llz. We confine ourselves to axial symmetry. In this case, the potentials cp and cp, appear as 1 4

cp = @ ( z )- -@"(z)(rr*)

1 + -@'"(z)(rr*)2 + ..., 64

1 cp, = @,(z) - -@l(z)(rr*) 4

1 + -@:(z)(rr*)2 64

(6)

+ ... ,

where @ ( z )= q ~ ( ~@,(z) = ~ = , cpmIr=o are axial distributions of corresponding potentials. By substituting these expansions into (4) and by retaining the terms not larger than the third order of smallness with respect to the set E ; / ' , r,,, we obtain the aberration equation r"

+ 21 (@ - @' + c,) r ' + -41 (0 -

~

@"

+ c,) r

162

V. P. IL'IN c't (I/

The function 9is determined by t r' F = P 2 (@

-

0' Do + 6;)3'2

rr*'r'

Q'

-_____

2

+ rr*r' + rr*r

(Q, -

= -@;,

+ c,)1'2 Q,'"

8(Q, - Qo

+

Q," -

0"+ 6;

rr*r' 4

-~

(Q,

-

32(@ - Do + E,)'"

D'D; -D o + t;z)3'2

-

a'' + &;)I

+ &:)3 *

S(a, -

EJ"2

Q'"

+-rgr' 8 (Q,

where J

Epr +-4 Q,

-~

16(@- Do + E

+-rgr 16 (D

-

= Q,(kj(zo),k =

r

=

: ) ~

@"a; Do + E

~ ) ~ "

0,1,2. By means of substitution. ue'*,

and Eq. (7) and the initial conditions ( 5 ) may reduce to

E M I S S I O N - I M A G I N G E L E C T R O N - O P T I C A L SYSTEM D E S I G N

where, in accordance with (lo),

163

-

Let us introduce the functions E, I?, which are solutions of a linear uniform differential equation @”

u”

+ 2(@

0‘ - @o

+ E Z ) u‘ + 4(@

+ e -

2m

@o

+

EZ)

u=o

(14)

with initial conditions

F(&i 2 , z o ) = 0,

G(ed’2,z0)= 1,

Using the method of variation of parameters, Eq. (11) with initial conditions (12) (to within the values of the third-order of smallness with respect to E : ’ ~ , yo) may read:

Note that the expansion coefficients (16) depend not only on the axial component of the initial E:” velocity but also on the coordinate of the takeoff point z o and the value of the potential m0 in it. If the curvature of the cathode and screen surfaces is nonzero (as well as if there is surface curvature of finestructured grids, which can be present in the optically active part of the field), solution (16) will depend on the shape of these surfaces and on the potentials’ distribution about them. The essence of further transformations is in “re-expansion’’ of (16), so that the final expansion coefficients would not depend on the above-mentioned values & : I 2 , z o , m0. In so doing, one should certainly specify the surface shapes of the cathode, grid, and screen to within the terms of the third order of smallness, as well as the relationship between the point coordinates on the surface and the electrostatic potential values. The existence of a singular point in Eq. (14) coinciding with the takeoff point at iz = 0 shows that it is most important to build up the asymptotics of

v. P

164

IL'IN et ul

the solution of (14) with respect to the small parameter E : " . This problem is detailed in Monastyrsky (1978) and Kulikov and Monastyrsky (1978), where asymptotic expansions of the functions ;(&,'I2, z), uniformly accurate in the interval zo < z , I z I z 2 , G ( E ~ / ~ ,$(E,"',z) z), with respect to the parameter E : ', and have been obtained on the basis of singularly perturbed equation theory: their properties in the boundary layer directly adjoining the cathode have also been analyzed (see earlier works by Artsimovich, 1944, and Vorob'ev, 1956). It is quite evident that the remaining coefficients in (16) do not contribute to the asymptotic functions with respect to &:I2 within the limits of the third order small values with respect to the set of parameters E:", c,li2, ro. Thus, in the third-order approximation, Eq. (4) will read

+ ro~,"2eiao(K+ ik) + E : " E , ~ ~ ' ~+( ip) P + ro&,eiflo(Q+ iq) + E ~ i z e i a n (+B ib) + c,r,[(G + ig)eipO+ ( F + i f ) e i ( 2 " o ~1B o ) where v(z) = lim G ( E ~ " , z ) , w(z) = lim G ( E : ' ~ , Zare ) solutions of the limit$2-0

EL 2 4 0

ing paraxial equation @"

u" + 1 0' u' 2CD-CD0 ~

~

1

+ -4

e 3 +-

2m

m-mo

2

u = 0.

The function and the coefficients B , . . . ,e have been obtained by the ~ &, B , . . .,Z. B, b, G,g, F , f , C , c , D , d , E, e are limiting transfer at ~ f +'0 from coefficients of geometric aberrations; H , h, K ,k, P , p , Q, q are coefficients of chromatic aberrations; specifically, B, b are coefficients of spherical aberration, G, g, F, f are those of coma, D,d those of curvature, C , c of astigmatism, E , e of distortion, K , k of second-order chromatic aberration, H , h of secondorder spherochromatic aberration, P, p of third-order spherochromatic aberration, Q, q of chromatic aberration of third-order position. In the case where the cathode and screen are flat and equipotential and where there are no fine-structured grids in the field of a cathode lens, the geometric aberration coefficients B, . . . ,e coincide with these aberrations but, if in the latter, cZ = Q0 = 0. Ignat'ev and Kulikov (1983) and Kulikov and Monastyrsky (1 978) generalize the expressions for these coefficients in the case of nonzero curvature of cathode surfaces and the screen and with

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

165

fine-structured grids present in the field of a cathode lens. The chromatic aberration coefficients P, . . .,q in Kulikov and Monastyrsky (1978) have been obtained for combined fields in the form that takes into account the curvature and nonequipotentiality of the cathode surface. Calculating chromatic aberration coefficients in cathode lenses with finestructured grid is dealt with in Monastyrsky (1980a). For the case of the cathode surface having the curvature l/Rc, it will be convenient to introduce the parameter w ; and the values F;'', & ! I 2 could be replaced by new small parameters E:

* = sin R,

&,1/'

= E'"

cos R,

(19)

where 6; is the projection of the initial velocity vector on normal n onto the cathode surface at the initial point, E : / ~is the projection of this vector onto the tangent plane at the initial point, w is the angle formed by the initial velocity vector projection onto this plane and the tangent to the meridian cross section [j = Po, and 0 is the angle formed by the initial velocity vector direction and the normal. It is easy to show that the coefficients K , Q, G, F, C, D in the old and new systems of variables are related as follows,

Hereafter, the coefficients of the aberration expansion of the trajectories with respect to & : I 2 , e : i 2 are denoted by the superscripted asterisk. The coefficients k , q, g, f , c, d are transformed similarly; the remaining expansion coefficients (17)do not vary. Expansion terms containing the powers E : / * act as chromatic aberrations in the new system of small parameters. The trajectory aberration expansion in the new variables will appear as

166

V. P. IL'IN et d.

Let us now consider some problems of computational technology arising in the numerical implementation of the approach described. First note that due to a singular point in the limiting paraxial equation (18), the numerical methods of the type developed by Runge and Kutt, Miln, Hamming, and others for building up u, w trajectories in the close neighborhood of the cathode can hardly be employed. In this region, usually with an extension of the order of lo&/@;, the trajectories indicated are conveniently presented as segments of rows with coefficients depending on the derivatives of @g'(k = 1,2,. . .), is., the potential variables in the cathode center. These expansions at zo = 0 are known to appear as c _

+

u = j@(z)(cxo XI:

+

X2ZZ

+ cx3z3 + ",),

"=po+plz+pzz2 +p, Z3+'.. .

(22)

The coefficients cx, p ( k = 0,1,2,3) for the case with no magnetic field are given in Section 1V.D The numerical experiments whose results are given in Section V1.D have shown that the formulae for aberration coefficients obtained in Ignat'ev and Kulikov (1978) possess a strong instability with respect to potential evaluation errors that are due to the singularities in the integrand expressions. To achieve a higher computing stability of the formulae, the singularities in corresponding integrand expressions have been removed in the following way: The integral of the function R(z), having the integrable singularity at z = 0, reads (23) where L ( i )is a Laurent series segment of the function R ( i )containing negative variable powers [. The first term on the right-hand side of (23) is calculated analytically; the second term does not have any singularities and can be calculated by means of an ordinary Simpson method. In the close neighborhood of the cathode, the major term in (23) is usually the first one corresponding to the axial potential approximation by a Taylor series segment whose coefficients can be represented here as finite functions depending on the z variable and the potential @g)derivatives ( k = 1,2,. . .) in the cathode center. At a large distance from the cathode, particularly in complicated multielectrode systems with apertures, the expansion in the close neighborhood of the cathode reflects the axial potential character inadequately; the second term in (23) contributes essentially into the value of aberration coefficients.

167

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

B. D$erentiution-Based Decelopment of an Aberration Model with Respect to the Parumeter

Here we shall dwell on the approach permitting the construction of an aberration model of cathode lenses that meets to a greater extent the requirements stated in the preface (compared with those described in Section LA). We shall present the electron trajectory r(z, $,I2, c i r o , ao, Po) and the transit time z(z, E ; ” , 8: r o , a,, Do) in the form of the asymptotic expansions

’,

‘,

(24)

cl

’,

c3

with respect to the set of the small parameters = c; l 2 = E;,’’, = ro, with coefficients depending on the space variable z, counted off along the field symmetry axis, and on the initial angles a o , &. The subscripts in (24), (25),and later in this section denote variables with respect to the parameters 5,. The idea of the approach presented here seems to have been first introduced in Kas’yankov (1966). It makes it possible to set up an explicit connection between the perturbations of solutions of the charged particle equations of motion [Lorentz equations (I)] and the perturbations of solutions of the general trajectory nonlinear equation (4), in principle permitting to solve the problem of “small denominators” typical for cathode lenses mentioned in Section 1I.A. The aberration coefficient calculation consists of two stages: integrating the differential equations for z-variations and recalculating the z-variations into z-variations according to algebraic formulae. A similar approach for the regions in the close neighborhood of the cathode has been used in Flegontov and Zolina (1 978). One can show (see Monastyrsky and Schelev, 1980) that for axially symmetric electrostatic cathode lenses, the following relations are valid,

= M’e‘iii’

-(I)

’3

= 0,

are linearly independent solutions of limiting

168

V. P. IL’IN et al.

paraxial equations, M[r,] having, with respect to z

---f

= @r” + -1W r ’ + -1W ’ r 2

4

=

0,

0, the asymptotics

0; w(z) = 1 - -z 2%

+ O(z2),

while the function zc)= z t ) ( z )satisfies the linear uniform equation 1 Ro[q] 3 @q’ - -wq = 0. 2

(29)

In a number of cases (e.g.,in the problems with a fine-structured grid in the close neighborhood of the cathode), it seems appropriate to pass from the paraxial trajectories u, w to the functions

satisfying the Picht equation: p”

+16

[“I’

p = 0.

-

@

The formulae of the recalculation to the linear approximation z-variations may be presented in the following simple form: r . = r!‘) I

I

)

One can readily see that the second relation in (32) at i = 2 gives the well-known Zavoisky-Fanchenko formula for the first-order chromatic aberration. Without giving a complete set of formulae here to calculate all the aberration coefficients rij, rijk, zij, we shall just show a general structure of these relations. The functions z$),r$), r$ satisfy linear nonuniform differential equations of the type

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

169

whose right-hand sides are finite algebraic forms at all z 2 0 depending on the arguments given in brackets and on their derivatives with respect to z. The initial conditions for Eq. (33)-(35) are zero, except when i = J = 3 in (35),and when z t i = I/R, ( R , is the cathode surface curvature radius). The formulae of recalculation into z-variations in the second and third approximations appear as 7 1J. . =

qIJ. . [ z:..I“ ‘19

(36)

9

r ?I. . = f..[r$), V rt’, rij’,@I,

r y. k.

= j :ilk. Cry),r g ) ,zls’, z:.?,

(37)

r!;;,

a].

(38)

The electron-optical classification of nonzero expansion aberration coefficients (24)-(25) was given in Section 1I.A (see the above-mentioned works on the aberration theory of cathode lenses). The specific type of relations (35) and (38) for the distortion coefficient E (i = j = k = 3) could be given as an example. The calculation formula reads

where the

~ $ function 4 is a regular solution (at z = 0) of the equation

with the initial condition lzyi(0)\ < x and the function r‘& being a regular solution (at z = 0) of the equation

with initial conditions r‘&(O) = 0, /ry!31 < sc. We shall now discuss the effectiveness of the given approach according to the criteria listed in the Introduction. First, the z-variation method permits us to obtain a closed system of relations for all expansion coefficients (24) and (25) in the given approximation in a simple algorithmic way that is quite correct from a mathematical point of view. By the same token, it allows the building of a rather universal aberration model of cathode lenses. The impact of fine-structured grids within this model is easily taken into account with the help of jump conditions (Section IILD), the effects connected with the curvature or nonequipotentiality of the cathode (screen) surface being considered by the type of differential equations and initial conditions for r-variations, and also by the recalculation formulae.

v. P. IL'IN

170

et ul.

C. The Main Criteria of Emission System Quality

As stated earlier, the emission system quality is evaluated, depending on its purpose, in terms of a certain set of functionals. Consider the typical functionals that could be based on the aberrational approach. In order to do this, we introduce the notions of the main and adjacent trajectories. The trajectory of the electron emitted by the cathode with zero initial energy (t: = 0) will be called the main trajectory. From the aberrational expansion (2 l), it follows that the main trajectory equation (to within the terms of the third-order of smallness) will read =

+ ri(E* + ie*)e$pO+i).

(42)

The trajectory of the electron with nonzero initial energy will be called the adjacent trajectory. In the general case (with regard to the curvature of the grid surface and image receiver), it is suitable to present the third-order geometric aberration coefficients as the sum of three summands

s*= s,*+ s,. + s,:

(43)

where the first term characterizes the contributions of the field of the cathode lens, the second one is connected with the fine-structured grid impact, and the third one determines the image contributions. To a certain extent, this presentation is, of course, relative. It is easiest to determine the limiting crossover plane, the limiting image plane, and the electron-optical magnification. The plane perpendicular to the symmetry axis and crossing it at the point I,,, where the electrons emitted with zero initial velocity from an infinitely small area in the cathode center are assembled, will be called the limiting crossocer plane. By this definition, the limiting crossover coordinate zcr satisfies the equation

(44) which is obviously similar to the equation H'(Z,,)

= 0,

(45)

where wfz) is a solution defined earlier of the limiting paraxial equation ( 18). The plane perpendicular to the symmetry axis and crossing it at the point z 9 . where the electrons emitted by the cathode center and with infinitely small velocities directed at a tangent to the cathode surface are assembled, will be called the limiting image plane. One can easily see that the limiting image plane coincides with the so-called Gauss plane (Kel'man et al., 1973) at c 0. For

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

171

this reason, we shall use the term “Gauss plane” without differentiating these two notions. In analogy with Eq. (45), the Gauss plane coordinate zg meets the equation

which is equivalent to the equation

4%)= 0

(zg)

f 0,

(47)

where u ( z ) is the corresponding solution of the limiting paraxial equation. Condition (47) will be called the fbcusing condition in the center. We further introduce the notion of the complex mean magnification, assuming M,

=

(48)

r/ro.

Then by virtue of (42), we arrive at

M

= e’*[w

-

Extracting aberrational terms in (49), we present M, as

M,

=

M,

1

+ rg(E* + ie*)

i&J&u

(49)

+ AM,

where

AM

=

e’*(E*

+ ie*)rc.

The value

will be called the magn$cution in the center. In the Gauss plane ( z = z g ) , we arrive at M,

=

M,

=

Iwgl.

(54)

Let argM, = AD = 0 - Po. Then by (51),

’I/

=

Po + $ - arctg

(4

-9,- - nn

(n = 0,1,2,...).

(55)

A g will be called the angle of’ image rotation in the center; AM being a small quantity of the second order indicates scale distortions.

172

V. P. IL'IN et al

We introduce the notion of complex distortion, assuming Ad = AM/Mo.

(56)

The real part Re A, = Aid is the isotropic distortion; the imaginary part Im A, = A,, is the anisotropic distortion. By (52) in the Gauss plane, Aid = ( E * / w ) r i ,

A,,

= (e*/w)ri.

(57)

In the case of E * / w > 0, we have pincushion-type distortion; when E* / w < 0, the distortion is barrel-type. Representing magnification and distortion in a complex form is rather convenient; it is seen to be equivalent to other presentations given, for example, in the work of Shapiro and Vlasov (1974). Note that in some cases, for example, when calculating the integral characteristics of emission systems, one must not use the mean but the local magnification, which characterizes the scale transformation of some physically small cathode region as a whole. We present the arbitrary point ro of the in the physically small domain G, containing the point r,M = rOM= rOMeiSO, form ro = T O M

+ so,

(58)

where so = ~ , e ' The ~ ~ rough . presentation with only linear terms along so is easily shown to be equivalent to the well-known isoplanatism condition (see Shapiro and Vlasov, 1974; Kulikov, 1975). The local magnification M, in the isoplanatism domain G reads

It follows from (59) that the local magnification depends on the so vector direction and not on the coordinates of the inner points of the G domain. For electrostatic cathode lenses at yo = Po (the meridian direction),

wn = IMhI = at yo

71

=-

2

+ PO (the sagittal direction), MIS

=

IMkI = ( w + E * r & ( .

(611

Formulae (60) and (61) compared with (49) show that the local magnification can differ considerably from the mean one. By extending the focusing condition (47) to the off-axis beams (r,, # O), we arrive at

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

173

In the case of electrostatic cathode lenses, this equation will read u

+ r i [ D * + C * e - 2 i w] = 0.

(63)

Equation (63) describes the parametric family of rotation surfaces depending on o,i.e., on the orientation of the initial velocity, being tangential to the cathode at the initial point. Since the image receiver (0: = C: = 0) is missing, it is not difficult to obtain an explicit expression for the surfaces of the family indicated by expanding (63) with respect to the powers z - z g and by assuming that r = rowg: -2

z = zq

+ -.2R

The curvature radius R of the specific surface depends on w , i.e., on the direction of the initial velocity component being tangential to the cathode. For meridian beams (w = 0), for example, z = z,, = zg

+

..2 rmr -;

2Rmr

where (with regard to the Lagrange-Helmholtz invariant, R,,

=

Z&W,&~

=

I),

wg

-

2(D*

+ C*)&'

Similarly, for sagittal beams lying on the plane o =

71

-,

2

where

The meridian and sagittal surfaces are limiting surfaces of the family indicated above. Between the limiting surfaces, there is a mean curvature surface whose equation reads

where

174

V. P. IL’IN et al.

On the mean curvature surface, the impact of the off-axis astigmatism on the image-defocusing does not depend on the family parameter, i.e., it is similar for all electron beams emitted tangentially to the cathode. Superposing the image receiver surface with the mean curvature surface allows elimination of the beam-defocusing related to the image curvature. The needed curvature radius of the image receiver R, may be obtained from (70) if we assume R , = R m d . The functionals listed above may be called differential, because they have been obtained by a particular limiting transition. Now we shall determine some integral characteristics. In general, the arbitrary surface lying on the way of the electron current can be treated as an image surface. In practice, choosing the position and shape of this surface depends on the requirements of the image accuracy and the permissible size of geometric distortions. A measure of image accuracy is its contrast. To evaluate the image contrast formed by the cathode lens, one usually employs a test object in the form of a shaded mira, which is a periodical rectangular or sinusoidal one-dimensional distribution of the current density in the chosen direction. By convention, the mira whose strokes are parallel to the meridian direction will be called the sagittal miru; and the one whose strokes are normal to it will be called the meridian miru (Fig. 1). The universal characteristic of the cathode lens from the point of view of contrast quality is the modulation transfer function (MTF). It may be defined in the following way. Let the current density distribution on the cathode along the vector so lying in the small domain G obey the law Aso) =

where so

=

j,( 1

+ cos 27rN0s,) 2

lsol and N o is the space frequency of the mira’s strokes. Then the

Y

FIG. 1. The definition of the meridian (1) and sagittal (2) miras

175

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

image quality of domain G in the so direction is determined by the modulation transfer function

are current density values represented by max and min where jma,and jmin points, respectively. Using the aberration approach, the works of Kulikov (1975) and Hartley (1974) describe the numerical method of determining the MTF of cathode lenses. Its essence is the following. The elementary electron current emitted from the infinitely small area with .yo and y o coordinates and dx,, d y , sides, whose initial energy lies in the interval (E,E + ds) and whose takeoff angles lie in the intervals (R, R dR), (to, co + dw), may be written as

+

d J = %(.yo, yO)SI(&, R, w )sin Q dc dR dw dx, dye,

(73)

where % ( s oy o, ) is the coefficient proportional to the cathode illumination. For the photocathode, the values E, R, o are usually independent of the distribution with respect to the angle w being uniform. Then the function S ~ ( E R, , 0 ) may be written as

si(s,R, w ) =

w,w, ~

2.n ’

(74)

where WE,W, are distribution densities of random variables E and 0. Let us divide the domain of definition s i ( ~ , R , winto ) elementary cells having the volume of h,h,h, with coordinates ci,Rj,w,. The image receiver surface will also be divided into small areas with the sides h,, h, and the coordinates x,, y,. Each set of three variables i, j , k will correspond to the trajectory of rijk,with the electron emitted by the point having the coordinates xo,y,, the energy E , , and the initial angles Rj, 0,. Without calculating the trajectory between the cathode and the image receiver, one may find the coordinates x i j k ,yijkof its crossing the image receiver surface by knowing the aberration expansion. If the trajectory gets into the small area with coordinates x,,y,, this small area will correspond to an elementary current density

which, in itself, is a relative number of electrons crossing the image surface together with the trajectory rijk. Summing up all A j corresponding to various combinations of i , j , k for the small area with x n , y mcoordinates, it is not

176

V. P. IL'IN et ul

difficult to show that j(x,,y,)

=

lim

C Aj,

(76)

iJ,k

at h,, h,, h,, h,, hJ + 0. The value j(x,,, y,) represents the current density by a point source situated at the point xo,yo, i.e., the point-spread function. By the method described, one may calculate the spread function for any cathode point where the thirdorder aberration theory is valid. Integrating the spread function in the chosen direction, we find the spread function of the line normal to this direction. The convolution of the ideal line spread function with the mira current density distribution allows us to determine the modulation transfer function. The method was applied to the cathode-lens model having uniform electric (@'(z)= a;) and magnetic ( B ( z ) = B0)fields. For the case of electrostatic cathode lenses under parabolic distribution with respect to the energies

and under the angular distribution of

W,

= k 2 C O S ~Q,

0I RI n/2,

(78)

typical of photocathodes in the long-wavelength region of optical radiation, a family of extended modulation transfer functions with regard to defocusing, second-order spherochromatic aberration, astigmatism, and curvature (Fig. 2), has been derived in Kulikov (1975). The parameter of the family is the size of a relative image defocusing (, counted from the image receiver surface for the given direction of strokes; the independent variable is a relative frequency R related to the absolute stroke frequency on the cathode by the relation N 0 -- - n% .

