H plasma? This question-and the search for its many answers-has

Transcription

1 HERSHKOWITZ: HOW DOES THE POTENTIAL GET FROM A TO B IN A PLASMA? 11 How Does the Potential Get from A to B in a Plasma? Noah Hershkowitz, Fellow, ZEEE (Invited Review Paper) AbsfructSolutions to the question of how the plasma potential varies from point A to point B, at which the values are specified, are discussed. Each answer depends on plasma and boundary conditions. Plasma potential structures measured by emissive probes are considered. Examples include potentials associated with sheaths and presheaths in collisionless and collisional systems, unmagnetized and magnetized double layers with single and multiple step transitions, semiconductor etching and rf self bias, tandem mirror plasma confinement and tokamak antenna impurity production. DC structures, time average structures associated with capacitive and inductive rf and the role of potential well pumping are considered. I. INTRODUCTION OW does the potential get from point A to point B in a H plasma? This question-and the search for its many answers-has been a recurring theme in plasma physics research since the time of Langmuir [ 13. It has important consequences for basic plasmas, plasmas in the earth s magnetosphere, plasmas in fusion devices, etc. In finding a solution to the problem, nature may have created auroras. Until recently, the answer made it impossible to carry out many rf experiments in tokamaks. The answer depends on whether the reference frame is moving or stationary and on the dimensionality of the problem. The question has been the recurring theme of my work. What some of the solutions look like and why, and how they are measured with emissive probes are discussed in this paper. Particular emphasis will be placed on recent laboratory measurements in which I have participated which confirm predictions of one dimensional phenomena (many for the first time) and also indicate the wide range of possible solutions. Examples include potential structures associated with sheaths and presheaths, double layers, semiconductor etching, tandem mirrors, rf self bias and tokamak antenna impurity production. 11. EMISSIVE PROBE POTENTIAL MEASUREMENTS Measurement of the potential between A and B is often nontrivial. Emissive probes are an extremely versatile potential diagnostic. By employing a variety of techniques, emissive probe measurements have been made from near vacuum conditions to plasma densities of 1013 cmp3. Emissive probes are superior to Langmuir probes as a potential diagnostic. In near vacuum conditions, Langmuir probes, which measure collected current, fail because there is too little current to measure. Neither are Langmuir probes satisfactory for drifting Manuscript received September 2, 1993; revised November 15, The author is with the University of Wisconsin, Department of Nuclear Engineering and Engineering Physics, Madison, WI IEEE Log Number plasmas, for anisotropic plasmas, within plasma sheaths, at very low plasma densities, etc. [2]. The use of emissive probes [3, 41 depends on the basic principle that electrons emitted from a hot wire probe biased below the plasma potential are lost to the plasma while electrons emitted above the plasma potential are reflected back to the probe, if space charge effects can be ignored. The temperature of a hot emitting wire is T ev. This effectively limits the voltage resolution of the technique to V. In some applications secondary electron emission can replace thermionic emission. This technique can work but the effective T and the associated voltage resolution is not as good. The spatial resolution is limited by the hot wire size. A spatial resolution of 1 mm can be achieved with a plasma potential uncertainty as small as 0.2 V. Such probes have been operated at plasma densities ranging from near vacuum with n < lo4 cmp3 to as high as ~m-~, with magnetic fields as high as 1 T and without magnetic fields (with n as high as ~m-~). Cold and hot emissive probe I-V characteristics are given for a relatively low density plasma in Fig. 1 [3, 41. A variety of techniques have been developed to interpret emissive probe data but all are based on the separation of the hot and cold traces that occurs near the plasma potential. The simplest approach is to operate the probe in strong emission [3, 41 and to identify the floating potential (where I = 0) as the plasma potential. This technique is accurate to within approximately 1 V but has the disadvantage that the strong emission can perturb the surrounding plasma, particularly in the presence of a magnetic field. An alternative approach is to track the voltage at which the hot and cold traces separate [5]. A third approach, which provides the most accurate measurement, is to monitor the inflection point in the I-V characteristic as a function of wire temperature and to extrapolate the result to zero electron emission [3, POTENTIALS IN NEAR VACUUM CONDITIONS The simplest problem we can consider is a parallel plate capacitor with large plates and small separation, filled with a vacuum. The problem is easy if we can neglect edge effects. In that case, the solution is a constant electric field between the two plates so the potential can be written v = (V2 - Vl)Z/S, (1) where VI and V2 are the potentials applied to the two plates and s is the plate separation (see Fig. 2a). While it is easy to calculate this solution, it is not so easy to measure /94$ IEEE

