ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS
VOLUME 78
EDITOR-IN-CHIEF
PETER W. HAWKES Laboratoire d’Optique Electr...
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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
ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York London Sydney Tokyo Toronto
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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
.
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 ) ,
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 > 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 > 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
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
88
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
THEORY OF T H E GASEOUS DETECTOR DEVICE I N T H t ESEM
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 > 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 -
))>
(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, 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 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
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 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 [ .
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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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