(79)

&O

It can be shown that for the meridian mira,

and for the sagittal mira, zq) < = tsq= @'b(D* + C * ) + @'b(z ,i, -

EA'20,

8

W,

(81)

'

The curvature analysis in Fig. 2 shows that there is a MTF for which the contrast in the high-frequency region is at a maximum (5 = 5, = 1.41). The surface at all the points where the condition = tSis implemented will

<

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

177

RI

0

I

1

' ' I ' ' ' ( 1

I

-

Of LO f0 n F I G .2. Modulation transfer function at various values of defocusing parameter [

0.U?

be called the best focusing surface for the given direction of strokes. It is obvious that the resolving power (the frequency at the given threshold control of the image receiver) on the best focusing surface is at a maximum and identical for all points r,,. Assuming tmr (or tm,) to be equal to t,(in Fig. 2, tS= 1.41),it is not difficult to find the equation of the image receiver surface providing the best focusing for the given direction of mira strokes. This surface vertex passes through the point whose coordinate could be derived by the formula

The correction of zg in (82) related to the best focusing surface position is essential only for bipotential EOS; in the rest of cases, the correction influence may always be removed by subfocusing. Therefore later, unless specially stipulated, the focused image surface will be considered to pass through the point z = zq. Figure 3 gives as an illustration the results of the resolving power

t

I

A4u1

X

X

X

i

,

I

,

1 O

l

-0.5

I

0

1

1

1

1

0.5

'

z-z ,mm S

3 . The dependence of the resolving power in the center on the image defocusing

F;ic;.

( ~-

l x

20

~

calculation; X

- experiment).

178

V. P. IL'IN er al

calculations depending on the image-defocusing in the working-field center. The theoretical curve corroborates experimental results we11 enough, with the difference with respect to the absolute value being accounted for by some discrepancy between the calculated and real values of the most likely energy E ~ which , is difficult to measure in real conditions. Individual coefficients of space and temporal aberrations, transit time, temporal resolution, etc., can also be treated as emission system functionals.

111. THEVARIATIONAL ANALYSIS OF CATHODE-LENS OPTIMIZATION AND

SYNTHESIS PROBLEMS The parametric optimization of design and electrode feeding conditions under the given initial approximation can be treated as an independent method to solve the cathode-lens synthesis problem. The advantage of this approach lies in the fact that with every iteration of the calculation process, we obtain a physically implementable design of a well-known electron-optical characteristics of cathode lens processing. At the same time, parametric optimization problems of cathode lenses have a number of singularities and as shown later, require the development of specialized numerical methods. Such methods along with some model problems are considered in Sections 1II.A- 1II.C. As mentioned in the Introduction, the axial-synthesis problem involving the calculation of axial distributions of electric and magnetic fields, which meet the given requirements of image characteristics, in the aberration approximation, is one of the major stages in the general synthesis problem. Its solution clearly enables us to distinguish the focusing-field classes able to implement some extremal image properties in a cathode lens. In Section III.D, a new nontraditional approach for imaging electron optics, which is based on reducing axial-synthesis problems to problems of optimal control over dynamic systems, is presented. A . The Formulation of Problems of Parametric Optimizution

OJ'

Cathode Lenses Let us accurately state the class of problems under study. Let x

=

(z,. . . . ,x,) be a vector whose components are the parameters identically

defining the design and electrode feeding conditions for a given cathode lens. In the aberration-theory approximation, we shall assume electron-image characteristics to be determined by a finite set of functionals { F k ) ,k = 0,. . . , I , depending on the axial distribution of the potential @ = @(z, x) (for simplicity,

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

179

we confine ourselves to electrostatic systems only). The functionals Fk may include such characteristics as the crossover and image plane positions, magnification, aberration coefficients, resolving power, etc., as well as the functions of these values. The nonnegative functional Fo will be called specific purposeful. The search for the optimal system meeting the given requirements may then be formulated as a mathematical programming problem. Problem A:

to minimize Fo(x) under the restrictions

F k ( x ) = O , k = 1 ,..., m, Fk(x) 5 0, k

= m f 1,. . . 1,

asxsb, where a, b are the given vectors. (The inequalities a < x 5 b mean that u, i x , I bi,i = I , . . . ,n.) The bilateral inequalities that the vector x should satisfy represent technical restrictions on the varying parameters usually present in practical problems. One of the most widespread versions of constructing program optimization complexes of technical systems is the direct “junction” of the program module responsible for calculating the system functionals at fixed values of varying parameters (analysis programs) with the program optimization module, which produced increments of varying parameters in the iterative descent process. For optimization modules, one often uses standard program sets that are a part of software support, or specially developed software packages designed to solve a large enough class of problems of conditional minimization. In this case, functional gradients are usually calculated by the difference scheme by means of repeated application of an analysis program, taking into account the necessity to correct differentiation steps when approaching the optimum. It should be mentioned that the attempts to use this approach for numerical optimization of cathode lenses face considerable difficulties, which, on the one hand are, due to the high requirements for the calculation accuracy of the electrostatic potential and electron trajectories (demanding a lot of computer time to calculate one version, i.e., 3-5 min. on an ES-1060 computer) and, on the other hand, result from the necessity to make a large number of calculations of functionals (of the order of 3 x lo3 and larger) in meaningful enough optimization problems. Thus, the numerical solution of the problems considered based on the direct “junction” of analysis and optimization programs is not effective enough, since it takes too much computer time. To reduce the computational expenditures considerably when solving cathode-lens optimization problems, one could use a technique based on partial or complete linearization of initial

180

V. P.IL'IN et al.

functionals. We have performed such a linearization with respect to the axialpotential distribution by means of integral equations in variations. We shall describe this in more detail. Let x = xo be a certain fixed vector of varied parameters. In the parallelepiped n,,(so)= {x: xo - E~ I x I xo + g o ) (the algorithm of choice E~ is discussed later), the axial distribution of the potential @(z,x), within linear terms with respect to x - xo, may appear as @@, x)

= Qz, x,xo) = @(z,xo) + (V,@(Z,

xo), x

-

xo?,

(84)

where @(z,xo)is the axial distribution of the unperturbed potential corresponding to x = x,; and V,CD(z,xo) = d@(z,x)/~x),=,,is the vector whose components are axial-potential perturbation functions evaluated in x = xo. The symbol (,) denotes the scalar vector product. If all vector components x are the potentials of boundary electrodes r,,the relation (84) will be correct. In this case, perturbation functions coincide with the so-called unit functions, which are contractions onto the symmetry axis of the boundary-value problem solutions, AcpL= 0,

q.1 l r, = s:,

(85)

where iJ = 1,. . . ,n; 6: is the Kronecker symbol. In a general case, the perturbation functions d,,@(z, xo) may be found with the help of integral equations in variations. The representation (84) permits us to substitute, in the vicinity of the point x = xo, the auxiliary problem A for the initial problem A, i.e., to minimize Fo(x)under the restrictions

x E nXo(&). Assuming that &(x)are functionals of the type Fk[@(z,x, x,)], problem 2 will be called a problem of quasilinear approximation with respect to the field. If the gradient V,F, is calculated with the help of V,@ and is assumed to be equal to Fk(x)= Fk(X0) (vxFk(xo),x - xo>, then problem A will become a problem of the linear approximation with respect to the field. It should be noted that calculating the functionals Fk(x)in the domain n,,(~,)does not require a repeated solution of the boundary-value problems; it is merely connected with integrating the trajectory equations in the linear approximation field (84). Therefore, the numerical solution of problem A, which can be carried out by means of one of the packages for conditional minimization problems stated above, does not take much calculating time. The iterations performed while solving problem & . will be termed internal. a5x I b,

+

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

18 1

After the internal iterations are completed, the recalculation of the boundary parameters and conditions according to the increment vector Ax,, obtained from the solution to problem A takes place, and the analysis program that calculates true values of the functionals Fk(xl) for the new vector of the varied parameters x 1 = x,, + Ax,, is used. If the true values of functionals satisfy the prescribed conditions accurately enough, the problem is considered to be solved and the calculations are stopped; otherwise, new perturbation functions Zx,@(z, xo) are calculated, a linear approximation with respect to the field is built again according to (84), problem is stated and solved, and so on. The transition from the vector x,- to the vector x, (s = 1,2,. . .) will be called the sth external iteration. It should be pointed out that in the approach used, the boundary-value problem solution requiring the largest calculating expenditures occurs only with external iterations. The following considerations lead to the choice of components of the vector 1, = { E , , ~ ) : = defining the quasilinearization (or linearization) region

A

n, =

nxs(&s).

On the one hand, E ~ , ;should not be too small, since it results in a large number of external iterations and increases the total calculation time. On the other hand, an extreme increase of c,,; in order to reduce the number of external iterations may give the opposite effect: Due to nonlinearity of @(z, x) with respect to x, the functional values Fk, Fkwill differ greatly; this ultimately will slow down the convergence to the optimum in problem A. It is advisable to choose the optimal value of E ~ , ~for , example, in terms of the 20-3004 agreement between the values of “the most nonlinear” functional Fk(x), k = 0,. . . , I and its approximation Fk(x) in the domain FIs (Fedorenko, 1978). It is not difficult to see that the central place in the given calculation procedure is occupied by calculating the perturbation function vector Vx@. B . Integral Equations in Variations. A General Three-Dimensional Case

The problem of calculating potential perturbations caused by small variations of the calculation domain boundary and boundary conditions may be formulated (generally enough) as follows. In the region G with the boundary r (Fig. 4), the Dirichlet problem for the Laplace equation is solved:

A ( ~ ~ = o P, E G ;

(plr=uQ, Q E ~ .

(87)

The region G can be unlimited, with its boundary r consisting of a certain finite number of connected components ri.In the case of the unlimited region

182

V. P. ILTN rt UI.

/

FIG.4. Example of the boundary variation

C, the boundary-value problem (87) is supplemented, as usual, by the infinity condition: ' p p -+ 0, P -+ m. Consider the boundary distortion r (Fig. 4), which is the displacement of points Q E r by the small vector 6rQdependent on the position of Q on r.The perturbation of the boundary values UQ will be given by the diagram

Thus, we shall obtain the perturbed boundary value problem AGp==O, P E E ;

- ~GF.

i$1==UQ,

(89)

We must find the perturbation of the potential 6 q pat every fixed point P of the space that is due to the distortion of the calculation domain boundary and the boundary condition alterations. Up until now, the well-known approaches to solve the stated problem were based on the statement formulated in the gravimetric theory by Bruns (1876) at the end of the last century and later developed by Bertein (1947) in connection with electron-optics problems. In brief, the statement runs as follows. Suppose that the boundary r of the region G is a regular closed surface and that under distortions, its boundary conditions do not change (6UQ= 0).Then, if 16rQ/is sufficiently small, the perturbation of the potential 6q, is the boundary-value problem solution A[dy]

= 0,

P

E

G;

=

- ( V C ~ Q , ~ ? Q ) ,Q E r.

(90)

EMISSION-IMAGING ELECTRON-OPTICAL Sk STEM DESIGN

1x3

The application of the Bruns-Bertein lemma to the problem of electronoptical system tolerance theory is considered in the works of Sturrock (1951), Glaser and Schiske (1 953), Der-Shwarts and Kulikov (1962), Janse (1971), and a number of other authors. Typically enough, although they use the Bruns-Bertein method, most of the authors mention its limited character: It is only possible to get correct results for the regions having a simple enough configuration. For this reason, we shall consider the well-known disadvantages of this approach in some detail. 1. When realizing the method numerically, it is necessary to calculate the gradient V, of the unperturbed potential on the charged surface r, and subsequently to solve the boundary-value problem (90). Even for regular surfaces, this problem is rather complicated and time consuming. Particular difficulties arise close to angular points and surface borders where the charge density and potential gradient grow infinitely. This is usually the case in practical problems. 2. The method is actually inapplicable to the surfaces whose sides are both in the optically active part of the field. In this case, a jump of the normal potential derivative leads to the boundary-value problem with ambiguous boundary conditions. The following simple example will clarify the point.

Consider a thin conducting disc r,whose symmetry axis coincides with the Oz-axis. It is necessary to find the potential perturbation at the point A caused by a small displacement of the disc in the positive direction of the Oz-axis. The thickness of the disc at every point Q ( r Q < R ) tending to zero, two values of the normal derivative (&p/dnQ)' satisfying a jump condition have been found,

On the disc borders at r -+ R , the charge density oQgrows infinitely, and co.Thus, in the problem considered, the boundary conditions under the Bruns-Bertein method are two-valued at rQ < R and are singular at rQ 4 R. It is obvious that the introduction of the artificial thickness h cannot solve the problem, since with small h, the boundary conditions practically remain singular, whereas with large h, the initial statement of the problem is altered. Below, a general approach will be given for the calculation of potential perturbations free from the deficiencies mentioned earlier. It is based on the integral equations for Lagrange variations of the charge surface density associated with an individual surface point. We shall show that this method is applicable to a large enough class of surfaces with angular points and integral boundaries, It also permits an effective numerical realization. c?cp/SnQ

--f

184

V. P. I L I N er ul

In this section, a general three-dimensional case is examined; it is specified in Section 1I.C in connection with perturbations retaining or weakly disturbing the axial symmetry of the calculation domain (see Monastyrsky, 1980a,b; Kolesnikov and Monastyrsky, (1983). The solution of problem (87) is presented in the form of a prime layer potential

with the surface charge density oQthat satisfies the Fredholm integral equation of the first kind,

Here

r=

u r, N

is a set of surfaces (possibly nonclosed) permitting the

1=1

parametric presentation

r,:rI = r,(u, u),

(u,u ) E D,,

(94)

with D(I = 1,. . . ,N ) being a plane region of the parameters (u, 0). Later, as a rule, we shall omit the subscript 1, assuming that the parametric presentation of means a set of smooth vector functions r = {rl(u, v)};"= with

u N

the domain of definition D

=

D,,

I=1

r

On the surface coordinates

r, the

=

r(u, r), (u, u ) E D.

(95)

mapping (94) sets a frame of curvilinear Lagrange

r = r(u, vn),

r

= (un,

v),

( u g , on) E

D,

(96)

with the basis vectors ru(un,uo), rv(un,u,,), respectively. As known from differential geometry, dSQ = JQ d u dv,

where JQ = Iru x rvl = J E G vectors rurr,,,

E

=

(ru,ru),

-

(97)

F 2 is the area of the parallelogram built on the

G

=

(rc,rv),

F

=

(rurrv),

(98)

being the coefficients of the first quadratic form of the surface r. For simplicity, we shall further assume the system of curvilinear coor-

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

dinates on I- to be orthogonal. Then F = 0 on

185

r, and

Designating

write the main integral equation (93) in Lagrange coordinates (u,u):

O n the original surface r, consider the arbitrary vector function br, = dr(u, c) having piecewise continuous derivatives with respect to parameters u, c in the area D and a small quantity in the sense of the norm

and also the arbitrary scalar function 6UQ, which is small with respect to the absolute value Prescribe the distortion r and potential perturbation of the boundary U, according to diagram (88). In this case it is of considerable importance that the unperturbed and perturbed surfaces turn out to be parametrized in exactly the same domain D. A certain new perturbed charge density distribution c ? ~is realized on the surface f . It will read as

Now changing Eq. (101)with regard to (103), we shall obtain

duQGpQdu d v

=

6Up -

oe 6 G p , du du; P, Q E r.

(104)

Consider the kernel variation 6Gp,. in detail, presenting it in the form

where d,, S, are variation operations with respect to the points P and Q, respectively. From (100) we get

With regard to the geometry (Fig. 5), it is not difficult to form

186

V. P. IL’IN eta/.

FIG.5. On reducing integral equations in variations

from which, with regard to the equation r;Q = ( r p Q , r p Q ) , follows

epQdenotes the unit vector r p Q / r p Q . Using the permutability of differentiation and variation operations, from (99) and (loo), we obtain

The subscript Q means that the corresponding value is calculated at the point Q E r. Substituting the expressions for 6( l / r p Q )and SJQ into (106), we obtain

If the transformation r + is isometric, i.e., if d(dSQ) = 0, then the complete variation of the kernel CpQtakes a simpler form: 6GpQ

= TJQ( PQ

6rp

-

6 r Q ,e p p ) .

( 1 1-21

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

187

In particular, the orthogonal space transformations-the transfers along a fixed vector and the rotations about some angle-are isometric. If a small distortion occurs only on some part of the boundary r,,, c r, assume that

Sr,

= 0,

P

E

r\rv,,,

Sr,

= 0,

Q

E

r\rV,,.

We shall later assume that the main integral equation (93) has been solved and that the distribution of the surface charge density aQon the unperturbed surface has been found. By setting 6rp = 0, P E r, and the perturbation of the boundary potential SUP,P E r, the right-hand side in Eq. (104)is uniquely determined by virtue of (105), ( 1 lo), and (11 I), with the correlation (104) being an integral Fredholm equation of the first kind on the unperturbed boundary with respect to the perturbation Sa,. From (92), (105), by assigning a large value to Sr, = 0, we obtain the perturbation of the electrostatic potential ' p p corresponding to the given boundary distortion at the arbitrary point P of the space:

If only the boundary potential is modified, the second term in the integrand expression (1 14) will be missing. Consider the geometric parameters of the boundary D k , k = 1,. . . ,n, and those of the boundary conditions c I , 1 = 1,. . . ,m, which uniquely define the values of the electrostatic potential Up on r. Introduce the extended vector of the parameters, x

= (x1,.

. . ,X,J,+

1,.

. . ,x,+,),

letting x k = j j k , k = 1,. . . ,n, x , + ~= c l , I = 1,. . . , m . Using the variation parametrization method, consider the case of boundary and boundary conditions 6rp, 6Up, P E r, perturbations representable in the finite-dimensional form:

where AF' are geometric variational functions (g.v.f.) defining small boundary are potential variational functions (p.v.f .). transformations, whereas similar to (113). Particular The functions A$),p'p")are determined on r\rvar types of g.v.f. for some practically important special cases are given in the next section.

V. P. IL’IN et al.

188

Two simple examples will clarify the meaning of the values 6ck, pg’. 1. Let the potential cI of some electrode varied. Then SUP = p p 6 c l , where

1, P E p = { 0, p I

rl c r be the parameter

that is

r, r,

is a characteristic function of the rl electrode. 2. Suppose that the optimal (in some sense) distribution of the potential m

r, is desired in the form of expanding 1 pL(pk’ckby the comk= 1 plete system of functions s = { p $ ) } ~ 1=.

on the electrode

In this case, 6ck are variations of expansion coefficients, with the potential functions coinciding on I-, with the corresponding functions pL(p’E S. By substituting correlation (1 15) into (1 10) and (11 l), we get SpGpQ

=

,f (P(Pk:,hxk,

m

6pGPQ

$gk:,6xk.

=

k=l

k= 1

(116)

where

Thus, the overall variation of the kernel parametric form

2

6GP Q (& -k=l

6GpQ

is presented in the

+ *$$fixk.

(1 18)

The perturbation of the surface charge density do is uniquely related to the k = 1,. . .,n, perturbations of the geometric parameters of the boundary and of the potential parameters c k , k = 1 , . . . ,m. By denoting the Lagrange derivative oQby akoQwith respect to x k , k = 1,. . . ,n + m, we have

a,

n+m

6oQ=

Zk@Q

k= 1

6xk.

(1 19)

By substituting (118) and (1 19)into (104) and by using the independence of variations 6 x k ,we obtain integral equations in variations for the distributions of the Lagrange derivatives on the unperturbed surface r: r

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

189 (121)

The function

Fg),P E r is defined by the relation F$’ = -

s

CTQ(~FA+ ~ g h ) d u d v , k = 1,. . . , n.

(122)

If we substitute the perturbation dcpP at the arbitrary point P in the form similar to (1 19), we get k = 1, ..., n,

(123)

n

- 1, ...,m.

(124)

It is easy to see that for Example 1 analyzed above, the derivatives 2 n + k ~ P coincide with “unit” distributions of the potential q$?,which are the solutions of the boundary-value problems (125) Now some circumstances should be pointed out that are rather important for the numerical implementation of the developed approach. 1 . Equations (120) and (121) are integral Fredholm equations of the first kind with respect to the distribution of the surface prime layer density Zku, k = 1,. . . ,n + rn, corresponding to the boundary values of the potential F ! ) , k = 1,. . . , n , or pg), k = 1,. ..,m, on r. This fact allows us to solve them numerically with the help of practically the same algorithms and programs used to solve the main integral equation (93). 2. Equations in variations (120), (121), and the main equation (93) have the same kernel G p Q ; therefore, the matrices of their discrete analogies, i.e., linear equations derived as a result of some approximation (e.g., spline , and quadrature formulae appliapproximation of distributions C J ~f?kOQ) cation also d o coincide. It permits us to reduce considerably the extent of calculations when solving cathode-lens optimization problems. 3. Due to the definition of the functions AF’,the integral in (122) extends over the whole domain D only in the case when Plies on the varied part of the boundary r,,, c r. Otherwise (if P E r\rVar), the intergrand function is different from zero only in the domain D,,,c D. Figure 6 directly shows that the extent of calculations required for solving equation (123) depends essentially on c( = mes R/mes(T x r)= 25 - C2,where = mes r,,,/mes r is

<

190

V. P. IL’IN et ul

FIG.6 . On the structure of the integral equation in variations. rvar- the modified part of the boundary G ; 9 - the region where the kernel is different from zero.

the relation of the area of the varied boundary part to its total area. The smaller the “a”, the more effective the calculation of potential perturbation functions, under otherwise equal conditions, by the method described above.

C. integral Equations in Variations f o r Axially Symmetric Surfaces and Surfaces with Weakly Disturbed Axial Symmetry 1. The Axially Symmetric Case

First suppose that the surface

r=

u r, N

is axially symmetric about

1=1

the Oz-axis. Then the parametric equation

r may be presented in the form r = r ( 0 , ~= ) z ( z ) e , + r(z)e,(O)

(126)

the subscript 1 is omitted. As Lagrange coordinates on r, hereafter, we employ the azimuth angle H E [0,271] in the x0y-plane and the scalar parameter T varying in the domain

u N

R

=

[ T ~ ) , T $ ) ] ,where [ 7 y ) , ~ $ ) is ]

a parametrization region of the genera-

i= l

tor of the smooth component r, c r. The vectors ex, e,,, e, represent basis vectors of the Cartesian coordinate system e,(O) = cos 0 ex + sin de,. The coefficients of the first quadratic form are easily calculated from (126):

-

E

= r2(T),

G

= r’*(T)

+ z‘~(T),

F

= 0.

(127)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

191

The prime in (1 27) and throughout this section means differentiation with respect to 7. By writing tQ = V’rG + z g , we obtain from (loo),

Consider the class of boundary perturbations retaining axial symmetry:

Sr(B, r ) = S z ( ~ ) e+ , 6r(7)er(8).

(1 29)

It is easy to see that in the axially symmetric case, the geometric variational functions introduced in the previous section are identically defined by the functions it’(z), j.t)(r), k = 1,. . . ,n, which correspond to the small perturbations

of the generator L on the surface G. Thus, I‘k’(B,r ) = Ap)(r)ez + i[’(z)e,(B).