2 12 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 22, NO. 1, FEBRUARY P I 1 I I I I I I I Fig. 3. Calculated and measured potential profile on the axis of a vacuum filled parallel plate capacitor. Measurements were made with emissive probes. A Vacuum ion species, unmagnetized and infinite. The Debye length AD is much less than the device dimension. Both electrons and ions have non-drifting Maxwellian distribution functions. In such a simple Maxwellian plasma, the plasma potential will be constant so there is no variation from point A to point B. Bulk plasma tends to be quasineutral, even in less idealized systems which are finite and have boundaries. At the boundary of an unmagnetized plasma or along a magnetic field, the electron thermal velocity we is much greater than the ion thermal velocity vi so the plasma charges positive with respect to the boundary to balance electron and ion losses. In order to simplify the problem of how the potential gets from A to B, most examples in this paper will involve structures that have one dimensional symmetry. For time independent situations with no magnetic field or along a uniform magnetic field, the plasma potential is determined by the solution of Poisson s equation. For one dimensional problems, Poisson s equation can be written: A Fig. 2. One dimensional schematic plasma potential profiles for: (a) vacuum, (b) sheath plus bulk plasma, (c) virtual cathode, (d) double layer. All correspond to identical values of boundary potential at A and B. Emissive probe measurements of vacuum potential using two different techniques have been made in a parallel plate capacitor [6, 71 see Fig. 3. The plasma density was estimated to be less than lo4 cmv3 so the Debye length AD > 100 cm compared to a plate separation of 10 cm. There is good agreement between data and lines representing constant electric fields between the electrodes. Results were found to be sensitive to surface cleanliness. IV. POTENTIALS LN SIMPLE MAXWELLIAN PLASMAS Addition of plasma complicates the problem. The simplest plasma is steady state, uniform, isotropic, collisionless, single where p is the space charge density, 4 is the potential, e the electron charge and n;_and ne is the electron density. Defining e$/te = 4, ni/no fii, where T, is the electron temperature and no is the density far from a boundary, Poisson s equation becomes This means deviations from charge neutrality only occur on a length scale determined by AD. At high density, AD is small (much less than the device size L) so significant deviations from charge neutrality only occur in sheaths (characterized by AD) at plasma boundaries. Hence, bulk plasmas are approximately neutral and are often called quasineutral and sheaths, because they repel electrons, are regions of net positive charge. (3)

3 HERSHKOWITZ: HOW DOES THE POTENTIAL GET FROM A TO B IN A PLASMA? 13 It is apparent from (4) that the c_urvature in the potential 4 (i.e., the second derivative of 4 with respect to 2) is proportional to 6, - 6i, so the curvature in sheaths is negative. Note that Poisson s equation does not depend on the sign or the magnitude of particle velocity so beam density contributes on an equal basis to background density. Bohm Presheath A. Boundary Sheaths Neglecting the contribution of electrons, the thickness s of sheaths with %&@ >> 1 can be shown to be (see Fig. 2b): where a = 3/4,2/3,0.6 and 0.5 for collisionless [2], mobility limited with constant mobility [8, 91 or constant cross section and for frozen ions respectively [lo]. For simple Maxwellian plasmas the potential drop across the sheath is [2] Te A4s % ( In p)- e where m is the ion mass normalized to hydrogen. This is the normal solution to how the potential gets from a boundary point A to a point B in the bulk plasma, but it is far from the whole story. B. Presheaths In order to match the boundary condition at the edge of the sheath, it has been found that ions must satisfy the generalized Bohm condition [ 113. For a general ion distribution at the sheath boundary and assuming absorbing walls, the generalized Bohm condition can be expressed [ll]: where cj is the relative concentration of positive ion species j, U; is the ion velocity component perpendicular to the wall at the sheath edge, and () gives the average over the ion distribution function at the sheath edge. Teff is defined: Note that if Ti 2 Teff, as is often the case in fusion plasmas, (7) is satisfied by thermal ions. If Ti << T,, as is usually true for cold laboratory and low pressure processing plasmas, the generalized Bohm condition becomes: (6) (7) vi 