`
`Plasma Etching
`
`A n Introduction
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`
`
`Edited by
`
`Dennis M. Manos
`Plasma Physics Laboratory
`Princeton University
`Princeton, New Jersey
`
`Daniel L. Flamm
`A T& T Bell Laboratories
`
`Murray Hill, New Jersey
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`£9
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`Academic Press
`San Diego New York Boston
`London Sydney Tokyo Toronto
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`ILLETTE 11 12
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`GILLETTE 1112
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` This book is printed on acid-free paper.
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`Copyright © 1989 by Academic Press
`All rights reserved.
`No part of this publication may be reproduced or
`transmitted in any form or by any means, electronic
`or mechanical, including photocopy, recording, or
`any information storage and retrieval system, without
`permission in writing from the publisher.
`
`ACADEMIC PRESS
`
`A Division ofHarcourt Brace & Company
`525 B Street, Suite 1900
`San Diego. California 92101-4495
`
`United Kingdom Edition published by
`ACADEMIC PRESS INC. (LONDON) LTD.
`24-28 Oval Road, London NW1 7DX
`
`Library of Congress Cataloging—in-Publication Data
`
`Plasma etching.
`(Plasma: materials interactions)
`Bibliography: p.
`Includes index.
`
`I. Manos, Dennis M.
`1. Plasma etching.
`Daniel L.
`III. Series: Plasma.
`TA2020.P5
`1988
`621.044
`87~37419
`ISBN 0-12-469370-9
`
`II. Flamm,
`
`Alkaline paper
`
`PRINTED IN THE UNITED STATES OF AMERICA
`
`97 BB 9 8 7 6
`
`

`

`
`An Introduction to Plasma Physics
`for Materials Processing
`
`
`
`
`
`
`
`Samuel A. Cohen
`
`Plasma Physics Laboratoty
`Princeton University
`Princeton, New Jersey
`
`I. Introduction ................................. , ..............
`II. The Plasma State .............................................
`
`III. Single~Particle Motion .........................................
`A. E = constant, B = 0 ........................................
`B. E = 0, B = constant ........................................
`C. Isl-perpendicular to B ........................................
`D. Non—Uniform Fields and Other Forces ...........................
`E. Time-Varying Fields ........................................
`F. Adiabatic Invariants ........................................
`G. Summary of Particle Drifts ....................................
`IV. Plasma Parameters ............................................
`A. Temperature. Density, and Pressure ..............................
`B. Debye Length and Plasma Frequency ............................
`C. Skin Depth and Dielectric Constant .............................
`D. Collisions ...............................................
`E. Summary of Plasma Parameters in Practical Units ...................
`F.
`Instabilities i .............................................
`G. Plasma Waves ........................................... -.
`.
`
`V. Discharge Initiation ...........................................
`A. DC Glow ................................................
`B. Microwave Breakdown .......................................
`
`VI. An Application—«The Planar Magnetron ............................
`Acknowledgements ...............................................
`References .....................................................
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`258
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`Plasma Etching:
`An Introduction
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`185
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`Copyright © 1989 by Academic Press. Inc.
`All rights of reproduction in any form reserved.
`ISBN 0-12~469370-9
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`techniques into a highly specialized art. Perhaps the basic foundation for
`this art is plasma physics,
`though chemistry, electrical engineering, and
`vacuum technology have defensible claims. Each must be understood and
`well-practiced for the material processing to succeed.
`
`’
`lengths, time scales, excitable modes (both stable and unstable), and atomic
`and surface processes important
`in the initiation and maintenance of
`plasma discharges. To discuss these topics in a practical way, we include
`information from a wide range of plasma configurations used in plasma
`processing, presenting material on dc- and rf-driven discharges with and
`without externally applied magnetic fields.
`
`Samuel A. Cohen
`
`developed in the later sections. Section III concerns single-particle motion;
`
`processing devices do not have even this simplicity, in large part due to the
`boundary between the plasma and the solid surfaces. It is at the boundary
`that our ultimate interests lie. However,
`the reader should find that the
`simplified situations described here will form a good understanding of the
`often counter—intuitive behavior of plasmas and will encourage improve—
`ments in existing equipment or processes.
`Plasmas are usually created in metal vacuum vessels, commonly used to
`attain the low pressures essential for particular plasma properties. Plasmas
`have a propensity to fill every crevice in these vacuum vessels. (The word
`“plasma” originates from a Greek root
`,
`'
`'
`efforts are made to constrain the plasma
`these are not completely successful. Device operation is considerably af-
`fected. To emphasize this, we shall use the label “containment” vessels to
`fully appreciate that some plasma reaches everywhere in them.
`We assume familiarity with college physics (especially Maxwell’s equa-
`tions) and introductory calculus. Most equations will be presented both in
`cgs and practical units to aid their easy application. Section II presents
`most of the basic ideas and definitions concerning plasmas. These are
`
`