( 1 30)

The functions A?), j.[) will be called geometric variational functions of the axially symmetric problem. The subscripts of independent vector components x and the sum symbol in appropriate expressions will sometimes be omitted for the sake of brevity. As is known (see Antonenko, 1964), in the case of axial symmetry r, the integral equation (93)may be reduced to read:

where G,*, =

4rQtQ

GPQ~ Q Q -X(kgQ), ppQ

PPQ = J ( r P

+ rQ)2 + (zP

- zQ)2

>

(133)

With X(kgQ)being an elliptic integral of the first kind with the modulus

192

V. P. IL'IN et a[.

and by using the operation permutability of variation with respect to x and integration with respect to 0, from (1 18), we obtain = ((PP*Q +

(135) 6x. : ~ can be obtained both from (134) by The specific expressions ( P and integration and directly from (133) by varying the kernel G & with regard to the well-known properties of elliptic integrals (Gradshtein and Ryjhik, 1956; Dvait, 1973). Both these methods lead to the same result: 6G,*Q

$;Q)

where

with X i Q ,E~~ being elliptic integrals of the first and second kind with the modulus k;Q. The equation in variations (120)for ?koin the case considered will read (the subscript k is omitted) f

doQG,*Qrl~ = F;;P, Q E L ,

where the right-hand side is defined by the correlation 9; =

-

Having solved integral equation (1 39) one could use (1 23) to calculate the desired perturbation function of the axial-potential distribution r

I

It can be easily shown that in the case where P is on the symmetry axis Oz(rp = 0, z p = z), with Q passing L ,

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

193

where

Provided that the parameter z on the generator L is normal (i.e., it coincides with the arc length to within the constant), tQ = 1 on L. Then, if the class of small boundary perturbations under study does not change (in the first order) the arc-length element (6(dl) = 0), the equation qQ = 0 holds everywhere on L . Such perturbations will be called quasiisometric. For instance, orthogonal transformations (transfers, rotations) of generators in the plane (r,z ) are quasiisometric. 2. The Weakly Disturbed Axial Symmetry Case

Now let us analyze integral equations in variations for surfaces with weakly disturbed axial symmetry. Consider an arbitrary nonaxially symmetric perturbation of the surface prescribed by the vector function 6r(B, z)

6z(H, z)ez

+ 6r(H,r)er(0),

(144) which is small in the sense of a norm C1{[0,2n] x Q}. For simplicity, we assume SUP = 0 in (105). Assuming that the surface distortion is due to small perturbations of a finite number of independent geometric parameters x l , . . . ,x,,, we shall reduce the problem to a finite-dimensional one by assuming =

As above, j.:’), ,if) will be called geometric variational functions. Using the periodicity with respect to O,, we present the g.v.f. as the Fourier series ii“(6,

7)

[cyi:i(z)

=

cos 18 + bjfi(.r) sin 161,

I

if)(fl,

7) =

C [ X ! : ~ ( T cos ) l B + b;lL(z) sin 101.

146)

1

The relations (144)-(146) define a rather large class of boui-lary perturbations weakly disturbing the axial symmetry. The coefficients at:,ctg,: in (146) correspond to the axially symmetric perturbations considered earlier. To illustrate this, consider the g.v.f. corresponding to the types of nonaxially symmetrc perturbations occurring in practice most often, viz., shift, axis distortion, and elliptic distortion.

I94

k

=

V. P. IL’IN et al.

1. The I(cos U , cos [I’, cosy) vector-directed shift by a small value 6.u:

CXti(T) = COSU,

h(;\(T)

=

COSP,

X(‘0.1 )

-

cosy.

(147)

The remaining coefficients in (146) are equal to zero.

k = 2. The Oz-symmetry axis rotation by a small angle 6 y with respect to the point (O,O,zc)in the plane n, which makes the angle qowith Ox-axis (Fig. 7): c~:*,\(T)

=

~x(lf)z(z)=

(z(T) - z,)c0s2q0, -

r ( r )cos cpo,

6f\(T)

h ‘ & ( ~ )=

-

=

(z(T) - zc)sin2qo,

(148)

r(r)sin cpo.

The remaining coefficients in (146) are equal to zero. k = 3. The elliptic distortion converting the circumferences of the radius T ( T ) in the cross section normal to the Oz-axis into ellipses with the semi-axes a(?) = r(z)(1 + ( ( T ) ~ X ) ,b(z) = r(r)(1 + ~ ( z dx), ) where ( ( T ) , V ( T ) are arbitrary smooth functions of the z parameter:

The remaining coefficients in (144) are equal to zero. The meaning of the angle cpo, which can also depend on T , is clear from Fig. 8. The problem now is to specify the integral equation (120) in connection with the perturbation class considered in (144)-(146). If we integrate the

“t

FIG.7. Example of rotation (axis distortion)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

195

FIG.8. A case of elliptic distortion

relation (1 18) deduced above with regard to (146), then for the overall variation of the kernel 6GpQ with respect to the variable OQ, we shall get JO2'

6GpQdbQ=

1

+ $$jk)COS

[(f$Ihhk

16,

I k=l

+ ( ( p $ J k + $$hk)sin I $ ]

6xk.

( 150)

The functions q k f , $ $ f ( i = 1, 2) are coefficients of the Fourier integrals

If we now assume ?ko

=

I

c , , k ( T ) cos !w

+ Dl,k(z)sin !w

(151)

and use the independence of the parameters x k from (120), we shall obtain a chain of integral equations concerning the coefficients C l , k ( ~Q) k, ( 7 ) ; jcl,k(TQ)G:b

In Df,k(7Q)G!&

d7Q

=

8!:i(p)? (152)

d7Q

=

%!:i(p).

V. P. IL'IN et ul.

196

It is easy to see that in (152),

The potential perturbation in an arbitrary point P of the space may be found by solving the integral equations (152) in the following way: Let ? k $ p be the potential perturbation function with respect to the parameter x k ,while ,41,k(rp, z p ) , B l . J r P ,z p ) will be its Fourier coefficients with respect to the variable dp. By integrating with respect to dQ, we obtain r r

The perturbation of the potential 6qPin an arbitrary point P may then be calculated to within the terms o(6xk) by the formula

i = 1,2; k = 1,..., n, 1 = O,l, ..., in (152) The kernels GgL, cpg$ + $$,: and ( I 53) are easily shown to have a logarithmic singularity for Q + P on r. As is known, near angular lines and edges, the surface charge density oQincreases as O ( t - & ) for z + 0 (0 I E 5 1/2); therefore, Eqs. (153) are single-valued functions of P limited everywhere on r (including near angular lines and edges). Thus, the integral equations (152) are equations with weak singularity. The theorems of asymptotic functions of the solution near specific boundary points as well as numerical methods developed enough to solve integral equations of the first kind are applicable to them, in particular. The analysis of perturbations of the higher-order potential with respect to dx may be performed by using a similar scheme, but it requires more complex computations and, therefore, is not given here. 3. The Types of Geometric Vuriations for Optimization Problems We shall briefly describe geometric variational functions specially designed to solve optimization problems of axially symmetric cathode lenses. Consider four types of g.v.f., with each being assigned the ordinal number s = 1,. . . ,4. The following notation will be used: x is a modified geometric parameter whose variation 6x may correspond to any type of g.v.f. considered (the existence of several perturbation types on one electrode at a time is permissible); I C ~(s = 1,. . ., 4 ) are connection coefficients of variations corre-

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

197

sponding to a given parameter x. (Connection coefficients maintain the boundary continuity in conjugation points of electrode generatrix.) Here are the formulae for g.v.f. of each type. s = 1. The transfer of the electrode generatrix [ I , 53 along the fixed vector l(L, I,) (Fig. 9a):

i."' =qlz,

i(*) =ql,.

(156)

s = 2. The rotation of the electrode generatrix [ I , J ] about the fixed point J by the angle q (Fig. 9b):

r

r

B

Z

C

FIG.9. Types of geometric variations.

d

V. P. IL’IN et ul

198

3. The electrode expansion (compression) along the rectilinear generator [ I , 51 (Fig. 9c):

s =

j”(Z)

=

j V W=

“3YznQ5

(158)

K3YrnQ.

+

Here y = ( y z , y r ) is a guiding generator vector; nQ = z,(l - z) (1 - z*)z are variable coefficients of linear distortion; z E [0,1] is a parameter on the generator [ I , 53; t* is z,the parameter value corresponding to the fixed point (point 1 in Fig. 9(c)). s = 4. The alteration of rJ = const [Fig. 9(d), (l)].

the spherical generator radius, provided that The modified parameter x = R is perturbed in such a way that r-the coordinate of the point J-remains fixed. The generator g.v.f. [ Z , J ] have the form

the parallel transfer (s = 1) with the guiding vector I = ( - 1,O) and the connection coefficient has been defined on the generator [ J , L]: K1 =

R

-

,I’R~ - rJ2

JR

-

r:

The linear distortion (s = 3) with the guiding vector y = I, the connection , the fixed point T, the selection of which usually coefficient ic3 = K ~ and depends on the convenience of describing the boundary of a specific EOS, has been defined on the generator [T,L ] . s = 5. The alteration of the spherical generator zj = const [Fig. 9(d), (2)]. The varied parameter x =

radius, provided that R is perturbed in such a way that the z-coordinate of the point J remains fixed. The g.v.f. of the generator [ I , 53 have the form ~ ( 2 = ) 0, j.(r) = (161) Q/ Q .

The linear distortion (s = 3) with the guiding vector y = (0, l), the connection coefficient t i 3 , and the fixed point L has been defined on the generator [ I , L ] . The last two types of the boundary perturbation given above are used when varied geometric parameters include the curvature radius of the cathode, fine-structured mesh, or screen surface. D. Vuriations of’ the Limiting Paraxial Equation Solutions In this section, we shall establish relations between small variations of electrostatic and magnetic fields and perturbations of some image character-

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

199

istics. These relations have the form of linear functionals depending on the potential perturbation functions. First, we shall consider the case of smooth fields and then extend these results to the case where fine-structured meshes are present in the optically operating part of the field. 1. Variations of Paruxiul Trajectories in Smooth Fields Let @ ( z ) and B ( z ) be axial distributions of the electrostatic potential and magnetic induction, respectively. As is known (see Shapiro and Vlasov, 1974; Kel’man et al., 1973; Monastyrsky, 1978), the limiting paraxial equation of the trajectories

has linearly independent solutions u(z), w(z), where regular function satisfying the equation

1;

=

&<, < = < ( z ) is a

e l

5 = 0. L o [ ( ] = a)(‘‘ + -aft’ + - 3@” + -@ (163) 2 4 2m Due to the fact that @ ( z ) is analytic in the vicinity of the point z = 0 and @(O) = 0, the functions <(z), w(z) are identically defined by the initial conditions

’(

By changing <,w, @, & inI(162)-(164), we obtain the following equations for the variations dw, S ( : Mo[dw]

=

-M;[w]

e 4m

- -BwSB,

(165)

and the initial conditions

(166) 6 W ( O ) = 0,

IdW’(O)( < m

(the superscript “el” denotes the electrostatic part of corresponding operators). We shall omit the calculations and shall give only the expressions for the regular solutions S<(z), S w ( z ) of the equations (165) satisfying the

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V. P. IL’IN et al.

conditions (166)

The function 6 F ( z ) is defined by the correlation

Taking the relation 6v

= &6(

1 6@ . + -5into account, we obtain

2 &

The correlations obtained make it possible to calculate the perturbation functions of the first-order characteristics of cathode lenses with combined fields. Using the parametric representation of the perturbations 6 0 , 6 B in the form

and passing in (167) and (169) from the variations to the derivatives with respect to xk, we obtain

where

If the magnetic field is fixed or if its variation is not related to the parameters xk,the term with a,$ in (171) is missing. In the “electrostatic” case (~= 4 0), expression (171) is essentially simplified:

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

201

Varying the identities with respect to X,

dx,z&))

= 0,

14% z&))I

=

Mg

>

which, depending on the x vector, define the position of the Gauss plane zq and the electron-optical magnification M,, we obtain (174) denotes the function gg =

sgnw,

=

1, w9 > 0, - 1, wg < 0.

(175)

By setting z = za in (171) with regard to the Lagrange-Helmholtz correlation in the Gauss plane L& = l/w,&,, we find from (174) the unknown derivatives of the first-order functionals with respect to the parameters xk:

Similarly, one can obtain a derivative of the image rotation angle ADS with respect to xk in the Gauss plane.

In (176) and (177), the following notation is used: Qq =

O(z,),

=

B(z,).

( 1 78)

It is readily seen that with respect to xk,the expression for the derivative of another practically important functional, i.e., the crossover plane position zcr found by the equation w [ x , z , , ( x ) ] = 0, is obtained from (176) as a result of the formal replacement z g + z,,, c -+ w : C'kZ,,

=

-

where vCr = ~(z,,),mCr= @(z<,) are written. 2. Jump Condition

Let us now examine the behavior of the variations of solutions for the limiting paraxial equation (2.80) in the case of nonsmooth axial distribution of the electrostatic potential @(z).

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A discontinuity in the field strength vector is characteristic of the class of electron-optical systems containing a fine-structured mesh, provided that the latter is interpreted as a continuous electrode “transparent” for electrons. The calculation of derivatives of electron-optical functionals with respect to the parameters varied in this approximation demands that additional correlations (jump conditions) on the grid surface be taken into account. First obtain a general jump condition for the variations of solutions of an arbitrary vector differential equation depending on the parameter. The righthand side of the equation is supposed to be discontinuous of the first kind on some prescribed plane also depending on the parameter. Consider a vector differential equation in the normal form P

= f ( 7 , P, x)

(180)

with the initial condition P(0, x) = P O ( X ) . (181) Here p = ( p l , . . . , p , ) , f = ( f i , . . . , f n ) are n-vectors, 7 is an independent argument, and x is an arbitrary scalar parameter. A dot above p means differentiation with respect to 7. Let the function f ( z , p , x ) have a finite discontinuity on some smooth surface S prescribed by the equation S(T,P,X) =

0.

(182)

Let us define the domains

and assume

f - ,f’ are supposed to be smooth enough functions of their arguments. For the sake of definiteness, let the initial point p o ( x ) be in the domain 0, for the x values, from some fixed neighborhood U = {x: Ix - xoI < r ) . The solution of the Cauchy problem will be denoted by p - ( z , x ) ,

d

=

f - ( z , P, x),

P ( 0 , x) = P&).

(185)

Let z* = z*(x) be the argument value corresponding to the hit of the phase trajectory (185) onto the surface S. Then the relation gC.r*(x), P-(t*(x)> XI, X I = 0

is identical with respect to x E U.

(186)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

203

Let us find the phase trajectory continuation (180) through the surface S, assuming p(7, X )

= p'(7,

X),

T

2 T*(X),

X E

u.

(187)

The function p ' ( 7 , x ) satisfies the differential equation P=f+(7,P,4

(188)

P + C ~ * ( X ) , X ~= P - [ T * ( X ) , X I .

(189)

with the initial condition on S,

Thus,

Studying the dependence of p ( 7 , x ) on the parameter x in the vicinity of the value x = x o , we shall call p ( z , x o ) the support trajectory. According to the well-known Poincare theorem of the smooth dependence of the equation solution (180) on the parameter x in the smoothness domain f ( z , p , x ) on the right-hand side (e.g., in D;),the solution p ( 7 , x ) = p - ( r , x ) , where T I T*(x), may be presented in the form P-(7,X)

=

P-(7,xo)

+ P i ( T , X o ) 6 X + -21p i x ( 7 , X o ) 6 X 2 + ....

(191)

The functions p ; , pLX entering the expansion (181) are derivatives of the corresponding orders of the solution p - ( 7 , x ) with respect to the parameter x when x = x o ,

and satisfy linear differential equations in variations. For example, p , ( ~ , x ) is the solution of the equation

Pi

=

~JOL

+ f;c71

( 193)

with the initial condition PL(Q x o ) = P,O(Xo).

( 194)

The matrix f,[z] and the vector f ; [ s ] are calculated on the support trajectory P - ( T , x o ) . It is readily shown that in the domain D:, the trajectory p ( ~x), = p ' ( 7 , x), for 7 2 z*(x,), may also be presented in the form P'(T, X)

= p ' ( 7 , Xo) f

P:(7, X o ) 6 X

+ -21P z x ( Z ,

Xo)6X2

+ . . .,

(195)

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V. P. IL‘IN et al.

. . satisfying equations in variations in the with the coefficients p ; , domain 0 : . For example, p:(z, xo) satisfies the equation

Li:

=

f,”tlP:

+ f:czl.

(196)

As in (193), the matrix f’l[z] and the vector j ’ ; [ z ] are calculated on the support trajectory P + ( T , xo). In order to continue the derivative with respect to the parameter p x ( z ,xo) in the domain D:, one must know the initial condition for equation (196) on the discontinuity surface S when T = z*(xo). Let

In general, ( p , ) ; # ( p , ) : ; nevertheless, there is a relation between these limiting values. Varying (186) with respect to x, we obtain g,&* -k g,[f-

6T*

+ p i 6x1 + g,6X

= 0.

(198)

Similarly, it follows from (187) that

f-6z* + p , 6x

= f’

6z* + p , 6x.

(199)

All the values in (198) and (199) are calculated on S when z = T*(x,), x = xo. The geometric sense of the variation operations performed is shown in Fig. 10.

FIci. 10. Jump condition derivation. 1 - the support trajectory; 2 - the perturbed trajectory: S - the discontinuity surface; 6 p - , 6p’ - solution variations before and after the discontinuity occurred; f - ,f’ - phase velocity vectors; 6;z* the time of the support trajectory arrival at the discontinuity surface. ~

EMISSION-IMAGINGELECTRON-OPTICALSYSTEM DESIGN

205

Omitting 67* from the last two equations (assuming that gr + ( g p ,f # 0), we obtain the conjugation condition ( p , ) S , (p,): on the surface S (jump condition):

->

In particular, if the right-hand side of Eq. (180) is continuous on S, the continuity of the derivative px(7,x,) follows from (200) when passing the discontinuity surface: ( P X G

(201)

= (P,)S'

Note that the continuity is also maintained in the case where f' # j - , with respect to Sz*(x,) = 0, which is equivalent to the relation Yx

+ (Yp,(Px)S)

= 0.

(202)

The condition (202) holds, for instance, if g, = g p = 0, i.e., the surface g = 0 is obviously independent of the parameter x and is a plane parallel to the space { P > . For the sake of definiteness, assume that the axial potential distribution has the form

As stated above, we consider the "outer" optical action of the grid, neglecting the microfield influence of its cells. Thus, the region "beyond the grid" represents an equipotential space with = @-(zg(x),x), whereas the equation of the discontinuity the potential surface of the strength vector z - zq(x) = 0 explicitly depends on some geometric parameter x. If there is a magnetic field ( B = 0), the trajectories of charged particles in this region are straight lines. In order to investigate the character of transfer of the derivatives d,v, ?,w through the discontinuity surface z - z,(x) = 0, we introduce the vector of the phase coordinates (p1,p2),p1 = r, p 2 = r' and the vector of phase velocity ifl3f213

and we reduce the paraxial equation (162) to the system of equations

Here a dot over ( p 1 , p 2 )means differentiation with respect to the independent argument z = z.

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V. P. IL'IN et al.

The equation on the right-hand side of the discontinuity surface (205) will be written in the form

Now use the general jump condition (200), noting as a preliminary that in the case considered,

f,

=f:lS,

1 @'

By substituting (207) into (200), we obtain

Returning to the variables r, r', z, we obtain transfer conditions of the derivative 13,u through the discontinuity surface S:

Similar relations also hold for d,w; one must only replace v by w in (209). Thus, in order to calculate d,v+(z, x),, dxw+(z, x) in the domain z 2 z,(x), it is necessary to use the relations of the previous section and the jump condition (209). Since in the domain z 2 z,(xo), @(z, xo) = @, = const, provided that there is no magnetic field, ZXE+(Z, Xo)

iiXW+(Z, xg)

= (d,u),' = (d,w),'

+ (d,u')s+(z

+ (d,w');(z

-

zg(xo)),

(2 10) - Zg(X0)).

The right-hand sides of the expressions in Eq. (174) for the first-order functional derivatives must be provided with the superscript (+):

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

207

The specific formulae for transforming aberration coefficients of cathode lenses on the discontinuity surface of the electrostatic field strength vector (on the grid) may be obtained from the general correlations (198) and (199) by the varying operation. The following jump formula for the distortion coefficient E ( z ) could be given as an example:

(a,+(a;> =

M!

(E’);

= (E’);- __ [(w”),’ - (w;)-],

2 4 where R, is a grid curvature radius.

E. The Problem of Axial Synthesis of Cathode Lenses

In this section, we shall discuss the problem of axial synthesis of cathode lenses, which in accordance with the classification given in the Introduction, will be later called Problem 1. There are two groups of publications devoted to it. The first one includes theoretical investigations of extremal properties of some types of aberrations and the appropriate stationary fields (e.g., Kas’yankov, 1950; Seman, 1953; Glaser, 1938). The problem of this type was first considered by Glaser and Sherzer in connection with the problem of correcting the electron microscope spherical aberration. The second group includes the works concerning the construction and implementation of numerical methods intended for solving Problem 1. Note the “direct” method (Kas’yankov and Taganov, 1964), in some cases permitting the reduction of Problem 1 to a nonlinear boundaryvalue problem for the system of ordinary differential equations. The essential limitation of the method is its fundamental inability to take into account the conditions of the inequality type connected with the technical realizability of the optimal solution. On the whole, the same limitation is also typical of another approach based on the employment of classical variational calculation. One should especially single out the work by Tretner (1959), which employs the method of conjugating Euler equation solutions (classical extremals) with the optimal parts of the solution lying on the permissible domain boundary. This method is conceptually close to the modern theory of extremal problems. Most of the works in this group deal with methods of reducing Problem 1 to a finite-dimensional problem of nonlinear programming by different parametrizations of axial potential distribution (Cheremisina and Kas’yankov, 1970; Orlov, 1967; Shapiro, 1964; Shapiro and Vlasov, 1974; Szilagyi et al., 1984).

208

V. P. IL’IN et a!.

The indicated approach is, without doubt, more universal than the approaches mentioned above; however, even here, it is connected with considerable difficulties in taking into account the conditions for the potential or field intensity (of the type U, 2 @ ( z ) I V z , l@’(z)l I E,,,, etc.), which occur in practice very often. The approach is also limited by the dependence of the found solution on the concrete form of parametrization, which frequently does not allow us t o single out explicitly the limiting properties of fields, which are physically inherent in the present problem. Besides, as was shown in Fedorenko (1978), the application of the methods of penalty functions (or their modifications) for the numerical solution of problems of nonlinear programming remains rather a complex problem in itself. Thus, the analysis performed indicates the necessity to substantially expand the complex of the employed computational methods with the aid of the mathematical apparatus that, in combination with results already achieved, would allow us to approach real physical statements and which could become a basis for the creation of applied program packages for the synthesis of cathode lenses. The primary objective of this section is to show that the apparatus of the contemporary theory of optimal processes, which represents a set of theoretical and numerical methods of analysis of general extremal problems, meets the specified requirements. The application of the theory of optimal processes in the most diversified regions of applied research is dealt with fairly well in monographs. Let us note that the indicated apparatus is, in a sense, natural for the problems of cathodelens synthesis, since it is convenient to treat the latter as the problems of controlling a certain set of trajectories of charged particles with the help of electric and magnetic fields (see Ovsyannikov, 1980). 1 . Reduction to Optimal Control Problems. Pontryagin’s Maximum Principle

It was pointed out in Section 1I.B that the set of differential equations both for t-variations and the conversion formulas (108),(114),and (115)-( 120)fully defines the electron-optical properties of the cathode lens in aberrational approximation. Regarding @(z) as the control (further on, the concept of control will be somewhat refined), this set of relations will be referred to as the controlled aberrational model of the cathode lens. Thus, the coefficients of expansions (106) and (107), as well as their arbitrary finite functions, are functionals of @(z) and may characterize the control quality. Functionals important from the practical point of view are: the position of the Gaussian (image) plane z g , the position of the crossover plane zcr, the electron-optical magnification Mg , the coefficients of spatial aberrations r I 2 ,

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

209

r133,r333,which characterize, respectively, the radius of the scattering circle in the center of the image field, the meridional and sagittal curvature of the image surface, and the distortions in scale; the temporal aberration coefficients z,, T ~ T ~~, ,t13, , zj3, upon which temporal resolution of the streak-tubes electronoptical systems depends considerably, and others. Let 2B = {@(z)} be some class of functions that will be considered as the axial distribution of the potential (this class will be treated in more detail below), and let F , [ @ ] , . . .,F,[@] be the continuous functionals on '113. Let us formulate a general statement of the problem of axial synthesis in the form of the following extremal problem: Find the function Qo E 2B that minimizes the functional FO[@] at the constraints.

