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Wiley Encyclopedia of Electrical and Electronics Engineering Electron Impact Ionization Standard Article S. K. Srivastava1 1California Institute of Technology, Pasadena, CA Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5907 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (203K)

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Abstract The sections in this article are Measurement of Cross Sections Ionization Properties of Radicals and Excited States Polar Dissociation Dissociation of Molecules into Neutral Species Ionization Properties at the Threshold of Ionization Additivity Rule Kinetic Energies of the Fragment Ions Conclusions About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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ELECTRON IMPACT IONIZATION

645

ELECTRON IMPACT IONIZATION For atoms and molecules the term electron impact ionization applies to the process in which one or more electrons bound to a target are removed as a result of collisions between a J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

646

ELECTRON IMPACT IONIZATION

free energetic electron and a target atom or molecule, thereby leaving the target species positively charged. The various processes, among many others, that may take place as a result of collision can be represented by the following relations: e− (E0 ) + A ⇒ An+ + (n + 1)e− (Eo − Ei )

(1)

or

e− + MN ⇒ MNn+ + (n + 1)e−

(2)

⇒ M+ + N or M + N+ +

−

−

⇒ M + N or M + N

(3) +

(4)

where A is an atomic species and MN a molecule with atomic and/or molecular components M and N, e⫺ an electron, E0 the kinetic energy of the incident electron, and n is the degree of ionization. The efficiency with which a projectile can remove bound electrons from a target is related to a quantity commonly called cross section and has the dimensions of area and is expressed in units of m2 (square meters). Its value mainly depends on the atomic and molecular structure of the target species and kinetic energy of the incident electron. The cross section represented by Eqs. (1) and (2) are traditionally called direct ionization or multiple ionization. Those represented by Eqs. (2), (3), and (4) are called direct or partial and dissociative ionization cross sections, respectively (conventionally represented by symbols ␴p and ␴d, respectively). A sum of all cross sections for one species is called the total ionization cross section of that species and is represented by the symbol σT = σ p + σd

(5)

It will be explained in the following paragraphs that the ionization cross sections for a specific species can be obtained either by measuring the ion current or by counting each individual ion when the number of ions produced as a result of collision is very small. If the ions are counted instead of integrated, then the total ionization cross section is called the total counting ionization cross section, ␴c, and is represented by the following relation:



σc = σp + σi, n (6) p

i, n

where p refers to the direct ionization process [Eq. (2)] for the removal of one or more electrons from the target, n is the degree of ionization of the dissociated ion, and i identifies the species resulting from the dissociation of the molecule. When the total ion current is measured for obtaining the cross section it is commonly called the gross ionization cross section, ␴g, and is represented by the following equation:



σg = σp + Zi σi, n (7) p

i, n

where Zi is the degree of ionization of the ith species and other symbols have been defined in previous paragraphs. Cross-section values are important for understanding properties of various plasmas (1). Their values are important for calculating abundances of elements observed in astrophys-

ical plasmas and for the interpretation of mass spectrometric data. Nowadays, low-temperature plasmas are being extensively employed for processing semiconductors. Properties of the ionosphere of planets, comets, and Earth can be better understood with the knowledge of electron-impact cross sections. They are also important for calculating the penetration depths of 웁 particles in biological samples. The values of cross sections for a species are important and must be accurately known for some applications. Their values can be obtained either by theoretical calculations or experimental techniques. It has been found that simple classical methods (1) of calculating cross sections do not predict their values accurately. Therefore, quantum-mechanical calculations (2) are employed, which are difficult because the two free electrons require continuum wave functions. The calculations involve coupled differential equations. Because of manybody interactions occurring in these calculations they tend to be very complicated. Several methods of approximations (3) have been developed in the past to make the calculations simple yet predict fairly accurate values. Therefore, semiempirical methods have also been developed. According to one semiempirical classical calculation [Lotz formulas (4)] the ionization cross sections can be calculated from the following formula:

σI =

Ns

ζn (a ln u/E0 In )

(8)

n=1

where ␨n is the number of equivalent electrons in the nth subshell, a ⫽ 4.5 ⫻ 10⫺14 eV2, u ⫽ E0 /In is the reduced energy of the impacting electron, In is the binding energy of electrons in the nth subshell, and ln(u) is the logarithm of the reduced energy u. Although cross-section values have been measured since the 1920s, the available data until 1950 were for simple and benign gases such as hydrogen, nitrogen, and oxygen. In the 1950s the activity related to measuring cross sections was dormant. However, in the 1960s, due to interest in lasers and fusion plasmas, the field revived and several groups (e.g., 5,6) designed and fabricated new instruments to carry out measurements of cross sections. MEASUREMENT OF CROSS SECTIONS There are different versions of experimental apparatus employed by various researchers for the measurement of cross sections. However, conceptually, most of them consist of components shown in Fig. 1. Each component will be briefly described here and references will be given for detailed understanding. Electron Gun The electron gun produces a collimated beam of electrons, the kinetic energy of which can be varied or fixed. There are several different designs of an electron gun (7,8). The simplest one uses a tungsten hairpin filament that can be heated in a vacuum to produce electrons. The electrons boil off the filament in the form of a cloud. The cloud consists of electrons that have an energy spread ⌬E. This spread is related to the temperature of the filament (9). Therefore, the temperature of the filament plays an important role in determining the

ELECTRON IMPACT IONIZATION

Vacuum chamber Chargedparticle detector Mass analyzer

Electron gun

+ – Pressure gauge

– +

Coil

–

.. ..

– +

Electrometer Faraday or cup Multichannel analyzer

Amplifier

P1 P2

Figure 1. A conceptual diagram of the experimental arrangement commonly used in ionization studies.

energy spread of the electrons. The cooler the filament, the smaller the energy spread. However, recently it has been shown (8) that the material of the filament is also important. Filaments made of irridium have less spread than the filaments made of tungsten. For most experiments smaller values of ⌬E are desirable. The electrons are subsequently collimated to form a beam. In general, two methods have been employed for collimating them. They are (1) the electrostatic method (8) and (2) the magnetic method (7). In the electrostatic method the cloud of electrons is pulled away from the filament region and converted into a beam by a series of electrostatic lenses to form a beam (7,8). A simple arrangement of lenses is shown in Fig. 2. The energy of the electron beam is determined by the potential difference between the filament and ground of the system. The magnetic collimation is based on the fact that electrons follow the magnetic lines of force in helical paths. In a magnetic electron gun the electron cloud is pulled away from the filament region in the same way as in the electrostatic gun and is subsequently directed along the axis of the electron gun. A magnetic field is applied along the axis of the gun. Figure 3 shows a simple design (10) of an electron gun that uses magnetic collimation. In this case the magnetic field is produced by a solenoid constructed of vacuum-compatible materials. The previously described simple electron guns (electrostatic or magnetic) generate electron beams with energy spreads varying from about 0.25 eV to 0.5 eV. Lower-energy spreads can be achieved by passing the electron beam

Einzel lens (F) W2

W3 L6

L5 L4 L3

Zoom 2

L2

Circular spring

;;;;; ;; ;;;;; ;;

+ MN

A1 A2

647

Filament (F)

W1 L1 An

Kh

Zoom 1

Figure 2. A simple electrostatic electron gun. The electrons are extracted from the filament region by lens W1. They are then collimated and accelerated by an Einzel lens and focused at the exit aperture of lens W3.

D

A

F

H

Figure 3. A simple magnetically collimated electron gun. The coil produces axial magnetic field. F, the filament; H, the cathode housing; A, the aperture for collimating electrons; D, deflectors for deflecting the electron beam; P1 and P2, deflector plates; A1 and A2, apertures.

through an energy-dispersive device. There are several different types of dispersive devices (11). Dispersion in magnetically collimated beams is achieved by passing them through a region of crossed electric and magnetic fields. The electron guns that employ this principle of dispersion are called trochoidal monochromators. There are several designs of these monochromators. Reference 12 and references contained therein will provide a good understanding of the design principles. Faraday Cup The beam of electrons, after leaving the electron gun, is made to collide with the target species under study. As a result of collisions, ions are produced and colliding electrons get deflected from their original path. Those electrons that do not collide with the target species keep on going on their path and are collected by a device called a Faraday cup, cylinder, or cage. The first description of this device dates back to 1895 (13), 1896 (14), and 1897 (15). The main requirement in designing these devices is that they should be able to collect all electrons without returning them back to the collision region. Figure 4(a) shows the simplest design used by Perrin (13). Figure 4(b) shows a design, used in the author’s laboratory, along with its wiring diagram. Ion Source As mentioned before, the beam of electrons is passed through the target species, the cross sections of which need to be measured. These target species can be generated by filling the entire vacuum chamber with the gas under study. As shown in Fig. 1, the region between the electron gun and Faraday cup becomes a source of ions. Therefore, this region is generally called the ion source. Instead of filling the entire chamber with gas the target species can also be generated in the form of a beam of atoms or molecules. In the case of gases a beam can be easily generated by flowing the gas through a capillary tube. For solids the material in the form of powder is filled inside a crucible that can be heated by electron bombardment or the resistive

648

ELECTRON IMPACT IONIZATION

;;;;;;;;;; ;;;;; (a)

v1

v2 (b) Figure 4. (a) Perrin’s Faraday cup. (b) A schematic diagram of the Faraday cup used in the author’s laboratory.

heating (16) method with a fine hole at the top through which the vapor of the sample efusses. Figure 5 shows a schematic diagram of the crucible used in the past by the author (11,17) for forming the beams of solid materials. If the target species of interest is in the form of a liquid (18) at normal temperature and pressure, then the liquid is usually filled inside a glass bulb that is heated to vaporize the material. The vapor is subsequently allowed to effuse through a hypodermic needle to form a beam. When the beam of electrons is made to collide with the target species by passing the electron beam through the gasfilled vacuum chamber, the experimental arrangement is called the static gas collision geometry. In the case in which

the target species is prepared in the form of a beam, then it is called the beam-beam or crossed-beam collision geometry. Ions produced as a result of collisions in the ion source are extracted by an ion extraction system. There are several methods of ion extraction (19). The ion current is usually measured by a sensitive electrometer that can detect currents in the picoampere range. For very weak currents the ions are counted individually. For this purpose, each individual ion is detected by a charged-particle detector (usually called a channeltron, spiraltron, or channel plate) that multiplies the charge of each ion by a factor of about 10 (20). Thus each ion is converted into a current pulse that is subsequently amplified by a fast electronic amplifier. The amplifier gives rise to a pulse of about 5 V amplitude corresponding to each detected ion. Each pulse is stored in the memory of a device called a multichannel analyzer as a function of electron-beam energy. The number of pulses counted per second is a measure of ionization current, which is directly proportional to the cross section. By varying the energy of the electron beam and counting the ions for each energy a plot is made between the count rate and electron-impact energy. A typical plot is shown in Fig. 6. Since this plot represents the efficiency with which ions are formed as a function of electron-impact energy, it is generally referred to as the ionization efficiency curve. It also represents the dependence of cross sections on the electron energy for the atom or molecule under study. If all ions produced in the ion source are collected from the ionization region, then the ionization cross section derived from this measurement is called total ionization cross section ␴c or ␴g as defined by Eqs. (6) or (7), respectively. The total ion current IT (total current generated by all ionic species irrespective of their degree of ionization) can be related to various measurable parameters through the following relation: IT = Nn σg LIe

(9)

1.0 D2+

;;;;

Cooling tubes Crucible (grounded)

Insulator base Filament supply

Magnetic shield

Heat shield

σ (×10–16cm2)

0.8

0.6

0.4

D+ ×4

Filament

High-voltage supply

Figure 5. Schematic diagram of a high-temperature crucible used for forming beams of metal atoms.

0.2

0

0

200

400 600 Electron energy (eV)

800

1000

Figure 6. Typical ionization efficiency curves (shown by a solid line). The ionization current is directly proportional to cross sections.

;;;;; C

649

Thus, it also distinguishes between the various multiply charged ions.

R

G

ELECTRON IMPACT IONIZATION

SP

ECP

G

2

ECS

ECC

3

Figure 7. Apparatus used by Rapp and his associates for the measurement of absolute values of cross sections for a number of atmospherically important atoms and molecules.

where Nn is the number density of the target species, L the path length (for the case of static gas collision geometry) of the electron beam in the gas under study, and Ie is the current of the electron beam. Equation (9) is used for measuring absolute values of cross sections ␴g. For this purpose one has to measure absolute values of all other quantities shown in Eq. (9). Nn can be obtained by measuring the gas pressure in the ion source under study, L can usually be accurately obtained by a calibration procedure employing a gas the cross sections of which are accurately known, and Ie is measured by the Faraday cup. By utilizing the static gas geometry Rapp, EnglanderGolden, and Briglia (5) measured accurate values of cross sections for a number of atmospherically important gases in 1965. Their apparatus is shown in Fig. 7. Instead of a hairpin filament they used an oxide-coated cathode as a source of electrons. Their method was to obtain relative values of cross sections first at different electron-impact energies by plotting the ionization efficiency curves. Then at a fixed electron-impact energy absolute cross sections were obtained by measuring all quantities of Eq. (9). For the case of crossed-beam collision geometry the relationship between the ion current and cross section is not so simple as in Eq. (9). It is then written (21) as  Im (E0 ) = K(m)σm (E0 ) f (r, E0 )ρ[r][r] dr (10) v

where Im(E0) is the ion current of mass m, K(m) is the massdependent transmission efficiency (19) of the ion extraction and detection system, ␴m(E0) is the value of the cross section for ions of mass m as a function of the electron-impact energy E0, ␳[r] and ⌬⍀[r] are, respectively, the target density and the solid angle subtended by the detector optics at a point r within the collision volume v, and f(r, E0) is a function of r and electron-impact energy (see Ref. 21). It is clear from Eq. (10) that accurate measurement of each and every quantity shown on the right-hand side of this equation is a very difficult task. Therefore, the crossed-beam collision geometry is not quite suitable for the measurement of absolute values of cross sections. Mass Selectors As a result of the dissociation of molecules, atomic and molecular species in their various ionic states are produced [Eqs. (3) and (4)]. For measuring cross sections for dissociative and multiple ionization processes a mass spectrometer is required for selecting a particular species of interest. A mass spectrometer actually measures the mass-to-charge ratio of an ion.

Measurement Procedure

The procedure for the measurement of cross sections normally proceeds through the following steps (5): (1) The energy E0 of the electron beam is fixed, (2) ions of specific mass-to-charge ratio are extracted out of the collision region and the ion current is recorded by an electrometer, (3) for the static gas collision geometry pressure of the gas under study and the electron-beam current is measured. These quantities are substituted in Eq. (9) and the value of the cross section is calculated. In the procedure for measuring cross sections by utilizing the crossed-beam collision geometry, the measurement of absolute values of quantities on the right-hand side of Eq. (10) is very difficult to obtain with any accuracy. Therefore, the measured cross section is highly unreliable due to following reasons: (1) A reliable estimate of the number density of the target species is very difficult with any accuracy, (2) the length L is almost impossible to measure, and (3) estimation of the size of the collision volume is highly unreliable due to uncertainty in the estimation of the part of the electron beam which actually interacts with the target. However, there are certain advantages of the crossed-beam collision geometry. The main advantage is that it presents a very well-defined ion source to the mass spectrometer and collection of all ions is possible. In the procedure for measuring cross sections by utilizing the crossed-beam collision geometry, the following steps are taken: (1) The electron-beam energy is varied continuously and the ion current is recorded as a function of electron beam energy E0. As explained in previous paragraphs the resulting plot is the ionization efficiency curve. (2) The next step is to normalize this curve by a known value of cross section. Therefore, if an absolute value of a cross section is available at one point of this plot, then the entire plot can be normalized to yield cross-section values at other electron-impact energies. There are several procedures of normalization. One of them was developed in the author’s laboratory by utilizing a method called the relative flow technique (20,22). This technique is applicable only to those species that are in a gaseous form or vapor state at normal room temperature and pressure. The relative flow technique depends on the fact that if a gas flows through a capillary tube and if the flow rate is very small, then the flow rate can be related to the spatial and velocity distribution of molecules in the beam (21). Thus, if a gas for which the cross sections are not known flows through a capillary tube and if we measure the ion current Iu(E0), then it can be related to unknown quantities shown on the righthand side of Eq. (10). Subsequently, if we stop the flow of this gas through the capillary tube and start the flow of a gas (such as He) the cross sections of which are accurately known in such small quantities that the electron-beam current and other experimental conditions do not change then the ion current, Is, for this gas will also be given by Eq. (10). We can relate the two ion currents, Is and Iu, through the following equation:

σu (Eo ) = σs (Eo )[Iu (Eo )/Is (Eo )](Ms /Mu )1/2 (Fs /Fu )[K(ms )/K(mu )] (11)

ELECTRON IMPACT IONIZATION

where the index u stands for a gas whose cross sections are unknown and s represents a standard gas, such as He, the cross sections of which are accurately known. K(ms) and K(mu) are the mass-dependent transmission and detection efficiency (23) of the apparatus. Ms and Mu are the molecular weights of the standard gas and the gas under investigation, respectively. Fs and Fu are the flow rates of the standard gas and the gas for which cross sections are not known, respectively. Various experimental details of the relative flow technique can be found in several publications (e.g., Refs. 20 and 22).

2.0

σ (10–19cm2)

650

0–/C0

1.0

IONIZATION PROPERTIES OF RADICALS AND EXCITED STATES 0

Studies related to stable species are readily available in the previously published literature. However, the same is not true for radical species. This is due to the fact that radical species are difficult to prepare in the form of a target and are short-lived. For example, species such as O, N, and C are difficult to generate in the form of a beam. Pioneering work in this area has been done by Hays et al. (24) and subsequently by Deutsch, Becker, and Mark (25). The apparatus of Hays et al. is shown in Fig. 8. In this apparatus the beam of radical species is prepared by forming a beam of ions of the species under study such as N⫹. This beam is accelerated to high energies and is then passed through a cell filled with the vapor of an appropriate material such as alkali-metal atoms. In the cell charge-exchange reactions take place, and a fast neutral beam of atoms emerges. This beam is then employed as the target beam for colliding an energy-selected beam of electrons.

10

20 30 Electron impact energy, eV

40

50

Figure 9. Dissociative attachment and polar dissociation cross sections for the production of O⫺ from CO.

tachment. Figure 9 shows a graph of O⫺ intensity resulting from low-energy electron collisions with CO. The first spectral feature is O⫺ formed by dissociative attachment. However, at higher impact energies the curve slowly rises above the background and goes through a maximum just like the ionization efficiency curves shown in Fig. 6. The continuum part of this curve represents the process of polar dissociation in CO. Ionic states are formed when two oppositely charged species are brought close to each other. The two charges experience the Coulomb force. Figure 10 shows typical curves con-

POLAR DISSOCIATION E(Y) X+ + Y

Equation (4) shows a situation in which the molecule dissociates into two component ions: one positive ion and one negative ion. This process is called ion pair formation or polar dissociation. This type of dissociation takes place through molecular states that are Coulombic (26) in nature. Negative ions are generally formed by the process of dissociative at-

X+ + Y– X** + Y X* + Y

DC discharge ion source Einzel Ion-beam lens deflector

Gas inlet

Hemispherical energy analyzer Neutral beam CEM monitor Einzel lens S

N

S

N

S

N

Charge transfer cell Ion-beam collector Vertical Horizontal Beam deflection deflection chopper plates Electron plates gun Figure 8. A schematic diagram of the apparatus used by Hays et al. (24) for the measurement of cross sections of radical species.

Potential energy

A I(X) X+ + Y– X+Y

D

Potential energy Figure 10. Typical potential energy curves which give rise to polar dissociation in molecules.

ELECTRON IMPACT IONIZATION

structed from a potential of the following form: V = −ke (e2 /r) + Be−r/ρ

(12)

These curves are generally crossed by (in the zero-order approximation) a number of their covalent excited states. These covalent states are essentially flat at large nuclear distances. Because the ion-pair formation takes place through the dissociation of neutral states of a molecule, positive ions begin to appear at energies lower than the ionization potential of the molecule.

old of ionization of acetylene is shown. The point where the curve begins to rise above the background is used to calculate the appearance potential of the species. The ionization efficiency curve in the neighborhood of the ionization potential is of interest because it is the region where for many species autoionization states are present. It is also the region that lies at the interface of classical mechanics and quantum mechanics. A relationship that is well known in atomic physics for ionization is called the Wannier (31) law. It describes the dependence of cross sections on excess electron-impact energy (E0 ⫺ Ei):

DISSOCIATION OF MOLECULES INTO NEUTRAL SPECIES Equation (4) represents dissociation of a molecule into a pair of ions. However, dissociation of a molecule into neutral fragments is a process of great interest to the subject of plasma chemistry. For example, e⫺ ⫹ N2 ⇒ N ⫹ N is a very important process for the Earth’s atmosphere. However, reliable experimental values for this process are not known. The main difficulty is due to the fact that methods to detect neutral fragments are difficult to implement. Methods such as laserinduced fluorescence (27) (LIF) and multiphoton ionization (MPI) are being developed, but both methods are either very selective or difficult to implement. Therefore, data on the production of neutral fragments by electron impacts on molecules are scarce at the present time. A method that has been pioneered by Winters and Inokuti (28) utilizes the technique of trapping of the neutral species from which one can infer the intensity of the species produced as a result of electron impact. More recently Goto et al. (29) have reported an experimental apparatus that is capable of measuring cross sections for the production of neutral particles. The apparatus is a dual-electron-beam device that is combined with a quadrupole mass spectrometer. This system consists of three compartments that are differentially pumped. The first compartment is a dissociation cell in which a primary electron beam dissociates the molecule of interest. The second compartment is a detection cell in which a probing electron beam (10 eV to 25 eV energy) emitted from a rhenium filament selectively ionizes neutral radicals that effuse from the first cell through a 4 mm diameter hole into the ionization chamber. As mentioned before, lasers have also been employed for detecting neutral fragments. It utilizes a tunable dye laser, which, when tuned to correct frequencies, produces fluorescence signal from the neutral particles. The technique is known as LIF. Although LIF is a powerful tool for detecting certain species, its implementation is extremely difficult and works only for those species that strongly absorb laser radiation in the wavelength range that the tunable dye laser can produce. IONIZATION PROPERTIES AT THE THRESHOLD OF IONIZATION The threshold ionization (30) potential can be defined as the electron-impact energy at which ionic species in the target begins to appear. The energy at which it begins to appear is generally called appearance energy or appearance potential (30). In Fig. 6 an ionization efficiency curve near the thresh-

651

σp ∝ (E0 − Ei )1.127

(13)

where E0 is the electron-impact energy and Ei is the ionization potential of the target species. This equation does not provide any information on how far above the ionization threshold this law is applicable. The ionization threshold region can be studied by photoabsorption as well as by electron-impact ionization. The photoabsorption method is superior to the electron-impact method from the point of view of high resolution, but in the latter case the resolution is poor. Therefore, the ionization potentials derived from photoabsorption data are considered to be more accurate than those obtained from electron impact. However, electron impact can excite all energy levels of the target near the threshold of ionization. Therefore, by electron impact one can probe those states of the target that are optically forbidden.

ADDITIVITY RULE Owing to complications related to many components present in a large molecule, theoretical calculations are very difficult. However, cross sections for ionization of large molecules can be ‘‘roughly’’ estimated by the application of the additivity rule (32). According to this rule the cross section of a molecule can be estimated by summing up cross sections of individual atomic and molecular components of the molecule. Thus, a cross section for a molecule MNP can be roughly estimated by summing individual cross sections for M, N, and P, i.e., ␴MNP ⫽ ␴M ⫹ ␴N ⫹ ␴P, where ␴MNP is the total ionization cross section for the production of MNP⫹ from MNP, ␴C is the cross section for the ionization of the C atom, and ␴O is the ionization cross section of the O atom. It was shown by Grosse and Bothe (32) and more recently by Orient and Srivastava (33) that this rule works well for organic molecules and for high electron-impact energies. Substantial progress in the application of the additivity rule has been made recently by Becker and his group (34). In general, almost all ionization curves (Fig. 6) have one common feature: they slowly rise above the background at the threshold of ionization, go through a maximum value, and then slowly fall to small values at higher electron-impact energies. The peak value of cross sections usually lies in the range of about 75 eV to 100 eV for most species. Franko and Daltabuit (35) have derived an empirical relation among the maximum value of ionization cross section, the energy at

652

ELECTRON MICROSCOPES

which it is maximum, and its ionization potential: σmax umax = const × ζ (R/I)2

(14)

where u ⫽ E0 /I, I being the ionization potential of the atom, and ␨ is the number of equivalent electrons in the outer shell. KINETIC ENERGIES OF THE FRAGMENT IONS It was first pointed out by Rapp, Englander-Golden, and Briglia (5) that molecular dissociation gives rise to energetic ions. The ions are created with energies ranging from almost 0 eV to larger values. These energetic ions can give rise to a wide variety of chemical reactions in a plasma. The information derived from the knowledge of kinetic energies of the fragment ions is important for constructing potential energy curves that perturb the stable electronic states and cause predissociation. Therefore an accurate knowledge of these energies is of fundamental importance. Studies on the kinetic energies of the fragment ions first began when Condon (36) predicted that the 30 eV energy loss in H2 was not due to the reaction H2 ⇒ H⫹ ⫹ H⫹ but due to H2 ⇒ H ⫹ H⫹ ⫹ kinetic energy. This was verified, later on, simultaneously by Bleakney (37) and Tate and Lozier (38). Obtaining accurate values of kinetic energies near the threshold of ionization is a very difficult measurement task due to low energy of ions. Therefore, these types of data are scarce. CONCLUSIONS In this article an effort has been made to familiarize the reader with various aspects of electron–atom or –molecule collisions that result in ion formation. The references provided here and references contained within these references give more detailed insight into this process. BIBLIOGRAPHY 1. S. M. Younger and T. D. Mark, in T. D. Mark and G. H. Dunn (eds.), Electron Impact Ionization, New York: Springer-Verlag, 1985, p. 1. 2. S. M. Younger and T. D. Mark, in T. D. Mark and G. H. Dunn (eds.), Electron Impact Ionization, New York: Springer-Verlag, 1985, p. 24. 3. Y.-K. Kim et al., J. Chem. Phys. 106: 1026, 1977. 4. W. Lotz, Z. Phys., 206: 205 (1967); see also Z. Phys., 232: 101, 1968. 5. D. Rapp, P. Englander-Golden, and D. D. Briglia, J. Chem. Phys., 42: 408, 1965. 6. F. J. De Heer and M. Inokuti, in T. D. Mark and G. H. Dunn (eds.), Electron Impact Ionization, New York: Springer-Verlag, 1985, p. 232. 7. J. R. Pierce, in Theory and Design of Electron Beams, 2nd ed., Princeton, NJ: Van Nostrand, 1954. 8. N. J. Mason and W. R. Newell, Meas. Sci. Technol., 1: 983, 1990. 9. K. Turvey, Eur. J. Phys., 11: 51, 1990. 10. M. A. Khakoo and S. K. Srivastava, J. Phys. E. 17: 1008, 1984. 11. G. Csanak et al., Elastic Scattering of Electrons by Molecules, in L. G. Christophorou (ed.), Electron-Molecule Interactions and Their Applications, New York: Academic Press, 1984.

12. M. I. Ramanyuk and O. B. Shpenik, Meas. Sci. Technol., 5: 239, 1994. 13. J. Perrin, C. R., 121: 1130, 1895. 14. J. Perrin, Nature, 53: 298, 1896. 15. J. Perrin, Ann. Chem. Phys., 7 (11): 503, 1897. 16. K. Fujii and S. K. Srivastava, J. Phys. B, 28: L559, 1995. 17. R. Boivin and S. K. Srivastava, J. Phys. B: At. Mol. Opt. Phys., 31: 2381, 1998. 18. M. V. V. S. Rao, I. Iga, and S. K. Srivastava, J. Geophys. Res., 100: 26421, 1995. 19. T. D. Mark, in T. D. Mark and G. H. Dunn (eds.), Electron Impact Ionization, Springer-Verlag, New York: 1985, p. 137. 20. S. K. Srivastava, A. Chutjian, and S. Trajmar, J. Chem. Phys., 63: 2659, 1975. 21. R. T. Brinkman and S. Trajmar, J. Phys. E., 14: 245, 1981. 22. S. Trajmar and D. F. Register, in I. Shimamura, and K. Takayanagi (eds.), Experimental Techniques for Cross Section Measurements in Electron Molecule Collisions, New York: Plenum, 1982. Also see J. C. Nickel et al., J. Phys. E, 22: 730, 1989. 23. S. K. Srivastava, US Patent, Apparatus and method for characterizing the mass transmission efficiency of a mass spectrometer, Patent No. 4,973,840, 1990. 24. T. R. Hays et al., J. Chem. Phys., 88: 823, 1988. 25. H. Deutsch, K. Becker, and T. D. Mark, in Proc. 20th ICPEAC, 1997, p. WE088. 26. S. K. Srivastava and O. J. Orient, in K. Prelac (ed.), Production and Neutralization of Negative Ions and Beams, New York: American Institute Physics, 1984. 27. D. R. Crosley, in G. W. F. Drake (ed.), Atomic, Molecular and Optical Physics Handbook, New York: American Institute Physics Press, 1996. 28. H. F. Winters and M. Inokuti, Phys. Rev. A 25: 420, 1982. 29. M. Goto et al., Jpn. J. Appl. Phys., 33: 3602, 1994. 30. H. M. Rosenstock et al., in Physical and Chemical Data, published by the American Chemical Society and the American Institute of Physics for the National Bureau of Standards, 1977, p. 6. 31. G. H. Wannier, Phys. Rev., 90: 817, 1953. 32. H. J. Grosse and K. H. Bothe, Z. Naturforsch., A23, 1583, 1968. 33. O. J. Orient and S. K. Srivastava, J. Phys. B: 20: 3923, 1987. 34. H. Deutch, K. Becker, and T. D. Mark, Int. J. Mass Spectrum. Ion Proc., 167/168: 503, 1997. 35. J. Franko and E. Daltabuilt, Rev. Mex. Fis., 27: 475, 1978. 36. E. U. Condon, Phys. Rev., 35: 1180, 1930. 37. W. Bleakney, Phys. Rev., 35: 658, 1930. 38. J. T. Tate and W. W. Lozier, Phys. Rev., 39: 254, 1932.