`

`ll. The Plasma State
`
`Plasmas are a state of matter that consists of a large group of electrons and
`ions with nearly equal numbers of opposite charges, each particle moving at
`a high rate of speed relative to the others. It is the precise electric field of
`the individual charged particles that gives the plasma its unique properties.
`The electric field of each particle influences the motion of distant particles,
`whether they have like or opposite charge. This action-at-a-distance causes
`a wide variety of waves and instabilities to be possible in a plasma. And
`because each particle is influenced by the electric and magnetic fields of
`many particles, the term used to describe the kinematicslS collective motion.
`The electric field of a single isolated electron1s proportional to r 2.The
`volume of a spherical shell a distance r
`from that electron increases
`proportional to r2. Thus, the product of the electric field times the volume,
`a measure of the effectiveness of the field at a distance, is constant (Fig. la).
`It
`is the same near the electron as it
`is far away, showing how the
`action-at—a-distance arises.
`
`An Introduction to Plasma Physics for Materials Processing
`
`187
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`
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`Section IV gives details of plasma parameters; Section V is devoted to
`plasma formation; and Section VI applies the previous four sections to the
`magnetron device.
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`What differentiates a plasma from a group of neutral atoms that also has
`equal and large numbers of electrons and ions? It is the distance at which
`the electric field is felt strongly. Neutral atoms (and molecules) have an
`electric field no stronger than a dipole. This falls-off proportional to the
`distance cubed or faster. Hence at large distances, it is weak compared to
`the Coulomb electric field of the bare electrons found in a plasma. Because
`of the very short range of their electric and magnetic fields, molecules
`interact with each other only by “hard” collisions, meaning close encoun-
`ters, typically at separations of about 1 A. Free electrons and ions in a
`plasma interact over much greater distances, typically 1000 A or more, as
`well as less, of course!
`Numerous distant interactions will change a charged particle’s trajec-
`tory more than the infrequent hard collisions (Fig. lb). For this reason close
`encounters may be unimportant to the charged particles in a plasma. (This
`is related to another reason why a group of neutral atoms does not behave
`like a plasma. The quantal nature of the electronic energy levels in an atom
`precludes the small changes in energy required by distant encounters so
`important to plasma behavior.) Hence plasmas are frequently termed colli-
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`

`

`Samuel A. Cohen
`
`\
`SPHEMCALSHELL
`VOLUME=47r2m
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`TRAJECTORY
`BENTBY“CLUMPS”
`OFPARTmLES
`ATiANDfi
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`'
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`trajectory of the test particle more than the infrequent hard collisions with nearby particles
`
`a) The electric field of an isolated electron falls OH proportional to r”. The
`FIGURE 1.
`volume of a spherical shell around that electron increases as r2. This shows the importance of
`distant particles to the motion of that single electron. b) When a charged test particle moves
`through a cloud of charged particles,
`the electric field of the distant particles alters the
`
`TEST
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`PARHCLEB
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`.
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`‘/’/_fl_______jMPROBABLE
`”HARD" COLLISION
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`