&[@I= 0,

k

Fk[@]
k = l + l , ..., m.

= 1,..., 1,

Let us give examples of two typical axial-synthesis problems that are of practical significance. Problem C,. The cathode lens is equipped with a flat cathode and a finestructure flat grid with the potential U, prescribed on it, which is placed near the cathode at a distance d (the field between the cathode and the grid may be considered homogeneous). It is necessary to find the axial distribution of the potential @(z)( z 2 d ) , which realizes the minimum length of the cathode lens (zCe-+ min) at a given magnification M 9 , the accelerating potential U,, and the field intensity on the axis that is limited in the absolute value: lW(z)\ 5 Em,,, Em,, is prescribed. Problem C,. We have to determine the axial distortion of the potential @ ( z ) that makes the mean curvature of the image surface (l/lRmdl + min) minimal in absolute value at the magnitude of distortion on the edge of the image field r = r o , which does not exceed the prescribed value of A%. the fixed first-order parameters zq and M,, and the accelerating potential U,. The cathode is supposed to be flat and equipotential; the field intensity in the cathode center is limited from below: 0;2 Emin,Eminis prescribed. Problem C , arises in developing cathode electron-optical systems with subpicosecond temporal resolution; problem C2 represents a high-priority task from the viewpoint of improving the quality of the image of cathode lenses with flat cathodes. Further, we shall show that the apparatus of the theory of optimal processes makes it possible to consider similar problems in a uniform way. Let us perform the reduction of axial-synthesis problems to the problems of optimal control of the dynamic systems. First of all, let us point out that the

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V. P. IL’IN et al

general problem is not completely defined, because the class of permissible functions YB is not singled out explicitly. In this connection, it is necessary to specify what is meant by “control.” Since also appearing in the equation of the controlled aberration model, along with the axial potential @(z), are the derivatives @ ( k ) ( z )k, = 1,. . .,n (in the approximation of the third-order aberrations, n is no greater than 4), it is natural to choose the higher derivative of the potential @(“)(z)as the control u(z). In the case of this selection of the control, the derivatives of the potential in the center of the cathode clk = k = 1,. . . ,n - 1 are the controlling parameters and have to be determined, along with u(z), as a result of solving the problem. Let us introduce the vector 2 = ( x k } $ = and refer to the complex ii = {u(z),a} as the generalized control. In the cases where it would not lead to misunderstandings, we shall use the term “control” for ii as well. It is easy to show that, at u(z) = @ ( “ ) ( z the ) , equality

@t’,

is valid. Certainly, one would like to use the class of analytical functions in the capacity of the class ‘113. However, in the theory of extremal problems, it is well known (see Ioffe and Tikhomirov, 1974)that in smooth problems of the type in (213), the lower bounds of the efficiency functionals are generally realized on the nonsmooth functions @‘(z), which can be approximated (in a weak sense) by a sequence of smooth (even analytical) distributions @“(z)to any preassigned accuracy. In this case, the following equalities

are accomplished. In conformity with the principle of expansion of the range of solutions for variational problems (Fedorenko, 1978), it is reasonable to discard the analyticity of the desired limiting distributions of the potential and to choose, as the space of controls, the class u = { u ( - ) } of bounded measurable functions L,. which is the most convenient one from the point of view of the existence of the solution. Note that at u ( - )E L,, the relation (214) ensures the continuity of the potential and its derivatives up to ( n - l), including, in this case, 2u = c”-1 . Problems of numerical modelling of a technically realized system of electrodes, meeting (within prescribed errors) the conditions of the problem, are closely associated with constructing the approximations @,,(z) in the class of analytical functions and their extension to the volume of the cathode lens.

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

2 11

The above problems are considered, in particular, in the works of Shapiro and Vlasov (1974) and Cheremisina and Kas’yankov (1970). c U in the following way: Define the class of permissible controls UCA)

u(,.l’= {u(.) E L,:

).I,

lu(z)l I

(216)

where i. is the parameter limiting the “power” of control. Assume that the vector ct varies in the closed set R that defines the range of values of the derivatives Or’ ( k = 1,. . . , n - 1) permissible under the conditions of the problem. Thus, the set of extended controls {i;} represents the direct product

6,= up x n.

Now, the class of permissible controls is chosen. In order not to complicate the presentation with insignificant details, we shall show the feasibility of a reduction of the axial-synthesis problems to the optical-control problems in the example of Problem C , . Considering the homogeneity of the field between the cathode and the grid, it is not difficult to obtain the initial conditions for the function p satisfying Eq. (31) in the domain z 2 d :

The symbol “+” designates the right-hand (at z -+ d + 0) limiting values of the corresponding functions on the grid surface. The microlens action of the grid is supposed to be negligibly small. By using the Lagrange-Helmholtz relation, we obtain in the Gaussian plane ( z = z ~ ) ,

~ “ ( ~=9 0,1

P X Z ~ =) -1/MS,

V,. (2 18) For definiteness, we shall further consider the case of the first focus. Let us introduce the dimensionless quantities t=-

Z-d d ’

@

’=-’u,

@(zg)

Uj’4 xl=- d

=

P C ,

(219)

In conformity with the theory of dynamic systems, we shall refer to the independent variable t as “time” and denote the derivatives of this variable by a dot. This will not lead to any confusion, because no derivatives with respect to the real-time T will be encountered from here on. Let us consider the phase space of the vectors x = {x,,x,,y) and the vector of the parameters ct = {ul}, which in the given case, consists of one

212

V. P. IL‘IN et al.

component. Set x2 = il,u = j , a, = u(O),and re-arrange Picht’s equation (31) in the form of the controlled dynamic system of a normal type x1 = x2,

xg = u,

with the initial conditions at t = 0, X,(O) = 2,

XI@) = 1

+ -,2 (x

x3(0) = y(0) = 1.

(221)

The boundary conditions (118) will assume the form

xl(T) = 0,

x 2 ( T )=

-W”4/M9,

X3(T) = y ( T ) = l / o .

(222)

It is convenient to present the equations of the controlled system (120) in a more compact and general form, .i = f ( x , u), x(0) = xo(m).

oa

The generalized control u” = { u(-),m} belongs to the class = Ug’) x Q, where Ug’ is defined in conformity with (216),whereas the set R is the segment 1x1 2 1, of the real axis. At the prescribed permissible control ii cod, the trajectory of the dynamic system (220),(221)unambiguously determines the electron-optical parameters of order unity of the cathode lens under consideration. Define the following functionals on the system trajectories (220):

and formulate the corresponding optimal-control problem. Problem C:. It is necessary to find both the optimal control Go E fiaand the optimal trajectory x o ( t ) that corresponds to it by virtue of (220) and (221), which ensure the functional F a minimum, when satisfying the conditions Fk = 0 ( k = 1,2).

EMISSION-IMAGINGELECTRON-OPTICALSYSTEM DESIGN

213

It is evident that the problems C and C* are equivalent. The latter represents a typical problem of optimal high speed and has a fairly simple geometric meaning: It is necessary in the course of the minimum “time” Tn to transfer the phase point x, in agreement with (220),from the segment N o N , (see Fig. l l ) , defined by the parametric equations (221), to the point P with the coordinates (0, - U ~ / ~ / M I/#).~ , Using the corresponding equations of the aberrational model, any axialsynthesis problems of the type (213) may be analogously reduced to the problems of optimal control of the dynamic systems (naturally, with different f’(x,u), xo(a), {J$}). Thus, this opens up the posibility of an active application of a fairly well-developed theory and effective numerical methods of optimal control based on the ideas of Pontryagin’s maximum principle for the solution of the problems of synthesis. We shall emphasize that a priori conditions of a technical nature, such as the presence of grids or apertures, the curvature and distributed cathode surface conductivity, limitations on the field intensity or on the potential, etc., may be taken into account in a natural way by a suitable choice of a set of admissible generalized controls. All this is fairly essential in order to realize the estimated optimal solutions in a practical way. Now we shall present the formulation of Pontryagin’s maximum principle i t . , the necessary requirement of optimality in the problem of optimal control with parameters. Let us consider Cauchy’s initial value problem for the controlled system

a?,

i = f ( x , u, a),

x(0) = xo(a).

% FIG. 11. Reduction to the optimum high speed problem.

(225) (226)

214

V P IL'IN

el

al

Here u ( t ) = { u l ( t ) ,. . . ,u,(t)} is the control vector, cr = ( a l , .. . ,aq)is the parameter vector assuming values in the closed bounded sets U c R' and R c R4, respectively; x = (xl,. . . ,x,) is the phase vector that characterizes the state of the system. In conformity with the adopted notation, the set of generalized controls has the appearance U, x 0, where U, = E L,: u(t) E U}. At prescribed ii E U,, the trajectory x ( t ) of the system (225) is defined uniquely. The quality of the control will be estimated through the system of functionals ( F k } ,k = 1, ...,HI:

ai

)a(.{

Fk[u('), a] = G k ( x ( T ) , a )

+

sb

g k ( X , U ,2)d t ,

(227)

under the assumption that the functions gk, Gk satisfy the conditions that ensure the differentiability of { F k ) in the sense of Ioffe and Tikhomirov (1974). Let us formulate the necessary conditions of optimality of control u' in the following problem: The minimization of the function Fo[u(-),cr], i.e.,

a,,

F,[u(.), a ] + min at {u(.), a ) E (228) on the trajectories of the controlled system (225) and (226) at the additional constraints F k [ u ( ' ) , U ] = 0, k = 1,. . . ,m. (229) Let us consider the set of canonically adjoint equations (n-system),

with a Hamiltonian (Pontryagin's function) m

2(x?

$9

u, = ( f , $)

+ k2= O Y k g k .

(231)

Theorem (Pontryagin's maximum principle). Let 'i = {u"(-),cr"} be the optimal control in the problem (228), (229). Then there exists a vector y = ( 1, y I , . . . ,),y such that the function X [ x ' ( t ) , $'(t), u, a'] on the optimal control u o = u'(t) attains a maximum over, and with respect to, the set U : ~

. 8 [ x o ( t ) ,$ O ( t ) , uo(t),a'] = max %[x'(t),

t,P(t), u, M " ]

utu

(232)

at almost all t E [0, T I . At the right-hand end of the optimal trajectory at condition for the adjoint vector $ is fulfilled:

t =

T, the boundary

(233)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

21 5

In this case, the optimal value of the parameter c1 = zo is determined from the so-called generalized conditions of transversality (Ioffe and Tikhomirov, 1974; Fedorenko, 1978):

where Q = xOais a matrix, n x q in size, and 8cr is the arbitrary permissible variation at the point ci = ci0 over the set SZ. The indices x, u in (233) and (234) denote the derivatives with respect to the corresponding arguments; the symbol (*) denotes transposition. The presence in problem (224), (225) of some functional constrains of 0 leads to an additional condition of the type of inequalities Fk[u(-),a]I nonnegativity of the corresponding components of the vector 7 . This condition is quite analogous to the condition of a “complementary nonrigidity” in mathematical programming (Ioffe and Tikhomirov, 1974). As is known, 2 = c = const on the optimal trajectory, where, as in problems with non-fixed time, c = 0. Hence, in the problems of optimal high speed with terminal functionals (Go = 0, go = 1, g k = 0 at k >_ 1) along the optimal trajectory, it immediately follows that the following equality is fulfilled: Ji? = (f ;$ ) - 1 = 0.

(235)

There is a proof of the maximum principle in the form similar to the one mentioned above in, e.g., the monograph by Fedorenko (1978). 2. The Problem of the Minimal-Length Cathode Lens We shall apply the maximum principle to solve problem C , , which was reduced above to the optimal high-speed problem in the third-order controlled dynamic system. In agreement with (227) and (224), we have

,

xo(aj =

(237)

216

V. P. IL'IN era!.

Let us introduce the conjugate vector t,h Hamiltonian (231) for this problem, setting

A' = x 2

*

1 -

3 u2

- 7xl*2

16 Y

= (~),,$~,p)and

+ pu

-

construct a

1.

Differentiating A' with respect to the phase and the conjugate variables, we shall write out the II-system:

x1*2 and present the part of the Hamiltonian X 16 y 2 that depends on the control in the form

Let us write A

=

--

~

HI = Au2 + pu.

(240)

In compliance with (216) the region of control U in the case under consideration is of the form {u: IuI 5 2). From the minimum condition S1 with respect to u and in the domain Iu( 5 2, we obtain the following dependence of the optimal control u o on the variables (x, $) (Fig. 11):

-; uO(x,*) =

-p/2A

in the cone K in the cone K', on the half-line K O , in thecone K, = K S u K T , on the half-line K,, undetermined at A = p = 0.

(241)

Proceeding from the physical meaning of the control u as the electrostatic field intensity reduces to the nondimensional form, we shall refer to the cones K and K as the cones of hard braking and hard acceleration, respectively, and to the cone Kf as the cone of weak focusing. The half-line K O will be referred to as the passive control section, and K , as the zero-overshoot response section. Taking into account that in the problem under consideration, Gka= f , = 0, Q = colon (0,1/2,0), from the conditions of transversality (234) for any variation 6cx feasible under the contraint JCIJ 2 I", we obtain ~

+

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

217

0 from which it follows that

the inequality $(0) 6u

1

~ z ( 0 ) d5a 0,

if SI = - 2 , if -1 < c( < i., if SI = A.

&(0) 6c( = 0, &(0)6a 2 0,

(242)

With the aid of the approach proposed by Gamkrelidze (1962)for studying the zero-overshoot responses, it is possible to demonstrate that the optimal trajectory {xo(t),$ O ( t ) } in the problem under consideration cannot remain in the zero-overshoot response region K , on the set { t } of the positive measure. It follows from further analysis that the optimal trajectory cannot appear on the passive section K Oeither. The physical sense of these statements, which follow directly from the maximum principle, lies in the fact that the optimal focusing conditions to be found contain neither sections of infinitely frequent reversals of field intensity from one extreme position u = A to the other - u = - 2, nor sections of equipotential space with u = 0. Likewise, it is not difficult to show that the optimum trajectory on the set { t } of the positive measure cannot satisfy the equalities A = p = 0 and, consequently, is uniquely defined by the maximum principle. Let us note that the absence of zero-overshoot response sections on the optimal trajectory is not a generally obvious consequence of the physical content of the problem under consideration. At present, a whole range of problems of applied nature is known in which optimal zero-overshoot responses occur (Gurman and Krotov, 1973). Thus, it follows from the maximum principle that the optimal focusing conditions in the problem under consideration may contain three regions differing in the nature of their electrostatic field; that is, two regions of homogeneous fields corresponding to the cones K - and K ', and the region of inhomogeneous field corresponding to the cone Kf = K ; LJ K / (Fig. 12). In terms of the theory of control, we shall refer to the transfer of the optimal trajectory from one region to another as a switching. It is possible to show that there is no weak focusing section on the optimal trajectory if the parameters w, M,, i. of the problem satisfy a certain relation. Here we should make a few general remarks concerning the properties of projections of the optimal trajectories { xo(t),rl/'(t)) onto the plane ( A ,p). The equation for the variable p in (239) may be presented as p

=

2AJy.

(243)

Since the nondimensional potential y ( t ) is positive, it follows from (243) that in the right-hand half-plane of the parameters ( A , p ) , the movement along the optimal trajectory occurs with increasing p ; whereas in the left-hand one, it occurs with decreasing p. Furthermore, by virtue of the boundary condition

218

V. P. IL’IN et al

I FIG. 12. The qualitative view the optimum trajectories in projection onto (A,p)-plane.

.x,(T)= 0 [see Eq. (222)], all the optimal trajectories terminate on the axis A = 0. Taking into account the above-mentioned situation and by using the transversality condition (242), it may be shown that the ( A ,p ) , i.e., projections of the optimal trajectories that do not get into the cone K , have the appearance shown in Fig. 12. In this case &(O) < 0 and a = - A in conformity with (242). We shall refer to these optimal trajectories with a single switching on the axis p = 0 as type 1. It is essential to point out that focusing on the type-1 trajectories occurs at the expense of the effect of an infinitely-thin electron lens, whose entire focal power, is concentrated at the point of switching. Indeed, for the trajectory of the first type, the optimum control is of the form uO(t) =

0 It It,, i., t , < t 2 T,

-A,

(244)

where t , is the “moment” of switching satisfying the equation p ( t l ) = 0. By differentiating (244) with respect to t in terms of the theory of generalized functions, we obtain ir0

=

23. s ( t - t ,)

(245)

(here 6 is Dirac delta function). Returning to the dimensional variables and substituting the expression for the second derivative of the potential

@”(z) = 2E,,,

6(z - z,)

(246)

into the paraxial equation (27), we obtain the condition of a jump for the derivatives with respect to z of the optimal paraxial trajectories at the

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

2 19

switching point z = z:

which has a fairly obvious physical meaning: The less the speed of the particle, the more could the slope of its trajectory can be changed at the expense of an abrupt change in the field intensity at a given “power” of the control E,,,. In terms of electron optics, Eq. (247) should be considered as a limiting relation corresponding to a sequence of the smooth electrostatic fields u,(t), which approximate the optimal field (244) (Fig. 13). It is noteworthy that, as was shown by Tretner (1959),the limiting field of type (244) is optimal in the problem of the electron lens with minimal chromatic aberration. Let us single out two more types of optimal trajectories. Trajectories of type 2 (see Fig. 12) originate in the lower quadrant of the right-hand halfplane, continue under the condition of hard deceleration u = - 2 as far as the entrance to the soft deceleration cone K / (the first switching), then get out of it (the second switching), and continue again under the conditions of hard deceleration until the condition of focusing x , ( T ) = 0 on the axis A = 0 is fulfilled. Type-3 trajectories are the most complex. They also originate in the lower quadrant of the right-hand half-plane under the conditions of hard deceleration, cross the axis p = 0 (the first switching), continue under the conditions of hard acceleration as far as the entrance to the cone of soft acceleration K ; (the second switching), emerge from it (the third switching), and then continue

I FIG.13. Approximation of the non-smooth optimum field u “ ’ ( t ) by the sequence of smooth fields { u n ( t ) ) .

220

V. P. IL’IN ct a / .

under the conditions of hard acceleration until the condition of focusing x,(T) = 0 on the axis A = 0 is fulfilled. The limited size of this book does not allow a detailed analysis of the optimality equations (239) that would permit, as may be shown, an exact solution in each of the regions K - , K’, K,.. The construction of the optimal trajectories qualitatively described above is, therefore, reduced t o joining the solutions of the equations of optimality at the points of switching and to determining the required constants. Without dwelling on the details of the numerical algorithms developed for this purpose, we shall present some end results concerning the type-1 mode. Type 1 trajectories are realized if the parameters w , M,, 2 of the problem are connected by the relationship

and satisfy the inequality (j. -

1)Mq < 1.

(249)

In this case, the optimal change-over potential y y and the optimal “time” of focusing T o are determined in terms of the expressions yy = (2”

+ 1 ) 2 (2M,M +i 1)’’

where A(M,) is the magnification factor,

By passing over to the dimensional variables in (248),(249),we shall obtain the condition of feasibility of the type-1 mode

and the corresponding minimal value of the Gaussian plane coordinate Z$

=

3[1 + (1 + yEmaxd ,)2A(Mg)] + d. Emax

22 1

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

It follows from (249) that, for 0 < i.I 1, the type-1 mode may be realized at any values of M9. The case i> 1 is more complex, and we shall not consider it here. Figure 14 shows the plots of optimal distributions of the potential y o , which correspond to the type-1 mode at i. = 0 , l for various values of electronoptical magnification of Mq.The abscissa of the right-hand boundary point of each distribution y o ( t , I&) yields the value T o ( M 9 ) . Plots of optimal modes of types 1 and 3 for 2 = 0 , l are shown in Figs. 15 and 16. Figure 17 presents the structure of the cathode lens that makes it possible to realize approximately the limiting optimal mode of type 1 (the

3

-

FIG. 14. The optimum modes of type I at i.= 0.1. 1 hf, = 1.5.

10

20

FIG 15. The optimum modes of type I1 at i.= 0.1. 1

~

-

.kf9= 0.5; 2

30 t Mq = 2.0; 2 -

-

=

M q = 0.7:

4.0

V. P. IL’IN et ul.

222

Ib

2b

327 FIG. 16. The optimum modes of type 111 at A

5b

40 =

0.1. 1

~

M,

=

2.0; 2 - M , = 2.4.

rj

R’

2

3

-:xxxxx x x x x x x x x x x x x x x xd

i UlJ

4

-A,

US 3

position of the fine-structure grid is marked by dots). The position of the aperture D coincides with the coordinate z = z1 of the point of switching of the optimal potential @‘(z); on the sections 1-2 and 2-3 of the boundary, linear distributions of the potential are prescribed. The radius a of the aperture is readily seen to act as a “smoothing” parameter, since at a -+ 0, the sequence of distributions of the potential mfl(z), which are analytical functions, approximates with any given accuracy (in a weak sense) the limiting nonsmooth function @‘(z) realizing the minimum of the functional z9 for the considered mode. Figure 18 lists the derivatives @b(z), @:(z) estimated for the values a = 1 and a = 0,2; and Fig. 19 presents the distribution of the potential Qfl(z) and the paraxial trajectories u,(z), w,(z) corresponding to the value a = 0.1. The calculations were made by solving the Dirichlet problem for the Laplace equation with the aid of the TOPAZ-1 (Ignat’ev et al., 1979). The values of corresponding design and electric parameters in relative units are given in Table I. The character of convergence of the realized electron-optical functionals z g ,z,,, M g their limiting values with decreasing the diameter of the aperture is shown in Table 11.

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN J!

a

-2

47''.

-4

223

@'.

J i

2

i'

f

-

+

5'

7

5

Z-d

FIG. 18. The first (0:) and second (0:) derlvatives of the axial potential in the cathode lens in the region z 2 d: a - curves 1.2 correspond to @:, @: at a = 1; b - 3.4 - at a = 0.2).

0 1-j W U

6

0.3

4

0.2

2

0.1

-2 -4 and the paraxial trajectories v,, wa in the FIG. 19. The axial distribution of the potential cathode lens in the region i 2 d at the region a = 0.1.

TABLE I

1.0

5.0

9.30

28.44

3.0

0.52

6.24

224

V. P. IL'IN et al

TABLE I1

1 .oo

0.8 0.6

80.54 64.3 1 52.13

16.99 16.50 15.99

4.24 3.39 1.76

0.4

42.6 I

15.48

2.27

0.2 0. I 0.05 optimum mode

34.94 31.76 30.33

14.97 14.72 14.60

1.87 1.71 1.63

28 44

14.28

1.50

The numerical experiments of the present section wcrc performed by M. V. Korneyeva, S. V. Kolesnikov, and V. A. Tarasov with the aid of a computer.

IV. IMPLEMENTATION OF

NUMERICAL COMPUTATIONAL METHODS SYSTEM OPTIMIZATION

THE

AND

The computational problems related to calculating the characteristics of the emission systems and their optimization for designing devices with specified properties within the framework of the aberrational approach taken by us can be divided into three principal parts: (a) computation of the potentials and their derivatives; (b) calculation of the paraxial trajectories and electron-optical characteristics of the image; (c) minimization of the functionals that determine the required properties of the electron-optical systems (EOS). Here, point (a) is the key step, since practical requirements concerning the precision in determining the electron-optical parameters of the devices make it necessary to calculate the electrostatic potential and its derivatives up to the fourth order with high precision. The implementation of point (b) is limited to solving the initial value problem for ordinary differential equations defining the paraxial trajectories of the electrons, and to computing integrals of a special form that quantitatively characterize the optical properties of images. Point (c) is related to the ultimate aim of designing electron-optical systems according to properties prescribed a priori, and it represents a standard problem of nonlinear programming, including both linear and functional constraints. Its computational complexity in optimizing real electron-optical systems consists of a great number of variable parameters,

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

225

considerable computer-resource consumption in finding one point, i.e., the computation of the current value of the functional, as well as the complexity of the objective function that has, as a rule, local extrema. Under these conditions, our aim in the present section is to investigate in detail the specifics of the problem for the possibility of minimizing the total volume of computations. This section also considers the problem of determining the perturbations of the potential that are stipulated by axially asymmetric small boundary distortions. In this case, for different harmonics of perturbations, it is necessary to solve integral equations with nuclei of a special form, and the boundary conditions that are determined from the solution of the original nonperturbed problem.