S. K. SRIVASTAVA California Institute of Technology

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Wiley Encyclopedia of Electrical and Electronics Engineering Electrorheology Standard Article Liang Fu1 and Lorenzo Resca1 1Catholic University of America, Washington, DC Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5903 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (235K)

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Abstract The sections in this article are Composition and Properties of ER Fluids Basic Requirements for Engineering and Commercial Applications of ER Fluids Functioning and Design of Basic ER Devices Newtonian Fluids and Elastic Solids Non-Newtonian Fluids ER Fluid Rheology ER Fluid Rheometry Theoretical Models and Simulations of ER Phenomena Theory and Computation of Electrical Interactions in ER Fluids About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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ELECTRORHEOLOGY

ELECTRORHEOLOGY Rheology is the science of the deformation and flow of matter. Electrorheology is concerned mostly with the electrorheological (ER) effect, which consists of a sudden change of apparent viscosity in a suspension of particles induced by an electric field. The first applications of this effect were patented in 1947 by Winslow after several years of detailed studies and observations (1,2). Hence, it is also named the Winslow effect. Winslow called his slurries electroviscous fluids, because electroviscous effects had been previously reported in other systems, although on a much smaller scale (3–5). In literature outside the US and Britain, the electroviscous term is sometimes still used as a synonym of electroheological, although this is not strictly correct from a microscopic point of view. In Russian literature, the ER effect is sometimes named the Quincke effect. A brief review of the early history and references on ER related phenomena can be found in Ref. 6. Although even some homogeneous liquids exhibit weak electroviscous effects (3–6), only concentrated dispersions of solid particles in dielectric liquids provide ER effects strong enough to be of technological interest. Large viscosity changes require the application of strong electric fields, of the order of kilovolts per millimeter (kV/mm). A significant increase in viscosity takes only milliseconds and is caused by the tendency of the micrometer-size suspended particles to form fibrous structures in the directions of the applied electric field. The individual particles become polarized, attract one another, and form chains, which bundle into filaments, spanning the gap between the electrodes. The strength of these filaments enables the ER fluid to sustain stress. Eventually, the liquid suspension turns completely into a viscoelastic gellike solid. To make this structure flow again, while maintaining the electric field, requires applying shear stress, which breaks the filaments. The minimum shear stress needed to cause flow, called yield stress, is typically proportional to the magnitude squared of the applied electric field. The amount of force or torque that an ER device transmits is related to the yield stress, whose effect persists as increased viscosity even when the shear stress exceeds the yield stress and flow is forced. The ER effect is fully and even more rapidly reversible. When the applied electric field is turned off, particle polarization and the corresponding attraction imme-

diately disappear, and the solidified or highly viscous fluid reverts almost instantly to the original low-viscosity liquid. The response of an ER fluid is completely reversible and much faster than that of conventional mechanical systems. Hence, it offers great potential for high technology applications to engine mounts (actively controlling engine shake) and other vibration isolators, shock absorbers and other tunable dampers, nonslip fluid clutches and variable-differential transmissions, brakes, valves without moving parts, flow pumps, and various other fluidic control devices. The ER effect typically provides better control of fluid flow and power transmission and reduces the energy loss and damage due to vibrations (for example, through optimal control of a vehicle’s vertical motion). In spite of that, commerical use of ER devices is still limited, in part because of the performance of currently available ER fluids. For example, the maximum strength of the yield stress seldom exceeds 5 kPa, which is insufficient for many applications. The corresponding need for large electric fields also requires that ER fluids be poor conductors of electricity, or power consumption and heat dissipation become excessive. This is a major problem for aqueous suspensions, most notably for dc applications. On the other hand, dispersed particles in low-conductivity liquids have a greater tendency to settle under gravitational and centrifugal fields (because of a lack of mobile charges that readily attach to particle surfaces and stabilize them against settling). Furthermore, for automotive use, ER fluids should remain stable and reliable for prolonged use and should sustain large temperature variations (from ⫺50 to 150⬚C), chemical action, mechanical attrition, and wear. A possible way to eliminate sedimentation is to match the densities of the particles and the medium. This is difficult for high-density particles, such as silica, but it may be achieved either by making insulating oils heavier by halogenation or by switching to lighter polymeric particles. Different thermal expansion coefficients between particles and medium typically allow a density match only at given temperatures, rather than over a wide temperature range (densities and thermal expansion coefficients can be matched only with special ternary liquid mixtures). In the absence of electrical charges or protective agents, particles generally tend to attract when close to each other (because of van der Waals forces) and may irreversibly coagulate when settling. Charge stabilization keeps the particles apart, but this typically requires media with unacceptably high conductivity, such as water. Steric stabilization provides substantial protection against cohesion through absorption or bonding of additives soluble in the fluid on the particle surfaces, such as surfactants or stabilizing polymer fragments. However, some surfactants act properly as stabilizers in the absence of electric fields but fail when high fields are applied (by allowing charge transfer between particles). Furthermore, surfactants are not readily available for anhydrous systems. On the other hand, block or graft copolymers appear more promising and versatile because their structures and chain lengths can be varied to sufficiently stabilize the particles, even when they are made of surfactant compounds (8). When stabilizing polymer fragments are relatively short, weak aggregates may still form, but they may be easily redispersed by stirring. The degree of reversible aggregation which can be tolerated depends on the ER application.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

ELECTRORHEOLOGY

Brownian motion counters settling, but its effect is significant only for small submicron particles. Such size is sometime taken to define rather restrictively the range of colloidal suspensions. However, reduction in particle size decreases the yield stress and increases the response time. In fact, when effective, random Brownian forces disrupt alignment of particles into chains and filaments, thus weakening the ER effect. Given that, it is generally preferable to use particles large enough (앑10 애m) that Brownian motion is negligible. Particles of such dimensions in dilute suspensions are visible under a high-power optical microscope, that allows direct observation of their shape and size distribution. Filamentation induced by a strong electric field is discernible in such conditions (9–15). However, concentrated ER fluids (with particulate volume fractions between 0.2 and 0.6) are generally opaque. Then, the degree of aggregation at zero field or the structures that form at high fields are usually inferred from rheological measurements. Other electro-optical techniques can be used to determine directly the morphology of ER systems at light wavelengths where they are transparent. For example, Chen et al. (16) studied an ER suspension of relatively large diameter glass spheres (about 20 or 40 애m) in a silicone oil by using the transmitted pattern of a laser beam. They found that the ER solid formed under the applied electric field has a columnar body-centered tetragonal structure. In a similar experiment, Martin et al. used a different optical configuration and much smaller spheres (about 0.7 애m in diameter) in an index-matched medium to further study the dynamics of chain formation and their aggregation into columns (17). Various models which attempt to interpret the results of these experiments and other basic ER phenomena are described in two popular articles (18,19). There are other systems analogous to ER fluids. Materials that are permanently ER solids (i.e., they do not flow under strain even at zero field) consist of particles frozen in elastomeric media. Application of an electric field increases the shear modulus of these composites. Various other electroviscous effects in sols, emulsions, and electrolyte systems are caused by internal electric fields produced by charged particles and free ions (20). Most notably, magnetorheological (MR) fluids consist of suspensions of ferromagnetic particles exhibiting under an applied magnetic field rheological changes analogous to ER fluids. In practical applications, MR fluids offer several advantages over ER fluids (21). For example, they are not limited by conduction or dielectric breakdown. Hence, conducting liquids, such a water, are not excluded. Purity is also not a problem. On the other hand, MR fluids have much longer response times than ER fluids, relax much more slowly, and retain residual magnetization. MR fluids also require typically larger and heavier power sources (magnets). Despite the great research and technological interest that ferrofluids currently enjoy, lack of space prevents any adequate treatment of this topic in this article. A general discussion can be found in Ref. 22. Recent advances are reported, for example, in Refs. 23 and 24.

COMPOSITION AND PROPERTIES OF ER FLUIDS The ER effect was originally discovered in suspensions of starch and silica gel particles in natural, mineral, or lubricating oils (1,2). Many ER fluids have been based on similar par-

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ticulates (such as cellulose, dextran, alginic acid, resins, metal oxides, clay minerals, talcum powder, gypsum) and dispersants (such as gasoline, kerosene, toluene). However, a significant amount of water (between 5% and 30%) is necessary for these dispersions to exhibit any appreciable ER effect. It is believed that water, repelled by the hydrophobic suspending liquid, collects on the surface of the hygroscopic particles. Dissolved salts in this envelope provide mobile ions that polarize in the (local) electric field. This substantially augments the particle’s intrinsic polarization, even though the aqueous envelope may be very thin. Unfortunately, the presence of such conducting water increases power consumption and heat dissipation. It further makes the ER properties sensitive to moisture content, hence, subject to chemical, electrochemical, and thermal degradation. A second generation of less abrasive ER fluids based on softer polymer particles has been developed since the late 1970s, but most of these suspensions still require water to produce substantial ER activity (12,25). Components frequently added to ER fluids to increase the ER effect are called activators. They presumably work by augmenting the intrinsic polarizability of the particles, although their effectiveness is not simply related to the resulting permittivities. The role of water as an activator has already been mentioned. Other additives that work by similar mechanisms are various alcohols, carbonates, and glycols. Although some of them are less volatile and less reactive than water, most of them are substantially less effective than water and also lead to undesirable conductivity and dissipation. Surfactants also play some role as activators, presumably through ionic conduction within the surfactant layer. The inherent relationship between charge mobility and ER activity has been demonstrated quantitatively in photoelectrorheological fluids, based on photoconducting particles, such as phenothiazine. These fluids have greatly enhanced ER activity when they are exposed to light of the proper frequency, which greatly increases the number of free electronic charge carriers (26, 27). Since the mid 1980s, a third generation of substantially anhydrous ER fluids, inherently more stable and effective at higher temperatures, has been introduced. The particles are made of polymers or semiconductors (28), or have a composite structure, with a conducting core and a chemically formed insulating thin-skin layer, which is needed to prevent electrical conduction between particles in contact in chains and filaments (29,30). Particle encapsulation may also serve to reduce particle attrition and chemical reactivity with the environment. Similarly, the particles may have a light polymer core and a double layer, consisting of an inner conducting shell and an outer insulating film (31,32). Carbonaceous particles and optically anisotropic spherules with insulating layers have also been used (33). Particles made of aluminosilicates dispersed in paraffin oil provide improved ER fluids substantially free of water (34). In this case, the polarization augmentation is produced by movement of intrinsic metal cations through cavities and channels within the highly porous zeolite particles, rather than by extrinsic ions of electrolytes typically dissociated by water (35). Polyelectrolyte particles, which commonly require water to function as ER fluids, presumably by dissociating the cations from the macroions, also function with greatly reduced amounts of water, if cations can

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move within the confines of a chain coil (27, 36, 37). ER fluids completely free of water have also been reported (38). Even truly homogeneous polymer solutions, such as poly (웂-benzyl-L-glutamate) in various solvents, are ER active. This highlights the great versatility and ubiquity of the ER effect and the variety and complexity of the microscopic mechanisms that produce it (27). In addition to the removal of water, the problems of settling and abrasion are eliminated with homogeneous solutions because there is no dispersed phase. However, some solvents are polar and cause high currents and power consumption, and some have disadvantages of greater toxicity, aggressiveness, and limited operating temperatures. These solutions may be in a liquid-crystal state, which may also contribute to their ER activity. Liquid crystals are generally more suitable for magnetorheological applications, but understanding the mechanisms of the ER effect in various types of liquid crystals is nonetheless important (39). Their ER activity results both from the high polarizability of polymer molecules in solution and from the alignment of their large permanent dipole moments with the electric field because the alignment of long rigid-rod polymer molecules perpendicular to the flow increases shear viscosity (40). Currently, the most common ER fluids in commercial use are silicas and zeolites in silicone oils, which are preferable to mineral oils because they are usable over a broader temperature range. In research, new materials are constantly produced and investigated for ER effects, e.g., fullerenes (41). A survey of the patent literature for material composition, that is, particles, dispersants, surfactants, and activators, of available ER fluids is provided in Table 2 of Ref. 6. A complete list of ER fluid and device patents issued up to 1991 is provided in Sec. 7.2 of Ref. 42. An earlier survey of ER fluids can be found in Table 1 of a review article by Block and Kelly (43). An update of that and a list of polymers used for the particulate dispersed phase are provided, respectively, in Tables 1 and 2 of Ref. 13. A chronology of ER fluid and device development is included in Ref. 44. BASIC REQUIREMENTS FOR ENGINEERING AND COMMERCIAL APPLICATIONS OF ER FLUIDS Given the advanced level of modern solid-state electronics and control circuitry, the development of reliable and inexpensive power supplies and computerized control systems for ER devices should not be an impediment to ER technology. However, low-cost commercial power/control systems that permit rapid (millisecond) switching at high voltages are unavailable. The design of specific devices is also not yet sufficiently refined to take full advantage of the unique characteristics of ER fluids, thus causing less than optimal performance in many applications. Nonetheless, the major obstacle to the development of ER technology seems to be the current unavailability of ER fluids capable of satisfying the demands for strongly competitive and commercializable units. Improvements of ER fluids for technological applications are needed in the following areas, roughly in order of their critical importance and corresponding difficulty to attain (see also Refs. 13 and 42): 1. High strength (yield stress and especially shear stress under flow conditions), at relatively lower applied fields

2. Wide operating temperature range (set by mechanical energy dissipation or environment, such as in shock absorbers or engine mounts) 3. Low electrical power requirement and consumption, hence, low conductivity (which is harder to maintain at higher temperatures and for faster responding ER fluids) 4. Stability (overcoming settling of particles, particularly when the fluid stays inactivated in the gravitational field for extended periods or when used repeatedly in centrifugal fields) 5. Resistance to thermal, chemical, and mechanical degradation in prolonged use 6. Zero-field viscosity less than a poise (to minimize drag and unwanted mechanical energy dissipation). 7. High dielectric breakdown strength (which depends on the microscopic composition and configuration in any state of the system and limits the maximum electric field that can be applied) 8. No contaminants (which may strongly affect, for example, conductivity and dielectric breakdown strength through ionic dissociation) 9. No electrophoresis, no dynamic separation, and low volatility 10. No abrasive, corrosive, electrolytic, or oxidative degradation (over repetitive cycle duties) of any component of the fluid, electrodes, seals, and containers 11. Fast response time (adequate for most applications, but still limiting high-frequency ac fields) 12. Nontoxic, nonhazardous, nonflammable, disposable substances No single ER fluid obviously meets all of these requirements for all possible applications. Each individual ER device has its own specifications, which leads to selecting a particular ER fluid. An overall measure of merit for an ER fluid is given by the Winslow number, which is the ratio between the power density inherent in the ER effect (defined as the ratio between the square of the yield stress and the zero-field viscosity) and the power density consumption (which is the product of the electric field times the current density). Current ER fluids achieve Winslow numbers over 103 but only within narrow temperature ranges. At higher temperatures, the power density consumption rises too much, and at lower temperatures the ER inherent power density becomes too low (45). The variety of requirements for ER fluids and devices is easily illustrated by automotive applications. For instance, anhydrous ER fluids are difficult to shield from aqueous contamination in shock absorbers exposed to harsh environments. Engine mounts require ER fluids that retain stability over wide frequency and temperature ranges, but only a moderate yield stress (1 to 2 kPa) is needed. Conversely, clutches and most other similar applications require a substantially higher yield stress. One possibility of attaining the latter is to manufacture spheroidal or elongated particles and optimize their aspect ratio, that is, the ratio between principal axes. This has been suggested by simple dipolar estimates and demonstrated more convincingly by experiment (14). Unfortunately, the increase in shear strength with aspect ratio is mostly static and disappears with increasing shear rate

ELECTRORHEOLOGY

(46), under which clutches must also operate. Furthermore, longer response times are definitely involved, because elongated particles must rotate against their inertia and friction into alignment with the electric field. However, a somewhat slower ER response may be tolerable and even desirable for clutches to reduce mechanical shock. For fluidic control systems there are two important concerns: (1) Filtration phenomena associated with pressuredriven flow in an ER valve may generate a difference between solvent and particle velocities, hence, cause dynamic separation. (2) Instabilities may arise in the flow of typically stable ER suspensions under electric fields that vary more rapidly in space and/or time. Finally, safety, environmental, and cost considerations must also be taken into account because they restrict the applications of materials otherwise particularly ER effective. For instance, halogenated hydrocarbons are ideal dispersants for stability, density and cost, but are toxic, difficult to contain, and hard to dispose of. Silicone fluids are not toxic, but they are not biodegradable, and even traces of silicone contaminants ruin the appearance of surfaces repainted in autobody shops. A safety concern is the high voltage required in ER applications. Thus, ideal fluids should operate with very low current densities (below 1 애A/cm2), so that current-limiting circuitry can be applied to protect against accidental shock to users. Unfortunately, current ER fluids cannot carry low current density and high yield stress at the same time (except, possibly, dry aluminosilicate particulate systems). For profitable commercial products, ER active dampers on engines are the best candidates, followed by shock absorbers based on similar concepts (but more demanding on ER fluid performance). The feasibility of ER clutches has been demonstrated in various prototypes (by Lord Corporation and various other centers), but competitive mass production of an automotive ER clutch of reasonable dimensions, low power consumption, and low cost still requires considerable torque enhancement (i.e., a yield stress up to about 20 kPa) and some zero-field viscosity reduction (to minimize open-clutch drag) in the development of ER fluids. On the other hand, low-torque applications, such as automotive alternators, air conditioners, and other accessory drive clutches are already viable and advantageous because they permit full decoupling when the accessory is not in use. Continuously variable ER transmissions are of interest because they would allow maintaining engine speed at an optimal value, producing major energy savings. However, this is a future application, because it requires at least an order of magnitude improvement in ER fluids. Current yield stresses are much too low to transmit the heavy-duty torques required, and a design to accommodate much larger multiplate coupling areas is not feasible (the corresponding power consumption with current ER fluids would be too high, anyway). Although the automotive industry is the most likely candidate for large-scale applications of ER fluids, one must also note the increasing involvement of the military industry, particularly in developing ER isolation systems for weapons, radar, and fuel tanks in helicopters and aircraft. Innovative use of ER technology is possible for many other feedback control systems and smart actuators with the most disparate purposes and complexities, such as variable-resistance exercise equipment (e.g., rowing simulators), rheoelectric motors, tracking devices for copying machines, ER clamps to fix work-

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pieces, joints and artificial muscles for robotic arms and prosthetic limbs, penile implants, reconfigurable Braille displays, and so on. ER fluid and solid materials also function in adaptive structures. For example, they could be included in highstress sections as active structural members of bridges, buildings and plants, to control and damp their dynamic responses to windstorms and earthquakes. Likewise, modulation of stiffness with speed in airfoils, helicopter blades, and hydrofoils could be optimally controlled with ER systems. Still other applications may exploit the effective thermal conductivity or convective heat transfer of ER fluids (which also increase with the magnitude squared of the applied electric field) for heat exchange systems, acoustic properties for designing ER filters, delay lines, and ultrasound finders, and optical properties for color films. Further reviews of ER fluid applications are provided, for example, in Refs. 42 and 47. Technical papers on ER devices are periodically presented in various ER conferences (23,24,48–54).

FUNCTIONING AND DESIGN OF BASIC ER DEVICES The functioning and design of basic ER devices are easy to understand, in principle. For example, an ER torque-transmitting device may consist simply of an inner cylindrical rotor mounted on a driving shaft, an outer cylinder coaxial to the rotor and mounted on bearings, and an ER fluid in between. Applying a potential difference between the rotor and the outer cylinder induces the ER effect in the fluid, which thus provides the rotational coupling. To increase the coupling area, the torque-transmitting device may alternatively consist of parallel coaxial (multiplate) disks rotating relatively to one another. See, for instance, Fig. 7 in Ref. 47. An ER torque-transmitting device that has either coaxial cylinders or rotating disks can function as an automotive clutch. In the former configuration, the car engine is connected to the outer cylinder, and the drive shaft powers the wheels through a reduction gear. The inertia of the outer cylinder hardly matters, given the high inertia of the driving motor, which runs at fairly constant high speed. On the other hand, the inner rotor must have low inertia to provide a fast response. That can be attained by using a lightweight nonmetallic material, coated with a conducting thin film to provide the electrical contact. When solidified, the ER fluid forces the drive shaft to rotate in connection with the engine. When liquified, the ER fluid allows the engine to disengage from the drive shaft and spin freely, as if in neutral gear. The almost instantaneous and continuously variable response of the ER fluid to the applied potential provides a prompter and smoother control than a conventional mechanical clutch. Clearly, the nonslip fluid clutch also has fewer parts to wear or fail. Attaching the outer cylinder to the car body, the same ER coupling device can function as a brake. Relative motion of the cylinders can be gradually stopped with optimal antilock feedback by rapidly modulating the applied voltage with a computerized control system. Heating must be removed by a cooling circulation system, or stability problems may arise in the ER fluid. More complex electrohydraulic brake actuators can be designed; see, for instance, Fig. 24.7 in Ref. 20.

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The same torque-transmitting device may also serve as an ER damper. Keeping the outer cylinder fixed, the torsional vibrations of the inner rotor can be damped out by controlling viscosity changes in the ER fluid. Alternatively, a design more similar to ordinary shock absorbers can be used. An ordinary shock absorber consists of a cylinder with a sliding piston inside, which pumps a viscous oil out of a small orifice when pushed in by an outside impact. The oil absorbs and dissipates the impact energy through its slow viscous flow and flows back more quickly into the cylinder through a large orifice after the shock, when the piston rebounds. However, the oil viscosity changes with temperature, which causes the shock absorber to perform unevenly depending on the weather and the road conditions. Repeated compression on a long bumpy ride may heat up the oil so much that it becomes very thin, and the shock absorber may soften and fail when needed most. The problem is best addressed by transforming such a passive device into an active one by introducing a fast electromechanical valve, which adaptively adjusts the size of a single orifice in response to the piston movement. Without moving parts, an ER valve provides a simpler and even better solution. The microprocessor that senses the piston motion modulates the applied voltage rapidly, which instantly thickens the ER fluid in midstrokes for maximal damping and thins it again immediately afterward for quickest flow-back. In this fixed-plate valve configuration, the piston forces the ER fluid to flow through a stationary annular duct, and the voltage is applied across this duct. In an alternative sliding-plate configuration, the piston itself acts as the grounded electrode, and the damping force originates from the controlled shear resistance between the piston’s cylindrical surface and an adjacent motionless surface, which acts as the other electrode; see Fig. 8 in Ref. 47. High shear stress (over 5 kPa) at high shear rates of the ER fluid is necessary for optimal functioning of ER shock absorbers. Prototypes have been larger than desirable to accommodate the necessary ER valve surface area. However, newly developed ER dampers within a conventional package size have been recently tested on a Ford Thunderbird and performed better than the standard suspensions of the SC and LX models, at least when operated within the optimal temperature range of the ER fluid (55). The basic design concept of an ER valve is simply that of an ER fluid flowing through a thin section of a pipe. At zero field, the corresponding low viscosity causes only a small drop in pressure between the beginning and the end of the pipe section (at least at low flow rates). When a potential difference is applied between two sides of the section, the viscosity increases, causing controllable resistance to the flow and a correspondingly large pressure drop. More complex additions of diaphragm seals may permit isolating the ER fluid from other hydraulic fluids. ER valves of various design can be used in a host of devices and most effectively in engine mounting. In an automobile, that must satisfy two contrasting demands. On the one hand, it must insulate the passenger compartment from engine vibrations and noise, which requires low dynamic vertical stiffness and low damping at high frequencies (앑100 Hz), to leave alone as much as possible the engine’s small-amplitude forced oscillations. On the other hand, the mounting must protect the engine from jolts caused by the pavement, which requires

high stiffness and high damping at the natural low frequency (앑10 Hz) of the engine-mount system, to prevent the persistence of large-amplitude excursions transmitted to the engine. Current fluid-filled passive mounts made of rubber and metal partially resolve this contradiction by adopting a twochamber design. When road shake occurs, the fluid flows back and forth between the chambers through an orifice or a short tube, dimensioned to provide substantial damping only at the troublesome low frequency. With the aid of a fluid decoupler, the mount can be further tuned to suppress two unwanted low frequencies or a narrow frequency range. At high frequencies, fluid inertia in the tube is carefully exploited to reduce the damping flow. However, that cannot be totally prevented and compromises noise isolation. Such a passive mount can be made active be adjusting damping in response to a sensor of engine motion. This can be achieved by adding a fast electromechanical valve designed to adaptively reduce the orifice at low-frequency jolts, while leaving it large and inactive at high-frequency constant engine vibrations. Such a design works effectively, but the valve mechanism itself adds noise, weight, complexity, and cost to the system. A better solution consists of replacing the hydraulic system (fluid and orifice) of the passive mount with an ER valve separating the two chambers, now filled with an ER fluid; see, for instance, Fig. 6 in Ref. 47. When transiently activated in response to a low-frequency jolt, the ER fluid quickly provides the high viscosity needed to damp it. Then, it immediately returns to zero-field low viscosity, providing good isolation to high-frequency constant engine vibrations (56). Because the ER valve has no moving parts, this allows for a simpler, quieter, less expensive, and faster design. Flow volumes and shear rates are relatively low, and a yield stress of only 1 to 2 kPa is needed for this type of application. Hence, ER fluid-filled engine mounts are already practical and advantageous for commercial use. Other simple ER devices (ER valve in a reciprocating piston system, safety valve, dielectric suspension pump, pulsating pressure generator, pulverizer, filter, settler) are sketched in Sec. 24.11 of Ref. 20. Illustrated therein are also membrane transducer devices (grinding-polishing tools, peristaltic pump) based on electrodilatancy, that is, the ability of certain ER systems to undergo rapid and reversible volumetric expansion in response to an electric field. More detailed descriptions and critical assessments of these and more refined ER applications are provided, for example, in Refs. 44, 47, and 57. NEWTONIAN FLUIDS AND ELASTIC SOLIDS The simplest model of a fluid (like water or air) is that of an isotropic continuous medium. An ideal fluid is further assumed to be nonviscous, hence, unable to sustain shear stress under either static or dynamic conditions. Then Newton’s second law for a fluid element of mass dm ⫽ ␳(x, t)dV is given by ρ

v dv = ρ f − ∇P dt

(1)

where ␳ and v are the element density and velocity, and P and f are the pressure and the applied force per unit mass acting on the fluid element. All of these quantities are func-

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tions of position x and time t, viewed as independent variables (Eulerian description). However, d/dt represents the total derivative d ∂ = +v ·∇ dt ∂t

∂ρ +∇· j =0 ∂t

shear flow shown in Fig. 1, where the fluid velocity increases with height. Take a horizontal plane y ⫽ y0 of constant flow velocity. The drag force Fx exerted on the fluid below the area element dA is given by

(2)

which provides the rate of change following the fluid element in its motion (Lagrangian description). Conservation of matter independently requires the continuity equation

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Fx = η

∂vx dA ∂y

(6)

where ␩ is the viscosity, measured in Pa ⭈ s [we generally adopt the International System of Units (SI) in this article]. In general, given an area element dA oriented with an outward normal unit vector n, the stress tensor is defined as

(3)

Fi =

3


Tij n j dA

(7)

j=1

where j ⫽ ␳v represents the mass current density. Conservation of linear momentum implies 3
∂Tij ∂ (ρvi ) = ρ f i + ∂t ∂x j j=1

where Fi are the components of the total force exerted by the outside fluid on dA. Then, in the configuration of Fig. 1,

(4)

where Tij is the stress tensor. For an ideal fluid, Tijid = −Pδij − ρvi v j

(5)

Note that ␳vivj represents convective transport of momentum (per unit area per unit time) and corresponds to the convective part of Eqs. (1) and (2). In fact, as physically expected, Eq. (1) and Eqs. (4–5) are completely equivalent, if Eq. (3) is satisfied. The vectorial dynamical Eq. (1) and the scalar continuity Eq. (3) provide four relationships for five unknown functions: v, P, and ␳. Thermodynamics may provide an additional equation of state, relating P to ␳ and entropy per unit mass in the energy representation, or P to ␳ and temperature in the freeenergy representation. Nonviscous flow is typically nondissipative, hence, isentropic, which leads to an additional equation representing conservation of kinetic plus internal energy. Equivalently, flow at constant entropy allows relating P just to ␳ through the thermodynamic equation of state. In principle, this yields a unique solution for all five unknown functions, v, P, and ␳, given appropriate boundary conditions. In practice, solution of the hydrodynamic equations (even for ideal fluids) may be very difficult, because of the intrinsic nonlinearity in the fluid velocity field, originating from the convective transport of momentum. However, considerable simplification occurs in the special but important case of isentropic irrotational flow, which leads to Bernoulli’s theorem. Full discussion and detailed derivation of all these standard results are readily found in many textbooks, for example, Chap. 9 of Ref. 58, or Chap. 1 of Ref. 59. Viscous fluids are much more complicated than ideal fluids because they sustain shear stress under dynamic conditions and involve dissipation. One can still retain Eq. (4) representing momentum conservation, provided that a viscous component is added to the stress tensor. An isotropic classical fluid which obeys a linear relationship, first postulated by Newton, between shear stress and shear rate is called a Newtonian fluid. Consider the basic configuration of steady laminar

τ = Txy = η

∂vx = ηγ˙ ∂y

(8)

where ␶ is the corresponding shear stress component and 웂˙ is the shear rate of strain. Conservation of angular momentum requires that the stress tensor be symmetrical. Assuming that its viscous part Tijv is still linear in the velocity gradients for the most general flow configuration, it follows that



Tijv

 ∂v j ∂vi 2 =η + − δij ∇ · v + ζ δij (∇ · v ) ∂x j ∂xi 3

(9)

where ␨ is called the bulk viscosity. For incompressible flow, ⵜ ⭈ v ⫽ 0, and ␨ plays no role, whereas for compressible flow, it contributes only to the diagonal compressive-tensile components of the stress tensor. Introducing the complete stress tensor Tij = Tijid + Tijv

(10)

of a Newtonian viscous fluid into the momentum-conservation Eq. (4), with simple manipulations (again involving the conti-

y vx (y) ^ n vx (y0) dA y

∂ vx Fx = η dA ∂y x

y0 z Figure 1. Viscous drag force exerted by the fluid above the y ⫽ y0 surface on the fluid below for steady laminar shear flow (58).