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` An Introduction to Plasma Physics for Materials Processing
`
`TEST PARTICLE
`
`
`
`
`
`__ _— _
`_
`—
`__+
`+.++ + + +_—+++ + —+ +
`+~+*+— +71 ++\
`+ __~
`_
`+
`_
`+ +++
`++-—_ ‘+
`—_+__+_\+ +
`+
`_+ L, _+,
`++
`+- -~ -
`—
`—~— +-—+* +‘“++ +
`++_+ :+‘ +-+_ ‘+‘+
`__+_+_+:+_ +_+ *+_:;_+
`i:_++— -+ +_+__ _.+
`+
`_+‘ +~_.+ + + +
`
`DEBYE LENGTH
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`
`
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`FIGURE 2. When a negative test charge is placed in a plasma,
`a volume around it, of radius
`equal
`to the Debye length,
`is partially depleted of negative
`charges. This results in the
`shielding of that test particle’s electric field at distances larger th
`an the Debye length. Outside
`that volume the negative and positive charges are more nearly e
`qual in abundance
`
`sionless, meaning that individual hard collisions are unimportant compared
`to the numerous distant soft ones. Exact criteria, which hinge on the
`characteristic lengths in the problem, must be examined before a particular
`plasma can be correctly labeled collisionless. For example, is the size of the
`containment vessel larger than the mean—free~path between hard collisions?
`When many charged particles are present,
`they alter their positions,
`like-charged particles being repelled and oppositely charged ones being
`attracted,
`to reduce the distance over which an applied electric field is
`effective. The source of this field could be external metal plates attached to
`a battery, or a single electron placed in the plasma as a test particle. The
`shielding (Fig. 2) occurs in a distance called the Debye length, whose size
`determines many properties of the plasma relevant for material processing.
`The plasmas typically used in materials processing have a Debye length in
`the range of .01 to 1.0 mm. Within a sphere of this radius there are still
`many (typically more than a million) charged particles to influence and to
`be influenced by the test charge.
`Shielding does not prevent the penetration of all fields into a plasma.
`Certain electrostatic and electromagnetic waves, for example, can penetrate
`into and propagate through plasmas. This is essential to many schemes for
`plasma heating.
`By extrapolation it is clear that, at too high a density, plasma particles
`may be too close together and thus “appear” like dipoles to the distant
`particles. Also, at room temperature electrons and ions will rapidly recom-
`bine to form neutral atoms and molecules. Hence, to sustain a plasma, its
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`190
`Samuel A. Cohen
`
`
`temperature must be kept above some minimum, about 10,000 K (about
`
`1 eV), which depends on the density. The higher the temperature, the higher
`
`is the allowed density. From this, one can estimate that most laboratory
`
`plasmas have densities in the range of 108 to 1012 cm‘3.
`
`In astrophysical situations, plasmas exist at much lower densities
`
`(10‘3 cm”) as in the interstellar medium, and at much higher densities
`
`(above 1020 cm”) as in certain stars (Fig. 3). Other systems, such as
`
`electrons in metals or ions in liquids, also have certain properties like those
`
`of our gaseous plasmas.
`
`The approximate equality between oppositely charged particles is termed
`
`quasineutrality. It is one of the most basic tenets of plasma physics. A 1%
`
`excess of either charged species in a plasma with parameters like a mag-
`
`netron planar etcher would cause an electric field in excess of about 1000
`
`volts/cm. This large field would cause the electrons to rearrange their
`
`positions to restore a more balanced distribution of charges. Small electrical
`
`imbalances do occur. The resulting restoring force causes the plasma
`
`electrons to oscillate internally at a frequency (naturally enough) called the
`
`plasma frequency. These occur, again for the magnetron etcher, at about
`
`1010 Hz.
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`Three phases of mattermgas, liquid, and solid—are commonly experi—
`enced first hand, i.e., they can be readily touched if not too hot or cold.
`Plasmas cannot. They are generally too hot, too tenuous, and too fragile,
`almost like a soap bubble. A further similarity to soap bubbles is that the
`plasma possesses at its boundaries a skin, called the plasma or Debye
`sheath. The sheath is about 5 Debye lengths in thickness. Trying to touch a
`plasma by penetrating the sheath can destroy the plasma or, if the plasma
`has enough stored energy, the finger! Hence one of our best ways to test
`material properties is unavailable. But visual inspection with the naked eye
`can reveal the presence and dimensions of the sheath, properties of which it
`has taken scientists decades to confirm and quantify accurately with other
`diagnostic equipment.
`As might be inferred from their densities alone, plasmas behave more like
`gases than solids. But their modes of internal motion are complex because
`of collective motion. One similarity is that both support
`longitudinal
`compressional (sound) waves. Another is that both expand to fill
`their
`containers. On encountering the walls of the containment vessel, the charged
`particles of a plasma are neutralized by attachment to charged particles
`from the solid, often metallic, surfaces. Thus, the rate at which the plasma
`expands will determine how long, and if, it can be sustained. The charged
`particles lost by contact with the wall must be replaced if the plasma is to
`continue in its existence (Fig. 4). The steady state thus achieved will depend
`
`