A . The Solution of Integral Equutions for the Axially Symmetric Potential of a Simple Layer

In this section, we shall consider the solution of the problem for which we shall present the following formalized mathematical statement. We have to find the solution of the Laplace equation in the cylindrical coordinate system

(256) in the domain R with the boundary S composed of the piecewise-smooth sections r,,on each of which is prescribed one of the boundary conditions of the first, seccond, or third kind:

or the conjugation condition at the interface between the two media with different dielectric constants e (the inner boundary): c?cp (PIr;

= qir[,

Ef-

?"

1/;

acp

= c-

XI.;

3

r, E S 4 .

(260)

Here c,, j , , g,, h,, E ; , E: are the prescribed numbers or the functions of the coordinates, and S , , S 2 , S,, S, are the sets of the boundary sections with the

226

V. P. IL’IN et u1.

boundary conditions of identical types. The estimated region may be bounded or unbounded, and the boundary S may be simply or multiply connected. We shall seek the solution of the considered boundary-value problems (256)-(260) in the form of the potential of the simple layer C P ( P= )

where

G(P,Q) =

b

a(Q)G(P,Q)dSQ, Q E S, P

4rI.X ( k ) ~

R

’

R = J(r

+ rO2 + ( Z

E Q,

-

(261)

zr12,

24rr’

k=-

R ‘

Here X ( k ) is the elliptic integral of the first kind, and r, z and r‘, z’ are the coordinates of the “point of observation” P and the “point of integration” Q, respectively. Now, in order to find the unknown density function a(Q) from the boundary conditions (257)-(260), we obtain the following integral equations with respect to o(Q): Js ~(Q)G(P, Q) dsQ = .UP),

p

E

r, E s,,

(262)

This set of equations is equivalent to the boundary-value problem (256)(260) and enables to fully determine the density o(Q); upon this, it is possible to compute the potential in any point P of the estimated region by the formula (261), and its derivatives, with the aid of the analogous relationships obtained by differentiating with respect to P of the integrand in (261):

We shall consider the formulae for the derivatives of the kernel with respect to the normal n, to the boundary S in (263)-(266), and with respect to the unspecified direction 1 below.

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

227

The integral equations (262)-(265) may be substantially simplified, if we proceed from the functions of the boundary point, which depend on two independent variables, to the parametric representation of the boundary. It will be represented by a set of straight or curvilinear “sections” of r,,each of which is described in terms of the parameter t having the meaning of length: r

= r(t),

z

= z(T),

( r , z ) E r,, a, I T I p,.

(267)

Here x,, p, are the limits of variation of T when moving along r, from its beginning to end. For the straight section, the coordinates of its points are determined in terms of t by the formulae Y = a,?

+ b,,

z =

cis

+ d,,

( r , z )E

r,,

xI I

T

I p,,

(268)

where the coefficients a,, b,, c,, d, are found from the values of the coordinates z(b,))]. The section of the ellipse is of the ends of the section [(r(cc,),z(cc,)),(~(p,), represented by the parametric relations r = u, sin T

+ yo,,

z

=

bl cos T

+ zo,,

x, I t 5

b,,

(269)

where a,, b, are the semi-axes of the ellipse, and (Yo,, zo,) are the coordinates of its center. The boundary S may have come singular points, in the vicinity of which the density function of the potential has a singularity determined by the size of the angle. To remove it on each section of r, in the vicinity whose ends such singularities occur, the following substitution is introduced (see Antonenko, 1964):

where C ( T )is the smooth function. The values ti1,ti2 characterize the orders of the singularities and depend on the magnitudes of the angles q,(02 between the adjacent portions at the end points of the section K~ = (71 -

wi)/(2n- wi), i

=

1,2.

(271)

For example, for the free end of the section, o = 0 and K = 0.5. If there are no singularities at the ends of the section (the first derivative at the point of joining of the adjoining sections is continuous), then ti1 = ti2 = 0 and O ( T ) = f?(T).

The numerical solution of the integral equations (262)-(265) is based on the sampling of the boundary, for which purpose each of the sections of r, is divided by the points Pi = P(T,) into N, parts (the easiest way to do this is to divide it regularly along T with the spacing AT, = (PI - cc,)/N,). The total

228

V. P. IL’IN el ul. L

number of the spacings will be equal to No =

2 N,, where L is the total

I=I

number of the boundary sections. The set of portions of r, that form a continuous boundary section will be referred to as a boundary branch, and the set of numbers of these portions will be designated by L,, m = 1,2,. . . ;M , (A4is the total number of branches). The parametric representation of the portions will be chosen such that the parameter z should vary in a continuous manner from zm to p, on the mth branch. O n each branch, the function Z(z) is approximated in terms of B-splines (Stechkin and Subbotin, 1976): k= - p

where N, is the total number of spacing intervals on the mth branch. In Eq. (272), ckare the unknown coefficients, and Bt)(z)is the elementary B-spline of the nth order defined in the following way:

W j ( 7 ) = (? -

If

i7

x =

T;-J(z

-

S;+l-a)“.(z

-

+ 1 is an uneven number, then x = n/2, fi = n / 2 + 1, p = a; otherwise, ( n + 1)/2, p = ( n - 1)/2. By 7;we shall denote the so-called nodal

p

=

points of the spline, which for the uneven n, coincide with the points t I ,and for the even ones, they lie in the middle between zi- and z i . The expression ( , f ( t ) )is + equal to f(z) at f > 0, and otherwise to zero. Since each of the splines Bg’(z) possesses continuous derivatives up to the ( n - 1)-st order, the corresponding approximation of Z(z) has the same smoothness, too. As is seen from (273),if the boundary branch is broken down into N, intervals, the function 5Jz) is expressed as a linear combination of N, 1 2p splines. To define unambiguously the spline approximation at the ends of the interval [a,, p,], the “boundary conditions” are assigned to the spline, for which purpose the following relations

+ +

Z$’(cx,) or

=

Ai,

Z:)(p,)

=

Bi, i

=

1,. . . ,p,

(274)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

229

may be used. Here A i , Biare the given numbers chosen on the basis of a priori considerations on the nature of the behavior of the approximated function in the vicinity of the end points of the interval [c(,,pm]. If, e.g., the boundary section is perpendicular to the symmetry line of the potential, then it would be natural to set 5:) = 0 on the corresponding end point. For the closed boundary branches their end points coincide, and then the spline is subjected to the periodicity condition = Z:)(b,,,)

::)(a,)

i

=

1,. . . ,p .

(276)

Approximation by the first-order splines is, naturally, the one most readily realized. Accurate to the constant l/h, all ck are the values of the interpolated function C ( 7 k ) in the nodes. At n = 1, there is no need for any boundary conditions on the spline; and in (273), 5(7)is virtually a common piecewiselinear approximation. We shall form the approximate solution of the set of integral equations on the principle of collocations in the following manner. By substituting in the integral equations (262)-(265), the desired function of Z(7) with its spline approximations on each of the boundary branches, and by demanding strict observation of the boundary conditions in the points of sampling Pi E S, we obtain the set of linear algebraic equations with respect to the coefficients c k :

i

=

1,2) . . . , No + M.

Here 5,bj are the boundaries of the range of variation of z,onto which those sections of the boundary that enter the j t h boundary branch are mapped; zi are the points of discretization of the boundary. For the sake of uniformity, we shall formally integrate all three equations (262)-(265). For the points lying on the boundary with the condition of the first kind, the term outside the integral on the left side of (277) is absent (y = 0), and for the rest of zi, we have 7 = 2n. The right side, f ’ ( 7 , ) , is equal to $(P), yj(P),or hj(P),if the corresponding point of observation P lies on the section of the boundary rjthat belongs to S , , S 2 , or S,, respectively; if P E rj E S,, then f(t,) = 0. The Jacobian J ( T ‘ )of the transformation of the variables when converting- the description of the boundary from the Cartesian coordinate system to the parametric representation is defined by the expression

230

V. P. 1L‘lN rt al.

Summing under the integral is carried out with respect to the elementary B-splines determined on all the M boundary branches. The number of equations (277) equals the total number of the points of quantization of the boundary of the region. For the portions of the boundary with different types of boundary conditions, the nucleus G(T:,T ’ ) is defined by the following formulas:

G0(zi,t ’ )=

R

4r(z’)X(k) R(Ti>T‘) ’

Ti E

rj,rjE s,,

+ r(2’))’ + ( ~ ( 7 ~ z(T’)”]’’’, )

= [(r(zi)

-

where X ( k ) , &(k) are the complete elliptic integrals of the first and second kinds. They can be approximated with high accuracy in terms of known polynomial representations. One of these approximations that ensures that the maximum error does not exceed 6 lo-’ is of the form (see Il’in and Kateshov, 1982):

-

X(k)= &(k) =

3

3

i=O

i=O

3

3

i=O

i=O

1 aiqi - Inq 2 biqi,

q

=

1 - k2,

1 ciqi - lnq C d i q i .

The values of the coefficients a i , bi, c i , di are determined by Table 111. Since computing elliptic integrals is a labor-intensive and repetitious operation in forming the matrix of the system and the calculation of the

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

23 1

TABLE 111 I

0

1

2

3

(4

1.386294361 0.5

0.097932891 0,124750742 0.44479204 0.249691949

0.054544409 0.060118519 0.085099193 0.08 150224

0.032024666 0.010944912 0.040905094 0.01382999

h, ct

1.o

d,

0.0

potentials, it is expedient for certain intervals of variation of the argument k to employ more efficient formulas. For example, at small values of k, it is efficient to use the series (Gradshtein and Ryjhik, 1956)

Jm,

where k’ = m , = (1 - k ’ ) ’ / ( l + k’)2. As follows from the form of the kernel of the integral equation at the point Q tending to P, the kernel has a logarithmic singularity that can be eliminated by a simple additive transformation. Let a certain expression under the integral sign be of the form w(z, z’)= 4 7 , ~ ’+ ) 47, z‘)lnlt - 5’1,

where the functions 4 7 , z‘),v(z, 7‘) have no singularities. Then the integral of w may be computed in the following way:

s

w(7,z’)dz’

=

s

+

(47, z’) [u(7,7‘) - u(z, T)] lnlr

+ u(z,7)

s

-

z‘l} dt‘

(279)

ln/z - 7’1 dz’.

The last integral is taken accurately, and the second-to-last one already contains a “good” integrand function, and it is substantiated now to make use of quadrature formulae in order to compute it. When searching for the matrix

232

V. P. IL‘IN ef al.

elements it is reasonable to perform the above-mentioned manipulations only when integrating over the boundary spacing intervals that adjoin the point of observation P. To redefine the set of equations, it is necessary to add 2pM equations approximating the boundary conditions for the splines, which are obtained as a result of a substitution of (272) into the corresponding equations (274)-

I

(276), to the equations (277). Let the boundary condition -

dx

=

0 for the

[,=am

spline be set at the end point ( z i = 2,) of the rnth boundary branch. The corresponding equation of the algebraic system will be of the form

where the sum is practically taken over those elementary splines whose derivatives of which are not equal to zero at the point x = CI,. In this case, we shall obtain an equation of the form c i - I - c i + l = 0 for the quadrature and cubic €3-splines. Thus, we obtain a linear algebraic system in which the number of equations and unknowns is equal to N = (2p l)M N o . Having introduced the notation c = {c l , c 2 ,..., c N ) , we may write the vector of the unknown coefficients of the spline in the form

+

Ac=

f.

+

(280)

Here .f = { .1;,1;,. . . ,f,.) is the known vector, and A is the quadrature matrix of the order N . If the kth equation of this system is obtained from the relation (277)for the point of observation P(z,),then fk = f ( z i ) ;otherwise, f k is equal to zero or is determined from the right sides of the boundary conditions (274)(276).The row elements of the matrix A that correspond to the points P ( r i )E S are obtained upon performing an integration in (277). Sometimes it may be done precisely, but frequently one has to employ numerical integration. In the latter case, one may recommend Gauss quadrature formulae because of the high accuracy in computing the integrand function for a sufficiently small number of auxilliary points (in practice, four to eight quadrature points on one integral of spacing of the boundary portion are quite sufficient). The matrix A is dense (a small number of nonzero elements is present only in the rows that correspond to the boundary conditions of the spline) and, in general asymmetric. The solution of system (280) with the aid of the iteration methods is not always feasible, since the spectral properties of the matrix A necessary for the optimization of such algorithms are, as a rule, unknown. In case of using direct methods, for example, the Gauss method of elimination, one may recommend

233

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

an iterative refinement of the solution, which consists of the following. When solving system (280), let us obtain. by virtue of the round-off error, not the exact result, but some approximation c('). Now we shall compute the vector of residual r'') = .f' - Ac") and find the correction for the solution z"): A:"' = r('),which we use to refine the initial approximation = c(') z" ) . If the new residual r ( 2 )= f - A c ' ~ is ) still large, then it is possible to , If compute another correction z(*)= A-'r'') and another refinement d 3 )etc. the initial approximation d') is not too bad and the conditionality of the matrix is not catastrophically bad, then the process of iterational refinement converges very quickly; and in practice, one correction is sufficient. The computation of the residual brings about an additional effect in that it allows control of the accuracy of the solution. At the software implementation of this algorithm, it should be kept in mind that the computation of the residual must necessarily be performed with double accuracy, since it is almost always obtained as a result of subtracting numbers that are close in magnitude, which results in a loss of correct significant digits. Also note that, when using the Gauss method, one step of refinement increases the labor input on the solution of the system approximately by a mere 25"/,, since when finding z ( I ) ,the system with the same matrix A as in the initial system is being solved; and this makes it possible to save a substantial portion of the computations.

+

B. The Computation qf Axially Asymmetric Disturbances by the Bruns- Bertein Method

The present section will present the numerical methods for the computation of perturbations of the axial potential and its derivatives occurring as a result of minor perturbations of the axial symmetry of the EOS. The algorithms described are developed on the basis of the Bruns-Bertein approach (Bertein, 1947; Bruns, 1876, the essence of which consists of reducing the three-dimensional boundary-value Dirichlet problem to the solution of a sequence of two-dimensional equations. Suppose we know the distribution of the potential cpo(r,z), which represents the solution of the boundary-value problem with axially-symmetric boundary r. If the boundary l- undergoes some axially asymmetric distortions, passing into a new position of r' then the respective distribution of the potential will become already the function of the three variables F', q ( r .z , Q), satisfying the equation

r ?r

234

V. P. IL’IN et al

with the condition on the perturbed boundary cp = (r,z , d ) I p = I/. Taking into account perturbations of the first order alone with respect to the parameters of the geometry, we shall reduce the problem to the axially symmetric boundaryvalue problems for separate components (harmonics) of perturbation in the region with unperturbed boundary r. Let the axially symmetric surface of with the potential V prescribed on it be subjected to the following axially asymmetric variations: shifts relative to the axes r, z by respective values; rotations about the point zo relative to the axes r, z , by the angles /I, y, respectively; and “elliptic”-distortion with the degree of ellipticity E . The latter quantity is defined in the following way: If a circle with the radius R is transformed into an ellipse with the lengths of the semi-axes a, b and the same perimeter ( a = R(l + E ) , b = R ( l - E), 2R = a + b), then E = (a - b)/(a + b); if the obtained ellipse touches the obtained circle from the inside (a = R, b = R(l - E)), then E = (a - b)/a. In the first case, the deformation is called isoparametrical. We shall only consider the perturbations of the first order relative to small parameters E , 6, y, K , /I, and as a consequence of this, we may write the approximate equality

cpP>z>

vo(r,z) + E$c

+ W 6 + Y$? + KcpK + Bvp.

By expanding further each of the functions t,bE, Fourier series of the kind

(281)

t,by, t+hK,t,bp into the

+ $ y ) ( r ,z)cos 8 + $i2)(r,z)sin e + $ y ) ( r ,z ) cos 28 + $ y ) ( r ,z ) sin 28 + . . . ,

$ ( r , z,0) = G0(r, z)

(282)

and by taking into account the dependence of r on E , 6, y, K , p, we obtain for the perturbations of the shift or distortion type, A1*1.6

=

Al$l,K

k 6 I r

=

*I,KIr

$L;fIr =

= Ai$L,y = Al$l.p =

(283)

= -qr>

$1.01 =

-(z

-

zo)~?,

or the isoparametrical elliptic deformation A242,E = 0,

$2,tlr

=

for nonisoparametrical elliptic deformation *z AO*O,F

=

$0.2

0,

+ *2.u

= A2*2,2 = 0,

-Rq,,

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

235

Here

and cpI

2%

= -(R,z).

In (283), (284), and (285), for any of the harmonics, it is

iir

understood that some homogeneous null boundary conditions of the first kind are prescribed on nonperturbed boundaries. If several kinds of perturbations are present in the variation of the boundary, then the corresponding disturbances of the axial distribution of the potential are computed independently; and in order to determine the complete distribution of the potentials, the principle of superposition is used. The solution of the equation of the form (286)for each m may be presented in the integral form *m(P)

=

Is

om(Q)Gm(P, Q ) ~ S Q , Q E S,

P

E Q,

analogous to (261). Here a,,,(Q) is some function of the “density of harmonic of the potential,” and the kernels Gm(P,Q) are expressed by the formula

Establishing, from the relations (283)-(285), the values of I ) ~ ( Pat) points P lying on the boundary, we obtain for each of the harmonics, an integral equation of the form (262). Computing the integral (287) in terms of the elliptic integrals at different rn, obtain the following relations:

a. =

21-2 1 - k 2 ___ 2 j - 1 k2 ’

~

3 - 2j

p. = ( 2 j - I ) k 2 ‘

(288)

236

V. P. IL'IN et a!.

In particular, for m

=

G,(P, Q ) =-4rQ R

2, the expression (280) is of the form

{

8

X ( k ) - 7[A'(/?)- 8 ( k ) ]

k

The use of the approximations (276) for the computation of the nucleus G,(P, Q ) at k I 0,15 (i.e.,at r p ---t 0) results in the appearance of large round-off errors due to reducing numbers close in magnitude. In such cases, it is expedient to compute X ( k ) , & ( k ) by using power series (Gradshtein and Ryjhik, 1956); in particular, for m = 2, obtain

+?r'2ynl)!!)k2n-8[p

k 4 4 - 16n 2 + 3(2n - 1) k 2

Analogously, at r p + 0, special approximations

Fr

~

+ 3(2n

-

1).

1)

(290)

s G(P,Q ) are used:

?r

r p k 2 4r A(k)- - + - - A B ( k ) c c R

where

For k < 0.2, the error in the formulae (290)-(291) for the utilized values of NH = 8 does not exceed 10 '. If the point of observation lies exactly on the axis of symmetry, then for the expression of the kernels and their derivatives, it is possible to simplify: Go(P,Q)(,,=o = 2n-, YQ

R

(292) (293)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

237

The kernels for the harmonics of perturbation (288) and (289) have the form G, = O(r:); therefore, for the points of observation lying on the axis of symmetry, the elements of the rows of matrix (280)become zero. In the case of numerical implementation, it is necessary to use the equation

The method of computating the right sides in the integral equations for perturbation harmonics expressed in terms of the derivatives of the nonperturbed potential at the points of the boundary is also of great algorithmic significance [see Eqs. (283)-(285)l. It is possible, in principle, to compute the values of cp, by the formula of the form (266);but the derivative of the nucleus, which is contained in it under the integral sign, has a singularity, and that requires to single it out analytically, similar to how it was done in the previous section when computing the diagonal elements of the matrix A in the set of equations (280). Another feasible way to computing is by means of the difference approximation of the derivative according to the values of the nonperturbed potential, which are determined numerically at several inner points of the estimated region. There are also special computational features here: If the estimated points are taken too close to the boundary, the accuracy of determining the potential decreases; whereas, if the point is removed to a certain distance, the error of approximation of the derivative increases.

C. T h e Algorithms for Computing the Derivatives of’ the Potentiul As was described in the previous sections, in order to determine paraxial trajectories, coefficients of aberrations, and other electron-optical characteristics, it is necessary to compute with high precision on the axis of symmetry, the potential and its derivatives up to the second order, including those in the vicinity of the boundary; on the cathode surface, it is required to know the derivatives of even the third and fourth orders. In conformity with the method of integral equations, the potential and its first derivatives may be computed by formulae (261) and (266), and the derivatives of higher order, by the analogous expressions obtained by differentiating the kernel of the integral equation. However, in the vicinity of the boundary, the accuracy of the potential and derivatives computed directly by applying numerical integration to such formulae is reduced considerably, due to the presence of singularities of the kernel at P Q. Therefore, to increase the accuracy of computations in the vicinity of the cathode and the screen, special algorithms must be employed. ---f

238

V. P. IL'IN et al.

Let the electrode cross the axis of symmetry at the point z = L , then it is necessary to make the potential more precise in the interval [ L , L + 61, where 6 is some distance from the electrode. If the values of the potential cp and its derivatives cp', cp" with respect to z are known on the axis at the point z T = L + 6, then, using the relations

R cp'(L)= 2 cp"(L),

8 cp'"(L) = - cp"'(L)R

(297)

which take place for any equipotential surface with the radius of curvature R , it is feasible to construct the polynomial of the fifth order, 5

q ( z )=

C ai(z

-

L)i.

i=O

The first coefficient of this polynomial is determined at once: a, whereas the rest are found uniquely from the set of linear equations

2a2

+ 6a36 + 12a4d2 + 20~2,d 3 = cp"(Z,),

a,

-

Ra,

= 0,

a4

-

2 -a3 R

-

=

cp(L),

(299)

a1

7= 0, R

where the latter two equations are obtained by substituting (298) in (297). In the considered method, it is assumed here that the distance from the electrode is sufficiently large in the sense that, for the points with the coordinates z 2 z,, the potential and its coordinates are determined accurately enough from the approximate computation of the integral along the boundary of the region. In general, the problem of optimizing the value 6 arises here, since at a large enhancement of this value, the error of approximation of the polynomial (298) will be considerable. Another independent computational problem is the estimation of the field near the screen of axially symmetric EOS. As a rule it is almost constant (equipotential). Yet, when employing the method of integral equations, we observe in the computations a nonmonotonic behavior of the axial distribution of the potential, though its values coincide with the potential of the screen V, with a high accuracy. This nonmonotonicity may strongly show in computing the derivatives of the potential and the characteristics of the electron-optical systems. The character of the behavior of the potential on the axis of an electron-optical system between the diaphragm and screen is known: The values of the potential converge monotonously to the potential of the screen cp(z,) = V,, and the derivatives in this case diminish monotonously. Hence, we

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGY

239

search for the approximation of q ( z ) for the points z E [ z b , z , ] at prescribed values of ( p ( Z b ) , q ’ ( z b ) , ( p ” ( z b ) , in the form of the function $(z) = u(z)

Here u(z) =

1

+ d(z

+ l(2).