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nuity equation),

and pours like whipping cream if the glass is tilted slowly. However, it immediately hardens upon stirring or shaking. v ∂v 1 2 v = ρ f − ∇P + η∇ v + + (vv · ∇)v η + ζ ∇(∇ · v ) (11) Furthermore, once hardened, the paste occupies a larger volρ ∂t 3 ume, looks drier, and returns to its initial condition only slowly. The combination of these effects is variously called which clearly reduces to Eq. (1) for an ideal fluid. For incom- dilatancy, rheopexy, or antithixotropy. Its opposite, namely, a pressible flow, the last term in Eq. (11) vanishes, and the cele- shear-thinning substance that recovers its full rigidity only brated Navier–Stokes equation is obtained. long after shearing has stopped, is called thixotropic. BentonAgain a unique solution requires an additional energy bal- ite gel is a classical example. Thixotropy makes paint easier ance equation, which is more involved than that for a nonvis- to apply and mud easier to drill. cous fluid. Now there are entropy changes caused by intrinsic Various suspensions and polymeric liquids often exhibit viscous dissipation and possibly heat conduction or heat both shear-thinning and shear-thickening behavior. Shear sources in the fluid. A full discussion and derivation of that thinning occurs at modest shear rates, followed abruptly by and of all the basic results for Newtonian viscous fluids can shear thickening at greater shear rates. In general, any matebe found, for example, in Chap. 12 of Ref. 58, or in Chap. 2 of rial that obeys an essentially nonlinear constitutive relation Ref. 59. between shear stress and shear rate is called a non-NewtonUnlike an ordinary fluid, an elastic solid can sustain shear ian fluid. Liquids that obey quantum mechanical, rather than stress in a static configuration. Then we have a linear rela- classical laws, such as liquid helium, are also called non-Newtionship between the stress tensor and the elastic strain ten- tonian: we are not concerned with such systems at all in this sor. In the isotropic continuum model, article. Non-Newtonian fluids may be anisotropic and obey tensorial constitutive relations, requiring definition of several   ∂u j ∂ui 2 (shear-rate dependent) viscosity coefficients. Typically, nonTij = G + − δij ∇ · u + Kδij (∇ · u ) (12) Newtonian fluids possess long relaxation times, compared to ∂x j ∂xi 3 the inverse of the velocity gradients that can be applied. where u is the local deformation and G and K are elastic con- Hence, constitutive relations may not depend only on the instants, called, respectively, the modulus of rigidity and the stantaneous shear rate but rather on its integration over the bulk modulus. Because the rate of deformation v(x, t) is specimen’s history. Thixotropy obviously depends on that. Ac⭸/⭸tu(x, t), Eq. (12) for an elastic isotropic solid corresponds tually, for most non-Newtonian fluids, constitutive relations to Eq. (9) for a Newtonian viscous fluid. In the shear configu- are unknown, depend on other poorly defined functions, or are ration corresponding to that of Fig. 1 for a viscous fluid, the too empirical and particular to provide much insight. In any such case, real progress in understanding can be made only shear stress ␶ is related to the shear strain component 웂 by through microscopic investigation or simulation of the structure and properties of the particular system under consider∂ux = Gγ (13) ation. τ = Txy = G ∂y The range and study of non-Newtonian fluids is vast and which corresponds to Eq. (8) for a viscous fluid. So, G is also complex and occupies most of modern rheology. In the following, we only consider a few basic aspects regarding ER suscalled the shear modulus. These observations are all that we need to introduce non- pensions. Further background in this important and fascinatNewtonian fluids, which often exhibit properties hybrid be- ing field can be obtained, for example, from Refs. 61–63. tween solids and liquids. A more adequate treatment of elastic continua can be found, for example, in Chap. 13 of Ref. 58, or in Ref. 60. ER FLUID RHEOLOGY 







NON-NEWTONIAN FLUIDS Despite their fundamental and practical importance, Newtonian fluids represent only a small class of all substances that flow under certain conditions (as opposed to solids). For all such substances, one may still define an apparent viscosity as the ratio between shear stress and shear rate, but that varies typically with shear rate. For instance, the apparent viscosity may be large or even infinite at a low or vanishing shear rate, but rapidly decreasing at higher shear rates. Common substances, such as yogurt, mayonnaise, white of egg, condensed milk, toothpaste, shampoo, foams, and chewing gum, display such behavior, called shear-thinning: they hardly flow under their own weight but flow readily when squeezed. Other substances do the opposite, becoming more viscous rather than less viscous at larger shear rates. This behavior, called shear thickening, can be simply demonstrated by mixing powdered cornstarch with water in a glass. The suspension easily flows

Most ER suspensions behave essentially as Newtonian fluids in the absence of any applied electric field. Their viscosity is only a few times larger than that of the dispersant fluid and is adequately described by relatively simple models and equations. In fact, the viscosity of a low-concentration suspension of N spherical particles of radius a in a volume V was first calculated by Einstein in 1906. It is just the viscosity of the dispersant times (1 ⫹ 2.5v), where v ⫽ 4앟Na3 /3V is the volume fraction of the spheres; see Sec. 22, Chap. 2, of Ref. 59. Appropriate generalizations for higher concentrations and/or nonspherical particles are reported, for example, in Refs. 62 and 64. When an electric field is applied, the overall behavior of most ER fluids resembles that of a field-induced shear-thinning Bingham plastic. The corresponding constitutive relation is τ (E) = τy (E) + η∞ γ˙

(14)

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where ␶y(E) is the yield stress, essentially proportional to the electric field’s magnitude squared, and ␩앝 is called the highshear limiting or plastic viscosity. In this basic model, ␩앝 is constant (at any given temperature) and coincides with the differential viscosity d␶ /d웂˙ . The plastic viscosity also coincides with the zero-field viscosity. Typically, both the yield stress and the zero-field viscosity increase with particle concentration, which determines an optimal concentration to maximize the ER effect and minimize the zero-field viscosity for any specific application. On the other hand, the effective or apparent viscosity η(E, γ˙ ) =

τy (E) τ (E) = + η∞ γ˙ γ˙

(15)

depends on both the electric field and the shear rate. It diverges at low shear rate (for nonzero field), and approaches ␩앝 at high shear rates. The parallel lines A and B in Fig. 2 represent Eq. (14) at nonzero and zero field, respectively. A more accurate description of ER behavior requires considerable corrections to the basic Bingham plastic model, particularly at low shear rates. A typical situation is shown as curve C in Fig. 2. Applying an electric field at zero shear rate, a viscoelastic solid forms. The corresponding static yield stress ␶sy(E) is defined operatively as the shear stress necessary to initiate flow from solid state equilibrium, regardless of whether the Bingham plastic model fits the ER fluid behavior at later shearing conditions. After ␶sy(E) is applied and flow is established, the shear stress immediately drops with increasing shear rate and quickly approaches curve A predicted by Eq. (14). Having attained that, if the shear rate is reversed all the way back to zero, the shear stress remains close to curve A. Therefore, ␶y(E) is called the Bingham or dynamic yield stress and is defined as the zero-shear-rate intercept of the linear Bingham plastic model, which generally provides a good fit at sufficiently high shear rates. Now, it takes only a short time before ␶sy(E) is fully recovered under static conditions. Hence, thixotropy is usually limited (barring structural alterations after prolonged shearing). However, what causes the usually positive difference between the static and the dynamic yield stress, an effect called stiction, is not well established. Further deviations and complications may occur. For example, the plastic viscosity may depend to some extent on the shear rate (typically shear thinning). Inclusion of this depen-

τ

τ sy(E2)

C C

τ y(E2) τ y(E1)

(E2 > E1)

A

(E1 > 0)

D

(E = 0)

C B

. γ

Figure 2. Shear stress vs. shear rate. Curve A: Bingham plastic model (nonzero field). Curve B: zero-field Newtonian fluid. Curve C: typical ER behavior. Curve D: pseudoplastic behavior.

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dence in Eq. (14) generalizes the Bingham model to that of a viscoplastic fluid. The Herschel-Bulkley model is a particular case, which adds an exponent n to 웂˙ (and renames its coefficient) in Eq. (14). The defining feature of all such models remains the presence of a finite dynamic yield stress. On the other hand, the static yield stress may turn out to be smaller than that, or it may even vanish at low fields; see curve D in Fig. 2. Furthermore, the dependence of either static or dynamic yield stress on the applied electric field often deviates from precisely quadratic, most noticeably at high fields. The Bingham plastic and related models are reasonably appropriate only in the post-yield regime, where flow is established. That is, of course, essential to most ER applications, which involve dynamic conditions and require knowledge of the corresponding shear strength. However, the pre-yield regime is also important, for example, in designing nonflowing devices, such as seals and safety valves, where the fluid is expected to remain solidified below a certain pressure threshold, or in flexible and adaptive structures, which must provide a controllable mechanical response or vibration damping without continuous alteration (47). In such a regime, the assumption of total rigidity as in the Bingham plastic model is obviously inadequate. Instead, we must at least consider an elastic relationship such as that of Eq. (13), for shear stress below yield and for shear strain below a corresponding yield strain. The next step necessary is to consider a viscoelastic model (62), which assumes a linear relationship between shear stress and both shear strain and its shear rate: τ = Gγ + ηγ˙

(16)

Applying an oscillating shear strain of small amplitude and sinusoidal time dependence ei웆t, we can transform Eq. (16) into τ = Gγ + iωηγ = G∗ (E, ω)γ

(17)

which again resembles Eq. (13). Now, however, G* is a complex shear modulus, whose real part G⬘, called storage modulus, provides the coefficient for stored elastic energy (quadratic in strain) and whose imaginary part G⬙, called loss modulus, accounts for viscous energy loss and relaxation. Study of the dependence on both the applied electric field E and the frequency of shearing 웆 for both G⬘ and G⬙ provides important information about the structure and properties of the ER solidified suspension in the pre-yield regime (15,27,65–67). The transition from the pre-yield to the post-yield regime involves larger strain amplitudes, hence, nonlinear relationships and a more complex Fourier analysis, whose precise interpretation and connection with experimental observations in such a yield regime is more difficult to establish. For simplicity, the discussion in this section has been limited to one-dimensional shearing. However, many ER applications involve more complex three-dimensional flows. Then appropriate tensorial generalizations of models and equations such as Eqs. (14)–(17) are required. These are relatively easy to obtain, but they cannot always be uniquely determined from purely theoretical principles. More serious difficulties occur if one abandons the common assumption of a symmetrical stress tensor, which is not quite justified in the presence of an electric field. That makes the ER fluid inherently anisotropic.

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Further complications arise when the flow or the electric field change along a streamline. These are called extensional deformations and occur, for example, in an ER valve of varying cross section. These issues are subjects of current and future research (27, 68). A further review of phenomenological models and comparison with experiments is provided in Refs. 6, 43 (see their Figs. 1–2, in particular) and Ref. 64. ER FLUID RHEOMETRY There are many techniques for measuring ER properties. Some are more suitable for general material characterization, and others are more focused on potential end uses. The two basic operating modes of ER devices define two broad classes of electrode configurations used to test ER materials. The first class comprises sliding-plate instruments, involving either parallel plates or coaxial cylinders. Parallel plates may be (circular and) rotating or (rectangular and) translating relative to each other. Coaxial cylinders may be rotating (Couette configuration) or axially sliding relative to each other. Rotational geometries are necessary for continuous (steady state) shear measurements and are generally easier to implement. Among them, coaxial cylinders with a narrow gap (compared to their radii) have the advantage of providing a nearly constant (radially) shear rate, whereas rotating parallel disks generate a shear rate increasing linearly with distance from the axis, which makes a rigorous interpretation of the data more difficult. Moving electrodes are electrically grounded with a carbon brush, a copper/beryllium spring, or a running contact via a conducting fluid (e.g., mercury). Coneand-plate viscometers, commonly used in other rheological measurements, are not suitable for ER fluids, because they involve largely nonuniform electric fields, especially around the cone apex. The operating modes of sliding-plate instruments involve either a controlled stress or a controlled shear rate. For example, in a Couette-type rheometer, stress is controlled by attaching the inner cylinder to a motor, while keeping the outer cylinder fixed. The motor supplies a given torque, and the angular velocity ⍀ that results in the inner cylinder is measured. For a Newtonian fluid, the viscosity is given by η=

R2 − R2 R − R1 τ = −τ 2 2 1 ≈ −τ 2 γ˙ 2R2 R2

(18)

where R1 and R2 are the radii of the inner and outer cylinders, (⫺␶ R1) is the positive (counterclockwise) torque exerted by the inner cylinder on the fluid per unit area, and 웂˙ is the (negative) shear rate at R1. This result is easily derived by solving the Navier–Stokes equation with no-slip boundary conditions at the cylinders; see, for example, Sec. 18, Chap. 2, of Ref. 59. The last approximate expression in Eq. (18) applies for R2 ⫺ R1 Ⰶ R1. In this case, the shear rate hardly falls radially, and the expression is also approximately valid for the apparent viscosity of a Bingham plastic, provided that the yield stress is exceeded all the way to the outer cylinder (61,64). Conversely, the shear rate is controlled by rotating one cylinder at a given angular velocity and attaching the other cylinder to a calibrated spring, whose deflection measures the

torque transmitted through the fluid (which is equal and opposite to the torque that must have been applied to the driving cylinder to keep it at the given angular velocity). The choice of driving the outer cylinder in the shear-rate controlled operating mode is often preferred for flow stability. In general, shear-rate controlled rheometers maintain flow stability at higher torques and shear rates, thus allowing better extrapolation of large dynamic yield stresses from low highshear-limiting viscosities, as desired in ER applications. Post-yield measurements are typically conducted in continuous. Full investigation of pre-yield and yield regimes requires various other conditions. The static yield stress is conveniently determined by stress-controlled rheometers by gradually increasing the torque from an initially static configuration until flow onset. On the other hand, full determination of the viscoelastic response requires further measurements conducted with constant strain, oscillatory strain at various frequencies and amplitudes, and sudden application of shear strain or shear rate. The second class of electrode configurations involves forced flow through fixed plates, as in ER valve applications. The fixed plates to which the potential difference is applied may be plane and parallel, as in slit or rectangular ducts, or they may be coaxial cylinders, as in annular channels. Capillary rheometers are not suitable for ER measurements, because they can hardly be configured as electrodes providing uniform fields. These instruments typically measure the pressure gradient for steady flow. The pressure gradient equals the yield stress divided by an appropriate fraction of the electrode gap (see, for instance, Eq. (4) of Ref. 47 for the basic slit configuration). Assuming Bingham plastic behavior (see, for instance, Chap. 3 of Ref. 61), the complete velocity profile in the channel can also be calculated. Solidification of the ER fluid starts around the pipe center, where shear is lowest, and proceeds outward with increasing electric field until a complete shut-off of the valve occurs. The total pressure drop between the inlet and outlet of the valve is proportional to both the yield stress and the length of the valve and can be as high as 6.9 MPa (1000 psi) (47). At zero field, all types of measurements typically give consistent results for the essentially Newtonian dispersions. Unfortunately, that is not necessarily the case when the electric field is applied because that makes ER suspensions inherently anisotropic, and some observations become dependent on the geometry. Even within the same geometry, such as flow through a rectangular duct, measurements of the apparent viscosity may differ, depending on the duct gap. In fact, a first-order theoretical analysis suggests a decrease in apparent viscosity at large gaps as a result of ER fluid anisotropy (68). Fortunately, at least the static yield stress is not affected and corresponding measurements are reasonably consistent and independent of the geometry. Further discussions and review of rheometric characterizations and testing of ER fluids can be found, for example, in Refs. 6,47,64. THEORETICAL MODELS AND SIMULATIONS OF ER PHENOMENA Electrical interactions are clearly at the origin of the ER effect. When an electric field is applied, the particles polarize

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and in turn generate nonuniform fields. In such fields, polarized particles, although neutral, migrate toward regions of higher field intensity, an effect called dielectrophoresis (69). Such regions are closer to other particles, which leads to further polarization, attraction, and aggregation. Ultimately, the strong polar interactions produce fibrous structures preferentially aligned with the applied field. In diluted suspensions, particles may have to travel over considerable distances. In weak applied fields, their movement may be slow. Hence, the response time for fibrillation may be considerable. However, for typical ER concentrated suspensions in strong applied fields, particles need to move a distance only a fraction of their size to form fibrous structures, and a fast (millisecond) response can occur. The subsequent evolution and reaction of the fibrous structures under static and dynamic shearing conditions (aggregating, stretching, tilting, breaking, and reforming) are ultimately responsible for all of the observed ER phenomena. Such evolution is continually determined by the balance of the microscopic electric and hydrodynamic forces acting on the particles. There is a relatively general agreement about this basic scenario of fibrillation induced by polarization, which was first envisioned by Winslow himself (2). However, which specific mechanisms are responsible for particle polarization in various types of ER fluids has not yet been fully established. Some models involve electrical double layers typically produced by ions in water surrounding the particles. Other models focus on the interfacial polarization resulting from the difference in permittivities between the particles and the dispersing medium. These two types of models may be equivalent in some instances, for example, for thin double layers and particles with thin coatings, or they may be essentially incompatible, for example, for double layers presumed to act through large distortions and overlap. In any event, the ER activity is controlled much more by interfacial effects than dielectric bulk properties of the particles. A more detailed and precise understanding of the microscopic structures and phenomena occurring at the particle surfaces is required for intelligent selection and development of more effective ER fluid components. In parallel with the uncertainties in characterization and microscopic description of electrical interfacial properties and partly as a result of those uncertainties, there is no definitive account of how fibrillation develops and evolves under shear, what relaxation times are involved at various stages, and various other dynamical characteristics. More detailed discussions and references on various proposed models and mechanisms for the ER effect can be found, for instance, in Refs. 6, 18, 19, 27, and 70. A complete and truly predictive theoretical account of ER phenomena must satisfy two essential requirements. The first is the proper and quantitative treatment of the underlying electrical interactions. The second requirement is the modeling and simulation of the fibrous structure statics and dynamics based on the adequate treatment of both electric and hydrodynamic forces. With regard to the first requirement, three crucial features must be considered. First, the electrical interactions in ER fluids are inherently multipolar and multiparticle. Because particles are in contact or very close to one another, the electric field that they contribute is very complicated and varies rapidly on the scale of the particle dimensions. The great strength of the resulting electrical interactions derives from

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many multipolar terms. A purely dipolar system is an inadequate model, which cannot consistently provide correct results for actual fluids with an ER effect strong enough to be of practical interest. For those, the higher multipolar terms provide a much stronger contribution than just the dipolar interactions, exhibit a quite different and more complex angular dependence, and decrease much faster with the distance between the particles. Likewise, pair interactions alone cannot provide a realistic description of the self-consistent polarization cooperatively induced by many particles at close range. Secondly, ER fluids operate between electrodes. The basic electrode configuration can be regarded as a parallel-plate capacitor (PPC), because the radius of curvature of the electrodes is typically much greater than their separation. Theoretically, the PPC configuration can be treated by considering image multipoles of the particles. The reflection of one particle on the right (or left) electrode results in one image, but the reflection of this image on the left (or right) electrode results in a new image further away, which in turn reflects on the right (or left) electrode, and so on. Hence, each particle generates an infinite set of images. The electrical interaction between a particle and the electrodes equals that between the particle and all of its images, as well as the images of all the other particles. The image multipoles are not equal to those of the real particles, although they are related [see Eq. (26) later]. Hence, the system cannot be equivalently described by an infinite system, consisting only of real particles immersed in a uniform external field (UEF), somehow produced by fixed external charges. Basic ER properties, such as yield stress and apparent viscosity, crucially depend on regions where the weakest electrical interactions occur, because particle chains typically break there first. Therefore, it is necessary to find out whether these regions occur near the electrodes or in the bulk, depending on the system and the static or dynamic shearing configuration (15,18,19,70). The UEF configuration may provide results equivalent to the PPC configuration in the bulk, but it cannot account for exact particle interactions near or with the electrodes. In fact, surface effects cannot be calculated ignoring the image multipoles, because their contributions near the electrodes are different but just as important as those of the particles. Thirdly, real particles in ER fluids cannot be treated as uniform spheres because they typically have a nonuniform electrical response and a complex structure, which required both to augment the ER effect and to stabilize the fluid (recall, for example, the corresponding roles played by activators and surfactants). A uniform electric field induces only a dipole moment on a spherical particle. In that case, the usual polarizability, which is the ratio between the induced dipole moment and the uniform electric field, is sufficient to describe the electrical response of the particle. However, a multipolar electric field induces multipolar moments on any particle (even on a simple homogeneous sphere). Therefore, to begin with, a realistic description of ER fluids requires determining the complete electrical response of isolated complex particles to any field. This involves generalized polarization coefficients, defined as the tensorial ratio between a multipole of given order induced on the particle and the inducing multipolar field component of any other order. These complexities in electrical interactions have limited most theoretical models and simulations of ER phenomena

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largely to dipolar approximations. As a result, even though those models and simulations may suggest real effects in some respects, none can be regarded as conclusive, nor is there any rigorous way to select one model over another. However, full treatments of multipolar interactions have been progressively developed (71–76), and a complete theory and computational scheme of electrical interactions in ER fluids, which includes all three crucial features just mentioned, has recently become available (75–76). Therefore, the first essential requirement for a truly predictive theoretical account of ER phenomena has in fact been fulfilled, even though the second still has not. The widespread use of dipole approximations also results from the fact that, occasionally, dipolar estimates are sufficient to suggest trends roughly agreeing with observation. For example, consider a uniform sphere of radius a and dielectric constant ⑀p in a host medium of dielectric constant ⑀h. A uniform electric field E acting on the particle induces a dipole moment (see, for example, Sec. 4.4 in Ref. 77) p ∝ βa3E

(19)

where β=

p − h p + 2 h

(20)

If two particles with such induced dipole moments interact, the corresponding force between them is proportional to p2 /a4 at contact. Then, the yield stress scales as τy (E) ∝ β 2 E 2 v

(21)

where v is the volume fraction of the particles in the fluid. Quadratic dependence on the electric field of the yield stress is basically confirmed by observations. However, we show in the next section that in a complete theory of electrical interactions this is just a consequence of linear response in a fixed or steady-state configuration, regardless of any dipolar model or approximation. Equation (21) further suggests a linear dependence of the yield stress only on the volume fraction of the particles, not on their absolute size. This is also confirmed by experiment, at least for particles not so small (in the submicron range) that Brownian motion effectively reduces the yield stress. Introducing the yield stress scaling in the Bingham plastic model, the apparent viscosity may behave as η=

τy (E) + η∞ ∼ η∞ γ˙



0 h vβ 2 E 2 +1 η∞ γ˙

 (22)

suggesting an approximate dependence of the apparent viscosity on the so-called Mason number, defined as Mn =

ηh γ˙ 2 0 h β 2 E 2

(23)

where ␩h is the viscosity of the suspending host medium. Evidently, the Mason number represents the ratio between the order of the hydrodynamic viscous forces microscopically exerted by the dispersant on the particles and the order of the

electrical forces between the particles. Then,  η ∼ η∞

κv +1 Mn

 (24)

where ␬ 앑 0.5 is a system-specific parameter. This scaling relationship agrees fairly well with observations in a rather wide range of fields and shear rates for any given system (64, 78). Encouraged by the apparent success of these and other simple dipolar estimates, several dipolar models and simulations have been developed to provide microscopic descriptions of ER phenomena. Despite their ingenuity and value in suggesting the possiblity of various mechanisms, none of these models or results can be regarded (nor has been claimed) as conclusive. The limiting factor essentially derives from the underlying dipolar approximations, which at the very least forbid any rigorous or truly quantitative prediction. Given the power available for current computation, now it is feasible to fully implement the rigorous treatment of the electrical interactions in similar molecular dynamics and Monte Carlo simulations. Accurate inclusion of all significant hydrodynamic forces is also possible. Therefore, a ‘‘second generation’’ of truly predictive simulations and microscopic descriptions of ER phenomena fully satisfying both essential requirements described previously is expected. This may finally provide a complete theoretical understanding of the underlying mechanism of the ER effect, which has been recommended by the DOE expert panel as a critical research need to guide the development of new ER materials and devices (42). Thus we refer, for example, to Refs. 6, 18, 19, and 78–80 for reviews of ‘‘first generation’’ models and simulations, and in the last section of this article we focus on the foundations and some key technical elements of the exact theory of electrical interactions in ER fluids, which is already well established. THEORY AND COMPUTATION OF ELECTRICAL INTERACTIONS IN ER FLUIDS The electrical response and activity of an ER fluid is ultimately determined by the microscopic structure and the complex dielectric functions of all of the fluid components, namely, the host liquid and the composite particles, including all sorts of coating layers. In turn, these complex dielectric functions result from the contributions of all mobile charges. Hence, they are generally frequency-dependent, and so is the resulting ER activity. The imaginary part of a complex dielectric function, associated with the conductivity of a particular component, may be dominant. Microscopically, that may be produced by light electrons, or by mobile ions at low frequencies. It is generally advantageous to have those components at or near the particle interfaces to enhance the electrical interactions and the resulting ER effect. The precise characterization of the microscopic structure and the complex dielectric functions of all of the fluid components may be difficult in practice, but is clearly required for any quantitative modeling and understanding of ER phenomena. Thus, a complete theory of ER fluids must consider particles of arbitrary structures, immersed in a dispersant host medium of dielectric function ⑀h, filling the space between two

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parallel electrode plates at a constant potential difference V0, and located at z ⫽ ⫺d/2 and z ⫽ d/2, respectively. Let us denote the position of the nth particle by rn, and the multipole moments (with respect to rn) by qnlm. The position of the kth image, resulting from the kth reflection on the right (k ⫽ 1, 2, 3, . . .) or left (k ⫽ ⫺1, ⫺2, ⫺3, . . .) electrode, is given by r nk = {xnk , ynk , znk } = {xn , yn , kd + (−1) zn } k

(25)

The corresponding image multipole moments are given by [see Eq. (10) of Ref. 81] qnlmk = (−1)(l+m+1)kqnlm

(26)

Obviously, for k ⫽ 0, rn0 ⫽ rn and qnlm0 ⫽ qnlm correspond to the particles themselves. The local electric potential acting on the nth particle is given by [see Eqs. (14) and (15) of Ref. 81]

Unlocal (rr )

4π E |rr − r n |Y1,0 (rr − r n ) 3 0 1
|rr − r n |l 1 Yl ,m (rr − r n ) + 1 1 0 l m 1 1 

l ,m Al 2 ,m 2 (rr n k − r n )(−1)(l 2 +m 2 +1)kqn ×

=−

n2 l2 m2 k

≡−

1

1

2

2 l2 m2



1

(27) where the coupling coefficients are l ,m

Al,m

 ∗  l ,m Yl+l ,m−m (rr − r ) , [Al,m ] +1) (l+l (rr − r ) = |rr − r |  0,

(rr = r )

(28)

(rr = r )

In Eqs. (27) and (28), Yl,m(r) are spherical harmonics, and



1/2 4π (2l + 1)(2l + 1)(2l + 2l + 1) 1/2  (l + l + m − m )!(l + l − m + m )! × (l + m)!(l − m)!(l + m )!(l − m )!