`

`An Introduction to Plasma Physics for Materials Processing
`
`191
`
`
`
`Magnetic
`Fusion
`Reactors
`
`Process
`Plasmas
`
`(cm
`
`26
`IO
`
`22
`IO
`
`I8
`
`IO
`
`I4
`,0
`
`IO
`
`IO
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`)
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`ELECTRONDENSITY
`
`Yohen
`
`bout
`
`gher
`Ltory
`
`;ities
`;ities
`h as
`hose
`
`'med
`l 1%
`
`nag-
`1000
`their
`rical
`isma
`
`1 the
`bout
`
`peri—
`cold.
`
`tgile,
`t the
`
`ebye
`lch a
`isma
`test
`
`1 eye
`ch it
`)ther
`
`: like
`:ause
`
`:linal
`their
`
`trged
`:icles
`isma
`
`tI'ng
`is to
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`
`10‘2
`
`too
`
`I02
`
`IO
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`ELECTRON TEMPERATURE (eV)
`
`FIGURE 3. Types of plasmas, categorized by their temperatures and densities. The corre-
`sponding Debye lengths are the diagonal lines.
`
`pend
`
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`

`
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`
`
`+aW
`
`all
`N“.
`/
`Containment Vessel
`
`strongly on the properties of the containment vessel, e.
`electrons and the right ions
`when and where needed.
`
`g., will it resupply
`
`Samuel A. Cohen
`
`9‘
`
`asma Source
`
`Plasma Loss To Wall
`
`