2 c, = E (9) which is usually referred to as the Bohm criterion. This means that the ions at the sheath edge must be in the form of a beam with directed energy of % >> Ti. The Bohm criterion is a curious result which requires some form of ion acceleration over distances which are large compared to the boundary sheath dimension, i.e., in the region of the quasineutral plasma. For the plasmas under consideration, electrostatic fields are the only source for such Fig. 4. Schematic potential profile for sheath and presheath. The sheath scale length is very enlarged compared to normal laboratory plasmas in order to permit a side by side comparison. (a) Collisionless presheath. (b) Collisional presheath. acceleration. Such considerations led to the idea of a Bohm presheath of potential drop at least 2 and size given by half the device dimensions in a collisionless plasma (if ionization is present) and determined by ion collisions in a collisional plasma. There has been much written over the years about Bohm presheaths but there have been very few experimental measurement of their characteristics. Presheaths are shown schematically in Fig. 4. If plasma production is included, the presheath electric field would not be expected to be constant. The idea of dividing the potential change at a plasma boundary into a sheath and a presheath is a somewhat artificial construction, invented before the development of high speed computers. In the limiting case AD/L -+ 0 the presheath is treated on a scale 7 = x/l while the sheath is treated on a scale = x/a0. The sheath edge (the boundary between the sheath and the presheath) in this approach is defined on the presheath scale by the location where dv/dv , while on the sheath scale by dv/d< + 0 as < The sheath edge represents the boundary at which quasineutrality breaks down. Measurements of collisionless Bohm presheaths are difficult to carry out because they are sensitive to the plasma production mechanism. In one such measurement [ 121, emissive probes were used to determine the potential profile in a multidipole plasma in which energetic primary electrons provided fairly isotropic and uniform plasma production. Data were compared to a model which depended weakly on the relative concentration of primary electrons np/ne (see Fig. 5). v. POTENTIALS NEAR BOUNDARIES IN OTHER PLASMAS A. Collisional Presheaths Presheaths become modified when the ion neutral collisional mean free path becomes smaller than L/2. The first laboratory measurements of collisional presheaths have recently been published [13]. As shown in Fig. 6, there appears to be a clear distinction between the quasineutral presheath (with a constant electric field) and the plasma sheath. Somewhat surprisingly, there is a fairly abrupt transition between the presheath and the bulk plasma. The presheath length A, was found to be

4 14 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 22, NO. 1, FEBRUARY Data Points I l l s=x/l = 60" Fig. 5. Comparison of experimental and theoretical presheath potentials for various primary electron densities in a "collisionless" multi-dipole device with bulk ionization provided by isotropic ionizing electrons Position from Wafer Stage (cm) I Fig. 6. F'resheaths in a collisional A'z plasma for several values of neutral pressure. proportional to the ion neutral charge exchange mean free path (with a proportionality constant of 0.7). B. Magnetic Presheath Magnetic fields can also influence presheaths. Sheaths with B fields oriented at an angle y to the normal to a boundary provide another system for which we can ask how the potential gets from A to B. For 1c, = 0, there is an electrostatic sheath whose thickness can be expressed s 01 (eq5/te)"ad. When 1c, # 0, a magnetic presheath (see Fig. 7) is found [14, 151 with length A, given by A, = r,, sin I, (10) (see Fig. 8), where the ion sound gyroradius r,, is defined: where wci is the ion gyrofrequency. Experimentally it is found that the magnetic presheath exists because ions gyrating about magnetic field lines, which run into a wall at an angle $I, are first lost when they reach the sheath boundary given by (10). There are three characteristic lengths, An, A, and res. Recent experiments [15] suggest that when collisions are present, two consecutive presheaths are present. The collisional presheath conforms to the boundary. As viewed from the bulk plasma, the collisional presheath is preceded by the magnetic presheath. Both presheaths are approximately quasineutral (see Figs. 6 and 7). Going from point A at a wall there is a thin sheath followed by two approximately constant electric field regions before reaching the point B in the bulk 0.0 $l I I I Position From Target Plate (cm) t indicates the electrostatic sheath boundary. 