(300)

-

(P(zb) - Zb)

+

e(Z - zb)2

+ a,

S

c k ( z - z,)* is a polynomial of the same type as in (298). In

and I(z) = 1=O

order to establish the unknown coefficients a, d, e, we employ the following conditions:

u(zJ

=dzsL

where ( p ’ ( z b ) , ( P ” ( z b ) are considered known from the computation of the integrals from the density on the boundary. Using the found values a, d, e, we obtain the equations for finding c k , analogous to the set (299): l’(Zb)

= 0,

l’’(Zb)

= 0,

R

l’(Z,)

- 7I“(Zs) =

- u’(z,)

L

24 R3

R

+ 7 u”(zs), L

8 R

24

- _ “(z,) + - l”’(z,) - l’”(z,) = 3 v’(z,) R

8 R

- - l,l’’‘(Zs)

+ U’”(Z,).

Due to this construction, the function $ ( z ) = u(z) + l(z) satisfies the relations of the kind in (297) for the derivatives on the surface of the screen and the following conditions: $’(Zb)

$”(Zh)

= q’(zb), =

IcI(zs) = $(zb)

cp”(z),

(303)

dZ,X

= (P(zb),

The same singularities occur when using the potential of a simple layer for the computation of the geometric perturbation functions in the vicinity of the

340

V. P. IL’IN rt al

boundary. The algorithm of their refinement is analogous to the method described by formulae (297)-(298), if the potential and its derivatives are substituted by the values of the perturbation function @a and its derivatives @$). Assuming that the requirements (297)hold for a linear approximation to a perturbed field on the axis of symmetry @ = cp uQa, it is possible to obtain the relations at the point z = L of the intersection of the considered electrode with the axis:

+

@:

@a

-

_

R

Rhcpg R 2 ’

-~ ~

2

Here cph is the value of the ith derivative with respect to z at the point z = L for the initial (nonperturbed) boundary-value problem, and the magnitude R: = i R / & is equal to unit, if the given perturbation function is determined by variation of the radius, and otherwise to zero. The function @): is the value of the ith derivative of the perturbation function at the point z = L . At all other points of the axis of symmetry where the considered algorithms of refinement are not applied, the computations of the potential and derivatives are performed by the formulae

with the expressions for the derivatives of the kernel at r and having the following form:

?C (?z

--

?4G ~~

?z4

-

=0

being simplified

27rr‘(z’ - z ) R?I ’

27rr’{ 9 - 9 0 [z R5 ~

R;]’+

I05 [ ( z H;‘)]4] ~

D. Inteyrution of’ Puruxiul Equations for Electron Trujectories

In this section we shall present the method of computation of paraxial trajectories u, w, described in the work of (Tl’in and Popova, 1983). As a rule,

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

24 1

the computation of trajectories is performed by numerical methods of Shtermer or Numerov type, in which, with successive recurrent computation of points, there is a risk of accumulating errors. We shall also consider the algorithm based on the replacement of the Cauchy problem that occurs in the course of the solution with a linear combination of solutions of two boundary-value problems of a special type that are solved with the aid of a stable sweeping method. The problem of determining the trajectories v, w in the interval [O, z b ] has already been stated in Section I.A. It finds the solution of a homogeneous differential equation (27): M o [ r ] = Qr”

with asymptotes at z

+0

+ 21 Q‘r’ + 41 W ’ r = 0, ~

(307)

~

[see Eq. (28)], u(z) =

“(Z)

=

2 a 4; +

~

@; 1 - -z 2@b

0(z3’2),

+ 0(z2).

In the vicinity of the cathode (0 < z i,,z , is a sufficiently small quantity), the functions v(z) and w ( z ) may be defined accurately enough in terms of the series (22): v(z) =

t(z) + a&), = %o

w(z) = 1

+ plz +

p2z2

a,z

+ Cf2z2+ % 3 z 3 ,

+ p3z3,

(309)

whose coefficients are established from the condition satisfying Eqs. (308):

33

39 E 2 E 3 140 E :

=-

~

p 3 = p7- - 3E,E, ---120 E l

where

57 E i 280 E ;

-

E, 14E;’

p

3 (E2)3 80 El

~

E, 60E,’

-~

-

~

242

V. P. IL’IN et al.

To construct the numerical solutions of u, w on the whole, we shall create a uniform net with the coordinates of nodes computed by the formula z , = z o + nh, n = 1,2,. . . ,N , h = (zn - z,)/N, 0 < zo < z,. The finite-difference approximation is applied not directly to (307), but to the equation U“

= f(z)U,

(310)

obtained upon introducing the notation U ( z ) = $%$jr(z),

f(z)=

Difference approximation yields the set of algebraic equations

which, as is easy to verify, has the error of approximation O(h4). After the substitution of the variables

un=

Y, 1 - h 2 / 12fn’

(313)

Eq. (312) is reduced to the form Yn-1

(314)

-

Note that if y o , y1 are somehow found, further computations using the formulae Y n + l = SflYfl+ Y n - I (315) correspond to Numerov’s method. Since the computation using the recurrent formulae (315) is not immune to the accumulation of errors, and since the Eqs. (314) themselves are linear, we shall solve them, using for their solution the sweeping method, thus representing the solution of the Cauchy problem as a linear combination of solutions of two boundary-value problems of the first kind (assuming that y o and y k for some integer k > 1 are known):

Pn

1

YnPn

~

~

g,q,

+ Pn+1 = 0, + q , + l = 0,

0 < n < N, Po 0 < n < N,

=

1,

qo = 1,

PN =

0,

q , = 0,

(316)

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

243

In this case, we assume that the point z , coincides with the node of net zk, and the values u(z,), w(z,), u(z,), w(z,) are computed with the aid of the series (308).Then the desired values u(z,), w(z,) are determined by the formulae

w ( z , ) = w, = w ( z , ) p ,

+

w(zk)

- PkW(ZO)

qk

(317)

9, >

n = k , k + l , ..., N .

The values p n , qn from the sets (311) are computed with the help of the economical formulae of sweep which, in our case, have the following simple form: 1 PN-1=-

P,

7

YN- 1

PI

=-

1

91 - P 2

1 21 = -,

,

gN-2

1

1,

(3181 , n

=

2,3,..., N - 1,

gn - @,-1

1 =

,..., 1,

, n=N-2,N-3

Sn - P,+1

P ~ = P , , ~ , - ~ ,n = 2 , 3 ,... , N -

u, =

Y1

Y'+l

1 =

-

,

g, = ~ ( , g ~ + ~n ,= N - 2, N

-

3, ..., 1.

'N-2

To refine the solution, we shall perform Richasrdson's extrapolation utilizing the solutions of U hand U Z hon two nets, i.e., the basic one with step h, and the auxiliary one with step 2h. In common nodes of the nets, we take the linear combination.

whereas in the other nodes of the net, we find the solution by using the difference equation (314): (320)

The numerical solution obtained in this manner has the error O(h6). After computing u,, w, in the nodes of the net, it becomes possible to establish (without losing any accuracy in the order) the values u(z)and w(z)and

244

V. P. IL'IN et al.

their derivatives for the intermediate values of z by using an interpolational spline of the fifth order whose error of approximation is 0 ( h 6 )for the functions themselves, and O(h5)for their first derivatives. E . Numerical Solution of the Problem of Optimization of Electron-Optical Systems The problem of optimization has already been formulated and considered in detail in the third section. In this section we shall look at some computational features of its implementation. Let x = ( x l , .. . ,x,) be the vector of the parameters at fixed values of the components for which the geometry of the electrodes of the EOS, the potentials prescribed on them, and consequently, all the characteristics of the electronic image considered in the first chapter and which we shall denote by f k , k = 1,2,. . . ,N , are determined unambiguously. We shall reformulate the basic problem A, considered in Section IILA, in the following more concrete way: Find the values of xy, . . . ,x: which ensure the minimum of the functional F,, i.e., Fo(xl,.. 0 . ,x,") = min F,(x,,. . . ,XJ, (321) x L .. . . . X n

under the prescribed conditions for the permissible variations of the parameters a
Fk(x)I 0, k

=

Fk(x) = 0, k

=m

+ l,.. .,l.

Here a = ( a l , .. . , a p ) ,j? = (pl,. . . ,&) are the vectors of the dimensions p , p I n, and A is a rectangular matrix with the number of columns and rows n and p , respectively; the functionals F,, . . . ,FLare expressed directly in terms of the characteristics of EOS: Fk =

&(Il,.. . , I N ) ,

k

= 0, 1,.

. . ,/.

For example, F, may be represented as the quadrature form of I k : F0

=

1 uk(fk k

-

k

E

J,

(325)

where ak are the prescribed nonnegative coefficients, If are the desired values of the characteristics, and J denotes a certain set of indices from the set

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

245

[ 1,. . . ,N } . The limitations (323), (324) may be taken in the forms

Here J1, J , are the sets of indices that intersect neither each other nor J . Under the conditions (326), there may exist no more than two (consistent) inequalities for each k', which corresponds to the one-sided or two-sided constraints on the characteristics (a typical condition: Ilk.- 1:,1 < &k at some prescribed value of 12, and sufficiently small & k , ; in this case, I:, = 1," k & k , ) . The solution of problems (321)-(324), as was pointed out in Section I K A , is reduced to a successive solution of the auxilliary quasilinearized problems. This process may be described in the following way. Let x" be some approximation to the desired vector of parameters x*, to which the axial distribution of the potential corresponds: @"(z)= @(z,x"). Let us suppose that, for the values of the vector of the parameters x close enough to xs,the distribution of the potential with a good approximation is described by the function linearly dependent on the vector of increments of the parameters ijx" = x - x s . &(z)

= @(z, XS)

+ (VX@(Z,XS),fixS),

(328)

where Vx@ is the vector function of perturbation of the potential. Let 7;. denote the values of the characteristics of EOS determined by the distribution of 6', and let F; = @,(r"S,,. . .,pv)denote the corresponding values of the functionals. Consider an auxiliary problem of the ''local'' minimization, namely, finding the values of x;*, . . . ,x:*, which ensure the minimum of the functional F : , i.e.,

-

F",x;*,. . . ,xp*)= min F",x,, . . . ,x,), XI

,..., X"

(329)

at the constraints

Pi(x) 5 0,

-

k

=

1,..., m,

Fi(x) = 0, k = m + 1,. . . , l , a" 5 Ax 5 fl". Here a" and fls are the vectors of the constraints of the sth auxiliary problem that have the same dimensions as for a, fl (322); in this case, it is natural to subject their choice to the condition a I a" I fl" 5 fl. The vector of the parameters xs*found from the solving problems solution of problems (329)-(332) is taken as the next approximation, i.e., xs+I

=

xF*

(333)

246

V. P. IL’IN et al.

For the values of the potentials and the electrode configurations corresponding to it, another distribution of the potential, @(z,x s c‘), is found; and from this, the new values of the characteristics of I:+’ are found. Subsequently, the linearization of the form (328) is performed once a new problem of “local” minimization is solved. The procedure proceeds in this manner until the values of the characteristics on the adjacent iterations of such a “global” minimization coincide (with the desired accuracy). As noted in Section IILA, the introduction of such two-level minimization is aimed at reducing the total volume of computations and is related to the specific character of the problems under consideration. The point is that, when solving the initial problem of minimization (327)-(324) directly for a sufficiently large number of variable parameters (at least more than three) and a complex nonlinear dependence on them of the estimated functionals (this is exactly what happens when optimizing EOS), it will require a considerable number of iterations, each of which require the computation of the values of the characteristics, each time, with another distribution of the potential. And it is precisely finding this (together with the corresponding perturbation functions) that represents the most labor-intensive operation.

V. AUTOMATION PRINCIPLES I N DESIGNING ELECTRON-OPTICAL SYSTEMS The present-day level of development of computational facilities, mathematical methods, and technology of simulation on electronic computers raises the problem of creating an automated design system (CAD) of EOS. The main objective of CAD is to provide a considerable increase in the quality and a reduction in the time for designing new devices with the required characteristics by replacing the expensive and time-consuming process of physical mocking-up by operative mathematical modelling on a computer. We shall only discuss problems of software provision for CAD of EOS, confining ourselves to problem statements and algorithms of their solutions discussed in the earlier chapters, and leaving aside some aspects essential for designing emission and image systems, such as research on the physical properties of the cathode, detailed analysis of the grid field structure, the problem of tolerance in part-fabrication tolerance, etc. The range of problems under study, however, appears to be fairly important in terms of the ways of achieving the main goals in creating EOS, i.e., realizing the required imagequality characteristics. Section 1V.A of this chapter will deal with the general principles of mathematical simulation on an electronic computer and of its software

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

247

technology, whereas all the other sections will treat examples of problemsolving that occur in the program package (EFIR) developed at the Computer Center of the Siberian Division of the USSR Academy of Sciences (see Kateshov, 1984). A . Software Requirements for C A D EOS

The starting point is a request to design an object, which in the present case is a set of requirements for the main performance characteristics, types of electrode configurations, dimensions, and other technological and operational conditions of the EOS. For the object to be designed, a physical model is constructed to determine which of its physical effects are to be taken into account and which are to be ignored. For example, in the case at hand, an EOS model is based on the assumption that the initial emission rates of electrons are null, the effect of their space charge is negligible, the grid represents a smooth equipotential surface transparent to electrons, and so on. Also to be determined are the configuration of the device being designed and the design features whose effect can be neglected or must necessarily be taken into account. The next step is to state the problem mathematically, i.e., to describe integral or differential equations, the form of the computation domain boundary, the boundary conditions, and the initial data, and to indicate the required solution accuracy. In simulating simple objects, the stages of physical and mathematical simulation can virtually merge, whereas in complex situations, the model adequacy has to be thoroughly verified. The art of choosing a model consists of the ability to achieve the minimal complexity (or maximal simplicity) in describing an object and at the same time, to ensure for practice a high enough accuracy in the determination of its characteristics. It is further necessary to work out (or to choose from among those known) some numerical methods for solving the stated mathematical problem. This implies specifying a way of discretizing the computation domain, approximating the original differential and integral equations and algorithms for solving the obtained algebraic problems. The main point here is to select computation parameters (e.g., step of the grid) so as to ensure the necessary accuracy of solution with economical use of computation resources. One of the most tedious processes is setting up a program. The programming matters grow in gravity as the program complex grows in bulk. An independent significance is attached here to the following stages: preliminary planning of work, writing the program proper, debugging it, testing it, and documenting it and its subsequent specification. We shall only touch upon the application software connected directly with the selected class of problems and dictated by the mathematical model of the object designed. True,

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complex application designs require good knowledge of and expert use of further developments of system software components. An independent and important stage for the user of application programs is the computation itself. In the trivial case, it involves routine punching-in of initial data, assembly of the package of punched cards and its transfer for computation, analysis of the digital printouts obtained from the computer. Soon, however, this “technology” reaches saturation. With rising computational powers making it possible to sharply increase the number and complexity of problems solved, the “manual” method of preparing the data and analyzing the results leads to an equally dramatic increase in the number of personnel and thereby reduces to zero the advantages gained from using an electronic computer. Thus, along with the efficiency of the mathematical methods, of no less significance is the timely assignment of initial information and the visual presentation of computational results. On the basis of the analysis of the numerical computational results, a decision is taken as to further operations. The computations themselves can be performed either in single versions or in series with the initial parameters recomposed. The most effective computations for solving design problems are optimizing computations with the automated (partially, at least) process of searching the values of variable values, ensuring the required characteristics of the object under investigation. Such an operation is feasible only if special hardware and software are available to ensure the control of the computation process in the user-computer dialogue. The results of numerical simulation are, as a rule, carefully tested either on methodological examples with known accurate solutions or on the data of physical experiments. Naturally, mathematical simulation does not imply a complete denial of physical breadboarding, which has the final say in estimating the compliance of the device designed with the initial requirements. In the framework of CAD, there arises an independent technical problem of automating a physical experiment and setting up its interaction with the technological chain of numerical computations. The computational experiment scheme under discussion is an iteration process with a large amount of feedback, because as new data are obtained and analyzed, it may become necessary to check and modify the following technological process partially or wholly: changing the physical model, replacing the mathematical statement, correcting the algorithms, and rewriting the program. For the above scheme to function normally, a sufficient CAD information basis has to be formed including the evidence on real objects, their models, algorithms, programs, and so forth. The ultimate goal of the entire process is a design solution that can be produced by the computer as design documentation. Due to sufficiently complicated mathematical statements, the EOS CAD

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software may amount to scores of thousands of operators and commands, the body of simultaneously processed data, to hundreds of thousands of numbers; and the computation requires a large number of versions to be run through, consuming significant amounts of time and computational resources. The development and operation of CAD software are a long and continuous technological process requiring that special attention be paid to the methodology of building, operating, and maintaining a program. In view of its industrial character, the CAD software must satisfy rather stringent requirements, which can be formulated as follows: a ) completeness of physical and chemical models covered by the set of the problems built; b) efficiency of realized algorithms, which is determined, as a first approximation, by the length of the machine time needed to obtain a solution with a prescribed accuracy (for more rigorous estimation, account should be kept of the cost of all computational resources and labor spent on program development); c) simplicity and convenience of operation, i.e., swift running of problems on a computer, graphic and economical presentation of initial information and of computation results, easy communication with the computer (dialogue aids or, with package programs, messages about errors and intermediate results), and complete and precise documentation; d) reliability of algorithms and programs, i.e., a high degree of debugging, failure-free performance, provision of required accuracy, and availability of various tests; e) possibility of expanding algorithms and programs, of changing or expanding a mathematical model quickly without appreciable software modernization; and f ) adaptability of software to changes in the configuration of the computing system. The CAD application software satisfying, to a certain extent, the imposed requirements is virtually a so-called package of application programs defined as a body of programs compatible with respect to data structure and control technique and united by the common functional assignment as a means of solving a class of problems by a particular circle of users. The main components of program packages are as follows: a ) a library of modules determining the program orientation of a package; b) means of controlling the process of solving a problem; e) means of handling the data; d) means of communication between the user and the package.

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More roughly, the APP structure can be divided into two parts: functional filling and system filling. The first part is the direct realization of algorithms and consists of computational modules, whereas the second is made up of various service, operational, and specialized system components constituting in total the medium or operational situation in which the computational modules must be efficiently worked out and must function. Note that the APP may have no system filling of its own and that the necessary software is performed by the basic means of the operational system. The creation of such a package would seem the cheapest. In practice, however, the contrary has been the case in recent years: an increase in the intelligence of APP, in the level of its communication with the user, and in the extent of computational process automation compels the designers to create their own self-subsistence systemic “farm” of growing proportions. The ideal software for the general system of a computer must comprise a sufficiently complete set of tools to provide a sharp increase in the productivity of labor as a process of design, exploitation, maintenance, and inevitable development of APP. An essential peculiarity of the EFIR package is its focus on the userexperts in the field of EOS development-and on the active application of new devices in practical designing. This makes certain demands on the applicability of algorithms to an arbitrary number of electrodes of various configurations, for their adaptability to particular EOS variants in order to provide accuracy, reliability, and economic efficiency of computations, and for the simplicity of operation and the convenient communication between the user and the package. All this largely determines both the organization of the computation process and the general structure of the EFIR APP. B. Automation of Algorithm Construction in the E F I R Program Package The problem of automating mathematical simulation is divided into two parts: “internal” automation of the computational process consisting on directing the algorithms toward universality and flexible adaptability to particular problems; “external” automation, ensuring the “intellectuability” of the program package-user interface, which means the possibility of using a natural language and of providing a quick two-way exchange of information from the user to the package, and vice versa. It is the realization of these two aspects of the problem in the EFIR APP that are to be discussed below. A package of application programs EFIR is a system for automated computation of the following problems grouped according to the character of the computational process: a) computation of potentials and their derivatives for axisymmetric boundary problems;

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

25 1

b) computation of fields with insignificant axial symmetry violations; c) computation of EOS characteristics; d) optimization of EOS; e) multivariant computations making it possible to solve in one step a series of the above problems in which each variant is distinguished by some initial data. The first of the above groups of problems can be regarded as “elementary” in the sense that it is a component of all the other problems. It is precisely this problem that we shall use to consider the general peculiarities of the numerical realization of the integral equation method. In other words, we shall carry out a module analysis of a given class of algorithms. In the computational process under study, one can point out the following principal stages.

Input and Primary Processing of Initial Data. Here the problem is that, as required from the user’s interface, the initial information is given in a natural input language. However graphic and laconic from the user’s viewpoint, or rather due to these properties, this language is formally based on symbol texts with rather complicated semantics. To make this information “understandable” for the computation modules of the package directly realizing the solution algorithms of the boundary problem, the initial data written in Fortran must be represented by a collection of appropriate objects, i.e., simple variables or files of real, integer, and logical types. This form of initial data will be referred to as internal representation, and the conversion of information from the input language into the internal language (to use the terminology of system programming) as the stage of translation. The program performing this translation is referred to as the translator or convertor. In the internal language, the initial data must not only be unambiguously represented, but must also be sparingly processed during automated synthesis of an algorithm. The function of the translator, therefore, involves not only the abstract conversion of one symbol into another, but also solving a number of problems of analytical geometry and algebra of logic. To cite a simple example: A circumference can most readily be given by the coordinates of its center and by the value of its radius; but in terms of programming, it is reasonable to have the coefficient values of the appropriate equation. In the case at hand, according to its content, the input information can be broken up into the following parts: description of the computation domain boundary and of boundary conditions, specification of numerical algorithms and of their different computation parameters (the number of intervals in dividing boundary fragments, the order of approximating B-splines and quadrature formulas, etc.) indicated depending on the type of problems to be solved, and the planned computation process. Characteristic of the input language of the EFIR APP is the use of the default principle: In case the user

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does not give information of a certain kind, then the preliminarily provided values of the language are selected via programming. Note also that the concept of input data processing comprises, among other things, control of its correctness as well as diagnosis and localization of errors made by the user. Discretizution qf Domain Boundary. This stage involves computation of collocation points on boundary contours according to the given numbers of fragment-division intervals (or to the default). The number of such points largely determines both the accuracy of numerical solution and the required computer resources (depending on the type of computer, the limiting factor may be either machine time or operative memory capacity). Theoretically, the problem of the optimal discretization that can be connected with a minimum number of collocation points providing a required accuracy for a particular problem remains open. Thus, in order to efficiently use a package, it is necessary to take into account the experience of using it. Approximation of Potential Density. After finding the points of collocation, this procedure consists merely of selecting coefficients of B-splines depending on their order. Some algorithmic nuances are connected here with the transition to the parametric presentation of connected boundary fragments, i.e., the piecewise analytic form of presenting a curve. Computation of Auxiliary Integrals. This is the problem of determining the elements of the algebraic system matrix. The problem branches out depending on the order of the used Gauss quadrature formulae, on the presence and the form of the characteristic feature of the desired potential density within a particular boundary fragment, and on the kind of integral equation kernel which, in turn, is determined both by the type of fundamental solution of the initial problem and by the character of the boundary condition. Formution of u System of’ Lineur Alyebruic Equations. It is this stage that realizes the discretization of the initial problem. Taking into account Sections 1V.C and IV.D, the present module fulfills mostly logical and organizing functions. The main differences in the formation of matrix lines are due to the fact that the calculated point belongs to the inside or to the end of the boundary segment; in the latter case, the corresponding equation approximates the boundary condition for the B-spline rather than the integral correlation. Solution of a System of Linear Equations. The present problem is rather bulky with respect to the number of arithmetical operations to be performed, but it is the simplest logically: The only thing to do here is to turn, by instruction in the input prescription (or by default), to one of the subprograms

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available in the package. In terms of content, an alternative may consist here of the preference: either the efficiency or the reliability of computations or their reliability. For instance, after solving the system numerically, one can, by way of checking, calculate the residual and, if necessary, make one or two iterative corrections. For some “run-in” problems, however, where they are reliably stipulated, such stages can be omitted. Another example of economical computation is that of solving systems of linear equations with different right-hand parts but identical matrices (such a situation occurs in computing axisymmetric perturbations of the potential). In this case, if use is made of, say, the Gauss method of elimination, it is sufficient to perform only one elimination run, thus storing the results of factoring the matrix into triangular components. Computation of the Potential and its Derivatives. In simulating EOS, these values must be determined on the symmetry axis. The number of arithmetic operations at this stage is roughly directly proportional to the number of calculated points on the axis, to the number of computed functions, of collocation points on the domain boundary and quadrature nodes on each boundary subinterval. A special case consists of the points in the vicinity of the cathode and the EOS screen, where special approximation algorithms described in Section 1II.C are applied, taking into account the behavior of the potential in the vicinity of the electrode. Problem (c) includes, besides all the stages listed above, the problem of numerical integration of paraxial trajectories and finding EOS characteristics, most of which reduce to computing integrals from rather complicated expressions. For the latter, the application of the Simpson method has been shown by the available calculation to be fully economical and sufficiently accurate. Here the computation parameters are the number of calculated axis points and the length of the cathode zone within which the trajectories are determined by presenting the solution of paraxial equations in the form of series. The stage of optimization [problem (a)] consists of the efficiency functional minimization controlling with the restrictions imposed by the user in the input language. As compared to the “elementary” problem discussed at the beginning of this section, the input task here contains additional information with a rich enough content load; and, accordingly, the fulfillment of stage “ A becomes rather complicated. As a rule, practical optimization problems are solved in several computation runs with the intermediate results stored on external (storage) media and with the minimization process control parameters corrected subsequently on the basis of the expert analysis of the data obtained. Before each run, a trained user can change either the initial values of the variable parameters and the limits of their admissible variations

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or the form of the objective function and of the functional restrictions, or the local minimization accuracy criteria for solving auxiliary quasilinearized problems, or the type of the minimization methods used. It appears impossible, at least within the realistic machine time, to fully automate the process of minimization, due to the complex ravine behavior of the minimized functional and to the “antagonistic” functional restrictions arising in designing EOS. The solution of this problem is left to the skill of the expert user; and it is only by working out some empirical recommendations for certain types of instruments that some compromise paths become discernible. The last type of problems-multivariant computations-are associated with automating the search for initial data according to a law specified by the user in the input language, permitting some small auxiliary programs to be written for these purposes in FORTRAN. This procedure can be used most advantageously in solving problems (a)-(c), but it can also be effectively applied to tentatively search for initial approximations, and to problems of optimization. The indispensable stage of solving all problems is that of treating and visualizing the computational results obtained. Without this, the efficiency of the entire numerical simulation may be lost. The typical representations of the summarized data are drawings of equipotential or field lines (with adjustment to the geometry of the EOS itself), axial distributions of potential derivatives of different orders, graphs of various functional dependences, and digital or tabular information with necessary text explanations. The corresponding EFIR APP software relies on the application of the SMOG computer graphics system and on standard service means of computer operation systems; however, it involves a significant number of its own specialized subprograms carrying out the interface between computational modules and the standard means.