`

`
`
`nuel A. Cohen
`
`An Introduction to Plasma Physics for Materials Processing
`
`
`
`193
`
`Electron Impact Ionization
`
`IUPCE
`
`e‘ + Atom :> 2e” + Ion
`
`FIGURE 5. The impact of a plasma electron on a neutral atom may result in ionization of
`that atom.
`
`temperatures and densities of these difl‘erent species. Though the interaction
`between the charged particle pairs is much stronger than between a charged
`particle and a neutral, if the neutral density is very high, then electrons will
`collide more often with them, and, for example, the electrical resistivity will
`be aflected. The neutral particles in materials processing plasmas typically
`outnumber the charged ones by more than 104 to 1. At the extreme, if the
`neutral to charged particle ratio exceeds about 10“, the neutral collision
`frequency will exceed the plasma frequency, collective aspects of the motion
`will be lost, and the plasma state destroyed.
`
`III. Single-Particle Motion
`
`Bulk plasma motion is dominated by collective effects. Yet the motion of
`each charged particle under the influence of the local electric and magnetic
`fields is the correct description of what is happening at a microscopic level.
`And so the Lorentz force law and Maxwell’s equations provide the proper
`way to predict a single test particle’s motion. What this approach lacks is
`the back-effect of the test particle’s own motion on the local fields as caused
`by its fields acting on the distant particles. To improve the single-particle
`picture, one could start with a description of the plasma as a fluid or as an
`ensemble of particles. Then fluid equations or the kinetic equations would
`be used to describe the plasma’s evolution. These approaches are usually
`reserved for the more advanced students. Several references are listed at the
`end of this chapter that should satisfy the more ambitious. Many results
`achieved by a kinetic or fluid analysis can be reproduced in a single-particle
`description by choosing the proper initial conditions based on the known
`answer. This is often done for pedagogical reasons because the single
`particle picture is so easy to visualize and hence to remember. We use this
`
`: vessel. There
`ial source.
`
`it resupply
`
`me devices
`on into the
`be used to
`
`lensity gas.
`picking up
`usual way
`1e contain’
`act of the
`3 ions thus
`3k and the
`
`0 types of
`1e plasma
`I a dozen
`s electrons
`
`, processes
`
`)n plasma
`lC relative
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`194
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`Samuel A. Cohen
`
`approach here.
`the motion of an
`In the single-particle approach to plasma physics,
`individual charged particle under the influence of externally applied electric
`and magnetic fields is examined. The fields are allowed to vary in space and
`time but do not change to reflect the subsequent motion of the charged
`particle. The motion of the single particle is readily obtained from the
`Lorentz force law, which (in cgs units) is
`
`mdv
`v X B
`
`where c is the speed of light (3 X 1010 cm/s), m is the mass of the charged
`particle (in gm), q its charge (—4.8 X 10‘10 statcoul for a single electron),
`0 its velocity (cm/s), and E (in statv/cm) and B (in gauss) are the applied
`electric and magnetic fields, respectively. We now discuss several simple
`cases of this equation.
`
`A. E = CONSTANT, B = 0
`
`The application of a constant and homogeneous electric field, but no
`magnetic field, results in the constant acceleration of a charged particle. The
`particle gains energy from the field at an ever increasing rate. If allowed to
`continue, the particle would eventually reach relativistic speeds. Then the
`simple Newtonian description fails and the particle gains mass rather than
`speed, and also energy and momentum. This relativistic limit is beyond the
`scope of processing plasma physics. Integrating Eqn. (1) for E parallel to
`the x-direction gives
`
`2
`
`(2)
`,
`x(t) = x0 + 0th + (12E;
`where x0 and 0x0 are the initial x-position and x-velocity of the particle. If
`the initial velocity U0 is zero or is in the direction parallel to E, this motion
`is in a straight line. Otherwise it is parabolic (Fig. 6). Though in most
`
`Applied Electric Field
`————————-———>
`
`@—-—-———-> Trajectory if Vyo=o
`
`
`
`
`Trajectory if vyo¥ 0
`
`FIGURE 6. Trajectories of charged particles in a constant electric field.
`
`