4 indicates the magnetic presheath boundary. Fig. 7. Plasma potential normal to a planar electrode as a function of position for several values of magnetic field. These data correspond to a magnetic field with an angle of incidence to the electrode of 60 degrees. The arrows indicate the edge of the electrostatic and magnetic sheaths. plasma. Understanding the details of such sheath structures could have important consequences in the design of divertors in tokamaks. C. Negative Potential Wells If electrons are emitted by a plasma boundary, for example, by thermionic emission, secondary electron emission or photo emission, the potential no longer needs to be monotonic. The excess electrons generated at the boundary can result in a net positive curvature near the emitting boundary. The combination of positive curvature at the plasma boundary with negative curvature near the boundary results in a double sheath. As shown in Fig. 2c, the plasma potential can develop a local minimum, which can serve as a virtual cathode, in front of the positively biased electrode. Nonemitting positively biased electrodes can also have potential undershoots. For example, consider the data shown in Fig. 9. A nonemitting positively biased electrode was inserted in a multidipole plasma [ 16, 171. The front side of the electrode was electrically conducting while the rear side was insulated. The measurements of the plasma potential along the axis of the probe show a dip 0.5 cm in front of the conducting surface.

5 - HERSHKOWITZ: HOW DOES THE POTENTIAL GET FROM A TO B IN A PLASMA? i 0. I I I I I I I + $=30(He) m $=60(He) Q $=80(He) A $=20(N2) Cs/wci sin$ I 1 I / Fig. 10. Equipotential contours near a collecting plate. The back of the plate and edges were covered with an insulating ceramic. the solution to both problems is to provide some form of ion pumping. The role of the ion pumping is to pump, i.e., remove the trapped ions from the well. In tandem mirrors, which have an axial magnetic field, the pumping can be provided by charge exchange on energetic neutral beams (for which the resulting ions are not trapped by the well) or by employing time varying electric fields [19, 201. Icm 0 -km -2cm Fig. 8. Magnetic sheath thickness vs. [-/El] sin($) for helium and nitrogen plasmas. B-field was varied from 0 to 170 gauss and angle from 0 to 80 degrees. Fig. 9. Axial plasma potential profile measured near a disc biased at +20 V. The back of the disc was coated with an insulator. The electron temperature was measured to be 3.5 ev. This separation is much smaller than the electrode radius of 2.5 cm. Potential structures such as those shown in Fig. 2c and Fig. 9, which have a local minimum in potential not attached to a wall, often do not exist as steady state structures in the laboratory because of ion neutral charge exchange. Charge exchange converts free ions traversing the potential wells into energetic neutrals and cold ions which are trapped in the well. This reduces the curvature in the potential (see (4)) and eventually eliminates the well. This problem is also found in the thermal barrier cells in tandem mirrors [18] where the plasma potential structure resembles the potential shown in Fig. 2c. A major difference between the two situations are the scale lengths. Nevertheless, D. Pumping Laboratory Plasma Wells For laboratory plasmas, pumping can be achieved by periodically emptying the well or by providing a small leak in the well. The easiest way to periodically empty the well is to employ a time varying electrode bias. In this case Fig. 2c gives the time average potential. Trapped ions are emptied when the bias of the electrode is negative with respect to the potential minimum. Biasing the electrode negative with respect to the plasma potential far from the electrode normally has little effect on that potential. If the probe is sufficiently large (comparable to the other loss area multiplied by JG, that potential will normally adjust to be more positive than the electrode bias potential. Under some circumstances it is possible that electric fields associated with turbulence in the well can provide the pumping. Here the well is only a potential dip in the time average sense. The data shown in Fig. 9 are an example of continuous pumping by a small leak. The leak path becomes apparent when the equipotential contours in the neighborhood of the biased electrode are examined. (see Fig. 10). While a potential well is seen on axis, it is not seen in the transverse direction. The equipotential curves associated with the minimum terminate on the insulating surface on the back side of the electrode. This provides a small leak for trapped charge exchange ions. Removal of the insulating layer from the back of the electrode, eliminated the pumping and resulted in a plasma potential that was everywhere more positive than the electrode bias potential. A curious example of how a small leak can come about is given in Fig. 11. In this case a conducting electrode, which did not have an insulated back, was biased at +20 V. A potential minimum was found between the electrode and the plasma [ 16, 171. Careful examination of equipotential contours found the island structure shown in Fig. 11. Examination of the electrode, when the chamber was let up to air, led to the discovery of