,m [All,m ] =(−1)(l

+m )



(29)

are numerical coefficients. There are clearly three contributions to the local potential in Eq. (27) that derive from the applied field E0 ⫽ V0 /d, the field produced by all of the other particles, and the field produced by all the images (including those of the nth particle under consideration). All orders of multipole moments (l ⫽ 1, 2, . . ., m ⫽ 0, ⫾1, . . ., ⫾l) are included. The nth particle may have an arbitrary shape or structure, but if its electrical response is linear (within the range of the applied field), the multipole moments induced on such a particle must be a linear combination of the multipolar components El1m1 of the local potential, namely, qnlm =


l m √ 1 1E 12π 0 λnlm l l1 m1

1 m1

1

(30)

1 0

where 兩0 means that the derivatives are evaluated at El1m1 ⫽ 0, for all l1, m1. These polarization coefficients are generalizations of the usual (dipolar) polarizability and depend on the structure and orientation of the particle. Symmetries therein may restrict the number of terms contributing to Eq. (30). For example, spherical symmetry requires a diagonal response, that is, only terms with l1 ⫽ l and m1 ⫽ m may not vanish (see, for example, Eq. (42) for coated spherical particles). For nonlinear particles, additional (and more complex) nonlinear polarization coefficients must be similarly introduced (83). From Eq. (27) for the local potential and Eqs. (30)–(31) defining the polarization coefficients, we obtain a self-consistent set of linear equations for all of the multipole moments:
qnlm + 3 qn l m



4π
E |rr − r n |l 1 Yl ,m (rr − r n ) 1 1 3 l m l1 m1 1

These equations provide the general definition of the linear polarization coefficients (82), that is,  ∂qnlm  1 l1 m1 λnlm = √ (31)  12π 0 ∂El m 

n1 l1 m1



745

×


l2 m2

1 1

2 m2 λlnlm

1


k

 (−1)

(l 1 +m 1 +1)k

1 (r rn k All 1 ,m 1 2 ,m 2

=

− r n )

√ 12π 0 λ10 nlm E0

(32)

This shows that the electrical response is completely and uniquely determined by the polarization coefficients of the particles and the microscopic configuration of the system (determined by the positions and orientations of all the particles). In principle, the set of linear equations for all of the multipole moments is infinite. In practice, the results are well behaved and converge at finite orders of images (k0) and multipole moments (l0), although l0 is typically large for particles at close range. Diverging exceptions occur at most for extreme cases of infinitely conducting and perfectly touching particles, where classical theory itself becomes inadequate. This is not a problem in practice because highly conducting particles must be insulated with thin coatings, to prevent charge transfer between them at contact and to avoid a corresponding large conduction through the ER fluid (29–32). Now let us consider the total electrostatic energy stored in the system, which is given by [see, for example, Eq. (4.83) of Ref. 77]:  1 W= ρ (rr )U (rr )dV (33) 2 V f where V, ␳f (r), and U(r) are the volume, free charge density, and electric potential of the system, respectively. The contribution from the right electrode plate (at z ⫽ d/2) vanishes because Uz⫽d/2 ⫽ 0. From now on let us assume that the host medium is nonconducting (further consideration of a conducting medium can be found in Ref. 76). Then, the contribution from inside the system also vanishes, because there is no free charge density in either the medium or nonconducting particles, and there is no net free charge on equipotential surfaces of conducting (neutral) particles. Hence, the only contribution to Eq. (33) derives from the left electrode plate and amounts to the free charge Qf on the electrode plate times V0. Qf in-

746

ELECTRORHEOLOGY

cludes two parts, one from the applied potential and the other induced by the particles. Using a standard result [see, for example, problem 1.13, p. 51, of Ref. 77],
V Qf = 0 h E 0 + h pnz d d n

(34)

where pnz ⫽ 兹4앟/3qn10 is the z-component of the dipole moment in Cartesian coordinates. Hence, W=


V 1 0 h E02 + h E0 pnz 2 2 n

(35)

The first term in Eq. (35) is the same constant that we have when the medium alone fills the capacitor, and the second term derives from the particles (including their interfacial polarization with the medium). When the positions and orientations of the particles change, the induced dipole moments change. That in turn produces a change in Eq. (34), corresponding to a charge transfer between the two electrodes. Then, the work done by the electrical source (battery) in response to a change in the microscopic configuration is given by

 Wb = V0 Q f = h E0 




 pnz

n

(36) V0

where the subscript V0 indicates that change must be done while keeping the potential difference between the electrodes constant. Subtracting Eq. (36) from the change in Eq. (35), we obtain the work mechanically done on the system to change its configuration:

 
1 E = − h E0  pnz 2 n

(37)

V0

From this, we can define the interaction energy as
1 E = − h E0 pnz 2 n

(38)

Then, the total electrostatic force exerted by the system (including the effect of the battery) associated with the displacement of a generalized coordinate ␰ is given by



∂E Fξ = − ∂ξ

 V0

1 ∂ = h 2 ∂ξ

 E0




 pnz

(39)

n

V0

These equations allow us to compute the exact electrostatic interactions in ER fluids. For instance, if we need to compute the electrostatic force on any given particle, we take ␰ as a coordinate of that particle, or ⭸/⭸␰ 씮 ⵜn, and

f n = −∇n (E )V

0

 
1 = h ∇n E 0 pnz 2 n

(40)

V0

On the other hand, if we wish to compute the shear stress, we take ␰ as the angle ␪ for tilting parallel chains times the

system volume:



∂E τy = − ∂ (V θ )

 V0

∂ 1 h = 2V ∂θ

 E0


n

 pnz

(41) V0

Its maximum, varying ␪, provides the yield stress. Clearly, the results depend on the system configuration. A few other specific examples are provided later. The previous equations indicate the evolution of the system from an initially disordered configuration when a potential difference is suddenly applied. Driven by the electrical source, the system undergoes a continuous change until it reaches a microscopic configuration of maximum total dipole moment (hence, polarization). The total electrostatic energy W is maximized, and the interaction energy E is minimized. The work done by the electrical source splits equally into mechanical work done by the system (negative of the interaction energy change) and total electrostatic energy stored in the system. Thus, the ground state of the solidified ER fluid has maximum total dipole moment and total electrostatic energy and minimum interaction energy. We may note that in the uniform external field (UEF) configuration, the electrostatic energy formally coincides with Eq. (38) (see, for example, Eq. (4.94) and Sec. 4.7 of Ref. 77) and is minimized in the ground state. Then results equivalent to those of the parallel plate capacitor (PPC) configuration may be obtained for forces in the bulk, formally corresponding to a Legendre transformation (84). On the other hand, of course, no surface effect can be properly treated in the UEF configuration. An important fact shown by Eq. (39) is that forces do not depend on the absolute values of dipole moments in a certain configuration but only on the variations of such dipole moments when the configuration changes (virtually and infinitesimally). This shows that the ER effect is purely a local field effect. It is caused by the change induced on the local field acting on each particle by a virtual variation of microscopic configuration. That changes the polarization of the particles, hence, the interaction energy. The fixed-dipole approximation considers forces between dipole moments induced on the particles only by a constant (applied) field without any regard for local field changes. That approximation is unrelated even in principle to the real ER effect. Furthermore, it important to note that although Eqs. (34)– (41) depend explicitly only on dipole moments, these must in turn be determined with Eq. (32), that is, with full self-consistent coupling to all higher multipole moments. The dipole approximation consists of truncating Eq. (32) just at the dipole level (i.e., l0 ⫽ 1). This includes part of the local field effect, but the approximation remains largely inaccurate, because many more multipole moments are actually coupled when the particles are at close range or aggregate. To understand the dependence of the yield stress and other generalized forces on the applied field E0, we recall that all of the multipole moments induced on the individual particles are assumed to be linear in the local field [Eq. (30)], and that the coupled equations [Eq. (32)] are linear in the multiple moments. However, Eq. (32) also involves the system configuration, which is generally affected by the applied field (as well as shearing and other conditions). Therefore, the response of the system is generally a nonlinear function of the applied field, especially over a wide range. However, if the

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λllmm =

m (2l+1) 1 l(2l + 1)δll δm b 3

0

Electrostatic force from the electrode

system configuration is fixed, as in a solidified ER fluid, or reaches a steady-state equilibrium under shear flow, it becomes independent of E0, and the dipole moments become proportional to E0 even through the coupled Eq. (32). Then, the yield stress and other generalized forces become proportional to E02 through Eq. (39) (the change in configuration is only infinitesimal therein). This by no means validates any dipolar approximation, which may also predict such quadratic dependence, because that still ignores coupling to higher multipoles and local field effects. Now we discuss a few basic examples of exact results that can be obtained for different forms of single chains. In these numerical calculations, multipole moments up to l0 ⫽ 300 and images up to k ⫽ ⫾20 have been included, with a corresponding estimated error below 1%. We consider identical spherical particles (hence, omit the particle index n) with a linear response in a core of radius a and dielectric constant ⑀c and another linear response in a coating layer of outer radius b and dielectric constant ⑀s. Because of symmetry, each multipolar component of the local potential can induce only one multipole moment of the same order on spherical particles. For coated spheres, the complete set of polarization coefficients is given by (76,85)

747

–5 –10 –15 –20 –25 –30 –35 –40 –45

0

0.01

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 x

Figure 3. Electrostatic force between the left electrode and a single particle (solid squares), a two-particle chain (crosses), and a five-particle chain (asterisks), as functions of the distance (x) between the left electrode and the (leftmost) particle surface. The particles have a conducting core of radius a ⫽ 0.95, plus an insulating coating shell of outer radius b ⫽ 1.0 and dielectric constant ⑀s ⫽ 10.0. Force values are in millidynes for an electric field of 1 kV/mm and an outer radius of 10 애m.

Figure 3 shows the electrostatic force between one electrode and chains with various numbers of particles as a funcb ×  a (2l+1) tion of distance from the left electrode. The figure shows clearly that the images attract the particles toward the elec[l c +(l +1) s ][l s +(l +1) h ]+l(l +1)( c − s )( s − h ) b trodes (the UEF model obviously cannot account for such at(42) traction to the electrodes). Notice that the attractive force decreases rapidly as the distance from the electrode increases, For a metal core, one can take the limit ⑀c 씮 앝 in Eq. (42) a clear manifestation of the multipolar nature of the interacand obtain tions. Now we consider the crucial issue of the strength of the ER effect and its dependence on the properties of the ER fluid 1 m (2l+1) λllmm = (2l + 1)δll δm b constituents. An indicator of such strength is the longitudinal 3  a (2l+1) force required to break a chain whose particles are in contact (43) l( s − h ) + [(l h + (l + 1) s ] b ×  a (2l+1) [l s + (l + 1) h ] + (l + 1)( s − h ) 0 0 b

 a (2l+1)

For uniform spheres, one can immediately take ⑀s ⫽ ⑀c (or let a ⫽ b) in Eq. (42). The following numerical results have been obtained with ⑀h ⫽ 1.0 for simplicity. As units for the electric field and the particle outer radius, we use the typical values E0 ⫽ 1 kV/mm and b ⫽ 10 애m. The corresponding forces are given in Figs. 3–7 in units of millidynes. Recalling that forces scale with E02 for linear response and with b2, if all distances in the system (including a) are scaled proportionally to b, force values for different fields and particle radii can be immediately obtained from the same figures. On the other hand, these calculations are confined to individual chains and their breaking at single points. Therefore, no precise value for any component of the yield stress tensor can be immediately obtained. However, dividing by the area 4 ⫻ 10⫺6 cm2 of a particle diameter squared, we can estimate that a force of 100 millidynes translates into a yield stress tensile component of the order of 2.5 kPa, which agrees with stress values typically observed for E0 ⫽ 1 kV/mm.

–50 Binding force of a chain

( s − h )[l c + (l + 1) s ] + ( c − s )[l h + (l + 1) s ]

–0.5

–100 –150

–1

–200 –1.5

–250 –300

–2 –350 –2.5 0

10

20

30

40

50

60

70

80

90

–400 100

c

Figure 4. Binding force of a chain of uniform spheres of unit radius and dielectric constant ⑀c, stretching between the two electrodes, as a function of ⑀c. The left ordinates refer to the dipole (crosses) and fixeddipole (asterisks) approximations, whereas the right ordinates refer to the exact results (solid squares).

748

ELECTRORHEOLOGY 0

0

Binding force of a chain

–0.2

–10

–0.4

–20

–0.6 –30 –0.8 –40 –1 –50

–1.2

–60

–1.4 –1.6 0

10

20

30

40

50

60

70

80

90

–70 100

S

Figure 5. Binding force of a chain of spheres with an insulating core of radius a ⫽ 0.95 and dielectric constant ⑀c ⫽ 10.0, plus an insulating coating shell of outer radius b ⫽ 1.0, as a function of the coating dielectric constant ⑀s.

0

0

Binding force of a chain

–100 –0.5

–200 –300

–1

–400 –1.5

–500 –600

–2

–700 –800

–2.5

–900 –3 0

10

20

30

40

50

60

70

80

90

–1000 100

S

Binding force of a chain

Figure 6. Same as Fig. 5 but for particles with a conducting core.

0

–10 –20

–0.5

–30 –40

–1

–50 –60

–1.5

–70 –80

–2

–90 –2.5

–100

–3

–110 –120 0

0.1

0.2

0.3 0.4

0.5 a

0.6

0.7

0.8

0.9

1

Figure 7. Same as Fig. 6, but with fixed coating dielectric constant ⑀s ⫽ 20.0 and varying core radius a.

and span the gap between the two electrode plates. Such a force, whose negative we call binding force, is directly related to the static tensile yield stress, except that a bundle of chains or a columnar structure should be considered in a more realistic microscopic configuration. Because of symmetry, the binding force is independent of the chain length and is the same for breaking the chain anywhere. On the other hand, the UEF model, lacking the images, yields binding forces that artificially weaken toward the chain ends (75). The experimental situation is more complicated. For example, one may observe that thin fibrils fracture in the middle, whereas thick columns break at the electrode (15, 70). This indicates (1) that surface adhesion plays an important role, and (2) that forces between chains are significant when they form thick columns. Surface adhesion may be included, at least phenomenologically, in the modeling and simulation of specific systems, and interchain electrical forces can be calculated accurately within the theoretical framework outlined (although no such calculations have yet been performed in detail). Here we compute the binding force only in single chains of uniform spheres, spheres with a dielectric core and a dielectric coating layer, and spheres with a metal core and a dielectric coating layer. The corresponding results are plotted in Figs. 4–7. Figure 4 shows that the ER effect for uniform spheres increases with contrast, which is the ratio between the dielectric constant of the particles and that of the medium. For coated dielectric spheres, Fig. 5 shows that the ER effect increases with the dielectric constant in the coating. This is consistent with the observation that activators with large dielectric constants surrounding the particles dramatically increase the ER effect, even when the particles themselves are not very ER active. Figure 6 shows that also for particles with a metal core, the larger the coating dielectric constant, the greater the ER effect. Figure 7 further shows that the thinner the coating layer (as long as it is adequate to keep the metal core insulated), the greater the ER effect. These results demonstrate that the greatest binding force is obtained with coated metal spheres. This supports the search for anhydrous ER fluids based on conducting particles surrounded by thin and highly polarizable insulating shells. Note, however, that metal spheres with thick or low-polarizability coatings are still not particularly ER effective. This may be one of the reasons why insulated metal particles have not yet conclusively demonstrated substantial ER activity independent of water (37). Improved stability against gravitational and centrifugal settling is attained by reducing the weight of structured particles. Thus, one may fabricate doubly coated spheres, consisting of a light dielectric core, a thin metal coating, and an outer dielectric coating (still needed to prevent conduction through the system) (31,32). At least at low frequencies, the electrostatic interactions among such doubly coated spheres should be independent of the core material. In fact, they should be the same as those of singly coated metal spheres (29,30), because the metal layer completely screens the penetration of the electric field into the inner core, and thus the polarization coefficients are the same as in Eq. (43). Figures 4–7 clearly show that the fixed-dipole (asterisks) and the dipole (crosses) approximations underestimate by two or more orders of magnitude the value of the binding force, a fact typically confirmed by comparison with experiments (14,

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15, 21, 86, 87). and even fail to describe the correct trends. In fact, the dipole approximations exhibit systematic saturation, 10 as expected from the dipolar polarization coefficient ␭10 , which saturates when ⑀c and ⑀s increase. However, the exact binding force never saturates, and grows faster with increasing ⑀c and ⑀s, as higher and higher multipoles contribute. Experiments have shown that the ER effect does not saturate with increasing particle–medium dielectric mismatch, contrary to dipole predictions (87). Our exact calculations demonstrate that the surprising strength of ER fluids indeed depends by and large on the multipolar interactions among the particles. As already noted, Eq. (39) shows that a large electrostatic force requires a large variation of the system’s total dipole moment in response to a (virtual) change of microscopic configuration, rather than a large absolute value of the total dipole moment itself. That occurs only if the dipole moments of the particles are strongly coupled (through the local field) to many higher multipolar terms, because only those change rapidly with the change of distance among the particles. Thus, only in systems with strong multipolar interactions, hence, sufficiently large polarization coefficients of higher orders, the total induced dipole moment, hence, the interaction energy, is very sensitive to the change of microscopic configuration. Systems with limited multipolar interactions should not be expected to exhibit a strong ER effect, even if the dipole moments of the particles are very large. Experiments indeed suggest that ER activity correlates with interfacial, rather than orientational polarization. This may explain why systems with ferroelectric particles do not necessarily yield significant ER activity when rigorously dried, despite the large dielectric constant of the particles. For example, a system of TiO2 particles in paraffin oil completely loses its ER activity if carefully dried (27, 37), even though TiO2 is an incipient ferroelectric with a dielectric constant between 70 and 200 (depending on the precise structure). The same occurs with electrets, that is, particles with attached permanent dipoles. However, other experiments indicate that ferroelectric particles may very well exhibit strong ER activity, increasing with the particle–medium dielectric mismatch, without saturation (87). That definitely indicates strong multipolar polarization coefficients, possibly resulting from a different ferroelectric domain structure or perhaps aided by even minimal water surroundings. The prototypical examples provided previously serve only as illustrations. However, complete and accurate computations of realistic systems and configurations based on this method are entirely feasible, although obviously much more demanding. For a system of n0 particles and required highest order l0 of multipole moments, one has to solve for n0 dipole moments qn1m in a set of N linear Eqs. (32), where


l0

N = n0

(2l + 1) = n0 l0 (l0 + 2)

(44)

l=1

For systems with symmmetry, N can be significantly reduced. For example, ground-state structures, static yield stress, interactions between pair of chains, and other ordered properties can be determined exactly with only moderate computation. On the other hand, full implementation of multipolar interactions in computer simulations of dynamic configurations and evolution, which also require a large number of par-

749

ticles, may very well need super or parallel computing resources. However, this is also the case for other presently developing fields in computational physics. Therefore, complete theoretical understanding and predictive computation of ER phenomena and materials may be attained in the near future. BIBLIOGRAPHY 1. W. M. Winslow, Methods and means for translating electrical impulses into mechanical force, U.S. Patent No. 2,417,850, 1947. 2. W. M. Winslow, Induced fibration of suspensions, J. Appl. Phys., 20: 1137–1140, 1949. 3. W. Ko¨nig, Bestimmung einiger Reibungscoe¨fficienten und Versuche u¨ber den Einfluss der Magnetisirung und Electrisirung auf die Reibung der Flu¨ssigkeiten (Determination of some friction coefficients and investigation of the influence of magnetization and electrification on the friction of the fluids), Ann. Phys. (Leipzig), 25: 618–625, 1885. 4. A. W. Duff, The viscosity of polarized dielectrics, Phys. Rev., 4: 23–38, 1896. 5. G. Quincke, Die Klebrigkeit isolirender Flu¨ssigkeiten im constanten electrischen Felde (The stickiness of insulating fluids in a constant electric field), Ann. Phys. Chem., 62: 1–13, 1897. 6. K. D. Weiss, J. D. Carlson, and J. P. Coulter, Material aspects of ER systems, J. Intell. Mater. Syst. Struct., 4: 13–34, 1993; also in Ref. 7, pp. 30–52. 7. M. A. Kohudic (ed.), Advances in Electrorheological Fluids, Lancaster, PA: Technomic, 1994. 8. J. D. Carlson, Surfactant-based ER materials, U.S. Patent No. 5,032,307, 1991. 9. H. Conrad, M. Fisher, and A. F. Sprecher, Characterization of the structure of a model ER fluid employing stereology, Proc. 2nd Int. Conf. ER Fluids, Raleigh, NC, 1989, 1990, pp. 63–81. 10. J. C. Hill and T. H. van Steenkiste, Response times of ER fluids, J. Appl. Phys., 70: 1207–1211, 1991. 11. See Fig. 3 of Ref. 6 and Fig. 3 of H. Conrad and Y. Chen, Electrical properties and the strength of ER fluids, Proc. Am. Chem. Soc. Symp. ER Mater. Fluids, Washington, DC, 1994, 1995, pp. 55–85. 12. J. E. Stangroom, ER fluids, Phys. Technol., 14: 290–296, 1983. 13. M. T. Shaw and R. C. Kanu, ER fluids (role of polymers as the dispersed phase), in J. C. Salamone (ed.), Polymeric Materials Encyclopedia, New York: CRC Press, 1996, vol. 3, pp. 2023–2028. 14. R. C. Kanu and M. T. Shaw, Effect of dc and ac electric fields on the response of ER fluids comprising cylindrical PBTZ particles, Proc. Am. Chem. Soc. Symp. ER Mater. Fluids, Washington, DC, 1994, 1995, pp. 303–323. 15. T. Jordan, M. T. Shaw, and T. C. B. McLeish, Viscoelastic response of ER fluids. II. Field strength and strain dependence, J. Rheol., 36 (3): 441–463, 1992. 16. T. J. Chen, R. N. Zitter, and R. Tao, Laser diffraction determination of the crystalline structure of an ER fluid, Phys. Rev. Lett., 68: 2555–2558, 1992. 17. J. E. Martin, J. Odinek, and T. C. Halsey, Evolution of structure in a quiescent ER fluid, Phys. Rev. Lett., 69: 1524–1527, 1992. 18. T. C. Halsey, ER fluids, Science, 258: 761–766, 1992. 19. T. C. Halsey and J. E. Martin, ER fluids, Sci. Amer., 269: 58– 64, 1993. 20. P. S. Neelakanta, Handbook of Electromagnetic Materials, Boca Raton, FL: CRC Press, 1995, chap. 24, pp. 527–548. 21. T. C. Jordan and M. T. Shaw, Electrorheology, MRS Bull., 16 (8): 38–43, 1991.

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22. R. E. Rosenweig, Ferrohydrodynamics, Cambridge, UK: Cambridge Univ. Press, 1985. 23. Proc. 6th Int. Conf. ER Fluids, MR Suspensions Their Applications, Yonezawa, Japan, 1997, to be published. 24. W. A. Bullough (ed.), Proc. 5th Int. Conf. ER Fluids, MR Suspensions, Associated Technol., Singapore: World Scientific, 1996. 25. J. E. Stangroom, Electric field responsive fluids, U.S. Patent No. 4,129,513, 1978; Improvements in or relating to electric field responsive fluids, G.B. Patent No. 1,570,234, 1980. 26. L. M. Carreira and V. S. Mihajlov, Recording apparatus and methods employing photoelectroviscous ink, U.S. Patent No. 3,553,708, 1971. 27. F. E. Filisko, Overview of ER technology, Proc. Am. Chem. Soc. Symp. ER Mater. Fluids, Washington, DC, 1994, 1995, pp. 3–18; Materials aspects of ER fluids, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sec. 5.9. 28. H. Block and J. P. Kelly, ER fluid containing continuous phase halogenated aromatic liquid and semiconductor or unsaturated fused polycyclic compound or a poly(acene-quinone) polymer as disperse phase functioning when anhydrous, U.S. Patent No. 4,687,589, 1987. 29. A. Inoue, Study of new ER fluid, Proc. 2nd Int. Conf. ER Fluids, Raleigh, NC, 1989, 1990, pp. 176–183. 30. W. C. Yu et al., Design of anhydrous ER suspensions based on I2-doped poly(pyridinium salt), J. Polym. Sci. B, 32: 481–489, 1994. 31. A. Inoue, New electroviscous fluid used for electrical controls, prepared by dispersing dielectric fine particles containing core organic solid particle, conductive- and outer-film layers, in oil medium, JP Patent No. 63,097,694, 1988. 32. M. Prendergast, ER fluids, EP Patent No. 396,237, 1990. 33. Y. Ishino et al., Electroviscous fluid with anhydrous carbon particulates, dispersed in insulating oil usable over wide temperature range, with low power consumption and with AC or DC, EP Patent No. 361106, 1990; U.S. Patent No. 5,087,382, 1992; also see Proc. Am. Chem. Soc. Symp. ER Mater. Fluids, 1994, 1995, pp. 137–146. 34. F. E. Filisko and W. H. Armstrong, Electric field dependent fluids, U.S. Patent No. 4,744,914, 1988; U.S. Patent No. 4,879,056, 1989. 35. F. E. Filisko and L. H. Radzilowski, An intrinsic mechanism for the activity of aluminosilicate based ER materials, J. Rheol., 34 (4): 539–552, 1990. 36. U. Y. Treasurer, F. E. Filisko, and L. H. Radzilowski, Polyelectrolytes as inclusions in ER active materials: Effect of chemical characteristics on ER activity, J. Rheol., 35 (6): 1051–1068, 1991. 37. F. E. Filisko, Rheological properties and models of dry ER materials, Proc. 3rd Int. Conf. ER Fluids, Carbondale, IL, 1991, 1992, pp. 116–128. 38. J. D. Carlson, Water-free ER fluid: Has dispersed particulate phase of lithium hydrazinium sulfate and dielectric liquid phase, e.g., silicone oil, U.S. Patent No. 4,772,407, 1988. 39. K. Wissbrun, Potential application of liquid crystals as ER fluids, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.10. 40. I-K. Yang and A. D. Shine, Electrorheology of a nematic poly(nhexyl isocyanate) solution, J. Rheol., 36 (6): 1079–1104, 1992. 41. B. R. Powell, Preparation of ER fluids using fullerenes and other crystals having fullerene-like anisotropic electrical properties, U.S. Patent No. 5,445,759, 1995. 42. U.S. Department of Energy, Office of Energy Research, Electrorheological Fluids: A Research Needs Assessment, Final Report, DOE/ER/30172, Washington, DC: Natl. Tech. Inf. Service, 1993. 43. H. Block and J. P. Kelly, Electrorheology, J. Phys. D, 21: 1661– 1677, 1988.

44. J. R. Wilson, ER fluid devices and energy savings, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.1. 45. J. W. Pialet and D. R. Clark, The dependence of shear stress and current density on temperature and field for model ER fluids, Proc. Am. Chem. Soc. Symp. ER Mater. Fluids, Washington, DC, 1994, 1995, pp. 251–262. 46. R. C. Kanu and M. T. Shaw, Studies of ER fluids featuring rodlike particles, Proc. 5th Int. Conf. ER Fluids, MR Suspensions, Assoc. Technol., Sheffield, UK, 1995, 1996, pp. 92–99. 47. J. P. Coulter, K. D. Weiss, and J. D. Carlson, Engineering applications of ER materials, J. Intell. Mater. Syst. Struct., 4: 248–259, 1993; also in M. A. Kohudic (ed.), Advances in Electrorheological Fluids, Lancaster, PA: Technomic, 1994, pp. 64–75. 48. R. Tao and G. D. Roy (eds.), Proc. 4th Int. Conf. ER Fluids, Singapore: World Scientific, 1994. 49. R. Tao (ed.), Proc. 3rd Int. Conf ER Fluids, Singapore: World Scientific, 1992. 50. J. D. Carlson, A. F. Sprecher, and H. Conrad (eds.), Proc. 2nd Int. Conf. ER Fluids, Lancaster, PA: Technomic, 1990. 51. H. Conrad, J. D. Carlson, and A. F. Sprecher (eds.), Proc. 1st Int. Symp. ER Fluids, Raleigh, NC: Univ. Eng. Publ., 1989. 52. K. O. Havelka and F. E. Filisko (eds.), Progress in Electrorheology: Science and Technology of ER Materials; Proc. Am. Chem. Soc. Symp. ER Materials and Fluids, New York: Plenum, 1995. 53. D. A. Siginer and G. S. Dulikravich (eds.), Developments in ER Flows, New York: Amer. Soc. Mech. Eng., 1995. 54. D. A. Siginer et al. (eds.), Developments in ER Flows and Measurement Uncertainty, New York: Amer. Soc. Mech. Eng., 1994. 55. T. R. Weyenberg, J. W. Pialet, and N. K. Petek, The development of ER fluids for an automotive semi-active suspension system, Proc. 5th Int. Conf. ER Fluids, MR Suspensions, Assoc. Technol., Sheffield, UK, 1995, 1996, pp. 395–403. 56. N. K. Petek, R. J. Goudie, and F. B. Boyle, Actively controlled damping in ER fluid-filled engine mounts, Proc. 2nd Int. Conf. ER Fluids, Raleigh, NC, 1989, 1990, pp. 409–418. 57. E. V. Korobko, The state of the art of ER studies in the former USSR, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 6.1. 58. A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua, New York: McGraw-Hill, 1980. 59. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., vol. 6, Oxford: Pergamon and Butterworth-Heinemann, 1987. 60. L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., vol. 7, Oxford: Pergamon and Butterworth-Heinemann, 1986. 61. W. L. Wilkinson, Non-Newtonian Fluids, New York: Pergamon, 1960. 62. H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology, Amsterdam: Elsevier, 1989. 63. T. E. Faber, Fluid Dynamics for Physicists, Cambridge, UK: Cambridge Univ. Press, 1995, chap. 10. 64. J. W. Goodwin, Rheological characterization of ER fluids, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.6. 65. D. R. Gamota and F. E. Filisko, Dynamic mechanical study of ER materials: Moderate frequencies, J. Rheol., 35 (3): 399–425, 1991. 66. D. R. Gamota and F. E. Filisko, High frequency dynamic mechanical study of an aluminosilicate ER material, J. Rheol., 35 (7): 1411–1424, 1991. 67. T. C. B. McLeish, T. Jordan, and M. T. Shaw, Viscoelastic response of ER fluids. I. Frequency dependence, J. Rheol., 35 (3): 427–448, 1991. 68. A. B. Metzner, Fluid mechanics, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.7.

ELECTROSTATIC DISCHARGE 69. H. A. Pohl, Dielectrophoresis: The Behavior of Neutral Matter in Nonuniform Electric Fields, Cambridge, UK: Cambridge Univ. Press, 1978. 70. T. C. Jordan and M. T. Shaw, Structure in ER fluids, Proc. 2nd Int. Conf. ER Fluids, Raleigh, NC, 1989, 1990, pp. 231–251. 71. Y. Chen, A. F. Sprecher, and H. Conrad, Electrostatic particleparticle interactions in ER fluids, J. Appl. Phys., 70: 6796–6803, 1991. 72. R. Friedberg and Y. K. Yu, Energy of an ER solid calculated with inclusion of higher multipoles, Phys. Rev. B, 46: 6582–6585, 1992. 73. H. J. H. Clercx and G. Bossis, Many-body electrostatic interactions in ER fluids, Phys. Rev. E, 48: 2721–2738, 1993. 74. H. J. H. Clercx and G. Bossis, Electrostatic interactions in slabs of polarizable particles, J. Chem. Phys., 98: 8284–8293, 1993. 75. L. Fu and L. Resca, Exact treatment of the electrostatic interactions and surface effects in ER fluids, Phys. Rev. B, 53: 2159– 2198, 1996. 76. L. Fu and L. Resca, Exact theory of the electrostatic interaction in ER fluids and the effects of particle structure, Solid State Commun., 99: 83–87, 1996. 77. J. D. Jackson, Classical Electrodynamics, 2nd ed., New York: Wiley, 1975. 78. C. F. Zukoski, Mechanisms of ER effects, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.3. 79. A. M. Kraynik, Modeling and simulation of ER fluids, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.5. 80. P. L. Taylor, Cooperative aspects of ER phenomena, in Electrorheological Fluids: A Research Needs Assessment, Washington, DC: Natl. Tech. Inf. Service, 1993, sect. 5.4. 81. L. Fu and L. Resca, Analytic approach to the interfacial polarization of heterogeneous systems, Phys. Rev. B, 47: 13818–13829, 1993. 82. L. Fu and L. Resca, Electrical response of heterogeneous systems with inclusions of arbitrary structure, Phys. Rev. B, 49: 6625– 6633, 1994. 83. L. Fu, Electrical response of heterogeneous systems of nonlinear inclusions, Phys. Rev. B, 51: 5781–5789, 1995. 84. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., vol. 8, Oxford: Pergamon and Butterworth-Heinemann, 1984, sec. 11, chap. 2. 85. L. Fu and L. Resca, Optical response of arbitrary clusters of structured particles, Phys. Rev. B, 52: 10815–10818, 1995. 86. A. P. Gast and C. F. Zukoski, ER fluids as colloidal suspensions, Adv. Colloid Int. Sci., 30: 153–202, 1989. 87. T. Garino, D. Adolf, and B. Hance, The effect of solvent and particle dielectric constants on the ER properties of water-free ER fluids, Proc. 3rd Int. Conf. ER Fluids, Carbondale, IL, 1991, 1992, pp. 167–174.

LIANG FU LORENZO RESCA Catholic University of America

ELECTROSTATIC CHARGES. See ELECTROSTATIC PROCESSES.

751

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Wiley Encyclopedia of Electrical and Electronics Engineering Field Ionization Standard Article W. Schweizer1 1Universität Tübingen, Tübingen, Germany Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5905 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (213K)

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Abstract The sections in this article are Field Ionization Photoionization About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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FIELD IONIZATION

415

FIELD IONIZATION Electric fields play an important role in understanding the observed physical properties in both laboratory and natural plasmas. Electric fields are not only due to external fields but also to ions and free electrons in the environment of the observed object. We will first discuss the basic definition of field ionization. Electric fields allow the valence electron of an J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

FIELD IONIZATION

atom to ionize by tunneling through the combined electric and Coulomb potential. Hence in external electric fields all bound states become quasibound due to tunneling. Then we will discuss tunneling through a potential barrier in general, followed by a basic discussion of the Stark effect. Because not only static external electric fields, which give rise to the celebrated Stark effect, are of principal interest, we will also discuss the behavior of a Rydberg electron under subpicosecond electric pulses. These studies are not only of fundamental interest but give also a better understanding of problems such as the transport of ions and atoms through solids or for fusion plasma modeling and diagnostics. To get a better feeling for the characteristic times involved we will add some remarks about wave-packet propagation due to the coherent excitation of Rydberg states via an initial laser packet. Impurities can be regarded as solid-state atoms. We will also briefly discuss ionized impurities before turning to the fundamental questions of spectral line broadening. Then we will discuss some plasma aspects, like the electron density in some natural plasmas and the distribution among the ionization stages of an atom in the ionization equilibrium. Ionization is not only due to external electric fields but also to single- or multiphoton ionization. These topics will be the subject of the chapter ‘‘Photoionization.’’ Historically important for the development of quantum mechanics was the photoelectric effect, which will be explained later. After that we will discuss the basic definitions of photoionization and selection rules for radiative transitions due to the dipole approximation. If ionization is not possible by absorption of a single photon, ionization could be also achieved by multiphoton absorption. In such processes, not only the least number of photons necessary for ionization will be absorbed by the valence electron. The absorption of photons in excess gives rise to the above-threshold ionization, as discussed later. This article will be completed by a discussion of impact ionization, which is due to the collision of an atom or ion with charged particles. FIELD IONIZATION Basic Definition The motion of an atomic electron in an external electric field is subject to two fields of force: an attractive Coulomb force due to the positive atomic core and the applied external electric field. Hence the potential energy of the electron is given by Vt (rr ) = −

Zq2 + qFz, r = (x, y, z) |rr|

6 4 2 Vt (z)

416

Coulomb potential

0 –2

Electric field

–4 –6

Total potential

–8 –10

–4

–2

0

2

z

4

Figure 1. Effect of the electric field on the potential energy of an electron in the Coulomb field of a proton in atomic units. (The electric field strength is 5.14 ⫻ 109 V/cm).