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`
`An Introduction to Plasma Physics for Materials Processing
`
`195
`
`an
`
`trio
`and
`
`ged
`the
`
`ged
`3n),
`lied
`
`iple
`
`no
`
`The
`i to
`the
`ban
`the
`1 to
`
`(2)
`
`:. If
`tion
`lost
`
`situations it is adequate to ignore a particle’s past history, i.e., x0 and 00,
`this is not true in the initiation of a discharge or in most microwave-driven
`discharges.
`
`B. E = 0, B = CONSTANT
`
`As seen from the Lorentz law, if there is no electric field and no initial
`velocity, the particle remains at rest. More precisely, the magnetic force acts
`only if the particle’s motion is perpendicular to B. The result is that a
`particle moving initially along B is unaffected by B while a particle moving
`perpendicular to B has its trajectory turned into a circle. For a particle
`moving at an angle to B the net result is a corkscrew or helical motion.
`There is no gain or loss of total particle kinetic energy from a static or
`slowly varying magnetic field (Fig. 7). Integrating Eqn. (1) for this choice of
`parameters gives
`
`where
`
`2(t) = 20 + vzot
`
`x(1)= x0 + rL sin(th/mc)
`
`y(t) = y0 + rLcos(th/mc)
`
`mu C
`
`a: q;
`
`,
`
`(3)
`
`(4)
`
`v i is the component of the velocity perpendicular to B, and we assumed B
`parallel to z, B = Bz.
`Notice that the circular motion has a characteristic radius, rL, called the
`Larmor radius, and a characteristic frequency,
`036 = 27rf = qB/mc, called
`
`'1‘
`
`B
`
`-—«>
`Vu
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`
`
`
`
`
`
`
`
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`FIGURE 7. Trajectory of a (negatively) charged particle in a constant magnetic field.
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`Electron-Genera ted
`Magnetic Field
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`Samuel A. Cohen
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`Externally Applied
`Magnetic Fieid
`
`
`
` Expelled
`
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`Field Net
`Magnetic
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`inside the orbit.
`
`the Larmor-, cyclotron-, or gyro-frequency, which are related by
`v
`rL= i'
`
`(5)
`
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`
`
`The circular motion of the charged particle produces a dipole magnetic
`field in such a way as to reduce the strength of the field inside the circular
`orbit and to increase it outside (Fig. 8). Then one can View the circular
`motion as being caused by a higher magnetic pressure on the outside of the
`orbit. This property of a charged particle or a plasma to partially expel
`magnetic fields from their vicinity (interior) is called diamagnetism. The
`dipole field has a strength called the magnetic moment equal to
`2
`
`__ quirL _ mUJ.
`4‘“
`2c
`‘ 213'
`
`(6)
`
`Another convenient picture of the motion is to separately consider the
`circular (cyclotron) and linear motions. By ignoring the cyclotron motion
`one has the guiding center motion remaining. This nearly linear motion is a
`worthwhile concept if the cyclotron radius is small compared to the other
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` An Introduction to Plasma Physics for Materials Processing
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`characteristic dimensions. And so it is of most use for electrons whose
`Larmor radii are small because of their small mass.
`
`C. E-PERPENDICULAR TO B
`
`If E and B are parallel to each other, there is .little change in a particle’s
`motion from the above description. That is, a particle will accelerate along
`E and B and have an unchanging circular motion around them.
`If E and B are perpendicular, however, a new effect takes place that is
`contrary to most everyday experiences, except that of the common toy top.
`What happens is that the charged particle starts to accelerate parallel to E
`until its velocity is large enough for the magnetic force to bend it. The bend
`becomes so large that the particle ends up moving in a direction that is
`perpendicular to both E and B (Fig. 9). Similarly, a spinning toy top, if
`pushed by a finger or by gravity, responds by precessing perpendicular to
`both the applied force and its axis of rotation, and not by falling over.
`So the motion of the charged particle is the sum of three distinct
`motions: cyclotron motion around B; linear motion along B; and a drift
`
` Y
`
`FIGURE 9. Drift of a charged particle in the presence of crossed E and B fields. Both
`electrons and ions drift in the same direction and at the same speed. Note that the Larmor
`radius changes as the particle gains and loses energy from the electric field. It is this difference
`in Larmor radius that causes the drift.
`
`Net
`Magnetic
`Field
`\
`
`iirected to
`18 reduced
`
`(5)
`
`iagnetic
`circular
`circular
`e of the
`
`.y expel
`'m. The
`
`(6)
`
`der the
`motion
`ion is a
`e other
`
`
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`
`
`
`

`

`
`
`198
`
`Samuel A. Cohen
`
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`
`
`
`
`
`
`perpendicular to both B and E. The B-parallel and B-perpendicular (drift)
`motions now comprise the guiding-center motion.
`Integrating the equation of motion for this case is more complicated than
`in the previous paragraphs. Instead, one can take the vector cross—product
`of B with the Lorentz law, Eqn. (1),
`ignoring the dv/dt
`term, which
`describes the cyclotron motion. Then one obtains for the transverse compo-
`nent of the velocity (the drift velocity)
`
`
`
`(7)
`
`This drift velocity is independent of the particle’s energy, charge, and
`mass because the electric and magnetic forces are both proportional to a
`particle’s charge and independent of a particle’s mass. Though a heavy
`particle has a larger gyroradius, its cyclotron frequency is slower by the
`same amount so the gain and loss of energy from the electric field during
`each cycle are balanced.
`
`D. NON-UNIFORM FIELDS AND OTHER FORCES
`
`The same derivation could have been carried out for a different force F
`acting on a charged particle in a B field. The result can simply be obtained
`by replacing E by F/q, the equivalent electric field. The classic example
`most quoted is that of a charged particle in crossed magnetic and gravita-
`tional fields (Fig. 10a). The result is a drift velocity of magnitude cmg/qB,
`where g is the gravitational acceleration. This drift does depend on charge
`and mass because the gravitational force does not depend on charge but
`does depend on mass. Charges separate and the heavier particles drift
`faster. Hence, a net current flows, in contrast to the E X B case.
`Another drift would occur if the particle experiences a force due to a
`pressure gradient, vp. This would arise in a plasma with temperature or
`density gradients. The resulting drift is called the diamagnetic drift and has
`a velocity equal to
`
`UD—
`
`ch X B
`7?
`
`<8)
`
`
`
`Similarly, the drift resulting from gradients or curvature in the B field
`itself can be written down immediately after identifying the forces they
`cause. A charged particle moving along a curved field line (Fig. 10b) feels a
`centrifugal force of magnitude mvfi/Rc , where RC is the radius of curvature
`of the field. And magnetic field gradients (Fig. 100) cause a force equal to
`
`