6 16 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 22, NO. 1, FEBRUARY f I I I I I I I I II r(cm) O/- Fig. 11. Equipotential contours near a circular conducting plate biased at +20 V. The location of a fingerprint and the associated potential well is at T = 1 cm PLASMA POTENTIAL mr A TYPICAL DOUBLE LAYER I I I l l In the double layer frame, ions enter the high potential side with at least Bohm velocity-require a Bohm presheath acceleration or external acceleration to 2 c,. This resembles a sheath condition but also can be interpreted as a supersonic velocity in a moving frame and could simply be provided by cold ions encountering a moving electrostatic shock. In many respects, a double layer can be thought of as a double sheath which forms in the plasma interior rather than at the edge. The distinction between a double sheath (double layer) and a single sheath becomes clear when one considers the ion and electron phase space associated with both types of structures in Fig. 13. For sheaths electrons are reflected (i.e., trapped, while free ions are accelerated. Only a single species (i.e., ions) is present over most of the structure. On the other hand, double layers consist of three or four species (two free and two trapped). There are an infinite number of different double layer solutions possible depending on the amount of trapped and free particles [23]. The structures shown in Figs. 2(a), (b) and (c) are BGK solution [24] for which the solution is entirely determined by specifying the distribution functions on the boundaries. Are laboratory double layers really disconnected from the boundaries? Although they are not located at a boundary, the answer is usually no! They certainly can occur many Debye lengths from the boundary. However, ions at the double layer boundary must satisfy a Bohm criterion so it reasonable to II I assume that a Bohm presheath length separates the double AXIAL POSITION IN un GRDS AT*o layer from the boundary. Since Bohm presheaths can extend to- the characteristic size of a system in collisionless plasma, Fig. 12. Representative laboratory double layer obtained in a triple plasma device. a fingerprint (which acted as an insulator) on the electrode at the location of the island structure. VI. SHEATHS DISCONNECTED FROM BOUNDARIES-DOUBLE LAYERS An examination of Fig. 2c leads to the questions: Does the double sheath have to form at a plasma boundary? Does the positively biased electrode have to be electron emitting? The answer to both questions is no! Double sheath structures known as double layers can form far (> AD) from boundaries. An electron rich region with positive curvature adjacent to an ion rich region with negative curvature can give rise to the double layer structure shown in Fig. 2d. A representative laboratory double layer is shown in Fig. 12 [21,22]. In general, laboratory double layers tend to be one dimensional structures when there is no dc magnetic field. They come in two varieties, stationary and moving. The stationary structures can be stable for many hours. Similar to sheaths, their thickness can usually be written as a function of AD. A. Particle Acceleration In collisionless, nonturbulent systems, particle acceleration is localized to the double layer potential step(s). Ions fall down the double layers and electrons fall up the double layers. In between they drift at constant velocity. Ions reflect from or are slowed by potential increases and electrons reflect from or are slowed by potential decreases. the double layer, while still dependent on the location of the boundary, can occur in the middle of the device, or almost anywhere else. Sheaths at boundaries often are established to balance electron and ion losses from the bulk plasma so no net current is present. Double layers have more flexibility since the same solution can correspond to many different distributions of free particles. For example, free ions can enter from the right and be accelerated by the potential or from the left and be decelerated by the potential. If the corresponding ion phase spaces for the ion beams were mirror images about the z axis, both beams would correspond to the same spatial ion beam density profile nb(2). In that case, they would correspond to the same solution to Poisson s equation (see Fig. 14). B. Moving Double Layers Double layers moving at c, satisfy the Bohm criterion by encountering cold plasma ions. Ion acoustic double layers evolve out of nonlinear ion acoustic waves as shown in Fig. 16 [ A potential dip forms, grows and reflects electrons and in the laboratory eventually evolves into a stationary double layer. Once it is recognized that the double sheath can be disconnected from the wall, it is no longer obvious that here must be only one double sheath. In fact, the transition can occur in two steps (see Fig. 15) [25] or more [26]. It is worth noting that these are no longer BGK solutions because they depend on modifications to the distribution functions resulting from turbulence within the plasma.