C. Basic Components und Characteristics of the APP E F I R The APP EFIR consists of the following basic components: a) an input language compiler; b) a set of modules realizing numerical algorithms; c) a service system for processing and the output of computational results; and d) a data base. The package structure may be presented as a block diagram (Fig. 20), where the alternative of the program block realization is shown with the help of their parallel connections. The control program of the package carries out the “global strategist” functions of the computing process. The task to be solved is determined here.

ControL program of packege

InitiaC date recazcu Gatiun for muztiuariant cumputZnp

-

-

STOP

I

FIG.20. The programs package EFIR structure.

It may be either calculation of the EOS field characteristics or calculation of the perturbations resulting from the axial symmetry violations; computation and optimization of the EOS optical characteristics, or service system processing of the obtained results, execution of the multivariant computations permitting us to solve series of tasks of the same type during one launching. The program also organizes the dynamic load sequence of the corresponding

256

V. P. IL’IN rt ul.

blocks realizing numerical functions. While conducting optimization calculations, it also controls the number of global iterations and the process of resetting the information into the external storage at several checkpoints of the computational process. It makes it possible, in case of abnormal termination of a computer, to continue the process of optimization in the following session, which is especially important when solving problems requiring more time for computations. K O P F is the control subroutine for solving integral equations. On the basis of the input information, it determines the type of boundary-value problem to be solved (calculating the potential, determining the nonaxially symmetric perturbations for EOS optimization); it chooses the method of computation and the sequence of lower-level program block loadings: BK, G, R, L, EX are organized. Block BK is the input language compiler. Block G yields the formation of matrices and free-term vectors of the algebraic system, approximating the initial integral equations. Block R realizes the computation of the system of linear equations with the help of the program complex KOPLA (Lyapidevskaya, 1980) or the standard procedure MATIN1 (Mazniy, 1978). Block L, depending on the task to be solved, calculates the values of potentials, the perturbation functions, and their derivatives at the points marked in the input task. Block EX is loaded into the main storage at its invocation from the segments G or L. It calculates the auxiliary integrals by means of Gaussian algorithm in intervals obtained from curve segments partition. Block AB, while solving optimization problems, directly organizes the process of the functional local optimization and the calculation of the EOS characteristics. Block G L serves only the tasks of optimization and, having completed the local step of minimization, executes the recalculation of the initial data for the boundary-value problem. All the information exchanges between blocks are organized by the control program, by means of the peripheral units (magnetic drum storage, magnetic disk storage). The service system provides calculation and construction of the calculated domain contour of equal potential lines (equipotential lines), the equal strength module lines, power lines, different vector fields, and functional dependences. The requirements for the data base are as follows: 0 possibility of multiple access, i.e., information storage and access to it by different users;

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

251

0 independence of programs from the database logical struclxe changes; independence of the logical database from the physical one; 0 0 maintenance support (duplication, restoring, etc.).

In the package of applied programs EFIR, the base provides execution of these functions for the following data types: problem descriptions in the input language, the results of field problems solutions, optimization results, and intermediate information.

VI. NUMERICAL EXPERIMENTS This section presents the results of numerical solutions for some simulated and practical problems, which show the efficiency of applying the methods and algorithms considered above.

A . Calculation of the Potential f o r the Perturbation Function 1 . Culculation of' the Potentiul f o r the Cylinder Condenser

Let us consider the problem of calculating the potential in an axially symmetric domain, the boundary of which consists of two coaxial cylinders of length 1 = 5 and the radii R , = 1 , R , = 2. At the internal cylinder, the potential q 1 = 1.0; at the external cylinder q2 = 0. End walls of the cylinders are covered by flat disks, where the boundary conditions of the second order with the zero value of the normal derivative are given. The given task is in fact one-dimensional and has the exact solution In r cp,(r) = 1 - -. In 2 The calculated domain is a rectangle with the sides A B , BC, C D , D A , that are divided uniformly by the collocation points into 4, 20, 4, 20 Intervals, respectively. In the calculations given further, the number of the Gaussian quadrature nodes was equal to four while computing the auxiliary integrals over every interval, and the density approximation of the prime layer was produced by the second-order splines. From Table IV, where for different values of r, the values of the solutions for the exact qe and the numerical (P, in the plane z = 2.0 are given, it follows that the relative error 6 of the potential calculation does not exceed in presence of the mixed boundary conditions. 5

-

258

V. P. IL'IN et a/.

TABLE IV

1.05

1.10 1.15 1.20 1.25

0.929615 0.862478 0.798355 0.736950 0.678057

0.929613 0.862498 0.798366 0.736965 0.678073

1.30 1.35 1.40 1.45

0.025 0.23 0.14 0.20 0.24

0.621471 0.567022 0.514554 0.463927

0.621489 0.567041 0.514575 0.463945

0.29 0.33 0.4 1 0.39

2. Cdculution of the Potentiul with B-Splines of Different Orders Let us consider the results arrived at by analysis of the errors in the numerical solution of the integral equation of the first kind depending on the B-spline order, the number of collocation points, and the quadrature nodes in the calculated intervals. In the domain representing the meridional section of a torus, three problems with boundary conditions of the first kind are considered. The exact r2 solution for the first problem is cpl(r, z ) = z , for the second cp2 = z 2 - -, and 2 2 for the third cp3(r,z) = r 2 z - - z 3 . 3 These problems are characterized by the fact that their solutions have no singularities and that the conditions of spline periodicity used in the process of density approximation fully correspond to the closure of the boundary contour. All this allows us to single out the influence of the main calculating parameters separately. The results of the carried-out numerical experiments are shown in Tables V and VI, where k is the number of the problem, N is the spline order. Table V corresponds to twenty intervals of boundary partition and to four nodes of Gaussian quadratures in each of them; Table VI shows ten intervals and four quadrature nodes; Table VII gives ten intervals and eight quad-

TABLE V N k

1 2 3

-1

3

4

5

6

7

46 48 142

14 16 49

2.7 3.0 8.0

2.4 2.4 4.0

2.3 2.2 3.3

2.3 2.2 3.1

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

259

TABLE VI N k

I

3

3

-1

3

4

5

h

7

2120 1381 3402

1356 704 2026

842 289 751

730 216 316

682 186 93

666 176 66

TABLE VII N k 1

2 3

2

3

4

5

6

7

1453 1253 3537

65 1 550 1074

154 142 821

82 68 383

40 38 162

21 28 73

rature nodes. Data given in Tables V-VII represent the maximum values of the absolute errors of the solution, calculated along the symmetry axis at 0 5 z 5 1, with r = 2,O enlarged lo7 times. The given results testify for the fact that the accuracy with increase in the approximation order of a solution increases besides application of the third- and fourth-order B-splines reduces the error by a factor of two, three, and more, as compared with the second-order splines. However, for splines of sixth and seventh order, the value of an error has no notable changes, which can be explained by the dominating rule of the numerical integration errors. With the number of quadratures being doubled, the process of convergence to an exact solution can also be observed for the splines of sixth and seventh order. Thus, at a relatively small number of boundary partitions, it is useful to combine application of the high-order Gaussian quadratures with highorder approximating B-splines O n the whole, the problem of optimal consistency for all the algorithm parameters (in order to achieve maximal efficiency of the method), as is clear from the tables, is nontrivial and requires special investigation.

3. Calculation of the Potential and its Derivatives f o r a Spherical Condenser Assume the potential of the external sphere (cathode) to be equal to zero, and the internal sphere (anode) to be equal to one. In the coordinates (r,z ) , the calculating area for the given task is bounded by two semicircles with the radii R, = 180, R, = 60 (Fig. 21).

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Z

FIG. 21. The spherical condenser

Thirty intervals of partitions are given over the external circle and ten over the internal; the number of quadrature nodes in the intervals is assumed to be equal to four. The surface density of the charge is approximated by the secondorder splines with boundary conditions of the form ? 0 2 / & = 0 at the ends of the semicircles. Calculating the potential and its derivatives in the neighbourhood of electrodes directly on the basis of integral representation leads to large errors. With application of the Hermitian interpolation described in Section IV.C, errors in the calculation of the potential and its two first derivatives near the boundary decrease accordingly to 6 4. and 8 The calculation values of the third and fourth potential derivatives in the center of the cathode appear to be equal to 5.14495. lo-’ and 1.14353 respectively, which satisfies the condition that an error should not exceed 2 10-20/,.

-

-

-

-

B . Calculations of’ the Nonaxial Symmetric Perturbations by the Bruns-Bertein Method

In Section IV.B, it is shown that the problems with small violations of the axial symmetry are reduced to solving integral equations for the perturbation harmonics. It is of particular importance to provide high enough precision for the first- and the second-order harmonics (rn = 1,2), because they make the major contribution to the perturbated field. The calculation algorithms for the harmonics of nonaxial symmetric perturbations were tested on the basis of the following simulated problems. Consider a sphere with the radius R = 1 and center at the origin of the coordinates, at the boundary of which the condition $,Jr,z) = I,(r)sinz is given, where I,(r) is a modified Bessel function. The boundary of the cal-

26 1

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN TABLE VIII m = l

0.5 0.4 0.3 0.2 0

2.4002 1.9495 1.4794 9.9456 0

111

2.4001 1.9495 1.4794 9.9459 1.lo- l 4

=

2

6.010 4.880 3.702 2.488 0

m=3

5.998 4.872 3.697 2.485 1.10. l L

1.23. 10 9.30. lo-' 6.61 . 10 I 4.23. 10-l 0

1.10 9.01 . lo-' 6.36. 10- ' 4.16. lo-.' 1.10-'2

culation domain represented by a circle with the radius R = 1, was divided by collocation points into 10 different intervals, with the number of Gaussian quadrature nodes in every interval being equal to 6. In Table VIII, values for the exact numerical solutions of Eq. (286) for m = 1,2,3, at r = 0 and different z , are given. As can be seen from the table, the numerical results at rn = 1 coincide with the exact solution by three to four significant digits. Note that as the number of the harmonics increases, the error also increases. For a spherical condenser (Fig. 21), consider the problem of calculating the potential perturbances caused by a small shift of the external sphere in the direction perpendicular to the symmetry axis. The results of the equation computation for the first harmonic ( m = 1) are given in Table IX, where cpr denotes the analytical values of the first derivative with respect to r when r = 0, taken from the paper by Dashevsky et al. (1979). Here q1,?, qZ,* are numerical values of the first derivatives of with respect to r, corresponding to the assignment of boundary conditions for harmonic equations, calculated by central differences with a step h = 1.8 and with the help of integral representation qo,rare the analogous values obtained by

TABLE IX 103. vn

1.5432 1.5321 1.5181 1.so00 1.4022 9.2648 5.9337

1.5202 1.5462 1.5417 1.5252 1.4189 9.2818 5.9686

1.5 0.92 1.6 1.7 1.8 0.18 -0.59

1.5439 1.5324 1.5208 1.5032 1.4051 9.2643 5.9641

0.04 0.02 0.18 0.21 0.21 0.006 0.51

1.5432 1.5327 1.5212 1.5036 1.4052 9.2648 5.9644

6.5. 10 ' 4.1 * 2.1 . l o - ' 2.4. lo-' 2.2. lo-' 4.3 . 5.2. lo-'

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V. P. IL’IN et a/.

numerical solution of the harmonic equation with an analytical assignment of the boundary condition. For every solution, the error percentage is indicated. C. Computation of‘ the First-Order Parameters and the Aberration Coeficients

In this section, we shall treat the results of some numerical experiments dealing with the computation of the first-order parameters and the aberration coefficients on the basis of a model problem for the cathode lens “spherical condenser.” In Table X, exact and numerical values of the indicated characteristics corresponding to R, = 180, R, = 60, (ps = 1 are given. The electrostatic potential and its derivatives were calculated at 24 points of the axis by solving the integral equation for the computation parameters indicated in Section VI.A.3. The data given in Table X speaks for reliable enough calculation of electron-optical characteristics in the model problem under consideration. Let us now consider the results of analyzing the algorithm stability with respect to the choice of method and some calculation parameters of numerical integration of the limiting paraxial equation, and also with respect to the error of the axial potential assignment and its derivatives. Table XI shows the value positions of the first-order functionals of the Gaussian plane z g and magnification M 9 , computated by the Runge-Kutta and Milne methods for two positions of the transition point (“starting” point) z,, from decomposition (22) to numerical integration of the limiting paraxial equation. Parameter N represents the relative error order of the potential and its derivative assignments that were computed on 120 points of the interval 0I zI R , - R, by the formula suggested by V. Ivanov. +,cn)(~) =

@.‘“’(~)(l + (5

-

0.5) *

TABLE X functionals

zy

calculated exact

360.0 360.0

functionals

6

calculated exact

6.000 6.000

H

P

Q

B,

180.0 180.0

720.1 720.0

-720.1 -720.0

4.000 4.000

- 2880

G,

D,

6

4

G.?

4

6.000 6.000

0.0056 0.0055

-6.001 -6.000

-6.001 -6.000

-0.008337 -0.008333

.& -

1,000 1.OOO

=,

-0.1547. -0.1543.

-

2880

TABLE XI ~

N

functionals

-51

5 I0

1

352.92 382.55

2

3

4

5

m

359.29 362.09

359.93 360.21

360.01 360.05

360.00 360.00

360.00 360.00

329.01 334.86 0.9962 1.0116

370.68 370.76 0.9996 1.0012

358.75 358.76 1.0001 1.0002

360.04 360.04 1.0000 1 .0000

360.03 360.04 1.0000 1.om0

- ,q

method Runge- K u t t a

360.00 5

10 5 10 MV

exact values

52.45 54.77 0.9625 1.1242

Runge-Kutta I .0000

~. ~

5 10

Milne

0.4340 0.2917

0.8729 0.8485

1.0407 1.042 I

0.9950 0.9948

1.0002 1.0002

1.0002 1.0002

Milne

264

V. P. IL'IN et al

where @("I ( n = 0, 1,. . .) is the exact value of nth derivative of the potential, 5 is the presido-random number uniformly distributed on [0,1] and produced by computer independently for all the values of z, n, N. From Table XI, it follows that in case of a rather small potential and derivative error (N > 4), the Runge-Kutta and Milne methods are approximately equivalent and give a rather small error. For N I 4, the Runge-Kutta method is preferable because in equal conditions it requires 1-2 orders less accuracy in field calculation compared with Milne's method. At the same time, dependence of the functionals z q , M,, on the choice of the parameter z,, appears to be negligible and practically similar to both methods. The method of paraxial equation integration on the basis of successive substitution given in Section 1V.D appears to be approximately as stable as the Runge-Kutta method but more economical. For comparison, Table XI1 represents relative errors of the computation of the curvature coefficient D with various N . In column 1, are the results obtained by V. Ivanov on the basis of the corresponding formula in the paper by Ignat'ev and Kulikov (1978), without distinguishing singularities in the integrand; in column 2, with distinguishing the singularities; in column 3, by the method of z-variation (see Section 1I.B). Analysis of Table XI1 shows that the formulae with singularities possess strong instability to error in the potential and the derivatives. True calculation of the abberations coefficients according to these formulae makes it necessary to determine the potential with an error tolerance equal to which today seems to be inaccessible in real multielectrode systems. The elimination of singularities in the integrands of the aberration coefficients with the help of the method suggested in Section 1I.A leads to a

TABLE XI1 calculation of hD,,

"()

n!

without distinguishing singularities

with distinguishing singularities

by method of T-variations

1

z loo

70 50 21 2.5 0.5 0.5 0.5

1.5 0.6 0.3 0.3 0.25 0.25 0.25

2 3 4 5 6

7

120 20 6 4

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

265

decrease of approximately one order in the requirements for accuracy of the potential calculations, still leaving them at a rather high level. When realized with efficiency, the integral equation method of the potential theory allow us, in general to satisfy these requirements and to obtain results that agree with the experiment; still, the solution of the analysis problems requires considerable time. Numerical experiments, carried out by Kolesnikov, showed that the method of z-variations combined with the adaptive computation of quadratures, at solving integral equations for the potential allows a decrease in the total time for solution of the analysis problems by approximately one order. It makes it possible in practise to apply widely the interactive mode for the computer simulation of the emissive system. D. Solution of’ the Practical Problem 1 . Aberration of the Cathode Lens with or Quasi-Spherical Field As was shown, the cathode lens model “spherical condenser” in a certain sense posseses the following unique properties: All its position abberations, including curvature and distortion, are equal to zero. Construction of a real cathode lens with a spherically symmetric field encounters considerable technical problems. The cathode lens considered below (Fig. 23) can serve as an approximation to the ideal one in Fig. 22 (Dashevsky et al., 1979). The potential c1 of the subfocussing electrode allocated outside the lensworking region, is chosen on the condition of the best approximation of the lens axial to potential the axial potential of the corresponding “spherical condenser.” The lens anode is a spherical fine-structural grid with curvative radius R, three times smaller than the curvature radius of the cathode, which due to (87) corresponds to the first-order parameters Mq = 1, zg = 2R,, for the cathode lens “spherical condenser.”

FIG.22. Ideal model of cathode lens-“spherical condenser.”

266

V. P. IL’IN et al.

In Table XIII, the calculated and experimental values of the first-order parameters and aberration coefficients are given (in pure units). The relations between measurable and pure values of the coefficients D and E are of the form e

=

ER:,

d

=

DR,&,

(334)

where, according to (43),

+ e, + e,, d = d, + d, + d,. e

= r,

(335)

The terms with subscripts I, g , and s conventionally characterize the contribution of the lens, the grid, and the image receiver. Note that the experimental values do not refer to the Gaussian plane but to the plane of best focussing (Shapiro and Vlasov, 1974). Evaluations show that the very negligible divergences between the calculation and experimental data in the first-order parameters are explained by this very thing. Aberration coefficients depend only to a small degree, on the installation plane of the image receiver, and that is why the contradiction between calculation and experiment with respect to the curvature coefficient should have another explanation. It is explained by the above-mentioned instability in the formulae used in this computation for aberration coefficients with a singularity in the integrands. Further corrections based on the formulae with an eliminated singularity allow a reduction of the error in the image curvature computation to the level of error tolerance. 2. Computation of the Cathode Lens with Variable Magni$cution (Potential Variation)

Problems for which with fixed geometry only the electrode potentials are being varied seem to be the simplest among the problems of emission system

TABLE XI11 functional -9

1%

-‘r C’

(I,

+ d,

calculation

I I8 0 88 0.950 -006 -1

1

experiment 1 80 0 87 0 952 0 (bounds of measurement error) -09

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

267

parametric optimization both with respect to the realization of the numerical algorithms and the requirements of the computer time. The necessity to recalculate the perturbation functions on the external iterations ceases to exist, because the potential representation (Section 1II.B) appears to be exact and can be extended over the entire range of the varying parameters. As an example, let's consider the problem of calculating the system with a variable magnification, assuming that there are two varying potentials e l , c 2 . Let's formulate it in the form of the following boundary value problem, i.e., to find values of Fl and C; such that trajections of the limiting paraxial equation c and w corresponding to them would satisfy in the plane of the screen z = z , the boundary conditions

u ( ? ~ , ? ~ , z=~ 0, )

IW(F~,?~,Z,)I = Mg.

(336)

With the solution of the assigned boundary-value problem for a series of values M9 for fixed zs, we can get the relations cl(M,), C;(M,) and can consequently point out the conditions in which various magnifications for the invariable position of the image plane are used. Let us solve the boundary-value problem (336) by the Newton-Rafson method, having written it for the first focus in the form of the following iterative scheme: c(P+ 1)

2

1

= c ( 1P ) - __ A(P)C v ' P ) d c 2 w . ( P )

=

+ __[,'P),? A(P)

p P )

+ M)a'c 2 v(P)],

(337)

1

c(P)

-

c1w ( P )

+ M)2clU(P)],

-

-

0,1,. . . , is the iteration number, A(p) = (?c 1d P ) (7c,u"p)IS . the functional determinant (all the derivatives are calculated in the plane z = zs). Iterations according to the formulae (337) go on until the following conditions are satisfied:

where

2

c2

p

@.(j

=

c, w ( P )

I w y + M,I

< E l , (usp)I < E 2 ,

(338)

where c , ,E~ are the given. Numerical realization of the Newton-Rafson method assumes realization of the conditions A ( p )# 0 ( p = 0,1,. . .), which speaks for the independence of (336) A small modification of the stated algorithm allows us to take into account constraints of the form: Cl,min

5

c1

5

C1,m ax,

C2,min

5

c2

5

C2,max7

(339)

and also provides searching for the solution that corresponds to the required focus number.