`

`rift)
`
`:han
`duct
`hich
`
`1pc—
`
`(7)
`
`and
`to a
`
`:avy
`the
`
`ned
`
`1ple
`'ita-
`
`irge
`but
`[rift
`
`o a
`z or
`
`(8)
`
`An Introduction to Plasma Physics for Materials Processing
`
`(a)
`
`Z
`
`X
`
`
`ring
`has
`
`
`
`
`. 1d
`[e
`hey
`
`
`
`FIGURE 10. Drift motion of positively charged particles under the combined influences of a
`magnetic field and a) gravity, b) curved magnetic field, and c) magnetic field with a gradient
`(continued on next page).
`
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`

`

`
`
`Samuel A. Cohen
`
`
`
`,4
`
`FIGURE 10c. Continued from previous page.
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`,uV B. Hence the drifts are:
`
`and
`
`U r B X VB
`00: i4?
`
`mvfiRCXB
`0-w-
`
`(9)
`
`(10)
`
`In contrast to the E X B drift, these depend on a particle’s mass, charge,
`and energy. The more massive a particle, the faster it drifts. This is another
`counter-intuitive property of plasmas. Generally one expects the electrons,
`the lighter particle, to carry all the electrical currents. Not so. The ions can
`play a dominant role in current flows.
`A non-uniform electric field will alter the drift velocity by giving weight
`to the time the particle spends in the regions of different field strength and
`direction. The result for a sinusoidally varying electric field with a periodic—
`ity distance d is that the drift is proportional to the usual E X B drift times
`(1 —- rLZ/4d2). This is called a finite Larmor radius correction because of
`the presence of rL in the equation. This is different for each species and
`may cause charge separation and thus plasma waves.
`
`
`
`

`

`An Introduction to Plasma Physics for Materials Processing
`
`E. TIME-VARYING FIELDS
`
`
`
`
`
`When the electric field varies in time the drift is modified. One can again
`understand the motion from a microscopic picture. Each time the field
`“turns-on” the particle slowly accelerates parallel to E and then the E X B
`drift develops. When the electric field reverses the processes again occur,
`but with the drifts in the opposite directions. This can be seen quantita—
`tively from the following derivation (Fig. 11a and b). The magnetic field is
`along the z—axis;
`the electric field is along the y-axis. The equations of
`motion for the x and y directions are:
`x = we)?
`
`(11a)
`
`and
`
`y‘ = ‘17? — we.
`
`(11b)
`
`
`
`
`
`
`
`
`
`
`
`
`la-‘l Cyclotron Orbit
`—'l
`TIME
`
`FIGURE 11. Polarization drifts for positively charged particles when E increases monotoni-
`cally with time. If E increases slowly, the Larmor radius is unchanged Two drifts develop: the
`E X B drift and the polarization drift (parallel to E). When E increases at a fast rate, relative
`to the period of the cyclotron motion, the Larmor radius grows as the particle gains energy
`from the field, Again two drifts develop. Note that when d E/d! = 0 the polarization drift also
`is 0.
`
`
`
`
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`