7 HERSHKOWITZ: HOW DOES THE POTENTIAL GET FROM A TO B IN A PLASMA? 17 a a d Fig. 14. Possible ion phase spaces associated with a double layer. (a) Double layer plasma potential. Note that if an ion beam which enters the high potential side (b) with U 2 cs gives a double layer, the mirror image(c) about ti = 0 also gives the same solution to Poisson's equation. Combination of such ion beams (d) can also be a solution. f t ve Axial Position (em) Fig. 13. (a) Plasma potential together with (b) ion and (c) electron phase spaces associated with a double layer. (d) Plasma potential together with (e) ion and (0 electron phase space associated with a plasma-wall sheath. C. Solitons Under other circumstances a pulse can evolve into a stmcture, stationary in a moving frame, known as a soliton [30]. Fig. 15. Representative two step, stairstep double layers. From the point of view of Poisson's equation, the edges of the soliton are electron rich, while the peak is ion rich. Cold plasma ions, are slowed by the potential increase and cold ion flux conservation combined with energy conservation increases

8 18 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 22, NO. 1, FEBRUARY 1994 I I I I Stationary peak i- -i - e I I I 0 IO DISTANCE (cm 1 Fig. 17. A Schematic of ion acoustic soliton comparing moving and stationary structures. Fig. 16. Evolution of rarefactive pulse (indicated by arrows) into a stationary double layer. the ion density. ni = (12) where M is the Mach number 2 1. Electron density is approximately described by the Boltzmann relation ne = no exp ($) so the electron density also increases with increasing potential. For 1 I M I 1.6 the curvature, which is proportional to ne - ni, is negative for small values of q5 and becomes positive for larger values (see Fig. 17) so it can satisfy the nonlinear evolution of an ion acoustic soliton. Creation of such a structure in the laboratoy frame is difficult and never been accomplished because the appropriate plasma density perturbation in the pulse must be created simultaneously with the appropriate ion beams. D. Double Layers in the Presence of Magnetic Fields Double layers have also been studied in the presence of uniform axial magnetic fields [31]. Most experiments have employed systems with radius small or comparable to the ion gyroradius. Equipotential contours near conducting boundaries tended to conform to the boundaries and behaved like double layers along the axis. This resulted in radial double layer structures. Recent laboratory measurements [32] on magnetized double layers in a system which contains many ion gyroradii have found multiple step two dimensional double layers, oriented with respect to the magnetic field. The characteristic dimension of these structures appears to be proportional to 10 in the parallel direction and to have a distribution in LA in the perpendicular direction which satisfies rci I LA I rcs (14) It is interesting to note that LA depends on T,,, the same parameter that appears to determine I, the magnetic presheath thickness. E. Space Double Layers It is currently argued that nature has employed double layers to provide magnetospheric acceleration of the energetic electrons which produce aurora [ Others argue that wave-particle acceleration is more important [36, 371. Double layer observations by satellites has suggested the structure shown in Fig. 18. There is no question that weak double layers are present, but whether they are present in sufficient numbers or sufficiently often. No consensus yet exists on this question. VII. POTENTIALS ASSOCIATED WITH RF A. Potentials Associated with Capacitive rj-rf Self Bias Plasma etching and deposition are sensitive to the energy of bombarding ions. This presents a bit of a problem if the substrate is an insulator or semiconductor. A relatively simple solution to this problem is to mount the insulator on a conductor and to apply rf to the conductor. The dc value of the insulator potential then floats negative to balance the electron and ion loss flux. If the substrate area is small compared to the area of the other system electrodes, the resulting rf self bias voltage V& will be negative and equal in magnitude to approximately half the applied peak-to-peak voltage. This solution works if the insulator impedance is much less than the sheath impedance. For frequencies much greater than the ion plasma frequency, the time average potential seen by ions in going from B to A will resemble the dc sheath potential shown in Fig. 2b. For the same conditions, electrons will see a time varying potential and will be lost to the substrate only a small fraction of the rf period.