Potential Barrier One of the most fascinating quantum phenomena is tunneling. A classical particle with energy E smaller than the potential height V of a potential barrier cannot penetrate this wall. In quantum mechanics the continuity of the wave function in space implies that the amplitude of a wave function will only decrease inside the wall and hence there is a finite probability of finding the particle beyond the classical turning point. Therefore a quantum particle has a finite probability to penetrate any potential barrier, and thus tunneling phenomena are strictly quantum mechanical with no counterpart in classical dynamics. For the potential wall, Fig. 2, the transmission coefficient or probability for tunneling is Pt 앜 exp[(2d/h)兹2m(V ⫺ E)], where h is the Planck’s constant, d is the width and V is the height of the potential wall, and E is the energy of the particle. For an electron, for example, the probability Pt is 앒 0.1 for tunneling through a barrier of width d ⫽ 0.2 nm and V ⫺ E 앒 1 eV. In nature there are numerous examples for tunneling, for example, field ionization, nuclear fusion and fission, impurity tunneling, and so forth, which give rise to many applications, for example, tunneling microscopes (2), tunnel and Esaki diodes (3), and Josephson junctions. Details about tunneling can be found in any quantum-mechanic textbook, for example, in Cohen–Tannoudji, Diu, and Laloe¨ (4).

(1)

if the electric field axis points into the z direction and the electric field strength is measured in atomic units, that is, in units of F0 ⫽ 5.14 ⫻ 109 V/cm. An example is shown in Fig. 1. The total potential Vt has a saddle point for x ⫽ 0 ⫽ y, z ⫽ 兹q/F and here the potential energy takes the value Vs ⫽ ⫺2q兹ZqF. Moreover, for Z ⫽ 1 the contribution to the potential energy of the two fields is identical (see Fig. 1). A classical electron will be bound for energies smaller the saddle-point energy Vs inside the potential walls as the walls are impenetrable barriers. As suggested first in 1928 by Oppenheimer (1) the application of electric fields enables bound electrons to escape by tunneling in a process called field ionization.

Figure 2. Potential barrier; the energy is such that a classical particle would be totally reflected by the barrier. On the left-hand side: The superposition of the incoming and the reflected wave and on the right-hand side of the barrier the outgoing wave.

FIELD IONIZATION

As tunneling phenomena are strictly quantum mechanical the tunneling process is described by the Schro¨dinger equation i~

∂ ~2 ψ (rr, t) = − ψ (rr, t) + V (rr )ψ (rr, t) ∂t 2m

(2)

where V(r) is the potential, ⌬ is the Laplacian operator, r is the coordinates, t is the time, ប ⫽ h/2앟, and m is the mass of the particle. During the tunneling process the particle is behaving like a wave but can be detected in the region beyond the barrier as a localized particle. Because the wave function is a continuous function the stationary wave function ␾(r) can be computed by matching the solution in different areas together: φout (rrs ) = φin (rrs )

(3)

∇φout (rrs ) = ∇φin (rr s )

(4)

with ␾in and ␾out the solutions in- and outside the barrier and rs the border position of the barrier. In the WKB approximation the tunneling probability Pt for one-dimensional systems is proportional to  
  b 2m Pt ∝ exp −2 [V (r) − E] dr (5) 2 a

~

where a and b correspond to the classical turning points, and the proportionality prefactor depends on whether the potential changes continuously or not. If the potential changes continuously at both turning points this prefactor is 1, and for a discontinuity it becomes energy dependent. Solutions for wave packets can be obtained by Fourier-transforming the stationary solution. In this context the question arises: Is there a time analog to the classical time for a particle tunneling through a potential barrier? This is a current controversy under discussion (5) because some experimental results show superluminal velocities (6). A critical review can be found, for example, in Refs. 7 and 8.

417

the distance from the atomic center the linear Stark effect vanishes if the unperturbed eigenstates of the physical system under consideration are not degenerate. Due to the symmetries of hydrogenlike atoms the eigenstates are degenerate with respect to the angular momentum and the magnetic quantum number. Therefore the linear Stark effect can be observed only for those atoms and ions. For nonhydrogenic atoms, for example, alkali-metal atoms, the angular momentum degeneracy is removed due to the multielectron interaction of the atomic core and hence only the quadratic Stark effect can be observed. For stronger electric fields perturbation methods are no longer applicable and hence more advanced numerical methods have to be used to obtain the correct spectrum. One of the most challenging numerical methods is given by using complex coordinate rotations of the space coordinates in combination with computations of the Hamiltonian matrix. By this method resonances are mapped onto bound states in a complex extension of the state space. In this context the question arises for which electric field strengths perturbation methods are no longer applicable. A rough estimate is given by the atomic electric field strength, for which the electric field potential equals the Coulomb potential. This field strength depends on the atomic system and on the principal quantum number of the excited state under consideration. For the hydrogen ground state this value is given by the previously mentioned ‘‘atomic units’’ and is equal F0 ⫽ 5.14 ⫻ 109 V/cm and scales with n4, n being the principal quantum number. Therefore for the n ⫽ 500 excited states the critical value is already 82 mV/cm. The Hamiltonian of a hydrogenlike atom placed in an homogeneous static electric field directed along the z axis reads in atomic units V (rr ) = −

1 + Fz r

(6)

and hence the Schro¨dinger equation becomes separable in parabolic coordinates ζ = r + z,

η = r−z

(7)

The Stark Effect Figure 1 shows the effect of an electric field on the total potential of an electron in the Coulomb field of a positive charge. Due to tunneling there is even for energies smaller the saddle-point potential Vs a nonzero probability of the electron to penetrate through the potential barrier. Thus the classically bound region is always coupled to the dynamical unbound region. As the tunneling probability depends on both the height and the width of the barrier, the effect of field ionization becomes most important close to the saddle-point energy. In general, the effect of a static electric field on atomic spectra is called Stark effect. The corresponding behavior for an external magnetic field is termed the Zeeman effect. A collection of recent articles that present the state of the art in the theoretical, computational, and experimental studies of atoms, molecules, and quantum nanostructures in strong external fields can be found in Schmelcher and Schweizer (9). For sufficiently small electric fields the electric field potential can be treated as a perturbation of the Coulomb potential. The first-order corrections to the energy are called the linear Stark effect. As the electric field potential is proportional to

The resulting equations for ␨ and ␩ are


d2
d2 dη2

F E m2 − 1 β − + − ζ+ dζ 2 4ζ 2 ζ 4 2 −

F E m2 − 1 1 − β + η+ + 2 4η η 4 2

 

f (ζ ) = 0

(8)

g(η) = 0

(9)

with E the energy, m the conserved magnetic quantum number, and 웁 the separation constant, the so-called fractional charge. The uphill equation (8) has only bound solutions, whereas the downhill equation (9) describes the tunneling solution. The eigenstates f and g are characterized by the parabolic quantum numbers n1 and n2, which are related to the principal quantum number n via n = n1 + n2 + |m| + 1

(10)

and parabolic quantum states are usually represented by 兩nkm典, with k ⫽ n1 ⫺ n2. The fractional charge is given by

418

FIELD IONIZATION

2웁 ⫺ 1 ⫽ (n1 ⫺ n2)/n and the energy eigenvalue in first-order perturbation theory by E=−

1 3nkF + 2 2n 2

(11)

In the field-free atom the n-manifold is degenerate and, neglecting the electron spin, consists of n2 states corresponding to different values for the angular momentum l and the magnetic quantum number m. This degeneracy is broken by the electric field. To describe the Stark effect in Rydberg spectra of bound and autoionizing states Harmin (10) has developed a quantum defect model in which the Stark effect is only taken into account for large distances from the atomic core. All states are quasibound since they can tunnel through the potential barrier into the continuum and hence the eigenenergies are associated with a complex energy value the real part of which Er, gives the energy and the imaginary part the halfwidth, ⫺⌫/2, of the decaying state. For time-independent quantum systems the time development of a state is given by 兩r, t典 ⫽ exp(⫺iEt/ប)兩r典 with E ⫽ Er ⫺ ⌫/2 and hence ⌫ is the transition rate given by Fermi’s golden rule (4). The complex energies can be directly obtained by the complex coordinate rotation (11). r → r exp(i)

(12)

As shown by Seipp, Taylor, and Schweizer (12) tunneling below the classical ionization limit Esp is relatively weak and even above Esp some resonances exist with negligible small widths. Qualitative differences occur for hydrogen and alkalimetal atoms due to the separability of hydrogen in electric fields. The spectra for alkali-metal atoms are much more disordered and no closely bound resonances above the classical ionization limit occur. If we turn on a magnetic field parallel to the electric field all resonances get shifted in the direction of positive energy, as is expected from the positive quadratic Zeeman effect. For more details see Ref. 9. Strong electric and magnetic fields are not only of interest under laboratory conditions but also in quest of observed optical absorption structures of compact astronomical objects. For example, for magnetic white dwarf stars additional electric fields (13) of the order of 108 V/cm occur, which give rise to an observable shift of the spectral lines. Stark effects are not only due to external electric fields. Atomic beams with the beam direction not in coincidence with the magnetic field axis of an external magnetic field give rise to the so-called motional Stark effect. In external magnetic fields the center-of-mass coordinate is no longer separable from the relative coordinate and hence in the Schro¨dinger equation of the relative coordinates an additional potentiallike term occurs, which depends linearly on the relative coordinates and on the wedge product of the magnetic field with the center-of-mass momentum. Therefore this additional term shows the same effect as an external electric field and is therefore called motional electric field. Studying the external fields for the hydrogen atom gives also some information about shallow donor states, as the Hamiltonian for many systems are equivalent. For example, for the donor in GaAs the effective electron mass is 0.0665, and the dielectric constant 12.56. In atomic units these two values are equal to 1. The Rydberg constant becomes 46.1

(11 ⫻ 104) and the Bohr radius 9.96 nm (0.0529 nm), where the values in parentheses are the corresponding hydrogen values. Hence an electric field of about 1010 V/m will have the same impact on a H atom as a field of 2 ⫻ 104 V/m on a donor. More details can be found, e.g., in Refs. 14 and 15. Pulsed Field Ionization Recently various experimental and numerical methods for investigating the dynamics of Rydberg electrons in external fields have been developed. The Rydberg wave packets have been created by photoexcitation of ground-state atoms using ultrashort laser pulses. The bandwidth of these laser pulses exceed the level spacing in the Rydberg-energy region. Therefore the Rydberg wave packet is built by several dipole-allowed coherently excited Rydberg levels. Time-resolved spectroscopic experiments show radial and angular oscillations of the electronic Rydberg wave packet. The period of the radial oscillation of the Rydberg wave packet is given by the Kepler time τrad = 2π




1 −2E

−3/2 (13)

with E the energy expectation value. Typical Kepler times are of the order of 2 ps. The static external electric field produces a torque that causes the angular momentum to progress from its initial value towards higher angular moments. The angular return time depends on the electric field strength and is given to first order by τang =

2π 3




1 −2E

3/2 F

(14)

this yields typically values of 10 to 50 ps for electric field strengths of 200 V/cm to 900 V/cm; see Ref. 16 and references therein. Very recently, experimental (17) and theoretical (18) work has explored the evolution of Rydberg wave packets of alkalimetal atoms subject to subpicosecond ‘‘half-cycle’’ electromagnetic pulses. The characteristics of half-cycle pulses are very similar to electric field pulses generated by the passing-by projectile in an ion–atom collision. Thus the study of the dynamics of Rydberg atoms subject to those pulses is in addition of practical importance in problems such as transport of ions and atoms through solids or plasma modeling and diagnostics of high-temperature plasmas, like fusion plasma. In contrast to the laser-excited wave packets the bandwidth of the initial laser wave packet is so small, that only a single stationary state will be populated. This Rydberg alkalimetal state will then be subject to a kick from a unidirectional, pulsed electric field. The pulses in this experiment are short compared to the Kepler period of the Rydberg electron. Approximately the only effect of this impulse is to change the electron’s momentum in the field direction. If the electric field points into the z direction the electron’s change of energy is given by E = ppz + p2 /2

(15)

where ⌬p is the change of the momentum of the electron in the field direction. If this change in the energy is greater than the binding energy EB ⫽ ⫺
(n ⫺ 웃l)2, with n the principal

FIELD IONIZATION

quantum number and 웃l the quantum defect of the alkalimetal atom, the electron will leave the atom. Therefore the percentage of the atoms ionized equals the percentage of the electrons with the z component of the electrons larger than pz ≥

−2EB − p2 2p

(16)

Note, that in a dc electric field the energy levels are energetically split due to the Stark effect and all states become quasibound due to the finite tunneling probability in an electric field, whereas due to the short electric pulse in pulsed field ionization tunneling becomes unimportant. Ionization in pulsed electric fields are only due to the momentum change of the Rydberg electron due to the electric field kick. This interpretation is supported by both the experimental results and the theoretical computations. Therefore the quantum dynamics is adequately described by a single electron in the combined field of the effective one-electron atomic Hamiltonian and pulsed electric field. For more details see Ref. 18 and references therein. Ionized Impurities Real crystals are imperfect and contain a variety of types of defects. Defects at the atomic level are point defects. Point defects can be native defects, such as vacancies and interstitials of the crystal atoms, as well as foreign—impurity— atoms, which are generally intentionally added to the material. Impurities give rise to two kinds of energy levels, which are often denoted as shallow and deep energy levels. Deep energy levels are levels towards the middle of the energy gap, whereas shallow impurities have levels close to the band gap. Shallow impurities are classified as donors or acceptors depending on whether they produce electron or hole conductivity. An impurity breaks the crystal symmetry of a perfect crystal and will give rise to different electric properties. In a simple picture but with sufficient accuracy shallow donor states can often be described by a solid-state analog (previously noted) of the hydrogen atom with rescaled energy, Bohr radius, effective charge, and mass (19). Under this assumption, the donor electron will have hydrogenlike bound states with energy En =

−R 2n2

(17)

but with a Rydberg constant R much smaller and will to some extent show the same qualitative spectroscopic behavior under external fields as the hydrogen atom. Note that since the system is in a medium the Coulomb potential have to be changed to e e → r r

(18)

where ⑀ is the dielectric constant of the material. We will now consider the ionization of a neutral impurity in a graphical way:

I 0 → I + + e− for a donor and I 0 → I − + e+ for an acceptor

(19)

419

with I0 the neutral impurity. The donor ionization reaction corresponds to an emission of an electron to the conduction band and the acceptor ionization reaction to an emission of a hole to the valence band, which is equivalently described by the capture of an electron from the valence band. These processes could also be generalized to multi-ionized donors and acceptors. The single ionization energy is 10⫺4 to 10⫺2 times smaller the hydrogen ionization energy of 13.6 eV. Therefore at room temperature most impurities are ionized and contribute to the conductivity because their ionization energies are comparable to kBT, kB being the Boltzmann constant. (1 eV is equivalent to a temperature of 11,400 K. Hence for hydrogen the ionization energy is equivalent to about 150,000 K, whereas for the previously mentioned shallow donor state of GaAs we get about 65 K.) Therefore even at low temperatures a finite nonzero concentration of ionized impurities exist. These ionized impurities create electric fields due to their charge and hence neutral donors are embedded in local electric fields with a randomly distributed field strength, and this causes Stark shifting in the level structure. Because in electric fields the angular momentum is no longer conserved, donor states with different quantum numbers are mixed and forbidden transitions (discussed later) are observed. Moreover, this leads to an inhomogeneous line broadening (14). For more details see, for example, Refs. 20 and 21. Spectral Line Broadening Any observed spectral line shape has a finite width not only due to the instrument used in the creation and observation process but due to fundamental processes. In general we distinguish between three different sources of these broadening processes: the natural broadening, the Doppler broadening, and the pressure broadening. The natural broadening of spectral lines is due to the finite lifetime of the corresponding states. Due to the uncertainty principle ⌬E ⌬t 앒 ប, the energy spread of a transition is inversely proportional to its life time. In the optical region the natural broadening is of the order of 10⫺7 to 10⫺8 of the frequency of the observed line. Classically this can be also understood by the damping of an harmonic oscillation. Doppler broadening is due to the Doppler effect, which is the shift in the wavelength caused by the relative velocity between the atom (source) and the observer. In astrophysics the so-called redshift of atomic spectral lines is used to measure the velocity of far stars compared to our earth and estimating from this value the distance of the observed star. In addition, for extremely compact astronomical objects the gravitational redshift, due to the strong gravitational potential of massive stars, could be observed. Atoms, ions, or molecules are never isolated single objects. This gives rise to effects due to neighboring objects, which is called pressure broadening. An emitting atom within a certain distance of a neighboring ion or electron is perturbed by an electric field, and the interaction between the atom and the field is described by the Stark effect. The electric field due to plasma ions and electrons near the radiating object is called the plasma microfield. A fundamental role in the microfield theory is played by the Holtsmark model, which approximates the effect of neighboring charged particles by an aggregate of randomly disposed immobile ions and electrons. Hence it is obvious that the width of the spectral lines depend on

420

FIELD IONIZATION

the temperature, pressure, and electron density of the environment of the radiating object. For more details, see Ref. 22. Plasma Aspects Plasma, generally speaking, is an internal electric neutral gas that consists of neutral and ionized atoms and molecules and of electrons due to the ionization processes. Typical temperatures of plasmas are above 10,000 K, in which the fastest particles in the thermal Maxwell distribution undergo collisions energetic enough for ionization processes. Plasma can also exist under much lower temperatures. In the vacuum of outer space ionization of hydrogen in hydrogen clouds is due to photoionization by starlight. The density in these clouds is extremely small and hence collisions are rare. Consequently nearly all matter in the universe is in the plasma state. On earth, however, the atmospheric density cannot support such plasma. Nevertheless plasma can also exist at room temperature in metals and semiconductors due to the extremely small band energies. One of the basic characteristics of plasmas is the importance of collective effects. Every charged particle interacts with several neighbors because of the long-range electrostatic forces. Hence the motion of the plasma particles are correlated. One important criterion is the Debye shielding radius, which measures the distance to which the electric fields of an individual charged particle extends before it is effectively shielded by its opposite charged neighbors. One of the fundamental plasma parameters is the electron density. In simple models the Debye shielding radius as well as the Holtsmark electric field is given by the electron density. For the interstellar matter the electron density varies from parts up to a few tens electrons per cm⫺3, for gaseous nebula this is of the order of a few hundred cm⫺3, in the solar corona 104 cm⫺3, for flames of the order of 108 cm⫺3 and in the stellar atmospheres it is up to roughly 1018 cm⫺3 (see Ref. 23). The Maxwell relation describes the distribution of the velocity among particles and the Boltzmann relation the distribution of the population of the discrete energy levels. The distribution among the ionization stages of an atom in the ionization equilibrium due to the process (20)

is described by Saha’s equation (23):

N (n) j

=

(n+1) 2(2πmkB T )3/2 gi exp h3 g(n) j

PHOTOIONIZATION The Photoelectric Effect and Ionization Potential The emission of electrons from the surface of a metal was discovered by Hertz in 1887. Later experiments by Lenard showed that the kinetic energy of the emitted electron was independent of the intensity of the incident light and that there was no emission unless the frequency of the light was greater than a threshold value typical of each element. Einstein realized that this is what is to be expected on the hypothesis that light is absorbed in quanta of amount ប웆. In the photoelectric effect an electron at the surface of the metal gains an energy ប웆 by the absorption of a photon. The maximum kinetic energy of the ejected electron is 1 mv2max 2

= ~ω − e

(22)

where ⌽ is the contact potential or the work function of the surface measured in electron volts (see Table 1). Einstein’s theory predicts that the maximum kinetic energy of the emitted photoelectron is a linear function of the frequency of the incident light, a result which was later confirmed experimentally by Millikan (1916) and which allowed to measure the value of the Planck constant h. The lowest level is called the ground state; higher levels are called excited states. The least energy E needed to remove a single electron from an atom or ion in its ground state is called the ionization energy of the atom or ion. This energy corresponds to a potential difference—the ionization potential—for a bound electron being removed to infinity. This ionization could be due to particle scattering (impact ionization) or due to photoionization. Photoionization

A + + e− ↔ A

Ne Ni(n+1)

equation involve absolute temperatures. If all these temperatures are equal the plasma is in the local thermodynamical equilibrium, which is usually assumed to compute from spectral data the plasma densities and single ion densities. Because of these additional limitations the overall accuracy is often poor. (For more details see Ref. 24.)


κ + E

i

− Ej

kB T

 (21)

where Ne is the number density for the electron, N(x) k is the xtimes ionized atom in the kth excited state, g are the corresponding weights due to the degeneracy of the state under consideration, E0 is the corresponding energies, ␬ is the ionization energy, m is the electron mass, I is the temperature, and kB is the Boltzmann constant, and h is the Planck constant. Thus by measuring the absolute spectral intensities the particle density in a certain excited state can be obtained and hence the plasma densities. All the different thermodynamical processes—radiation, kinetic distribution of the particle, distribution of the atomic excitation and ionization stage— described by the Planck, Maxwell, Boltzmann, and Saha

Although investigations of photoionization started already at the end of the nineteenth century when the photoelectric effect was discovered, photoionization still remains an active field in research that got an additional stimulus from laser physics during the last two decades. In the following we will describe some photorelated processes in a graphical way. In this process descriptions A is either an atom, or ion, or molecule but could also stand, e.g., for a metal cluster. A* is the excited object, A⫹ is the same Table 1. Contact Potential ⌽ and Corresponding Wavelength of the Incident Light, the So-Called Red Border Wavelength ␭r

Li Na K Rb Cs

˚) ␭r (A

⌽ (eV)

5280 5300 5460 5800 6400

2.38 2.33 2.26 2.13 1.93

Cu Ag Au Pt W

˚) ␭r (A

⌽ (eV)

2880 2610 2600 1960 2720

4.29 4.73 4.76 6.37 4.57

FIELD IONIZATION

object but one-time-higher ionized than A. Therefore if A stands, e.g., for the three-times ionized nitrogen N IV, A⫹ is one times-higher-ionized, hence in the example N V. (In this notation the Roman numerals stands for the ionization stage, I for neutral, II for single ionized, III for double ionized objects, and so forth.) Graphically the excitation of an object A can be described via hν + A → A∗

(23)

In this photoabsorption process the photon h␯ is annihilated. When h␯ exceeds the ionization energy of A, the process of photoionization can occur: hν + A → A+ + e−

(24)

The inverse process is known as radiative recombination. The photoionization reaction hν + A− → A∗ + e−

(25)

involving a negative ion (excited or not) A⫺ is called photodetachment. Photoionization cross sections are dominated by resonances in certain energy regions. In this energy area not only direct photoionization process, like the one described by Eq. (24), are of importance, but in addition ionization processes via an intermediate resonance state hν + A → A∗ → A∗,+ + e−

(26)

The interference between the direct and the intermediate photoionization process leads to the so-called Fano line profile in the photoionization cross section ␴(E), which can be parameterized by a simple formula

~k

2 2

(28)

2m

where Wi and Wf the eigenenergies of the target in the initial and final state, h␯ the energy of the photons, and the photoelectrons are emitted with momentum k. The ionization cross section is given by the coupling between the target and the electromagnetic field, described by the vector potential A in the Coulomb gauge, ⵱ ⭈ A ⫽ 0. Using Fermi’s golden rule in the dipole approximation (4) the transition probability for a transition 兩␺i典 씮 兩␺f典 is given by wif(q) =

4 3


1 E

e2 4π0 ~

i

− Ef

~c

l = l − 1 = ±1,

m = m − m = 0, ±1

(30)

hold or in the j ⫽ l 丣 s coupling scheme, with l the orbit angular momentum and s the spin but

j = 0 → j = 0 is forbidden

(31)

(27)

with ⑀ ⫽ (E ⫺ Er)/(⌫/2), Er the resonance energy, ⌫ the resonance width and q the shape parameter. In all the processes previously described the energy is conserved; hence in the nonrelativistic limit hν + Wi = Wf +

For atoms with two and more electrons several approximations have been developed, see, e.g., Bartschat (26) for the R-matrix approach and Fano and Rau (27) for multichannel quantum defect theory, to name only two very successful methods that have also been used in various combinations. The basic idea of the R-matrix approach is to separate the full space into subspaces in which solutions can be numerically obtained and to merge these subsolutions together to get the correct solution; the idea of quantum defects is to describe the deviations from an exact Coulomb potential in the asymptotic limit. Simplified, this leads to a linear combination of the regular and irregular Coulomb function in which the relative contribution of the irregular Coulomb function is weighted by the tangent of the quantum defect. For radiative transitions, whether photoabsorption, photoemission, or photoionization, several selection rules hold. The interaction with the electromagnetic wave can be expanded as a series in order of decreasing strength: electric dipole, magnetic dipole, electric quadrupole, and so forth. Most computations are done in the electric dipole approximation. This is justified as the dipole transitions are the most probable and hence all other transitions are called forbidden. The selection rules are due to the angular momentum coupling of the initial state with the electromagnetic wave in the dipole approximation. The dipole operator is a spherical vector operator; hence we have to couple in the orbit momentum frame the angular momentum 1 with the angular momentum l of the initial state. Therefore radiative transitions between states l, m and l⬘, m⬘ are possible only if

 j = 0, ±1

q+ σ (E) ∝ 1 + 2

421

3

a20 dif(q)

(29)

(q) 0 2 (q) with the dipole strength d(q) (q ⫽ 0, ⫾1) if ⫽ 兩具f兩r /a 兩i典兩 , and r the spherical component of the relative vector, Ei and Ef the initial and final energy. Only for hydrogenic atoms and ions transition probabilities can be calculated analytically (25).

Insofar as l and s are good quantum numbers, there are additional selection rules l = 0, ±1

but l = 0 → l = 0 is forbidden and s = 0 (32)

(For details about angular momentum coupling see Ref. 4 and about selection rules see Ref. 25). Of course in external electric or magnetic fields additional transitions occur due to inter-l- or inter-m mixing (9), which means that either l or m or both quantum numbers are no longer conserved quantum numbers. This has an important effect, e.g., in the interpretation of observed absorption spectra from magnetic white dwarf stars (13) or the broadening and occurrence of forbidden transitions in shallow donor states (14). Above-Threshold Ionization If the energy of a photon is smaller than the ionization potential of the atom in its initial state, ionization will only be possible by absorption of several photons. This process is called multiphoton ionization. Experiments showed that the energy of the emitted electron was larger than the total energy of the least number of photons necessary for ionization. The explanation for this unexpected behavior is that the already emitted electron absorbs additional photons and hence gains a larger kinetic energy. This process is called above threshold

422

FIELD IONIZATION

ionization or sometimes excess photon ionization. Hence the kinetic energy of the electron is given by 1 mv2max 2

= n~ω − e

(33)

with n the number of absorbed photons and ⌽ the ionization potential of the atom and the sufficient number of photons for ionization is smaller than n. The probability for photon absorption in excess by the electron increases with increasing laser intensity and becomes for high laser intensities larger than the probability to absorb only the least of photons necessary for ionization (for further details see Ref. 28). Impact Ionization Ionization processes caused by the interaction with charged particles, like electrons, positrons, protons, or even ions, is called impact ionization. These collisions are of importance, for example, in plasma- and astrophysics. In a simplified model this ionization is solely caused by the Coulomb interaction between the charged projectile and the atomic or ion target. In a graphical description these processes can be described by A + B(q) → A+,∗ + B(q) + e−

(34)

with A the target atom or ion, B the projectile of charge q, A⫹,* the higher ionized and, in general, excited target object after the scattering process, B⬘(q) is the projectile after the scattering process and e⫺ the additional free electron due to the impact-ionization process. The most important impact-ionization processes are those caused by free electrons and hence are called electron-impact ionization. The following scattering processes are of fundamental importance: The electron-atom collision A∗ + e− → A∗ + e−

(35)

in which the energy of the projectile electron is too small to ionize the probably initially excited target atom, direct processes A∗ + e− → A∗,+ + 2e−

(36)

in which the energy of the projectile electron is sufficiently high to ionize (single ionization or even multi-ionization) the target atom. In addition there are processes with an intermediate resonance state and of course the excited atom could emit photons. Note that due to the selection rules of radiative transition in some cases multiphoton emissions occur. For further details see Ref. 26.

7. E. H. Hauge and J. A. Støveng, Tunneling times: a critical review, Rev. Mod. Phys., 61: 917–935, 1989. 8. V. S. Olkohovsky and E. Recami, Recent developments in the analysis of tunnelling processes, Phys. Rep., 214: 340–356, 1992. 9. P. Schmelcher and W. Schweizer, Atoms and Molecules in Strong External Fields, New York: Plenum, 1998. 10. D. A. Harmin, Theory of the Stark Effect, Phys. Rev. A, 26: 2656– 2681, 1982. 11. Y. K. Ho, Phys. Rep., 99: 1–68, 1983. 12. I. Seipp, K. T. Taylor, and W. Schweizer, Atomic resonances in parallel electric and magnetic fields, J. Phys. B, 29: 1–13, 1996. 13. P. Fassbinder and W. Schweizer, The hydrogen atom in very strong magnetic and electric fields, Phys. Rev. A, 53: 2135– 2139, 1996. 14. D. M. Larsen, Inhomogeneous broadening of Lyman-series absorption of simple hydrogenic donors, Phys. Rev. B, 13: 1681– 1691, 1976. 15. A. van Klarenbosch et al., Identification and ionization energies of shallow donor metastable states in GaAs : Si, J. Appl. Phys., 67: 6323–6328, 1990. 16. P. Faßbinder, W. Schweizer, and T. Uzer, Numerical simulation of electronic wavepacket evolution, Phys. Rev. A, 56: 3626–3630, 1997. 17. R. R. Jones, Creating and probing electronic wave packets using half-cycle pulses, Phys. Rev. Lett., 76: 3927–3930, 1996. 18. F. Robicheaux, Pulsed field ionization of Rydberg atoms, Phys. Rev. A, 56: R3358–3361, 1997. 19. A. K. Ramdas and S. Rodriguez, Spectroscopy of the solid-state analogues of the hydrogen atom: donors and acceptors in semiconductors, Rep. Prog. Phys., 44: 1297–1387, 1981. 20. M. Altarelli and F. Bassani, Impurity states: theoretical, in W. Paul (ed.), Band Theory and Transport Properties, Amsterdam: North-Holland, 1982, pp. 269–329. 21. J. J. Baranowski, M. Grynberg, and S. Porowski, Impurities in semiconductors: experimental, in W. Paul (ed.), Band Theory and Transport Properties, Amsterdam: North-Holland, 1982, pp. 330–357. 22. V. S. Lisitsa, Atoms in Plasmas, Berlin: Springer-Verlag, 1994. 23. K. R. Lang, Astrophysical Formulae, Berlin: Springer-Verlag, 1980. 24. G. V. Marr, Plasma Spectroscopy, Amsterdam: Elsevier, 1968. 25. I. I. Sobelman, Atomic Spectra and Radiative Transitions, Berlin: Springer-Verlag, 1979. 26. K. Bartschat, Excitation and ionization of atoms by interaction with electrons, positrons, protons and photons, Phys. Rep., 180: 1–81, 1989. 27. U. Fano and A. R. P. Rau, Atomic Collisions and Spectra, New York: Academic Press, 1986. 28. J. P. Connerade, K. Dietz, and M. H. R. Hutchinson, Atoms in strong fields and the quest for high intensity lasers, Phys. Scr., T58: 23–30, 1995.