`

`
`
`202
`
`Samuel A. Cohen
`
`the
`
`By differentiating (11a) and substituting in (11b) and vice—versa,
`equations can be made separable yielding,
`
`55 + wfx = 41;”
`
`(12a)
`
`and
`
`which have solutions
`
`and
`
`y + way = 1,5,
`
`)3 = —ui sin(wct) + 2:?
`
`
`
`(121a)
`
`(13a)
`
`(13b)
`)6 = +1)i cos(wct) + £5.
`It is now easy to identify the E X B drift motion in the x—direction and
`the polarization drift in the y-direction, mczE/qBZ. The orbit shapes will
`vary depending on the rate of change of the electric field. Figure 11 shows
`two cases, both for monotonically increasing fields, not oscillating ones. In
`Fig. 11b the rapidly changing intense field increases the Larmor radius
`because it adds an appreciable amount of energy to the particle in a time
`faster than the cyclotron period.
`The polarization drift is in opposite directions for oppositely charged
`particles. Hence charge separation occurs. Again the more massive particle
`carries the current.
`
`F. ADIABATIC INVARIANTS
`
`opposite change in parallel energy. So if a particle moves into a region of
`
`A similar derivation carried out for slowly varying magnetic fields shows
`that no additional drift arises. Instead the Larmor radius grows or shrinks,
`depending on whether B decreases or increases. This is associated with a
`decrease or increase of the particle’s transverse energy (see Eqn. (4)). The
`change is such that the magnetic moment is unchanged. So vi
`increases
`proportional to B.
`The magnetic moment is called the first adiabatic invariant. It is constant
`as long as changes in E or B occur slowly compared to a cyclotron orbit. Its
`constancy reflects the symmetry and periodicity of the cyclotron orbit.
`Sitting in the particle’s frame-of-reference, a change of B can come from
`a change in its position as well as a change in the local field strength. Recall
`that static magnetic fields add no energy to a particle. It is then clear that
`the change in perpendicular energy must be accompanied by an equal but
`
`

`

` An Introduction to Plasma Physics for Materials Processing
`
`
`
`'sa,
`
`the
`
`(12a)
`
`(12b)
`
`(13a)
`
`(13b)
`
`n and
`:s will
`shows
`tes. In
`radius
`l time
`
`arged
`trticle
`
`hows
`
`inks,
`ith a
`The
`eases
`
`stant
`t. Its
`
`:‘rom
`ecall
`that
`but
`it of
`
`FIGURE 12. Magnetic mirror formed by two coaxial coils with co-directed currents, I. As a
`charged particle approaches a region of higher B,
`its perpendicular energy grows at
`the
`expense of its parallel energy. Reflection will occur if B reaches a high enough value. Note that
`the Larmor radius shrinks as the particle moves into a region of higher B.
`
`increasing B, it will continue to gain perpendicular energy at the expense of
`its parallel energy. At some point it may have lost all its parallel energy.
`WithV B still causing a force on the particle it is reflected. This property of
`spatially varying magnetic fields is called the mirror eflect and forms the
`basis for many plasma confinement configurations (Fig. 12). For the mirror
`to “wor ,” the particle must start with sufficient perpendicular energy
`because the mirror force only acts on the magnetic moment (F = av B).
`From conservation of energy and p. we can show that a particle starting
`from a region of magnetic field strength BO, will only be reflected in the
`region of higher field if its pitch angle, 0 = 1)” /vi , is less than
`B
`_
`7° = sin20= (RM)
`M
`
`(14)
`
`1
`
`where
`
`R M = the mirror ratio, and
`
`BM = the maximum field strength.
`
`The underlying principles for these statements are the laws of conserva-
`tion of energy and angular momentum. These are very powerful tools when
`a physical situation has symmetry. For example, calculations of cosmic ray
`trajectories in the earth’s vicinity are easy because of the size of the earth
`coupled with the symmetry of its magnetic field. The symmetry is such that
`other invariants of the motion, the so-called second and third adiabatic
`invariants, help to solve the problem. They are based on the symmetry and
`periodicity of the large-scale motion around the earth (bouncing back and
`forth between the magnetic mirrors at the poles and circulating around the
`equator), not just of the cyclot