9 HERSHKOWITZ: HOW DOES THE POTENTIAL GET FROM A TO B IN A PLASMA?,/ Mayetotoil Fig. 18. Magnetosperic equipotential contours near the Earth, inferred from satellite data Recent experiments [38] have found that etch rate of a Si02 wafer in an Electron Cyclotron Resonant (ECR) etching tool depends on the ion energy flux which is proportional to ji(v, - VSB) if sufficient fluorine is present (see Fig. 19). Here ji is the ion saturation current density at the wafer being etched. This is a curious result since the etching is thought to be chemical etching. It provides yet another example of the importance of knowing and controlling V, and Vi?, the potentials at A and B. B. Potentials Associated with Inductive rf I) The Problem of rf in Small to Medium Sized Tokamaks: Studies of rf in small to medium sized tokamaks have been impeded by the problem of uncontrolled fueling and impurity production [39]. The source of the problem has been identified as an increase in the plasma potential in the edge region (the so-called scrape off layer (SOL)) of the tokamak. The increased SOL plasma potential results in increased sputtering and desorption from the tokamak walls. Representative results from the Phaedrus-T tokamak using a two (poloidal) strap antenna operated below the ion cyclotron frequency, are shown in Fig. 20. It is apparent that impurity generation and SOL potential both increase with rf power and that the increase is sensitive to the phasing of the two poloidal straps. The goal of Phaedrus-T is to study Alfven wave current drive which requires directional waves. Such waves can be achieved by phasing at other than 180". Such antenna phasing means that a net is forced into the antenna near field giving rise to an electromotance in current paths that connect to the antenna (and in current paths that connect to the walls). The electromotance means that E fields will be present along the high impedance parts of the path. The plasma-limiter impedance is normally the highest impedance in any current path so big E fields are produced there. With grounded antenna limiters, this means that the SOL plasma potential must increase with a subsequent increase in fueling Fig. 19. Etch rate of Si02 substrate in a CF4 ECR plasma vs. ion energy flux J, E,. and impurities. Phasing at 0" and 90" have similar effects on impurity emission while emission associated with 180" phasing is significantly reduced. Phasing at 180" results in an approximate cancellation of the net g. The solution to this problem is straightforward. The SOL potential increase can be eliminated if an impedance is added to the current path that is much greater than the sheath impedance. This is accomplished by the addition of a thick (0.6 cm boron nitride) insulator in front of the limiter. As shown in Fig. 20, data taken in the presence of insulating limiters at 0" fall slightly below the best 180" data without an insulating limiter. C. Potentials Associated with Propagating rj-plasma Potential in the Phaedrus Tandem Mirror The tandem mirror concept is based on taking advantage of ion and electron confining potentials to improve particle and energy confinement and a variety of techniques have been employed to control V,. A schematic axial plasma potential profile is given in Fig. 21 [40, 411. Axial ion confinement is provided by the positive potential peaks. The potential dips (known as thermal barriers) are provided to allow the electron temperature in the end cells to be larger than in the central cell. The simplest way to achieve potential increases (and the first that was successful) [42] is to inject neutral beams into magnetic mirror end cells. The local density is raised when the

10 20 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 22, NO. 1, FEBRUARY v S.S. C.L A *a. * I 1 h RF Voltage [kv] CC nf [lo ci ] 0.0 I I Anrenno current (A rms) I Fig. 22. The measured plasma potential in the end cell of the Phaedrus tandem mirror vs. rf power and plasma density. A = Self-Emissive Prone, r = 0 cm 0 = Differential-Emissive Probe, r = 0 cm LII = Oifferentiol-Emissive Probe, r = 7 cm loo1 I I 4 I I I z - Ave. Power (CW) Fig. 20. Variation of scrape-off-layer (SOL) potential and FeXVI with rf power coupled out of to a two strap rf antenna in the Phaedrus-T tandem mirror. Passing ions -/\ 0 cl -20 I I I Fig. 23. Axial plasma potential in Phaedrus-E profile on (T = 0 cm) and near (T = 7 cm) axis for two times: (a) end