W. SCHWEIZER

BIBLIOGRAPHY

Universita¨t Tu¨bingen

1. J. R. Oppenheimer, Phys. Rev., 31: 66–81, 1928. 2. G. Binning et al., Phys. Rev. Lett., 49: 57–61, 1982. 3. L. Esaki, Long journey into tunneling, Les Prix Nobel en 1973, Stockholm: Imprimerie Royale, P. A. Norstedt, 1974.

FIELD, MAGNETIC. See MAGNETIC FLUX. FIELD-ORIENTED CONTROL. See MAGNETIC VARI-

4. C. Cohen-Tannoudji, B. Diu, and F. Laloe¨, Quantum Mechanics, Paris: Hermann, 1977.

FIELD PROGRAMMABLE CIRCUIT BOARDS. See IN-

5. R. Landauer, Barrier traversal time, Nature, 341: 567–568, 1989. 6. A. M. Steinberg, P. G. Kwiat, and R. Y. Chia, Measurement of the single-photon tunneling time, Phys. Rev. Lett., 71: 708–711, 1993.

ABLES CONTROL. TEGRATED SOFTWARE.

FIELD-PROGRAMMABLE GATE ARRAYS. See PROGRAMMABLE LOGIC ARRAYS.

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Wiley Encyclopedia of Electrical and Electronics Engineering Fusion Plasmas Standard Article Martin Greenwald1 1Massachussetts Institute of Technology, Cambridge, MA Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5906 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (461K)

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Abstract The sections in this article are Controlled Fusion Plasmas: Basic Processes Magnetohydrodynamics Waves in Plasma Transport Heating, Current Drive, and Fueling About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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FUSION PLASMAS

45

FUSION PLASMAS CONTROLLED FUSION For half a century, scientists and engineers working in labs scattered around the globe have labored to turn the promise of controlled fusion energy into a practical reality (1–3). In fusion reactions, nuclei of light elements join to form heavier elements releasing enormous quantities of energy. Nuclear fusion powers the sun and other stars and is responsible for creating all the elements of the periodic table out of the ‘‘primordial soup’’ of protons and neutrons. The advantages of fusion energy seem clear enough: It offers the prospect of inexhaustible energy with few of the environmental drawbacks associated with fossil fuels or nuclear fission. The quest has been challenging: The problem of controlled fusion has turned out to be much more difficult than anyone had imagined. Progress has been steady, however, with recent experiments generating almost 20 MW of fusion power for extended periods (4,5), and workers in the field are confident that a fusion device can be built which could produce power at levels comparable to conventional power plants. The remaining questions concern the technological practicality and economic viability of current approaches. This is not to say that the problems that remain are ‘‘simply’’ a matter of engineering. Further progress will require the continued coevolution of experimental science, theory, and technology that has characterized the field since its inception. The reactions of greatest interest for fusion energy involve deuterium (1D2), a stable isotope of hydrogen whose nucleus contains one proton and one neutron: 1D

2

1D 1D

2

+ 1 D2 ⇒ 2 He3 + 0 n1 + 3.2 MeV

(1)

+ 1 D2 ⇒ 1 T3 + 1 p1 + 4.0 MeV

(2)

2

4

+ 1 T ⇒ 2 He + 0 n + 17.6 MeV 3

1

(3)

Deuterium occurs naturally, making up about 0.015% of all hydrogen. Separating the deuterium present in a glass of water and fusing it in a reactor would produce energy equivalent to 250 gallons of gasoline. Tritium, 1T3, is an unstable isotope, with a half-life of 12.3 years, and would be bred from reactions between fusion neutrons and lithium, another abundant element. The net result would be to burn D and Li, producing He neutrons and energy. Despite complications raised by the necessity of breeding tritium, Eq. (3) has the largest reaction rate and thus is the most likely candidate for a fusion reactor. The potential to use these fusion reactions to produce power was recognized almost as soon as they were discovered. Speculation began in 1944 among scientists developing the atomic bomb at Los Alamos (1). By 1951, these discussions spawned experiments in the United States and Great Britain. The work was initially classified, as all atomic research was in those times. In the same period, a parallel effort was takJ. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

46

FUSION PLASMAS

ing place in the Soviet Union, also within the secret confines of the weapons program. It soon became clear to scientists working on the project that the problem would not be solved quickly and that no particular military importance would be attached to its solution. The issue of declassification was debated by national governments for several years, with a favorable decision made in 1957. Contacts among scientists working on the project from various nations increased and culminated in a dramatic joint conference held in Geneva in 1958. All the nations involved in fusion research came together to compare experience and exchange information. Fully operational versions of some fusion devices were put on display along with large-scale models of others. Despite the Cold War, which raged for another 30 years, controlled fusion research became a model for cooperation between otherwise hostile blocs. Requirements for Fusion Energy The basic requirements for a fusion reactor follow from the nuclear physics of the reactions. Figure 1 shows cross sections for the processes listed above versus the kinetic energy of an incident nucleus. The reaction rates peak at energies above 100 keV and are negligible below 5 keV. A comparison of cross sections with those for coulomb (elastic) scattering is also shown. Unlike the scattering rate, which diverges at low tem-

10–24

Coulomb scattering

Cross section (m2)

10–26

10–28

τE = W/P

D–T fusion

10–30 D–D fusion

10–32

1

10

peratures, the fusion rates peak at very high temperature and are always smaller by at least a factor of 10. This last fact means that even at the optimum temperature, around 30 keV for the deuterium–tritium reaction, nuclei must collide and scatter many times before they are likely to fuse. (In the plasma sciences, temperatures are usually measured in electronvolts; kT ⫽ 1 eV is equivalent to 11,600⬚C.) At these temperatures, matter is ionized into its ion and electron components, becoming a substance called plasma. In order not to lose the energy invested in bringing the nuclei to these high energies, it is necessary to confine the plasma for multiple scatterings during the time it takes for them to fuse. Plasmas are ubiquitous in nature, making up virtually all of the visible universe. Stars, the aurora, lightning, and neon lamps are all examples of plasmas, though they differ enormously in their temperature and density. Plasmas represent a fourth state of matter; if heat is added to a neutral gas, its temperature will rise until a point around 1 eV to 3 eV, where it begins ionizing. The phase transition is sharp; at only slightly higher temperatures the gas becomes a fully ionized plasma with new physical properties. Perhaps the most significant property that plasma has is that it is an excellent conductor of electricity. Strong currents can easily be induced to flow in plasmas, and they support a rich variety of electromagnetic and electrostatic waves. In steady state, power input to the plasma must balance power outflow. The input power can be externally applied, as it is in present-day experiments, or can come from the fusion reaction itself. In most reactor schemes, charged fusion products (i.e., the alpha particle in DT fusion) would be contained and thermalized to heat the plasma, with the energetic neutrons used to drive a heat engine. Plasma losses can include thermal conduction and convection as well as various types of electromagnetic radiation. The most important radiation process is bremsstrahlung (‘‘braking’’) radiation, which is generated whenever electrons accelerate as they do when they collide with other charged particles. Conductive and convective losses are analogous to heat transfer in ordinary materials, though details of the transport mechanism are quite different for plasmas. In fusion research, it is customary to define an energy confinement time:

100

1000

Deuteron energy (keV) Figure 1. Cross sections for fusion reactions are shown and compared with the cross section for Coulomb scattering. The fusion curves peak at very high energies showing that high temperatures are required for fusion energy, At these temperatures, ordinary matter is ionized becoming a plasma. The cross section for elastic scattering is larger than that for fusion, even at the peak of the fusion curve. For a practical fusion energy device, the plasma must be confined for times long compared to the scattering time.

(4)

where W is the total kinetic energy stored in the plasma, and P is the power flowing out via conduction and convection. Lawson (6) showed that elementary power balance led to two separable requirements for net gain from a fusion reactor; these were a minimum ion temperature of about 5 keV and a density confinement time product (n␶E) of approximately 1020 s/m3. Together they define a boundary for ‘‘breakeven’’ shown in Fig. 2. The temperature requirement was first met in 1978 on the PLT tokamak (7) and the confinement figure exceeded in 1983 on Alcator C (8), while later experiments have approached the combined requirements (9–11). The Lawson criteria define the minimum performance for a fusion device to generate net thermal power. For a practical device, the power required to run the plant must be considered along with the inevitable losses that thermodynamics imposes whenever heat is converted to other forms of energy. It is also desirable if the energy to sustain the plasma comes entirely from charged fusion products. In this case, no external power source is re-

FUSION PLASMAS

1022 Break–even

Ignition

1021

Alcator C (1983)

niτ E (s/m3)

1020

Alcator A (1976)

1019

DIII–D (1996) JET (1986) TFR (1975)

1018 T3 (1968)

ST (1971)

JT60–U TFTR JET(1996)

TFTR (1986) PLT (1978)

1017 Model C (1966) 1016

0.1

force that objects much smaller than the sun won’t ignite. In a hydrogen bomb, a scheme called inertial confinement is employed. The fuel is heated and compressed by a fission explosion; the plasma is effectively confined by its own inertia and burns in less than a nanosecond. Of course, a hydrogen bomb is an example of uncontrolled fusion energy and is of no use for generating electricity. Researchers have instead attempted to use intense laser or ions beams to compress tiny pellets of fuel with the goal of producing fusion microexplosions. While most of this work has a military goal, that of exploring the physics of extremely high-density plasmas and validating computer codes which attempt to simulate nuclear weapons, it is possible that inertial confinement could be employed for energy production. The last approach for confining plasmas is by the use of strong magnetic fields; most of the research dedicated to fusion energy over the last 50 years has been in magnetic confinement. Magnetic confinement relies on the fact that magnetic fields exert a force on ionized particles which is perpendicular to their direction of motion: F =m

1.0

10.0

100.0

Ti (keV) Figure 2. The requirements for net energy gain and for ignition are shown. Good confinement and high temperatures are both required. Results from various experiments are shown, along with the dates that the results were achieved. In current experiments, fusion power output has reached 20 MW, only slightly lower than the power input.

quired and the plasma ‘‘burns’’ as long as fuel is supplied. The requirements for this condition, called ignition, are more stringent than the breakeven requirements as can be seen in Fig. 2. Plasma Confinement The requirements for fusion described above are mathematically separable; however, in practice, both are linked to a single parameter, namely, confinement. Perfect confinement would be a state where heat and mass transfer were reduced to zero. To understand why confinement is so important, consider a simple analogy, namely, heating a house. The goal is to attain a given temperature inside, perhaps 20⬚C, while heat is lost to the colder outside world. The better the house is insulated—that is, the more efficiently the heat is confined within the house—the less fuel is consumed. The analogy is imperfect of course: An ordinary furnace will function even in a cold house, but the nuclear furnace only works when it is very hot. Of course, ordinary insulation is no help at all when trying to confine a plasma at temperatures approaching 100,000,000⬚C. There are three ways to confine hot plasmas. The immense gravitational force of a star’s mass can overcome the natural tendency of hot particles to fly apart. For such a huge object, the ratio of volume, in which energy is produced, to surface area, at which it is lost, is immense. Confinement in these systems is so good that the fusion reactions can proceed at very slow rates and still keep up with losses. Energy is produced by nuclear fusion in the core of the sun at a rate somewhat lower (per unit volume) than is produced by metabolism in the human body. Unfortunately, gravity is such a weak

47

v dv E + v × B) = Ze(E dt

(5)

where v is the velocity of the particle in meters per second, m is its mass, the charge on an electron is ⫺e, Z is the net ion charge in units of e, and B is the strength of the magnetic field in tesla. The solution to this equation, called cyclotron or gyro motion, has circular orbits in the plane perpendicular to B; the radius of the gyro orbit is given by the expression ρc =

mv⊥ ZeB

(6)

The magnetic field has no effect on parallel particle motion so the full orbits are helical, with particles spiraling around magnetic field lines, and are effectively confined by them to within one gyro radius (Fig. 3). To confine high-temperature plasmas, whose particles have thermal velocities in the range of 106 m/s or charged fusion products that have velocities 10 times higher, in devices with finite size, strong magnetic fields are required. Experimental devices employ fields in the range of 1 T to 10 T;

B

–

+ Figure 3. Cyclotron or gyro motion for ions and electrons in a straight uniform field. The particles are confined in the plane perpendicular to the field but are unconfined parallel to it.

48

FUSION PLASMAS

for comparison, the earth’s magnetic field is on the order of 5 ⫻ 10⫺5 T. A deuterium plasma ion at 10 keV in a field of 5 T has a gyro radius 앑3 mm. In a configuration like that shown in the figure, particles would be well-confined in the directions perpendicular to the magnetic field, only crossing field lines as the result of collisions. Unfortunately, this configuration, which could be produced by a simple solenoidal coil, provides no confinement at all in the direction parallel to the magnetic field and hot plasma particles would stream freely out the ends. A variety of ideas for eliminating the end losses have been tried, but without much success. A more promising approach is to get rid of the end losses by getting rid of the ends, bending the field lines into closed circles (Fig. 4). The resulting geometry is a torus, a donut shape, which is defined when a circle or other two-dimensional shape is rotated about an axis lying in the same plane. This is, in essence, the approach used in most magnetic confinement experiments; however, the resulting physics is anything but simple. While particles are well-confined on straight magnetic field lines, they quickly drift off when the lines are curved. This particle drift can be canceled by introducing a ‘‘rotational transform’’—that is, by twisting the magnetic field lines as they circle around the central axis. A few definitions are useful at this point. The long way around the donut is called the toroidal direction, and the short way around is referred to as the poloidal direction. A quantity, q, is defined which is equal to the number times a field line circles in the toroidal direction for each time it circles in the poloidal direction. The magnetic field that we have been discussing so far is purely in the toroidal direction. The rotational transform is produced by a magnetic field in the poloidal direction. The two fields, when combined, form helices that wrap around the torus; the particles unrestrained motion parallel to the field is also helical. Since the particle drifts are vertical (with the torus oriented as in Fig. 4), they alternately drift toward the minor axis of the torus and away from it as they circle, cancelling the drifts on average. The poloidal fields that create the rotational transform can be generated by external magnets or from currents flowing toroidally in the plasma itself. A nearly infinite

VD ion

VD electron

BToroidal

BPoloidal Figure 4. End losses can be eliminated by curving the simple solenoid shown in Fig. 3 into a torus. A second component of the field is required to cancel the vertical drifts that particles experience in this geometry.

variety of toroidal devices are possible from these basic components. Some have been successful and are the subject of continuing research, while others have been abandoned as false starts. Later sections will describe some of the more promising schemes. Confinement of individual particles is a necessary but not sufficient condition for magnetic confinement. Plasmas can be thought of as electrically conducting fluids, capable of carrying current and exhibiting collective behavior, like waves. At sufficient pressure, and at the densities and temperatures necessary for fusion, plasmas exert considerable pressure; these collective effects can generate magnetic fields that compete in strength with those confining them. Unless great care is taken to create a stable configuration, the plasma can, in effect, push aside the bars of its magnetic cage and escape. A later section on magnetohydrodynamics (MHD), will elaborate on this topic. Confinement Experiments In such a small space, it would be impossible to do justice to the myriad of approaches to magnetic confinement that have been taken over the years. This section will review a handful of the more prominent schemes. The reader is referred to the bibliography for more detail. Stellarators. One of the earliest magnetic confinement schemes was the stellarator, invented by Lyman Spitzer in 1951 at Princeton University and named after the stars that it was designed to emulate. The earliest versions created the necessary rotational transform by literally twisting the torus into a figure-eight configuration. A plasma was formed in this oddly shaped vacuum vessel that was surrounded by solenoidal electromagnets. This approach was soon discontinued in favor of machines with toroidal plasmas and a magnetic transform generated by helical windings. Experiments continued at Princeton and at other labs throughout the 1950s and 1960s, but met with only modest success. Electron temperatures achieved never exceeded 50 eV and confinement times were no more than a few milliseconds (12). During this period, the science of plasma physics was developing quickly, however, the intrinsic three-dimensional geometry of the stellarator made accurate calculations of its properties impossible at the time. The lack of good confinement in stellarators was attributed to plasma instabilities, and the approach was all but abandoned as promising results from the Russian-invented tokamak were disseminated. Groups at Kyoto University and the Max Planck Institute in Germany continued work on stellarators on a smaller scale, while theory and computers advanced to the point where they could tackle threedimensional problems. Scientists recognized that great care needed to be taken in the design and construction of stellarator coils. The poor results of earlier machines were attributed to field errors coming from imperfect coils. By the 1980s, stellarators were producing plasmas with temperatures near 2 keV and with confinement times up to 30 ms, only slightly worse than tokamaks of similar size (13,14). Since stellarators have a natural ability to operate in steady state (tokamaks, the most developed confinement device, tend to work in pulsed mode), these results generated great interest and prompted the construction of more ambitious experiments. Currently two very large stellarators are under construction.

FUSION PLASMAS

Figure 5. A computer model of an advanced stellarator designed at the Oak Ridge National Laboratory. The large shaded torus represents the plasma surface which is surrounded, in turn, by the 24 magnet coils which are used to create it. The coil shapes allow them to carry both toroidal and poloidal currents. This so-called modular design allows design of a stellarator which would be easier to construct and assemble.

In Japan, the large helical device (LHD) will begin operation late in 1998; and in Germany, Wendelstein 7-X is scheduled to begin experiments about 4 years later. Both machines have major diameters over 10 m, more than twice the size of existing facilities, and are designed to test the scaling of stellarator physics to reactor like conditions. Figure 5 is a computer model of an advanced stellarator, showing the complex coil shapes required for these devices. Tokamaks. Parallel to stellarator development was the exploration of the tokamak in the Soviet Union. The acronym tokamak stands for toroidalnaya kamera magitnaya katushka or toroidal chamber and magnetic coil, a device invented by Igor Tamm and Andrei Sakharov (perhaps best known for winning the Nobel Peace Prize for his work on human rights). The early Russian work may have been inspired, in part, by information gathered from the British nuclear laboratory at Harwell. Scientists at Harwell had been working with a class of devices called ‘‘pinches,’’ named after the pinching force that a current carrying plasma exerts on itself. Regardless of its origin, the tokamak was a marked improvement over the existing pinches. In addition to the pinching fields, provided by toroidal plasmas currents, the tokamak added a very strong toroidal field to stabilize the plasma. Soon results from tokamak experiments far exceeded those from any other fusion experiment. Plasmas with electron temperatures near 1 keV, ion temperatures of 300 eV and densities over 5 ⫻ 1019 were reported from the T3 tokamak at the Kurchatov Institute in Moscow (15). Energy confinement times were up to 10 times better than the best stellarator results. Scientists outside the Soviet Union were at first skeptical of the claims; the plasma diagnostics available at the time were rudimentary

49

and the results seemed too good to be true. A second dramatic moment in the development of fusion energy came in 1969 when a team of British scientists traveled to the Kurchatov and used a newly developed laser scattering diagnostic to verify the Russian claims (16). The impact of this measurement rolled over the international fusion community like a thunderclap. Almost overnight, plans for new tokamaks were laid at all the major labs. The early tokamaks were relatively simple devices. A toroidal vacuum vessel was surrounded by a solenoid to produce the strong toroidal magnetic field. Threaded through the hole in the torus was an iron core, which, when powered by external windings, became the primary for an enormous step-down transformer. When current in the external windings was changed, a toroidal electric field was produced that was parallel to the magnetic field. A small amount of hydrogen that had been introduced into the vacuum chamber was quickly ionized as electrons were accelerated by the electric field. Soon a substantial current was flowing in the plasma, generating the poloidal field that provided the rotational transform. The plasma current also provided a source of heat as electrons, accelerated in the electric field, collided with the relatively stationary ions. Like the heating of a filament in an electric lightbulb, this process converts electrical energy to heat. These devices easily produced plasma temperatures in the range 100 eV to 1000 eV with confinement times of several milliseconds. Later, tokamaks replaced the iron core with an air core transformer, added additional forms of heating, and increased the size of the machine dramatically. The main coils of today’s large tokamaks stand 4 m high and have vacuum vessels large enough for a man to walk around in. These machines produce temperatures up to 40 keV and confinement times on the order of 1 s. The JT-60 tokamak in Japan has produced plasmas that are at the point of scientific breakeven (9), though they are run only in hydrogen and deuterium and do not produce substantial fusion power. The TFTR tokamak (Tokamak Fusion Test Reactor) in the United States and JET (Joint European Torus) in England produce similar plasmas and have been run with deuterium–tritium fuel. These devices produced 10 MW and 20 MW of fusion power, respectively (5,11). Researchers have learned how to optimize confinement in tokamaks, at least transiently, almost entirely eliminating the microinstabilities that normally drive transport in plasmas (17–20). Figure 6 shows the DIII-D tokamak in its test cell at General Atomics in San Diego. Other Toroidal Confinement Schemes. The reversed field pinch (RFP) first developed at Harwell, England in the late 1950s (21), is an axisymmetric device similar in many ways to the tokamak. It evolved out of work on toroidal versions of the stabilized Z pinch (see below). Like the tokamak, a poloidal field is produced by toroidal current flowing in the plasma and a toroidal field produced by external coils. However, in the case of the RFP, the two are of roughly equal strength and, most significantly, the toroidal field is reversed near the plasma edge, producing a ‘‘minimum energy’’ equilibrium that is particularly stable to certain MHD modes. In principle, the RFP can confine plasmas with a higher ratio of plasma pressure to magnetic pressure than stellarators or tokamaks. The field reversal can be obtained spontaneously or by programming the external currents, using the low resistivity of the plasma to freeze-in the field. The field reversal is sustained

50

FUSION PLASMAS

Figure 6. The DIII-D tokamak was built by General Atomics in San Diego. The tokamak, which is in the center of the photograph, is surrounded by neutral beam heating sources (the three large cylindrical chambers) and various diagnostic equipment. The most notable feature of the tokamak is its large toroidal magnetic field coils. The scale can be estimated from the technician standing in the upper left of the photo.

against resistive relaxation by a magnetic dynamo, where motion of the conductive plasma is converted to magnetic field energy, a process similar to the one that produces the earth’s field. Research on RFPs is continuing, with emphasis on transport and magnetic turbulence (22). Another family of confinement devices is the compact torus. Devices of this type attempt to create a configuration in a minimum energy state with respect to gross MHD stability. The torus-shaped plasmas do not encircle any coils (as they do in the tokamak, stellarator, or RFP), a significant engineering advantage for a reactor where stresses and heat loads on the inner cylinder of conventional toroidal devices can limit machine performance. Like the RFP, these devices should be capable of confining high pressure plasmas. Two variants exist: the spheromak, which has both toroidal and poloidal fields in a force free configuration (the plasma current is everywhere parallel to the magnetic field), and the FRC or field reversed configuration, which has only poloidal fields. Both have been produced only in short pulsed experiments that have concentrated on creation and verification of the basic equilibrium (23). Linear Systems. While it was clear that end losses would make simple linear devices unsuitable for reactors, many early experiments were carried in this geometry to test the basic principles of magnetic confinement. Two basic types of schemes were tested: the theta pinch and the Z pinch. Cylindrical in shape and named after the direction of the induced current in the standard coordinate system, these devices were operated in short pulses; end losses rapidly drained particles and energy from the plasmas that were created. Work on Z pinches actually predated controlled fusion research (24). At Los Alamos in 1951 and somewhat later at the Lawrence Berkeley Lab in California, researchers used capacitor banks

and low induction coils to compress plasmas to very high densities. In machines built after 1956, a small axial field was introduced that had the effect of somewhat reducing instabilities, but also lowered the density of the plasma produced. This so-called ‘‘stabilized Z pinch’’ was the ancestor of the tokamak and the RFP. The theta pinch was also studied extensively at Los Alamos, starting in 1958. The experiments had a low-inductance single turn coil in the theta direction, allowing current in the machine to be pulsed at very high rates. The resulting plasma was heated and compressed by a shock wave that propagated inward toward the machine axis and allowed production of very high densities (앑1022 /m3) and ion temperatures (앑1 keV) (25). Because of their low mass, electrons did not pick up much energy from the shock and remained much colder. Attempts to produce a toroidal version of the theta pinch could never overcome MHD instabilities, and the approach was eventually abandoned. At the Lawrence Livermore Lab, researchers began working on an idea that they hoped would allow them to plug the end losses in linear machines (1). This idea was the magnetic mirror, in which plasma particles are reflected by converging magnetic fields produced by high-field coils added to each end of a simple solenoid. The mirror works by exploiting the principle of adiabatic invariance; which will be described in a later section. It is the same phenomenon that causes charged particles from the solar wind to be trapped in the earth’s dipole field: the radiation or Van Allen Belts (26). The simple mirror was shown to be unstable by Ioffe et al. (27), who suggested an improvement, the minimum B mirror, an idea that was incorporated into all later experiments. Plasmas in a minimum B mirror were confined by coils shaped like the seam of a baseball and spread out in two large fans. Because mirror plasmas were confined by external fields only, they held out the prospect for steady-state operation. Their open fieldline

FUSION PLASMAS

geometry was seen as a virtue; it was proposed that energy from charged fusion products, many of which would be lost out the open lines, could be converted directly to electricity, bypassing the inherently inefficient thermal cycle upon which all conventional power generation is based. A magnetic mirror only traps particles whose motions are mainly perpendicular to the magnetic axis. Those with mostly parallel energy are rapidly lost. To maintain this highly non-Maxwellian velocity distribution, a mirror machine would have to run in a regime where collisions were rare. In the end, microinstabilities doomed this approach and despite a host of inventions to improve the concept, the mirror never succeeded in achieving good confinement for thermal plasmas (28). Plasma Wall Interactions Eventually, all the energy and power that is put into the plasma and all the energy from charged fusion products must come out onto an ordinary material surface. It is a measure of progress that these fluxes are now a major concern for fusion experiments. Hot plasmas were once thought of as fragile things, extinguished by even the lightest touch of material objects. Now, with plasma energies over 10 MJ, it is the material objects that are at risk. Plasma facing components must be carefully designed to avoid melting and vaporization. It is also necessary to prevent impurities from entering the plasma. Impurities, particularly low-Z ions like carbon or oxygen, can dilute hydrogenic fuel and reduce fusion rates. HighZ ions, from the vacuum vessel structure, are not fully ionized even at high temperatures and can radiate copiously and may be a significant energy loss channel. To minimize impurity generation in modern experiments, all components that come into contact with the plasma are usually made of refractory materials like graphite, molybdenum, or tungsten. Heat loads are not the only process which can generate impurities. Energetic ions can remove atoms from a surface directly, by a process called sputtering, even if the average heat load is too low to vaporize material in bulk. Serious attention must be paid to the design of the structures that form the plasma boundary and that will take the largest heat and particle loads. There are two approaches to this problem: limiters and divertors (29). A limiter is simply a solid object put into direct contact with the plasma edge. Hot plasma on magnetic field lines that intersect the limiter quickly gives up its heat via parallel transport along the relatively short connection length. The limiter sets a boundary condition for the plasma with near zero temperature and density (at least compared to the core plasma). To provide sufficient ‘‘wetted’’ surface to absorb the heat loads, limiters must be made conformal to the plasma surface; even small mismatches can lead to localized heating and melting. In a divertor, by contrast, the magnetic field lines at the edge of the plasma are diverted into a separate chamber, where all plasma wall interactions can be isolated. There are a number of advantages to this approach. First, because the divertor plates are located away from main plasma, it is easier to keep any particles and impurities that are generated away from the plasma core. Second, the connection length along open field lines can be made quite long. This allows for a larger temperature gradient between the divertor and main plasma, reducing the effects of energetic ions at the contact point and easing the boundary condition imposed on the main plasma.