268

V. P. IL’IN et a/.

FIG.23. Cathode lens with quasi-spherical field

The curves F1(Mg)and F2(Mg),shown in Fig. 24, are obtained as a result of the boundary-value problem solution (336) for the electrostatic cathode lens, whose geometry is shown schematically in Fig. 25. Curves 1 and 2 in Fig. 24 correspond to two different combinations of geometrical parameter values for the variation of Mq in the region 0.6 < Mq < 1.3. The calculation errors were assumed to be equal to = E~ = 9 The coordinate of the screen plane z, was assumed to be equal to 3.7 (in relative units). The values Zl and C; are also given in relative units and normalized by the accelerating voltage value U,. The calculations were carried out in the BESM-6 computer using APP TOPAZ (Ignat’ev et al., 1979). Let us analyze the obtained curves. Note that on curve 1, unlike curve 2 (Fig. 24a), there is a point K with the coordinates MgCu, for which the

-

condition

=

~

0 is fulfilled. The horizontal straight line passing higher

c;./0-2

B

Q

t

2.

I d6

Mi

f.0

M;

h

I My

0’6

MG

I Ib M$

I j My

FIG. 24. Dependence of the optimal potentials c , , c 2 of the magnification M, in the cathode lens

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

269

than the point K crosses the curve F,(M,) at two points, with the magnifications M & and MG corresponding to them. In Fig. 24b, the values of c"; and 7,of'curve 1 correspond to the values of M b and M ; . Thus, having two potential values 7; # c";'and fixed 7, > Flcu,we get two different magnifications M $ , M g ( M ' , < Mgcu< M ; ) for a stable position of the representative plane z = z,. In practice, this effect of double focussing and its application is considered in detail in the paper by Ignat'ev and Rumyantsev (1976). 3. Geometrical and Potentiul Optimization of the Cathode Lenses Let us consider examples of numerical optimization of the cathode lenses with simultanious variation both of the geometrical and potential parameters. To begin with, let's consider the solution of two optimization problems for the cathode lens shown schematically in Fig. 26. The calculating parameters entering the task conditions are chosen to provide a solution for the given constraints. The initial approximation coincides for both problems. 1. Obtain the given position of the image plane zg = 3.502 and the crossover plane zcr = 3.148 for the electron-optical magnification M , con0.07. tained in the region 0.06 < M g I

The vector of the varied parameters x = {xi}:=l has the following components: x1 is the potential of the subfoccusing electrodes corresponding to the boundary points 14-17, x 2 is the electrode potential corresponding to the boundary points 8- 13, x j is a curvature radius of the cathode surface, x4 is the distance from the symmetry axis to the boundary part that lies between points 14 and 15. To vary the component xj, r variations of the cathode node that correspond to s = 4 according to the classification given in Section 1II.C were

FIG.25

270

V. P. I L I N et al.

used; to vary the component x4, the variations of the shift along the Or-axis were used, which is a generator of the subfocussing electrode (s = 1). Being formalized, the task A (see Section 1II.A) consists of minimizing the functional F,,(x) = 103(z, - 3.502)2 + 103(z,, - 3.148)2 constrained by F,(x) = MS

-

F,(x) = - M y

0.07 i 0,

+ 0.06 I 0,

0 i x1 i 400,

< 1000, 2.0 5 x3 < 2.4, 1.26 5 x4 < 1.55.

-300 5 X Z

The process of optimization is shown in Table XIVa, where the values of the functionals under control and their disparities from the optimum are given for every external iteration; included are also some results of optimization with quasilinear approximation for the first external iteration (Table XIVb, where n is the number of an internal iteration). The disparities were computed by the formula

where F") is the functional for the external iteration with the number N ( N = 0 corresponds to the initial approximation), FoPtisthe optimal (required) value of the functional. The computations were stopped and the task was considered solved when the constraints (340) were fulfilled and the disparities dz,, SzCrwere not greater than 0.2%. To solve the task A in the region n, (E,,), 91 computations of the goal functional and the constraints were required (see Table XIVb). The perturbation function diagrams dXl@(z),i = 1,. . . ,4, calculated for the first external iteration by solving the corresponding integral equations in variations, are shown in Fig. 27. It is clear from Table XIV, that the solution of task A with the accuracy given above required three external iterations, which took ~ 1 . hours 3 of computing time for the ES- 1060 computer. At this, the minimized functional decreased by a factor of lo4. 2. It is necessary to provide the value given for the curve coefficient D = - 1.18 for the given position of the image plane zg = 3.502 and of the crossover plane z,, = 3.148. The first three components of the varied parameters vector are the same as in the first task, the component x4 is the length of the electrode corresponding to the points 9-10.

-

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

27 1

U

FIG.26. Cathode lens with variable magnification

TABLE XIVa

0 1 2 3

2.712 3.099 3.149 3.148

100.0 233.0 273.0 212.6

2.290 2.344 2.347 2.341

- 200.0

233.0 303.1 301.1

0.0583 0.0646 0.0666 0.0666

250.07 3.78 0.02 0.01

100.00 11.5 0.30 0.15

1.360 1.263 1.263 1.263

3.171 3.464 3.503 3.502

100.00 13.0 0.27 0.13

TABLE XIVb

12 23 36 91

233.0 233.0 233.0 233.0

-86.2 233.0 233.0 233.0

2.298 2.316 2.331 2.344

1.344 1.319 1.284 1.263

3.349 3.454 3.458 3.461

3.045 3.073 3.082 3.087

0.0563 0.0641 0.0642 0.0643

33.79 7.78 6.28 5.46

272

V. P. IL'IN rt ul

Task A consists of minimizing the functional Fb = 103(z, + 105(D + 0.00118)2constrained by

-

3.502)'

+

103(zcr- 3.148)2

0 I x1 I 400,

-300 I ~2

< 1000, (342)

2 I .y3 5 2.4, 0.5 4 'id

1.0.

As can be seen from Table XV, an approximate solution of the task A was already obtained at the second external iteration with disparities 6z, = 0.6%, dzC, = 3.50/,, 6, = 1.9%. The minimized functional F, decreased during this process by more than a factor of 20, compared to the initial appoximation. I t is essential to note that practical calculations completely confirmed the conclusion made in Section 1II.B that the method of integral equations is more economic in variations as compared with potential differentiation with respect to the parameters of perturbation with the help of the difference scheme. Numerical experiments showed that in rather complicated practical problems, the greatest difference in both results that refer to the maximum does not exceed 10-'04, with the calculation of the two perturbation functions by the integral equation method being produced two times as quickly as differentiation by the formula of central differences; and with an increase of the number of varied parameters, time economy increases. It is clear that, far from all the problems of the emission systems, construction designs can be solved as the parametrical optimization problems during one computer session. In many cases, the service facilities discussed in Section V.C are necessary: calculation interruption at the predetermined checkpoints in order to evaluate the obtained results; change in the

0

100.0 347.5 347.4

1 7

200.0 371.8 369. I

2.290 2.009 2.142

0.990 0.987 0.98 1

2.875 3.502 3.506

~~

-3.96. lo-' -4.94. 1.71 . 1 0 ~ -

810.1 2.557 0.368

100.0 0.03 1 0.64

IOO.0 9.65 3.46

100.0 2.41 1.91

2.629 3.098 3.166

273

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

optimization “scenario” by means of efficient replacement of the functionals; constraints; types of geometrical variations, and so on. It often follows from the statement of the problem itself, into what stages it should be divided in order to achieve a more effective solution. Below, we shall consider the problem of the geometry and the potentials of the cathode lens with variable magnification design, providing for two model magnifications ( M = 0.4, M = 1.2) high enough quality of the image to serve as an example. Analogy of the lense is considered in Chin-tao (1982) (see Fig. 28).

CI

0

LO

2.0

3.0

FIG. 27. The perturbations functions of potential in cathode lens represented in Fig. 26. a ) for parameters x , ( l ) ;x,(2): h) for parameter x 3 , cl for parameter xq.

274

V. P. IL'IN et al

Without going into detail, let us describe the main stages of this problem solution and show some results. The initial problem is naturally divided into two independent optimizational tasks. 1. Minimization of the curvature coefficient D at magnification M , = 0.4 and assigned constraints for the distortion coefficient and the Gaussian plane position z g , by variation in the assigned limits of the cathode surface curvature radius of the geometrical parameters, determining the position and form of the electrodes r,, r,, r, and also the potentials U,, Us of the subfocussing and scaling electrodes. 2. For the cathode lens, found in the result of the first problem solution, the determination of the potentials U, and Us, realizing the other assigned magnification M , = 1.2 at fixed (within the error tolerance) position of the Gaussian plane. Table XVI shows the values of the electron-optical functionals under control for both enlargement modes. In Fig. 29, the dependences Dlw,, Elw, of the magnification M , in the 1.2 are given. range 0.35 5 M ,

0.395 1.17

1

7

0.16.10.5 -0.13.

88.4 89.6

142 -506

-0.53.10-4

-0.66-

1505 15000

12500 12500

]: 0

I

I

I

I

I

I

I

I

I

I

40 6U 80 FIG.28. Scheme of cathode lens with variable magnification (above axial line trajectory and potential for magnification M = 1.2 are showed, below for magnification M = 0.4).

0

20

-

EMISSION-IMAGING ELECTRON-OPTICAL SYSTEM DESIGN

a

I

275

B

FIG.29. The dependence of normed curvature coefficients D/w.,(a) and distortion f ,wg (b) from magnification M9 in cathode lens, is represented in fig. 27.

C

d

5 10 5 10 5 , m m Fic;. 30. The calculated distribution of distortion and relative resolution along field for M y = 0.4 and M9 = 1.2 in cathode lens, represented in fig. 27. a , b b for distortion; c,d-for resolution, D-result of experiment.

It is interesting to note that the curvature coefficient D in the considered range of enlargements becomes equal to zero twice, and the distortion coefficient E is rather small. These conclusions of the calculation are reliably confirmed by the experimental data. Figure 30 shows the calculation distribution of distortion along the representation field; the experiment result is marked by a small square. The calculation results given above are obtained with the help of APP EFIR.

216

V. P. IL’IN e t a [ .

REFERENCES Antonenko, 0. P. (1964). Numerical Solution of the Dirichlet Problem for Non-Closed Surfaces of Rotation, Vychislitel’nye Sistemy, 12, p. 39, Novosibirsk. Artsimovich, L. A. (1944). Electrostatic Properties of Emission Systems. Izu. A N , Ser. Fizrka 8, 131. Berkovsky, A. G., Gavanin, V. A., and Zaidel’, 1. N. (1976). “Vacuum photo-electronic devices.” Energiya, Moscow. Bertein, F. (1947). Relation entre les Defauts de Realisation des Lentilles et la nettete des Images. Ann. de Radioelectricith 2, 379, (1948) 3, 49. Bruns, H . (1876). Ueber einen Satz der Potentialtheorie. CreLle Journ. Math. 81, 349. Cheremisina, N. S . , and Kas’yankov, P. P. (1970). Research of Aberrations of a Cathode Lens with the Axial Potential Presented as Sums of Exponents. Izv. LETI 89,222. Cheremisina, N. S., and Kas’yankov P. P. (1970). Application of Non-Linear Programming Methods to Minimization of Aberration of the Electron-Optical Systems. Izu. LET1 96,108. Chin-tao Fong, and Kung-lin Tung. (1982). Methods of Design for Multielectrode Image Tube. Proc. of CLEO-82, p. 112. Brighton. Chou Li-Wei, Ai Ke-Cong, and Pan Shun-Chen. (1983). O n Aberration Theory of the Combined Electromagnetic Focussing Cathode Lenses. Acta Phys. Sinica. 32, 376. Dashevsky, B. E., Ignat’ev, A. N., Ivanov, V. Ya., and Kulikov, Yu. V. (1979). Cathode Lens with Quasi-Spherical Field. Optiko-Mekh. Prom. 11, 41. Der-Shvarts, G. B., and Kulikov, Yu. V. (1962).O n the Theory of Tolerance of Electron-Optical Devices. Radiotekhnika i Elektronika 33, 2067. Dvait. G . V. (1973). “Tables of Integrals.’’ Nauka, Moscow. Fedorenko, R. P. (1978). “Approximate Solution of Optical Control Problems.” Nauka, Moscow. Flegontov, Yu. A., and Zolina, N. K. (1978). On the calculation of Features of Electronic Image in Electrostatic Cathode Lenses. Zhurn. Tekhn. Fiz. 48,2479. Gamkrelidze, R. V. (1962). O n Zero-Overshoot Optimal Responses. Dokl. A N . 143, 1243. Glaser, W. (1938). Die kurze Magnetlinse von kleinsten offnungsfehlern. 2. anyew. Phjsik. 109, 700. Glaser, W., and Schiske, P. (1953). Bildstorungen durch Polschuhasymmetrien bei Electro nenlinsen. Z . anyew. Physik. 5, 329. Godunov, S. K. (1980). “Solution of Systems of Linear Equations.” Nauka (Sib. otdel.), Novosibirsk. Gradshtein, I. S . , and Ryjhik, I. M. (1956). “Table of Integrals, Sums, Series and Products.” Fizmatgiz, Moskow. Gurman, V. I., and Krotov, V. F. (1973). “Methods and Problems of Optimal Control.” Nauka, Moscow. Hartley, K. F. (1974). On the Electron Optics of Magnetically Focused Image Tubes. J . Phys. D . I . 1612. Ignat‘ev. A. N., Ivanov, V. Ya., and Kulikov, Yu. V. (1979). In “Numerical Methods of Solution of the Electron-Optical Problems,” p. 16. Izd. VTs SO AN, Novosibirsk. Ignat’ev, A. N., and Kulikov, Yu. V. (1978). Theory of Aberrations of the Cathode Lenses of Third Order with Curvilinear Components. Radiotekhnika i Elektronika 23,2470. Ignat’ev, A. N., and Kulikov, Yu. V. (1983). In “New Methods of the Calculation of the Electron-Optical Systems,’’ p. 131. Nauka, Moscow. Ignat’ev, A. N., and Rumyantsev, N. G. (1976). Automatic Conception of Defocusing at the Change of the Energy of the Electron Beam. Electronnaya Tekhnika, Ser. 4,9,513. ll’in. V. P., and Ivanov, V. Ya. (1975). In “Typical Programs for Solving Mathematical Physics Problems,” p. 5. Izd. VTs SO AN, Novosibirsk.

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ll’in. V. P., and Kateshov, V. A. (1982). Program Package EFIR for Calculation of Potentials and their Perturbations. Avfometriya 4,67. ll’in, V. P., and Popova, G. S. (1983). In “Algorithms and Methods of Calculation of the Electron-Optical Systems,” p. 120, Novosibirsk. loffe, A. D., and Tikhomirov, V. M. (1974). “Theory of Extremal Problems.” Nauka, Moscow. Ivanov, V. Ya. (1977). Numerical Solution of Integral Equations of Potential Theory in Electron-Optical Systems,” p. 120 Izd. VTs SO AN, Novosibirsk. Janse, J. (1971). Numerical Treatment of Electron Lenses with Perturbed Axial Symmetry. Optik 33, 270. Kas’yankov, P. P. (1950). Conditions for the Correction of Aberrations of Astigmatism and Curvature of Image in the Electronic Lenses. Jhurn. Tekhn. Fiz. 20, 1426. Kas’yankov, P. P. (1966). A New Method of Calculation of Aberrations. Trans. Gos. Opt. Inst. 33, 62. Kas’yankov, P. P., and Taganov, I. N. (1964). O n the Direct Method of Calculation of Electromagnetic Systems. Optiko-mekh. prom. 11, 14. Kateshov, V. A. (1984). “Applied Program Package EFIR.” Preprint, VTs SO AN, N 473, Novosibirsk. Kel’man, V. M., Sapargaliev, A. A,, and Yakushev, E. M. (1973). Theory of Cathode Lenses. Jhurn. tekhtz. fiz. 43, 52. Kolesnikov, S. V., and Monastyrsky, M. A. (1983).A New Method of Calculating Perturbations of the Potential in Problems with Weakly Disturbed Axial Symmetry. Jhurn. Tekhn. Fiz. 53. 1669. Kulikov, Yu. V. (1975). On the Evaluation of Image Quality of Cathode Electron-Optical Systems with an Account of the Third-Order Aberrations. Radiotekhnika i Elektronika 20, 1249. Kulikov, Yu. V., and Monastyrsky, M. A. (1978). Theory of Aberrations of Cathode Lenses. Chromatic Aberrations. Radiotekhnika i Elektroniku 23,644. Lyapidevskaya, Z. A. (1980). “Complex of Procedures According to Linear Algebra.” Preprint, VTs SO AN, N 259. Novosibirsk. Mazniy G. L. (1978). “Programming on BESM-6 in the System ‘Dubna’.” Nauka, Moscow. Melamid, A. E., and Soboleva, N. A. (1974). “Photoelectron Devices.” Vys. Shkola, Moscow. Monastyrsky, M. A. (1978). On Asymptotic of the Solutions of Paraxial Equation of Electron Optics. Jhurn. Tekhn. Fiz.48, 1 1 17. Monastyrsky, M. A. (1980a). On a Jump of lnfluence Functions in Cathode Lenses with Non-Smooth Distribution of Electro-Static Potential. Jhurn. Tekhn. Fiz. 50, 1939. Monastyrsky, M. A. (1980b). In “Numerical Methods of Solving Electron-Optical Problems,” p. 121. Izd. VTs SO AN, Novosibirsk. Monastyrsky, M. A,, and Schelev, M. Ya. (1980). “Theory of Time Aberrations of Cathode Lenses.” Preprint, Fiz. Inst. AN. N 128. Moscow. Orlov. B. 1. (1967). On Numerical Solution of the Direct Problem of Electron Optics. Radiotekhnika i Elertronika 12, 2274. Ovsyannikov, D. A. (1980). Mathematical Methods for the Control of Beams. Izd. LGU. Leningrad. Recknagel, A. (194 1J. Theorie des Elektrischen Elektronen-mikroskops fur Selbstrahler. 2.anyew. Physik. 117, 689. Seman. 0. 1. (1953). On the Question of the Existence of the Extremum of the Spherical Aberration Coefficient for Axial Symmetrical Systems in Electron Optics. Jhurn. Tekhrr. 1.7:. 24. 581. Shapiro, Yu. A. (1964).O n the Direct Method for Calculation of Electromagnetic Lenses. Jhurn. Tekhn. Fiz.34, 1747.

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Shapiro, Yu. A,, and Vlasov, A. G. (1974). “Methods for Calculation of Emission ElectronOptical Systems.” Mashinostroenie, Leningrad. Sturrock, P. A. (1951). The Aberration of Magnetic Electron Lenses due to Asymmetries. Phil. Trans. Roy. Soc. London. A . 243, 387. Szilagyi, M., Yakowitz, S. J., Duff, M. 0. (1984). Procedure for Electron and Ion Lens Optimization. A p p l . Phys., Lett. 44, 7-8. Stechkin. S. B., and Subbotin, Yu. N. (1976). “Splines in Computational Mathematics.” Nauka, Moscow. Tretner, W. (1959). Exitenzbereiche Rotationssymmetrischer Elektronenlinsen. Optik 16, 155. Uchikawa, Y., and Maruse, S. (1969). Theory of Chromatic Aberration of Cathode Lenses. J . Electron. Microscopy, 18,26. Vorob’ov, Yu. V. (1956). Scattering Pictures in Electrostatic Emission Lenses. Jhurn. Tekhn. F i i . 10, 2269. Ximen Ji-Je, Chou-Li-Wei, and Ai-Ke-Cong. (1983). Variational Theory of Aberrations in Cathode Lenses. Optik 66, 19.

Index A

Aberration coefficient chromatic, 165 geometric, 164 model, 158 theory traditional, 159 based on differentiations with respect parameter, 167 Absorbed current, 15 Adjoint equation, 214 vector. 214 Amplification, 49,60 basic configuration, 66 cylindrical geometry, 69 formula, 75-78 limits. 60 parallel plates, 49 Angle of image rotation, 171 Aperture, pressure-limiting, 67 Approximation of elliptic integrals, 230 Attachment, electron, probability, 31-32 Automation principles, 246 Avalanche, 57, 68 Axial synthesis of cathode lenses, 207

B Backscattered electrons, 5 Best focusing surface. 177 Boundary distortion, 182 Breakdown potential, 54 Bruns--Bertein method, 182 B-spline, 228 C

Cascade. 41 Cathode lens, 158 with quasi-spherical field, 265 with variable magnification, 266 Charging. 14- 15 Collocation principle, 229 Computation of auxiliary integrals, 252 Contrast, 97

Counters Geiger--Moiler, 9 I proportional, 2, 94 proportional scintillation, 83 Criteria of emission system quality, 170 Cross-section, 52 Cylinder condenser, 257 tci

D Density distribution, 175 of current, 174 with respect to angles, 176 with respect to energies, 176 Derivative of the potential, 237 Diffusion back-diffusion, 47 coefficient, 25 Discharge, 42 Discretization of domain boundary, 252 Distortion anisotropic, 172 complex, 172 isotropic, 172 Drift velocity, 20

E Electrode, 67 Electron attachment, 31 current, 175 image. 158 microscope SEM, ESEM, 2 STEM, ESTEM, 89 mobility, 22 temperature, 16 Electron-optical emission-imaging system. 158 EFIR program package, 250,254

F Fano factor, 86 Functional of axial synthesis problem, 208

280

INDEX G

Gain derivation, 70 equations, 75-78 ;‘-process, 56 Gas, gaseous, 41 detector, I ionization, 37 scintillation, 80 Generalized control, 214 Geometric variational functions (g.v.f.), 187 Grid. 64

Magnification in thecenter, 171 local, 172 mean, 171 Mean free path. 24 Minimal-length cathode lens problem. 215 Mira meridian, 174 sagital, 174 Mobility, electron, ion, 20 Modulation transfer function, 176

N I Image surface meridian, 173 sagital, 173 Imaging parameters, 4 Induction, 7, 12 induced signal, 11 Insulators, 13, 15 Integral equation in variations, 181, 188 Fredholm, of the first kind, 184 lntegration of paraxial equation, 240

Ion mobility, 20 temperature, 16 Ionization coefficient, 52 energy, 34 luoplanatism condition, 172 Iteration external, 181 internal. 180 J Jump condition, 201

L Limiting image plane, 170 crossover plane, 170

M Magic gas, 59

Necessary conditions of optimality, 214 Noise, 4 Non-smooth optimum field, 219 Numerical computational method, 224 experiment, 257

0 Optimal control problem, 208 Optimal field, 219 Optimum mode, 221

P Parametric optimization of cathode lenses, 178 Paraxial equation, 163-164 Paschen law, 54 Perturbation of potential, 182 of surface charge density, 184 Photoemission. 158 Photons, 82 Pixel, 4 Point-spread function, 176 Pontryagin’s maximum principle, 208 function, 214 Potential electrostatic, 161 scalar magnetic, 161 Potential variation function (p.v.f.), 187 Probe, 41

Q Quasilinear approximation, 180

28 1

INDEX

T

R Recombination, 29 Resolution. 91 Resoi\lng power, 177

S Saturation current, 47 Scanning electron microscope, 2 transmission electron microscope, 89 Scintillation, 80 Secondary electrons. 5 Signal, induced, 1 I Signal-to-noise ratio. 4 Smooth field, 199,219 Software requirements for CAD EOS, 247 Solution of a system of linear equations, 252 Sparking potential, 54 Specimen current, 15 Spectroscopy, 84 energy resolution. 85 statistics. 85 Spherical condenser, 259 Stoletow constants. 54 Switching, 217 System optimization, 224

Temperature, electron, ion, 16 Terminology, 36 Theory gaseous detector. I induced signal, 6 Time response, 64, 79 Townsend factors, 17 20 Trajectory adjacent, 170 main, 170 Type of geometric variation, 196

Y Variation of paraxial trajectories, 199

W Walls, 41 Weakly disturbed axial symmetry, 193 Weak singularity, 196

X X-rays, 4 I , 89

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