cell rf on t = msec,(b) 2 msec after end cell rf shutoff, t = 14 msec. Fig. 21. A schematic axial plasma potential profile of a tandem mirror neutral beams charge exchange and become locally trapped. The Boltzmann relation then predicts that the plasma potential will increase by Ad=-ln I+- :( :) Experiments in the Phaedms tandem mirror explored a different approach. It was shown that rf electric fields parallel to the dc B field can expel end cell electrons resulting in net positive potential [43, 441. Unlike the density produced potential increases, the rf produced ion plugging potential increased as the local plasma density decreased and as the rf voltage was increased (see Fig. 22). Thermal barrier potential dips were also found, the result of a density depression associated with an expanding flux tube together with end cell rf pumping [45]. In general, the observed potentials depend on the details of the propagating waves (see Fig. 23). VIII. CONCLUSION How does the potential get from A to B? Clearly there are many answers to this question. Each answer depends on plasma and boundary conditions. In sheath problems, the potential is determined in part by the application of boundary potentials. For example, the profile shown in Fig. 2b can be achieved by biasing one boundary at VA and the other somewhat below VB. Potential wells (Fig. 2c) can be created, for example, by inserting small positively biased electrodes (either hot or cold) into a plasma with provision for pumping the wells. Laboratory double layers with single step transitions (Fig. 2d) can be created by imposing a combination of boundary potentials and boundary beams of charged particles. Application of rf to a boundary electrode results in a time average self bias potential. This is an example of capacitive coupling. Inductive rf can lead to a similar rf self bias of the plasma with respect to a conducting boundary. Another possibility is to take advantage of the characteristics of propagating waves in plasmas to modify the plasma potential. The answer matters. For example, nature may have chosen double layers to answer the problem in the magnetosphere. Ion confining potentials and thermal barriers make tandem mirrors work. RF produced potentials can confine ions or

11 HERSHKOWITZ: HOW DOES THE POTENTIAL GET FROM A TO B IN A PLASMA? 21 Control plasma etching or produce impurities in tokamaks. The possibilities are without limit. ACKNOWLEDGMENT The author would like to thank his many students and colleagues, whose names can be found throughout the references, for their valuable contributions towards the understanding of how potential gets from A to B in a plasma. Special thanks are due to D. Diebold for his reading of and suggestions for this manuscript and to G.-H. Kim for his assistance in preparing the figures. REFERENCES L. Tonks and I. Langmuir, A general theory of the plasma of an ARC, Physical Review, vol. 34, September N. Hershkowitz, Plasma Diagnostics, vol. 1, 0. Aucielo and D. L. Flamm, Eds. How Langmuir Probes Work, San Diego: Academic, 1989, Chapt. 3, pp J. R. Smith, N. Hershkowitz and P. Coakley, The inflection point method of interpreting emissive probe characteristics, Rev. Sci. Instrum., vol. 50, p. 210, R. F. Kemp and J. M. 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Fluids B, vol. 4, p. 3663, Noah Hershkowitz (F 89) received the B.S. degree from Union College in 1962 and the Ph.D. degree from Johns Hopkins University in He was a member of the faculty at the University of Iowa from 1967 to During that time he took sabbaticals at UCLA and the University of Colorado. In 1981, he joined the University of Wisconsin as the Irving Langmuir Professor Nuclear Engineering and Engineering Physics at the University of Wisconsin in Madison. He is an active member of the IEEE plasma science community. He has helped to organize IEEE Plasma Science Conferences by serving as a session organizer and as a Program Committee member. He has been a member of the PSAC ExCom and has represented the PSAC on the NPSS AdCom. He served as AdCom Vice President from , and is presently Chairman of the NPSS Awards Committee. He has served as Editor for Physical Review Letters and Physics of Fluids and is currently Editor-in-Chief of Plasma Sources Science & Technology. Prof. Hershkowitz received the NPSS Merit Award in He is also the 1993 recipient of the Plasma Science and Applications Committee (PSAC) Award.