51

Temperatures near the divertor walls can be a fraction of an electronvolt, while upstream temperatures, adjacent to the plasma, can be near 100 eV (30). Finally, neutral gas in the divertor chamber can be highly compressed, allowing impurities and helium ash from the fusion process to be efficiently pumped. PLASMAS: BASIC PROCESSES Although the underlying processes that determine plasma behavior are from well-understood branches of physics, classical electrodynamics, and classical mechanics, the behavior itself can be extremely complex. The motions (usually referred to as orbits) of individual particles are fundamentally important, however, plasmas are collections of enormous numbers of particles. A typical plasma fusion experiment may have over 1021 ions and electrons rendering a simple mechanical approach impossible. This is not only because of the large numbers of particles involved; even the dynamics of systems as simple as three mutually interacting charges cannot be solved exactly. Ensembles of plasma particles evolve chaotically; systems with almost identical initial conditions diverge exponentially over time. At best, only the evolution of macroscopic properties can be calculated. Because plasmas are typically dynamic and far from equilibrium, this approach requires the machinery of nonequilibrium statistical mechanics. Unfortunately, even with a statistical approach, most real-world plasma problems are intractable and require further approximations before they can be solved. Researchers develop the theory of plasmas by working out the simplest cases first, then gradually reducing the number and scope of the approximations as more realistic problems are tackled. Quasi-Neutrality An essential property of plasmas is quasi-neutrality, which means that the negative and positive electrical charges that make up a plasma are approximately balanced everywhere. Significant charge separation can occur only over a small scale, called the Debye length, defined by the balance between electrical and thermal forces: λD ≡




Te n e e2

1/2 (7)

where ne is the electron density in m⫺3, Te is the electron temperature in electronvolts, and e is the electron charge. The plasmas we deal with in controlled fusion are hot enough to achieve significant ionization and dense enough for ␭D to be much smaller than the plasma size. Electron thermal motions enforce quasi-neutrality on time scales longer than ␶ ⫽ ␭D /ve; thus ordinary electromagnetic waves cannot propagate in a plasma unless their frequency is higher than 1/ ␶ ⫽ ve / ␭D ⬅ 웆pe, the electron plasma frequency. (The reflection of radio waves off the ionosphere is a manifestation of this property of plasmas.) Particle Orbits Despite the intractability of a purely mechanical approach to plasma physics, it is still important to study the motion of individual particles in a plasma. First, the collective proper-

52

FUSION PLASMAS

ties of plasmas are made of these motions, and they contribute strongly to its behavior. Second, fusion plasmas contain high-energy ions, including charged fusion products which interact only weakly with the background plasma and which can be well understood by studying their orbits. Particle orbits are determined by elementary mechanics and the interactions of charged particles with electric and magnetic fields shown in Eq. 5. As discussed above, the solution, for the case of a uniform B field with E ⫽ 0, is free and constant motion parallel to B and circular motion in the directions perpendicular to B. The circular motion is characterized by a frequency 웆c and a radius ␳c given by ZeB mc v⊥ mv⊥ ρc = = ωc ZeB

ωc =

B × ∇B B) v⊥ ρ (B 2 B2

(9)

(10)

Note that for a simple toroidal solenoid we have 兩B兩 앜 1/R and ⵜB 앜 1/R2, so B ⫻ ⵜB/B2 is just 1/R. A drift of similar magnitude occurs when the magnetic field lines are curved. Because electron and ions drift in opposite directions, the motion can give rise to electric fields perpendicular to the magnetic field. These exist because the magnetic field inhibits the motion of electrons that would normally cancel the electric field. Electric fields can also cause electrons and ions to drift off of magnetic field lines, this time in the same direction and independently of their thermal velocity. This drift velocity is given by: E ×B vE = B2

As ions in a plasma approach each other, they are mutually repelled by the electrostatic force generated by the electric field that surrounds all charged particles. The interaction deflects the particles, scattering them elastically—that is, without any net change in the combined kinetic energy for the pair. The scattering angle is larger if the particles pass close to each other, or if the particles are moving slowly, which allows a longer period of interaction and for a larger relative effect of that interaction. The cross section for 90⬚ scattering was first calculated by Rutherford in 1911: σ90 =

(8)

where v⬜ is the component of the particle velocity perpendicular to the magnetic field and v储 is the parallel component. For a typical fusion plasma deuterium ion with a temperature of 10 keV in a field of 5 T, the cyclotron frequency, 웆ci ⫽ 2.4 ⫻ 108 and ␳ci ⫽ 2.9 mm. For electrons at the same energy, the lower mass results in 웆ce ⫽ 8.8 ⫻ 1011 and ␳ce ⫽ 0.047 mm. In this case, both types of charged particles are confined onto the magnetic field lines to within a gyro radius. For nonuniform magnetic fields, electrons and ions drift off the field lines in opposite directions. The drift velocities, which are proportional to the particles’ thermal velocity, are given by vd = ±

Collisional Phenomena

(12)

The attractive force between unlike charges has a similar effect, with the scattering dynamics of electrons and ions differing only due to their different masses and velocities. Unlike collisions between neutral atoms, which are due to the very short-range forces induced by mutual polarization, the scattering of charged particles is inherently long-range. This means that multiple small-angle interactions have an accumulated effect that are more important than the relatively rare large-angle collisions. It was shown by Spitzer that in a plasma the scattering cross section is larger than ␴90 by the factor log ⌳, the Coulomb logarithm, where ⌳ ⬅ ␭D ⫻ mv2 /e2 앜 兹(n/T3). ␭D appears in this expression because Debye shielding cancels the electrostatic field over larger distances. The scattering cross section for plasma particles is then of the order ␴ 앑 ␴90 log ⌳, with log ⌳ on the order of 15 for fusion plasmas. With a collision cross section ␴, a particle with velocity v will undergo collisions at a rate ␯ ⬅ n␴v 앜 n/(m1/2T3/2), where n is the density of the background plasma. At equivalent energies, the faster moving electrons will collide more often by a factor 兹mi /me. Collisions impede the ability of plasmas to carry current as the momentum of the charge carrying electrons is lost to the ions. Spitzer resistivity, appropriate for unmagnetized plasmas or for currents flowing parallel to the magnetic field, is given by Eq. (13). For a typical fusion plasma, at temperatures of 20 keV, the resistivity is about 100 ⫻ lower than for copper: η = 0.51

(11)

Note that the particles continue to execute gyro orbits as they drift. Since the gyro motion is fast compared to drift motions and the gyro radius is small compared to other scale lengths, the gyro orbits are often averaged over for all subsequent analysis. Also of importance are so-called adiabatic invariants. These are quantities that are approximately conserved by particle orbits; the first and most important of these is 애 ⬅ mv⬜2 /2B. Charged particles can be reflected by converging magnetic field lines since 애 conservation implies that perpendicular energy must increase as the field magnitude grows. To conserve energy, the additional perpendicular energy must come at the expense of parallel motion which is reduced to zero at a sufficiently high magnetic field. This is the principle behind the magnetic mirror discussed earlier.

πe4 (4π0 )2 m2 v4

2 m1/2 7.8 × 10−4 e e log  ≈ ·m 2 3/2 30 (2πTe ) Te3/2

(13)

A magnetic field alters the effects of collisions on a plasma. Since electrons and ions in a uniform magnetic field are tied to the field lines, a collision will move the particles at most by one gyro radius. A diffusion coefficient can be defined by D=

(stepwise)2 nm1/2 = ρc2 v ≈ 2 1/2 τcollision B T

(14)

We note that in this simplified geometry, diffusion is inhibited by the magnetic field by a factor that is roughly the ratio of collision mean free path to the gyro radius squared. A magnetized plasma can be defined as one where this ratio is large. Diffusion is also reduced as plasmas grow hotter—a favorable result since the goal is to confine very hot plasmas. It is also worth remarking that in a magnetized plasma, ions, because

FUSION PLASMAS

of their larger gyro orbits, tend to diffuse faster than electrons, while in an unmagnetized plasma the opposite is true. Typically, electron dynamics will dominate transport parallel to the field, and ions will dominate in the perpendicular direction. Collisions between like particles, electrons and electrons or ions and ions, will not lead to particle diffusion since the net momentum is unchanged and the particles merely exchange positions. This is the same reason that electrical conductivity is governed by electron–ion collisions. Since the magnetic field does not affect parallel motions, diffusivity and conductivity are anisotropic quantities, being much smaller in the perpendicular direction. The mass dependence of the coefficients suggests that electrons and ions would diffuse at different rates. In practice, this is not the case; as one charge species tries to leave regions of high density ahead of the other, an electric field is built up. This ambipolar field holds the faster species back, maintaining quasi-neutrality, and both diffuse together. The Fluid Picture Of Plasmas

MAGNETOHYDRODYNAMICS Magnetohydrodynamics (MHD) is the macroscopic fluid theory of plasmas. The equations that govern MHD are Maxwell’s equations and moments of the Vlasov equation. These moments yield separate equations for ions and electrons, which are coupled through the fields generated and through collisions. It is common to use a single-fluid set of equations that can be derived by ignoring electron inertia, assuming that electron motion is fast compared to time scales of interest. This is equivalent to restricting our interest to times long compared to the electron cyclotron frequency and plasma frequency. A further simplification, appropriate in many cases, is made by ignoring resistive effects—that is, by assuming the plasma is a perfect conductor. The MHD equations are Continuity :

∂ρ + ∇ · (ρvv ) = 0 ∂t

(18)

Momentum Balance:

v dv = J ×B − ∇p dt

(19)

dp = −γ p∇ · v dt

(20)

µ0 J = ∇ × B

(21)

B ∂B = ∇ ×E ∂t

(22)

J E + v × B = ηJ

(23)

∇ ·B = 0

(24)

Like ordinary gases, which are also large collections of particles, plasma can be treated like a fluid, although one that has significant electrical properties. The equations that govern the fluid-like properties are obtained by taking velocity integrals or moments of the Boltzmann equation, which describes the statistical evolution of a group of particles (the equation is essentially an equation of motion in phase space). In plasma physics, the relevant version of Boltzmann’s equation (often called the Vlasov equation when collisions are rare) takes into account the effects of electromagnetic fields: ∂f q ∂f E + v + B) · + v · ∇ f + (E = ∂t m ∂v


∂ f  ∂t

(15)

C

where f is the density of particles in six-dimensional phase space and (⭸f /⭸t)C is a collision operator that describes the effects of coulomb interactions. q, m, and v are the charge mass and velocity of the particles under consideration. In many cases in plasma physics, collisions are ignored, though the fluid picture is only valid formally, when collisions are frequent enough to keep the mean free path much smaller than the system size. The zeroth moment of the Vlasov equation corresponds to mass conservation or continuity, the first moment corresponds to momentum conservation or force balance, and the second moment corresponds to energy conservation. In simplified form, the first two moments can be written ∂nm v) = −∇ · (nmv ∂t

(16)

v) ∂ (nmv E + v × B ) − ∇P = nq(E ∂t

(17)

where nm ⬅ ␳, the fluid mass density, and v is the fluid velocity and P is the pressure tensor. Note that the equation for each moment refers to the next higher moment, and the fluid velocity is needed to complete the continuity equation for example. To be useful, the set of equations must be closed, typically by making simplifying assumptions, like an equation of state. The moment equations combined with Maxwell’s equations for electromagnetics form the basis for the fluid picture of plasmas called magnetohydrodynamics.

53

Equation of State: Ampere’s Law: Faraday’s Law: Ohm’s Law:

where ␳ is the plasma mass density, p is the plasma pressure in SI units, and 웂 is the ratio of specific heats. Equilibrium In the fluid picture, magnetic confinement is achieved through the balance between plasma pressure and the magnetic force. At equilibrium the time derivatives vanish, leaving J ×B = ∇p

(25)

Figure 7 shows the geometrical relations between these quantities for a straight cylindrical plasma and B field. From Eq. (25), it is clear that both J and B must lie on surfaces of constant p. These surfaces are usually called flux surfaces and labeled with the enclosed magnetic flux. For a confined plasma, p will be a maximum near the axis and close to zero at the boundary. The current and field are related by Ampere’s law, Eq. (21). For the case of a straight cylinder with a radial pressure gradient and no current parallel to B, J ⫽ J⬜ ⫽ B ⫻ ⵱p/B2. This is called the diamagnetic current, and it arises from the imbalance in gyro-orbiting particles that is created by the pressure gradient. The pressure profile and parallel current can be considered free parameters in this equilibrium. Note that the magnetic field can be aligned in the z (axial) direction, with theta, or in some combination of the two. The essential pressure balance between the confining field and the plasma can be seen by substituting J from Ampere’s law, which, after some vector algebra, yields the follow-

54

FUSION PLASMAS

lated by the Grad–Shafranov equation (31,32). An example of an equilibrium in a toroidal device is shown in Fig. 8. Bθ

Stability

Pressure

Jz

0.0

0.2

0.4 0.6 Normalized radius

0.8

1.0

Figure 7. Sample profiles for the magnetic fields, currents, and plasma pressure for a simple MHD equilibrium, in this case a linear Z pinch. Named after the direction of the plasma current, the Z pinch actually predates the fusion energy program. It was first studied in 1934 by Bennet (24). The equilibrium requires balance between magnetic and plasma pressures.

ing for the case of straight field lines:

∇ p = −∇


B2  2µ0

(26)

From the earliest studies into magnetically confined plasmas, researchers recognized that consideration of plasma equilibrium was not sufficient. Experimental plasmas could exhibit violent behavior sometimes losing their stored energy in a few microseconds. Further analysis showed that these plasmas were MHD-unstable: Like a ball sitting at the top of a hill, they were in a state of unstable equilibrium. Free energy for the instabilities comes from the plasma pressure and current. Pressure-driven modes exhibit ‘‘interchange’’ behavior: Parts of the fluid move toward the high-pressure region, while other parts move away. This phenomenon is analogous to the Rayleigh–Taylor instability that occurs when a glass of water is inverted. Current-driven instabilities often take the form of a kink; the plasma tries to twist itself into a corkscrew shape. Fortunately there are stabilizing forces that can come into play as the plasma moves away from equilibrium. Since the plasma is tied to magnetic field lines, these must be bent or compressed if the plasma is to move. Both processes require energy input and are thus stabilizing. Magnetic field curvature can be either stabilizing (when the pressure gradient is away from the center of curvature) or destabilizing. MHD stability is calculated by analyzing the effect of an infinitesimal displacement of the plasma. Destabilizing and stabilizing forces are summed up and found to move the plasma toward or away from equilibrium. Ideal MHD instabilities propagate at the Alfven velocity (see Table 1), which is on the order of 107 m/s for a fusion plasma, and therefore they must be avoided.

This relation suggests a definition for the normalized plasma pressure: β=p

2µ0 B2

The importance of plasma pressure in the dynamics of a system is determined by beta. As discussed previously, practical considerations require that a magnetic confinement device be toroidal. The toroidal geometry, illustrated in Fig. 4, adds two complications to the simple equilibrium just considered. First, the magnetic field must have both poloidal and toroidal components to cancel the single particle drifts. Second, in addition to radial force balance (from plasma pressure that tries to expand in the r direction), toroidal force balance must be considered as well. Two forces tend to expand the plasma in the R direction: one current-driven and one driven by the plasma pressure. Toroidal current exerts a hoop stress, as a result of the self-force between different current elements. The pressure imbalance arises because there is more surface area on the outside (large R) of the torus than the inside. Toroidal balance is achieved by the addition of a vertical magnetic field, which, when crossed with the toroidal current, produces a compensating force. In toroidal geometry, MHD equilibrium is calcu-

Figure 8. A poloidal cross section of MHD equilibrium in the Alcator C-Mod tokamak. The center of symmetry for the torus is on the left; the three-dimensional geometry is recovered by rotating the entire picture about that axis. The contours shown are for magnetic flux, which are also surfaces of constant magnetic pressure. In addition to the radial balance required in a linear system, toroidal force balance must also be preserved.

FUSION PLASMAS

WAVES IN PLASMA Just as ordinary fluids (like air) support sound waves, magnetized plasmas support a rich variety of wave phenomena. Waves are characterized by the displacements or perturbations that they produce in the medium that carries them. Plasma waves may perturb density and pressure as sound waves do, but are also able to alter electric and magnetic fields. A magnetic field breaks the isotropy of a plasma, with particle motions free in the parallel direction and inhibited in the perpendicular direction. Furthermore, there are at least two particle species in play (i.e., one or more types of ions and electrons), which can respond differently to waves owing to their very different mass. The general theoretical approach to describing waves in plasmas begins by linearizing the fluid equations about small perturbations, which are described as traveling waves, ei(kx⫺웆t), where the wavenumber k ⬅ 2앟/ ␭. This process reduces the set of coupled partial differential equations to a system of ordinary algebraic equations that are then solved for 웆(k). From this expression, called the dispersion relation, types of waves can be identified; also, group and phase velocities, regions of propagation, and resonances can all be calculated. The result is a complex family of waves whose description has filled many textbooks (33,34). Table 1 gives a greatly simplified summary of some of the more important wave types. Waves are important for a number of reasons. They can be used for heating plasmas by launching waves from an external antenna, which then propagate into the plasma and then deposit their energy, usually by interacting with a natural plasma resonance. Waves are also generated spontaneously in plasmas, driven by the nonequilibrium state of a confined plasma. Temperature, density, and pressure gradients along with non-Maxwellian velocity distributions can all cause waves to grow. The waves, in turn, try to push the plasma back to a state of thermodynamic equilibrium, relaxing gradients and so forth. This can be a powerful mechanism for driving plasma transport and for destroying confinement.

55

cally nonlinear. For example, consider self-diffusion where plasma particles diffuse via collisions with other plasma particles. The diffusion coefficients themselves are functions of the plasma density and temperature, rendering the diffusion of particles or heat nonlinear. Waves, when their amplitudes become large enough, affect the medium that is carrying them and thus change the propagation properties of the waves themselves. This mechanism also allows waves, which in linear theory are independent, to interact and to exchange energy. Finally, waves can interact with particles, whereby the wave fields strongly modify the particles’ motions; conversely, particle motion can be converted into wave energy. Nonlinear interactions allow large-scale, organized motions to be converted to smaller-scale, less organized motion. This cascade of energy between scales is characteristic of turbulence, a phenomenon which can be stimulated in any fluid. In a gas, a dimensionless quantity called the Reynolds number defines the regime where turbulence appears. R ⬅ UL/ ␯, where U is the fluid velocity, L is the system size, and ␯ is the fluid viscosity. For R Ⰷ 1, viscosity is not sufficient to hold the fluid motion together into organized flow. For plasmas, the analogous dimensionless quantities are the magnetic Reynolds number ⬅ 애0LVT / ␩, and the Lundquist number ⬅ 애0LVA / ␩. (VT is the thermal velocity and VA is the Alfven velocity, defined in Table 1.) These are closely related, differing by 兹웁, the square root of the ratio of plasma pressure to magnetic pressure. For a typical fusion plasma, the magnetic Reynolds number can approach 108. As a result, dynamically significant behavior can occur over a wide range in scale lengths, with collisional dissipation entering only on the smallest scales, where wave energy is degraded into thermal motion (heat). Typically, turbulent behavior can only be analyzed by simulation on powerful computers; however, some analysis is possible in terms of scaling relations. While the details of turbulent flows are terribly complex, overall, they must obey the basic conservation laws or symmetries of the the underlying physics. TRANSPORT

Nonlinear Effects

Collisional Transport

While the approach to problems in plasma physics usually begins with linear analysis, plasmas themselves are intrinsi-

Magnetic confinement can be understood from either the particle or fluid pictures of plasmas. In either case, nonideal ef-

Table 1. Properties of Some of the More Important Types of Waves Which Can Propagate in Magnetized Plasmas a Wave Type

Common Name

Direction of Propagation

Polarization

Approximate Phase Velocity

Electrostatic ion wave

Ion accoustic wave

k 储 B0

E1 储 B0

Electrostatic electron wave

Plasma waves or Langmuir waves

k 储 B0

E1 储 B0

Electromagnetic ion waves

Shear Alfven waves or simply Alfven waves Compressional Alfven waves or magnetosonic wave O-wave X-wave L-wave R-wave

k 储 B0

E1 ⬜ B0

k ⬜ B0

E1 ⬜ B0

cs ⫽ (kTe /mi)1/2 (ion sound speed) ve ⫽ (kTe /me)1/2 (electron thermal speed) vA ⫽ B/(4앟nimi)1/2 (Alfven velocity) vA

k ⬜ B0 k ⬜ B0 k 储 B0 k 储 B0

E1 储 B0 E1 ⬜ B0 E1 ⬜ B0 E1 ⬜ B0

c (speed of light) c c c

Electromagnetic electron waves

a

The common names given are for waves which propagate strictly parallel or perpendicular to the field; in general, propagation at arbitrary angles is possible and the nomenclature becomes less appropriate. The phase velocities listed are only approximate; the actual velocity, v␾ ⬅ 웆/k, is a function of frequency (웆) or wavenumber (k).

56

FUSION PLASMAS

fects will tend to spoil confinement. Collisions cause particles to diffuse across the magnetic field by a random walk process. In the MHD model, the equivalent process is described by the diffusion of a conductive fluid through a magnetic field. Note that in ideal MHD, where resistivity is assumed to go to zero, the diffusion rate also goes to zero; the magnetic field is said to be frozen into the fluid. Taking Ohm’s law from the equation for resistive MHD [Eq. (23)] and crossing it with B gives J ×B E × B + v⊥ |B|2 = ηJ

(27)

Combining this with the force balance equation [Eq. (19)] and solving for v⬜, we obtain v⊥ = −η

∇p E ×B + B2 B2

(28)

The first term describes plasma diffusion, while the second term is the fluid equivalent to the E ⫻ B particle drift. A little algebra will show, not surprisingly, that the diffusion rate for the fluid is the same as that derived above for particles. For a typical fusion plasma, this classical diffusion coefficient is on the order 10⫺3 m2 /s, far too small to present any problem for magnetic confinement.

The collisional diffusion coefficient, just calculated, is correct only for a plasma confined by a magnetic field that is straight and uniform. In a torus, diffusion is significantly enhanced by the particle drifts. The rotational transform forces the drifts to cancel when averaged over complete orbits; however, collisions can disrupt the orbit and cause the cancellation to be incomplete. The theory for collisional diffusion in axisymmetric toroidal geometry is called neoclassical transport and has been extensively developed for tokamaks (35). Figure 9(a) illustrates a tokamak cross section with the directions of the magnetic fields, plasma current (taken to be in the same direction as BT in this example), and particle drift shown. Consider a particle with its parallel motion in the same direction as the plasma current, which follows the field lines in a right-hand spiral. In the poloidal cross section, such a particle will travel in a clockwise circle. The ⵜB drift will move such a particle off its flux surface, toward the plasma center while it is near the bottom of its orbit and away from the center while it is near the top. The maximum displacements occur when the particle is on the horizontal midplane, resulting in an orbit that is displaced outward, away from the torus center. This result holds whether the toroidal field and plasma current are in the same or in opposite directions. Ions circulating opposite to the plasma current will always be

BT, IP into page

BT, IP into page

Vll = 0

Passing particles "counter" orbit

Trapped “banana” orbits

"co" orbit

+

+

BPoloidal r

Ion drift direction

Ion drift direction

R C

C (a)

(b)

Figure 9. (a) In a torus, the ⵜB and curvature drifts result in particle orbits that do not follow magnetic field lines exactly. In this poloidal cross section, the projection of one set of field lines (flux surface) is shown by the dashed circle. The center of symmetry is on the left, and the coordinates R and r are shown. Simple application of Ampere’s law shows that BToroidal 앜 1/R thus B ⫻ ⵜB is vertical, upwards in this case, for positively charged ions. The poloidal field component gives a twist or helicity to the field lines. The result is that ions with velocities parallel to the plasma current (co) have orbits shifted out in major radius, R, relative to the flux surfaces while those moving in the opposite direction (counter) are shifted inward. (b) In the same geometry part (a), ions in so-called ‘‘banana’’ orbits are shown. These ions are trapped in the magnetic mirror created by the gradient in the toroidal field. Trapped particles are shifted off of flux surfaces even farther than the passing particles and thus can take large radial steps when they collide. These processes are the basis for the neoclassical theory of transport.

FUSION PLASMAS

shifted inwards. The approximate size of that displacement 웃r, can be calculated: δr = v D τ

(29)

where ␶ 앒 qR/vT is the period the particle takes to complete a poloidal orbit. (In a low-aspect-ratio tokamak with a/R Ⰶ 1, q ⫽ rBT /RBP and is typically in the range 1 to 5.) Using Eq. (10) for the ⵜB drift velocity and noting that ⵜB/B ⫽ 1/R, the orbit shift is given by δr = v T ρ i

B qR ∇B = qρi B vT

(30)

Thus, the stepsize caused by collisions is increased by a factor of q and diffusion by q2. The effects of toroidal geometry are even more pronounced on another class of particles, those with enough perpendicular energy to be trapped by the tokamak’s inhomogeneous toroidal field. The magnetic field experienced by a particle in a tokamak varies by a factor on the order r/R as it moves. Those with v储 /v⬜ ⱕ 兹⑀ will be reflected by the stronger magnetic field on the inner part of their trajectory. These trapped particles execute banana shaped orbits as shown in Fig. 9(b). The width of the bananas are larger than 웃r by 兹(R/r), and they diffuse even faster than circulating particles. Overall, neoclassical diffusion coefficients are on the order 50 times larger than the classical values for a plasma in a tokamak. Plasma resistivity is reduced by neoclassical effects, since trapped particles do not complete circuits of the torus and cannot carry current. For nonaxisymmetric plasmas, collisional diffusion can be larger still (36). In an axisymmetric system, particles are governed by conservation of canonical angular momentum p⌽ ⫽ mv⌽R ⫹ eA⌽R, where A is the vector potential of the magnetic field. As long as the particle’s kinetic momentum is not too large, they are constrained to stay close to flux surfaces, which are also surfaces of constant RA⌽. (This is, in fact, an alternative picture for describing particle orbits in a torus.) This constraint is absent if the system lacks axisymmetry where certain classes of particles can make large radial excursions or leave the plasma entirely without undergoing any collisions. The loss rate for the plasma as a whole is then governed by the rate at which the hole in the velocity distribution function is filled in. These effects are most important in intermediate collisionality regimes, where particles on lost orbits can travel significant distances without colliding, but where collisions are still able to fill in the losses from the background plasma. For stellarators, whose helical field breaks the axisymmetry, these losses can dominate transport; modern stellarators are carefully designed to minimize their magnitude. The fields of nominally axisymmetric devices, like a tokamak, have a small asymmetry due to the finite number of toroidal coils. This periodic ripple can cause significant loss of energetic particles, particularly those with mostly perpendicular energy, since parallel motion tends to average out the asymmetries. Anomalous Transport Even with neoclassical corrections, collisional transport is not fast enough to cause serious concern (at least in axisymmetric devices); even small machines could have confinement times longer than 1 s. In experiments, energy confinement was al-

57

most always found to be much worse than that calculated from the collisional theories, often by a factor of 100 or more (37). At the outset it was suspected that this anomalous transport had its origins in small-scale plasma instabilities (usually called microinstabilities), but it took several decades before scientists could develop a quantitative understanding of the phenomenon, and even now it is far from complete (38). Experimentalists have developed techniques for drastically reducing anomalous transport, sometimes to nearly the neoclassical levels (17–20), though so far these have been only for transient situations. Waves, driven by the plasma’s free energy, can cause transport by two basic mechanisms. First, magnetic perturbations can break up magnetic flux surfaces, allowing particles to make radial excursions as they follow the field lines. While large-scale magnetic perturbations driven by MHD instabilities destroy confinement almost instantly, low-amplitude waves can result in enhanced transport. Because of their higher velocity, electrons follow magnetic perturbations more readily and would carry most of the heat in this case. The second mechanism, and the one believed responsible for anomalous transport in most experiments, results from the electrical component of plasma instabilities. These are typically low-frequency waves (웆 Ⰶ 웆c), which have a fluctuating electric field as part of their dynamics. Differences in electron and ion motion, due to their different mass, cause charge separation and result in regions of positive and negative electric potential, each of which is then surrounded by fluid rotating in the resulting E ⫻ B drift (Fig. 10). When combined with ~ φ

V

~ ~ n, φ

~ φ

(b)

(c)

(d)

~ n

E

E

(a)

Figure 10. Particle and energy can be transported by plasma fluctuations. This series of pictures shows how particle fluxes arise from electrostatic oscillations. The magnetic field is assumed to be in the direction directly out of the page. Part (a) shows the fluctuating potential ␾˜ , with positive ␾˜ shown by solid lines and negative ␾˜ shown by dotted lines. The potential gradients result in electric fields, which lead to E ⫻ B drifts and the fluctuating flow patterns shown in (b). In (c), ␾˜ is overlaid with a fluctuating density pattern, n˜, shown in dotted lines. A profile through the center of the pattern is shown in (d). With the relative phase of ␾˜ and n˜ as given, the flows are strongest to the right where the density is highest. The result is net particle flux to the right. If the relative phase of ␾˜ and n˜ were shifted by 앟, the net transport would be in the opposite direction.

58

FUSION PLASMAS

density or temperature perturbations from the instability, net particle and energy transport can occur. Theories of anomalous transport try to answer three questions. First, what is the nature of the plasma waves and the conditions that cause them to grow? Second, What is the spectral distribution of the waves in frequency and wavelength? Third, how much transport is caused? The source of microinstabilities is usually analyzed by linear theory—that is, by comparing destabilizing and stabilizing terms with respect to small-amplitude perturbations. Wave–particle interactions must often be added to get an accurate model for the dynamics. Linear theories can predict the conditions under which these waves will grow, as well as the exponential growth rate for small fluctuations. The growth rate can be determined as a function of k and 웆, suggesting a linear spectrum that is largest at those k and 웆 where the growth rate is highest. For finite-amplitude waves, nonlinear interactions will modify the instability as waves of different sizes exchange energy. On very short wavelength scales, linear and nonlinear damping mechanisms provide an energy sink as wave energy is converted to thermal particle motion. The fully evolved nonlinear, turbulent spectrum can be quite different from the one predicted by linear theory and requires powerful computers for its calculation. Particle and energy fluxes, driven by the turbulence, can then be determined by constructing appropriate phase-averaged quantities. For example, the particle flux from electrostatic instabilities is calculated from =

˜ n˜ 2 1/2 E˜ 2 1/2 sin θ n˜ E = B B

(31)

˜ are the fluctuating components of those two Where n˜ and E quantities, ␪ is their relative phase and 具 典 represents a phase average. While fluctuations in plasmas, similar to those predicted by theory, are readily observed, conclusive evidence linking them to anomalous transport has been difficult to obtain. In the plasma edge, electrostatic probes can measure all the required fluctuating quantities along with the their relative phases. In the plasma core, these measurements are much more difficult. Density fluctuations are measured routinely, though only over a limited range of wavenumbers, ˜ and B ˜ can scarcely be measured at all. The computawhile E tion of the turbulence spectra is also fraught with difficulties. In realistic geometries, three-dimensional (3-D) simulations may be necessary and substantial approximations must be made to allow the calculation to complete on even the most powerful machines. Runs taking over 1000 h on advanced supercomputer are not unusual. Thus progress is incremental, with experimental measurements and theoretical calculations used to guide each other to successively better models. HEATING, CURRENT DRIVE, AND FUELING The previous section considered the dissipation mechanisms for energy, mass, and current. In a fusion device, these must be balanced by sources, namely heating, fueling, and current drive. In an ignited reactor, the heat source would be internal, from the fusion reactor itself. However, even in this case, the reactor would need an external source of heat to bring the plasma to temperature. The present generation of experiments, of course, are entirely dependent on external heating.

Any device with finite particle confinement time needs a mechanism to replace or recycle particles lost at the boundary. Of course, a reactor would also need a source of fuel to replace hydrogen isotopes as they are converted to helium. Finally, those confinement schemes, which rely on currents flowing in the plasma, must find a method for sustaining that current against resistive losses or operate only in a pulsed mode. Heating Methods Ohmic Heating. For confinement schemes like the tokamak, which rely on large currents flowing in the plasma, one source of heat is readily at hand: resistive or ohmic heating. The plasma currents are driven inductively, with voltage appearing as the result of time-varying magnetic flux, which, in turn, is driven by external coils. Resistive dissipation of the plasma current provides the heating source. The local source rate is equal to ␩ j 2. The total plasma current is usually limited by MHD stability, resulting in heating proportional to ␩ 앜 1/T 3/2 e . Thus ohmic heating becomes less and less effective as the temperature increases. Tokamaks with ohmic heating alone can reach electron temperatures of several kiloelectronvolts, though the ions are typically much cooler. While there is no a priori reason why ohmic heating alone could not be large enough to reach ignition, extrapolation from experiments suggests that it would not be sufficient in any practical device. Neutral Beam Heating. Intense beams of neutral atoms have been used successfully for heating fusion plasmas since 1971 (39–41). Neutrals are used because charged particle beams, though they are easily produced, cannot penetrate magnetic fields. Neutral injectors (42) have three principal components: a plasma source, where gas (typically a hydrogen isotope) is ionized; an accelerator, where plasma ions are electrostatically extracted and accelerated; and a neutralizer, where the charged ions interact with neutral gas, picking up electrons to become neutral atoms. Atomic cross sections limit this approach to beam energies less than 150 keV. Beam currents are limited by space charge effects in the extractor/accelerator system to about 0.5 A/cm2. Injectors, however, can be quite large, enabling multi-MW systems to be assembled. Major fusion experiments may have more than 20 MW of neutral beam heating available. Neutral beam injectors have been used on virtually every type of magnetic confinement device. Once inside the plasma, the neutral beam is quickly ionized and the ion energy is converted to heat. Experiments have shown that these processes are essentially classical— that is, dominated by collisional rather than anomalous processes. It is believed that this is because the gyro radius of beam ions is much larger than the fine scale turbulence that is the cause of anomalous transport for the slower thermal ions. The large orbits effectively average out the fluctuations. Beam penetration is limited by atomic processes, principally electron and ion impact ionization and charge exchange, with ion impact ionization dominating at the highest energies available from conventional injectors, where the cross section is about 10⫺20 m2. Thus in a typical fusion plasma with a density of 1 ⫻ 1020 /m3, the beams will penetrate about 1 m. This would be problematic for proposed devices that might have cross sections which are several meters in radius, or for very

FUSION PLASMAS

high-density devices. Higher energies can only be produced in so-called negative ion sources, in which hydrogen atoms with two electrons are created, extracted, and accelerated. The second electron is only weakly bound, and the system is easily neutralized. Negative ion sources are experimental, but have produced beams with energies up to 500 keV (43). Radio-Frequency Heating. In radio-frequency (RF) heating, energy is added to the plasma via electromagnetic waves. A wide variety of approaches have been tested, differing mainly in the frequency of the waves employed. Each shares certain common features and faces similar issues, namely, RF generation, launching, coupling, propagation, and dissipation. Efficient, high-power RF sources are available over a wide range in frequency from a few hundred kilohertz to over 100 GHz. (The limitation at the high end is particularly important because this is the frequency range for electron gyro motion; f c ⫽ 28 GHz/T.) RF power can easily be transmitted by coaxial conductors or in waveguides to launching structures, situated near the edge of the plasma. For low frequencies, antennas are the appropriate launcher, with horns or windows used for microwaves. Coupling to the plasma is not as straightforward. Recall that due to rapid response of the electrons in a plasma, transverse electromagnetic waves below the plasma frequency are reflected rather than transmitted. In RF heating schemes, the goal is to launch a plasma wave that is able to propagate. Since most plasma waves do not propagate in a vacuum and since the antenna in general is not in direct contact with the plasma, there is typically an evanescent (nonpropagating) layer through which the wave must tunnel. The type of plasma wave employed will depend on plasma properties and the frequency of the launched wave. Finally, the waves must be made to deposit their energy into plasma particles. Depending on the heating scheme, this takes place via various wave–particle resonances, in which the particles are accelerated by the wave field and then give up their energy by collisions to the background plasma. Overall, the physics of wave propagation, mode conversion, and damping can be quite complex and it remains an active area of research. Three frequency ranges show the greatest promise for RF wave heating: ion cyclotron waves at frequencies of 20 MHz to 120 MHz; lower-hybrid waves at 1 GHz to 5 GHz; and electron cyclotron waves at 50 GHz to 250 GHz. In the ion-cyclotron range, antennas are generally used to launch the waves since the free space wavelength is 4m to 25 m, which is large compared to the size of the plasma being heated. The antenna, a current-carrying poloidal loop, drives a compressional Alfven wave that tunnels through a thin evanescent region at the plasma edge and then propagates across the plasma until it reaches the ion cyclotron layer (the radius where the RF frequency equals the ion cyclotron frequency). Efficient wave–ion coupling is realized only when the wave has the proper polarization relative to ion cyclotron motion. This is achieved by resonating with a minority ion species (minority heating) or by launching an RF wave at twice the cyclotron frequency of the majority ion (second harmonic heating). Both have been employed successfully in experiments (44,45). Figure 11 shows a set of ion cyclotron range of frequencies (ICRF) antennas installed in a tokamak. 2 2 /웆ce ), The lower-hybrid frequency is 웆LH 앒 웆pi兹(1 ⫹ 웆pe where 웆pi is defined by analogy to the electron plasma frequency as 웆pi ⫽ 兹(niZ2e2 /mi). The free space wavelength is on

59

the order of 0.1 m, so fundamental mode waveguides are typically used to carry and launch the waves, with the electric field polarized parallel to the magnetic field of the plasma. As in the previous case, the waves must tunnel through a thin evanescent layer before propagating in the plasma. To reach the lower-hybrid resonance, the waves must satisfy an accessibility condition:

k c > ω


s

1+

2 ωpe 2 ωce



(32) ω=ω LH

To produce this spectrum, waveguide arrays are used, with the waves’ phase appropriately shifted from one waveguide to the next. The waves can damp at the lower-hybrid layer by resonance with the ions or electrons, depending on details of the wave physics. Lower-hybrid heating experiments have been carried out with mixed results, mainly on smaller-scale experiments (46,47). The highest-frequency waves used for heating are in the electron cyclotron region. Here, strong RF sources are available only up to about 100 GHz, limiting the scheme to devices with fields less than 3 T. The wavelengths involved are only a few millimeters, and overmoded waveguides or quasi-optical techniques can be employed for transporting and launching the waves. Coupling and propagation are relatively simple: The plasma carries two types of electromagnetic waves that propagate perpendicular to the field at these frequencies, one polarized with E储B (ordinary or O mode) and the other with E ⬜ B (extraordinary or X-mode). The waves can be cut off when the wave frequency approaches the plasma frequency, so the scheme is limited to relatively low densities. By using higher electron cyclotron harmonics for the resonant absorption, the method can be applied to higher-density plasmas, but source availability and the weaker damping limit this to the first few harmonics only. Current Drive Plasma currents, which are necessary in many types of confinement schemes, are typically driven inductively, by swinging the flux in external coils. This approach is quite effective for pulsed experiments, but the flux swing is limited by the current carrying capacity of the coils. The induced voltage can only be sustained for a finite amount of time, after which the coils must be recharged. At high temperatures, the plasma resistivity can be very low and inductive currents can be sustained for quite long pulses—up to 30 s in some experiments. To achieve steady state, which is desirable because it reduces cyclic stresses, noninductive current drive methods must be employed. In certain circumstances, the plasma itself can generate most of the required current, tapping into free energy from the plasma pressure gradient. This phenomenon, called bootstrap current, can be understood by considering the banana orbits shown in Fig. 9(b). Note that the outer segment of these ion orbits is always in the direction parallel to the plasma current (co), whereas the inner segment is always in the antiparallel direction (‘‘counter’’). Since the density is higher near the center of the plasma than on the outside, the number of particles traveling in the ‘‘co’’ direction will outnumber those in the counter direction. The current carried by these particles can be estimated by first noting that for a

60

FUSION PLASMAS

Figure 11. An antenna used for ICRF heating installed in the Alcator C-Mod tokamak. The current straps, which generate the RF fields, can be seen behind a Faraday shield which keeps plasma from interfering with its operation.

collisionless plasma, the fraction of trapped particles is ⑀1/2 and that they have an average parallel velocity 具v储典 on the order ⑀1/2vT. The banana width is 웃 앒 ⑀⫺1/2q␳, from which the differential number of ‘‘co’’ and ‘‘counter’’ going particles can be inferred: n ≈ e1/2 δ

dn dn = qρ dr dr

(33)

Combining these results, the bootstrap current carried by trapped particles is j = ev n =

q 1/2 T dn B dr

(34)

Actually, somewhat more bootstrap current is carried by passing particles. This results from a distortion of the passing particle distribution function that is caused by their interaction with trapped particles (48). In theory, almost all the required current can be generated by this process, however, the bootstrap current profile may not be optimal from the point of view of MHD stability. Another method used for noninductive current drive is the application of RF waves with phase velocities parallel to the direction of the desired current. These waves can interact resonantly with electrons traveling at the same velocity, accelerating them in the wave field. Waves in the lower-hybrid frequency range can be employed for this purpose, though the launched spectrum must be modified from that used for heating (49,50). Electron cyclotron waves can drive current by differentially heating electrons traveling in the ‘‘co’’ and ‘‘counter’’ direction, using the resonance condition to separate the two populations via their Doppler shifts.

This effect is only strong for those electrons with velocities much higher than the average thermal velocity, leading to relatively low efficiency in most cases. Fueling In order for a fusion device to run much longer than a confinement time, some mechanism for replacing lost plasma must be found. Ionized particles leaving a plasma in a confined experimental volume are neutralized quickly when they touch the vacuum system walls. A substantial fraction are implanted in the surface layers of the wall, while at the same time other gas molecules are liberated by the impact of heat and particles on the surface. In equilibrium, there is a balance between these two processes, establishing a steady-state neutral gas pressure in the region between the plasma and the wall. Gas impinging on the plasma is disassociated and rapidly ionized, creating a strong source at the plasma boundary. The width of the source region varies with plasma density and temperature, but is typically a few millimeters to a few centimeters, much smaller than the plasma cross section. Most of these new ions are quickly lost again, but some are able to move up the density gradient into the core plasma, by processes that are only partially understood. A collisional process, the neoclassical or Ware pinch, can account for this inward particle flux in some cases; however, in many others, it is found to be too small. Anomalous pinches can be driven by some of the same processes that cause outward transport; in effect, a heat engine is created with outward energy flow driving inward particle flow against the gradient. Basic thermodynamic considerations put limits on how large these fluxes can be.

FUSION PLASMAS

While gas fueling works well in current experiments, the lack of a comprehensive theory for the process has motivated researchers to search for alternate techniques. Neutral beams, which are often used for heating, also supply particles to the plasma. In some experiments these can dominate the source from gas fueling; however, it does not seem to be a method that could extrapolate to a reactor. Deep fueling requires high-energy beams, but it is too expensive to supply all the plasma particles with that much energy. Another alternative is to inject fuel in the form of small cryogenic pellets (51). As macroscopic objects, they can penetrate much farther into the plasmas than individual molecules. Extrapolation is also an issue here, since injection velocity seems to be limited by the tensile strength of frozen deuterium, to a few kilometers per second. Pellets at these velocities are very effective for central fueling of current machines, but would not penetrate deeply into a reactor plasma. Recent experiments suggest that it may be possible to take advantage of certain MHD instabilities to augment this penetration, however (52). Central fueling, whether from beams or pellets, has additional benefits. Gas fueling tends to produce very flat density profiles, while fusion yields could be increased with peaked densities. Furthermore, it has been demonstrated that plasmas with peaked density profiles from central fueling can have dramatically reduced energy transport (17,53). BIBLIOGRAPHY 1. A. S. Bishop, Project Sherwood, the U.S. Program in Controlled Fusion, Reading, MA: Addison-Wesley, 1958. 2. J. L. Bromberg, Fusion: Science Politics and the Invention of a New Energy Source, Cambridge, MA: MIT Press, 1983. 3. R. Herman, Fusion, The Search for Endless Energy, Cambridge, UK: Cambridge Univ. Press, 1990. 4. K. M. McGuire, TFTR Team, Phys. Plasmas, 2: 2176, 1995. 5. A. Gibson, JET team, Phys. Plasmas, 5: 1997. 6. J. D. Lawson, Proc. Phys. Soc. B, 70: 6, 1957. 7. H. Eubank, PLT Team, Proc. 7th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Innsbruck, 1978, Vol. I, 1979, p. 177. 8. M. Greenwald, Alcator Team, Phys. Rev. Lett., 53 (4): 352, 1984. 9. K. Ushigusa, JT-60 Team, Plasma Physics and Controlled Nuclear Fusion Research, Proc. 16th Int. Conf., Montreal, 1996; Vol. I, 1997, p. 37. 10. JET Team, Proc. of 16th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Montreal, 1996; vol. I, 1997, p. 57. 11. K. M. McGuire, TFTR Team, Proc. 16th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Montreal 1996, Vol. I, 1997, p. 19. 12. A. S. Bishop and E. Hinnov, Proc. 2nd Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Culham, 1966, Vol. II, 1967, p. 673. 13. F. Wagner, Wendelstein Team, Plasma Phys. Controlled Fusion, 36: A61, 1994. 14. F. Sano, Heliotron Group, Nucl. Fusion, 30: 81, 1990. 15. L. A. Artsimovich et al., English translations of papers from Proc. 3rd Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Novosibirsk, 1968, Nucl. Fusion Suppl., 17: 1969. 16. N. J. Peacock et al., Nature, 224 (5218): 488, 1969. 17. M. Greenwald, Alcator Team, Proc. 11th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Kyoto, 1986, Vol. I, 1987, p. 139. 18. Y. Koide et al., Phys. Rev. Lett., 72: 3662, 1994. 19. F. M. Levinton et al., Phys. Rev. Lett., 75: 4417, 1995. 20. E. J. Strait et al., Phys. Rev. Lett., 75: 4421, 1995.

61

21. H. A. B. Bodin and D. E. Evans, Nucl. Fusion, 25: 1305, 1985. 22. D. J. Den Hartog et al., Euchimoto, Proc. 16th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Montreal, 1996, Vol. II, 1997, p. 83. 23. M. Tuszewski, Nucl. Fusion, 28: 2033, 1988. 24. W. H. Bennet, Phys. Rev., 45: 890, 1934. 25. E. M. Little et al., Proc. 3rd Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Novosibirsk, 1968, Vol. II, 1969, p. 555. 26. J. A. Van Allen et al., Jet Propulsion, 28: 588, 1958. 27. M. S. Ioffe et al., (English translation), Sov. Phys.—JETP, 13: 27, 1961. 28. T. C. Simonen, Nucl. Fusion, 25: 1205, 1985. 29. P. C. Stangeby and G. M. McCracken, Nucl. Fusion, 30: 1225, 1990. 30. G. F. Matthews, J. Nucl. Mater., 104: 220–222, 1995. 31. H. Grad and H. Rubin, Proc. 2nd United Nations Int. Conf. Peaceful Uses of Atomic Energy, United Nations, Geneva, Vol. 31, 1958, p. 190. 32. V. D. Shafranov, Sov. Phys.—JETP, 26: 682, 1960. 33. T. H. Stix, Waves in Plasmas, New York: American Institute of Physics, 1992. 34. R. A. Cairns, Radiofrequency Heating of Plasmas, Bristol, UK: Adam Hilger, 1991. 35. F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys., 48: 239, 1976. 36. L. M. Kovrizhnykh, Nucl. Fusion, 24: 851, 1984. 37. P. C. Liewer, Nucl. Fusion, 25: 543, 1985. 38. J. W. Connor and H. R. Wilson, Plasma Phys. Controlled Fusion, 36: 719, 1994. 39. K. Berkner et al., Proc. 4th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Madison, 1971, Vol. II, 1971, p. 707. 40. J. G. Cordey et al., Nucl. Fusion, 15: 441, 1975. 41. L. A. Berry, ORMAK Team, Proc. 5th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Tokyo, 1974, Vol. I, 1975, p. 101. 42. W. S. Cooper, K. H. Berkner, and R. V. Pyle, Nucl. Fusion, 12: 263, 1972. 43. K. Watanabe et al., Proc. 14th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Wurtzburg, 1992, Vol. III, 1993, p. 371. 44. Equipe TFR, Plasma Phys., 24: 615, 1982. 45. D. Q. Hwang, PLT Team, Proc. 9th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Baltimore, 1982, Vol. II, 1983, p. 3. 46. J. J. Schuss, Alcator Team, Nucl. Fusion, 21: 427, 1981. 47. C. Gormezano et al., Nucl. Fusion, 21: 1047, 1981. 48. R. J. Bickerton, J. W. Connor, and J. B. Taylor, Nature Phys. Sci., 229: 110, 1971. 49. S. Bernabei, PLT Team, Phys. Rev. Lett., 49: 1255, 1982. 50. M. Porkolab, Alcator Team, Proc. 9th Int. Conf. Plasma Phys. Controlled Nucl. Fusion Res., Baltimore, 1982, Vol. I, 1983, p. 227. 51. S. L. Milora et al., Nucl. Fusion, 35: 657, 1995. 52. P.T. Lang et al., Phys. Rev. Lett., 79: 1487, 1997. 53. B. J. Tubbing, JET Team, Nucl. Fusion, 28: 827, 1988. Reading List G. L. Baker and J. P. Gollub, Chaotic Dynamics, Cambridge, UK: Cambridge Univ. Press, 1990. A readable introduction to nonlinear dynamics. G. Bateman, MHD Instabilities, Cambridge, MA: MIT Press, 1978. F. Chen, Introduction to Plasma Physics, New York: Plenum Press, 1984. P. C. Clemmow and J. P. Dougherty, Electrodynamics of Particles and Plasmas, Reading, MA: Addison-Wesley, 1990.

62

FUSION REACTOR INSTRUMENTATION

R. O. Dendy (ed.), Plasma Physics, An Introduction Course, Cambridge, UK: Cambridge Univ. Press, 1993. J. Freidberg, Ideal Magnetohydrodynamics, New York: Plenum Press, 1987. R. J. Goldston and P. H. Rutherford, Introduction to Plasma Physics, Bristol: Institute of Physics, 1995. I. H. Hutchinson, Plasma Diagnostics, Cambridge, UK: Cambridge Univ. Press, 1987. An introduction to the experimental techniques used for making measurements of plasmas. N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, New York: McGraw-Hill, 1986. W. Lochte-Holtgreven, Plasma Diagnostics, Amsterdam: North-Holland, 1968. K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fusion, Tokyo: Iwanami Book Service Center, 1997. L. Spitzer, Physics of Fully Ionized Gases, New York: Wiley, 1962. An introduction to the field from one of its founders. T. H. Stix, Waves in Plasmas, New York: American Institute of Physics, 1992. A formal survey of wave phenomena in plasmas. D. G. Swanson, Plasma Waves, New York: Academic Press, 1989. J. Wesson, Tokamaks, Oxford: Oxford Science Publications, 1987.

MARTIN GREENWALD Massachussetts Institute of Technology

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Wiley Encyclopedia of Electrical and Electronics Engineering Plasma Chemistry Standard Article S. J. Pearton1 1University of Florida, Gainesville, FL Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5910 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (156K)

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Abstract The sections in this article are Model System: F -Based Etching of SiO 2

2

Etching of Si Etching of Al and Al alloys Etching of Refractory Metals and Silicides Etching of Photoresist and Polymers Loading and Aspect-Ratio-Dependent Etching Damage and Residues Plasma Analytical Techniques Compound Semiconductor Etching About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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504

PLASMA CHEMISTRY

PLASMA CHEMISTRY Many chemical reactions of neutral gases with semiconductor materials can be accelerated in the presence of a plasma (1). Examples include etching of patterns into single- or polycrystalline Si using F2- or Cl2-based chemistries, removal of photoresist films in O2 plasmas, etching of Al metallization in Cl2-based plasmas, and patterning of compound semiconductors in various halogen (Cl2, I2, or Br2) plasmas. The typical process takes advantage of the ability to tailor the vertical and horizontal etch rates of a film or substrate selectively masked by photoresist or dielectric and thus achieve highfidelity pattern transfer. Alternatively, in some applications one simply wants to strip a film selectively from an underlying layer of a different material, and in this case an isotropic etch process is employed. There are many variants of plasma etching apparatus (2–4), ranging from simple barrel reactors in which etching occurs only through reactive neutral species (leading to isotropic etching) to systems involving an additional, significant flux of energetic ions arriving at the sample surface simultaneously with the reactive neutrals. The effect of these ions is to enhance greatly the adsorption of the reactive neutrals by providing ‘‘active’’ sites, the subsequent reaction of the adsorbed species with the sample (for example, by breaking bonds in the materials), or desorption of the etch product by essentially sputtering it from the surface. Of course, the latter step exposes a fresh surface for the process to be repeated all over again, producing a synergy between the chemical and physical component of the etching process and leading to a material etch rate larger than the sum of the two-individual components. A schematic of a sample in an ion-assisted etch system is shown in Fig. 1. MODEL SYSTEM: F2-BASED ETCHING OF SiO2 The most common variant of the anisotropic (i.e., ion assisted) etching techniques is called reactive ion etching (RIE), and some typical characteristics of plasmas in this mode are shown in Table 1 (3). In Si device technology there are many dry etching steps involved, with requirements ranging from formation of deep (⬎6 애m), narrow (⬍0.5 애m) trenches for storage capacitors in the Si wafer to polysilicon/polycide gate definitions, where very high selectivity over a thin underlying oxide is required.

Plasma +

+

+

N

Te ∼ 10 eV Ti ∼ 1 eV ni = ne ≤ 1012 cm–3 vi = 106 cm ⋅ s–1 nN = 3 × 1013 cm–3 (1 mtorr)

+

Mask

Volatile product

Sheath ∆V ≈ 100 V λ D ∼ Few millimeters

Substrate Electrode Figure 1. Schematic of sample geometry in an ion-enhanced etch reactor showing impingement of ions and neutral reactive atoms.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

PLASMA CHEMISTRY

Table 1. Typical Characteristics of Low Pressure Plasmas Used for Reactive Ion Etching Quantity

Typical Values

rf Power density rf Frequency Pressure Gas flow Wafer temperature Gas temperature Electron temperature Ion energies Gas number density Ion density Electron density Ion flux Radical flux Neutral flux

0.5–1.0 W/cm2 10 kHz–27 MHz (commonly 13.56 Hz) 0.01–0.2 torr 10–200 sccm ⫺120⬚C up to 300⬚C 300–600 K 3–30 eV (bulk of plasma) 앑.05 eV (bulk of plasma) 10–500 eV (after traversing cathode sheath) 3.5 ⫻ 1014 –7 ⫻ 1015 cm⫺3 109 –1010 cm⫺3 Similar to ion density 1014 –1015 cm⫺2 s⫺2 1016 cm⫺2 s⫺1 3.6 ⫻ 1018 –7.2 ⫻ 1019 cm⫺2 s⫺1

Since a plasma is a partially ionized gas, there are a number of different processes occurring in the gas phase. For example, for a molecule xy, the following may occur due to electron collisions:


x+y+e
x + 2e; e + xy
xy + 2e e + x
x + e; e + xy
xy + e

dissociation e + xy ionization excitation

e+x

+

∗

+

∗

where x⫹, xy* are ions and x⫹, xy* are radicals or species whose energy is much larger than the ground state, and thus are very reactive because of their incomplete bonding state. The major application of dry etching of SiO2 is to pattern contact holes to underlying Si and to create via connection holes between various levels of metal in a multilevel metallization. The basic plasma chemistry for SiO2 is based on fluorocarbon gases. The etch products are therefore SiF4 and CO or CO2 (5–9). The etch reactions may therefore be written as

CF4 → 2 F + CF2 SiO2 + 4 F → SiF4 + 2 O SiO2 + 2 CF2 → SiF4 + 2 CO Atomic fluorine reacts rapidly with Si, but its reaction rate with SiO2 is orders of magnitude lower. The amount of free fluorine in the discharge may be increased by adding oxygen (4). That is, CF4 + O → COF2 + 2 F If the amount of oxygen addition is too great, the etch rates may decrease because of gas phase recombination through the reactions (4) O2 + F → FO2 FO2 + F → F2 + O2 By contrast, the concentration of free fluorine radicals may be decreased by addition of H2, rather than O2, to the fluorinebased plasma, the recombination of hydrogen and fluorine to form HF. As small concentrations of O2 are added to CFx, the

505

etch rates of both SiO2 and Si are increased, but not at the same rate. Thus it is possible to achieve high etch rates (or selectivity) for Si relative to SiO2. By contrast, as H2 is added to CF4, the etch rate of Si falls much more quickly than that of SiO2, producing high selectivity for SiO2 in the absence of ion bombardment, but CFx species can etch SiO2 through the reactions noted previously. The two main factors determining the selectivity for etching SiO2 over Si are the deposition of the carbon-based polymer residue and the role of oxygen in the etching process. If a thick, nonvolatile polymer deposits on a surface, it will inhibit chemisorption of reactive species and quench the etching process. However, a lesser amount of polymer forms on SiO2 than on Si because the carbon in the polymer can react to form volatile CO and CO2. Therefore, SiO2 can continue to etch even as the etching of Si is completely prevented by the polymer formation. Selectivities for SiO2 over Si greater than 20 are readily achievable by this method. The ion-induced reactions in a CF4 /H2 chemistry typically require a minimum ion bombardment energy of 100–200 eV (10–14). Much of the data on etching the Si/SiO2 system is explainable by considering the F/C ratio in the plasma chemistry. The Si etch rate is faster at higher F/C ratios, which can be created by altering the gas mixture (such as by O2 addition). At low F/C ratios, obtained with CHF3, C3F8, C2F6, or CF4 /H2, there is high selectivity for SiO2 over Si. Note that frequent chamber cleaning is required with polymer-forming chemistries to reduce particle counts. Most selective etching of SiO2 is now based on CHF3 with Ar or He as a diluent and a more fluorine-rich compound (e.g., C2F6 or CF4) as a moderator. A typical process might involve varying the F/C ratio as endpoint is neared to enhance selectivity. Generally, in the set of plasma etching conditions optimized for selective SiO2 removal, it is difficult to achieve selective etching of SiNx relative to either SiO2 or Si. This is due to the fact that the nitride has bond strength and electronegativity between the other two materials. It is possible to etch SiNx selectively over oxide by reducing the ion bombardment energy or by employing very H2-rich chemistries. There are three basic mechanisms for achieving selectivity of one material over another (3): 1. Selective formation of an etch-inhibiting layer on one of the materials (i.e., the situation where deposition is occurring on one material, while the other is etched under the same conditions) 2. Nonreactivity of one of the materials in the particular plasma chemistry employed, such as removal of resist films in an O2 plasma that does not etch the underlying Si or SiO2 3. Nonvolatility of a reaction product, such as formation of the nonvolatile AlF3 on the surface of AlGaAs upon removal of an overlying GaAs layer in BCl3 /SF6 or equivalent chemistry (15,16). Selectivity is nearly always decreased by increasing ion bombardment energy because this accelerates the adsorption, reaction, and desorption steps that initially produce the selective etching (17–20). ETCHING OF Si Table 2 shows the range of plasma chemistries typically employed for plasma etching of Si (21). There are two basic areas

506

PLASMA CHEMISTRY

Table 2. Etch Chemistries for Si (After Ref. 21) Category

Comments

Primary Gases CF4

SF6

CCl4 , CF2Cl2 SiCl4 Cl2 Br2 , HBr

Isotropic with added O2 , low etch rate without O2 . Relative low selectivity over SiO2 (with O2), compared to low-pressure, high-density Cl2 /O2 plasmas. Isotropic except at very low pressure and/or substrate temperature. Relatively low selectivity over SiO2 , compared to low-pressure, high-density Cl2 /O2 plasmas. Anisotropic under most conditions. O2 or Cl2 addition prevents polymer formation. Addition of Cl2 increases etch rate to acceptable levels. Anisotropic under all conditions, except for n⫹ Si. Slightly higher rates and selectivity over SiO2 , compared to Cl2 .

Additives Cl2 O2 HBr BCl3

Added to CCl4 , CF2Cl2 to increase etch rate, minimize polymer formation. Increases selectivity over SiO2 when added to Cl2 . Sometimes added to Cl2 to increase etch rate and selectivity. Sometimes needed to break through native oxide and to scavenge H2O.

of application. The first is etching of high aspect ratio (depthto-width ratio) trenches or grooves for vertical capacitors in dynamic random access memory integrated circuits, or for circuit isolation. The second major application is patterning of polycrystalline Si for gates and high-temperature interconnects in metal oxide semiconductor (MOS) technology. For some applications metal silicides or refractory metals have replaced poly-Si, and these materials are covered later. The trenches for isolation or capacitor formation are deep (3 to 6 애m) and need excellent verticality and clean, smooth sidewalls. The need for subsequent filling and planarization steps also requires a V-shaped bottom of the trench. Oxide masks are generally used because resist degrades at the high biases needed to produce high etch rates and vertical sidewalls. F2-based plasma chemistries are often not the best suited for trench etching because of the difficulty in obtaining anisotropy without extensive polymer-sidewall protection, which leads to difficulties in maintaining constant etch rates as the aspect ratio of the trench increases with etch time (this is usually called reactive ion etch lag, in which narrow features etch slower, or aspect-ratio dependent etching; see Ref. 22). Chlorine-based plasmas etch Si by an ion-assisted mechanism, so that vertical sidewalls without undercut are achievable (23–26). The SiCl4 etch product is less volatile than SiFx and hence requires ion assistance to desorb. The basic reaction is therefore (4)

e + Cl2 → 2 Cl + e Si + 2 Cl → SiCl2 SiCl2 + 2 Cl → SiCl4

Some have suggested that silicon subchlorides act to coat the trench sidewall and provide additional protection against undercut (4). Undoped Si etches very slowly in Cl or Cl2 in the absence of ion bombardment. However, heavily doped n-type Si etches spontaneously at high rates without ion bombardment in atomic Cl. This effect due to doping may be as high as a factor of 25 larger than undoped Si and is independent of the dopant species (27,28). This suggests that atomic Cl chemisorbed on Si does not break the underlying Si—Si bonds initially, but on an n-type surface will become negatively charged and can ionically bond with the Si. Undercut on n-type sidewalls can be eliminated through polymerization schemes, such as adding BCl3, SiCl4, CCl4, or a fluorinated precursor to the Cl2 plasma. A typical example of the latter is C2F6 /Cl2. Often the etch is initiated with BCl3 /Ar to remove the native oxide (29). The addition of O2 to Cl2 enhances selectivity over SiO2, and bromine-based plasma chemistries (HBr or Br2, mixed with Cl2) may provide higher etch rate and selectivity over SiO2 than for pure Cl2 in high-density plasmas. SF6 may be used to etch Si anisotropically by adding an inert gas such as Ar to increase the ion-to-F flux impingement ratio, by adding a sidewall passivant to inhibit attach by F atoms, or by lowering the substrate temperature so that the SiF4 etch product is longer volatile except where ions are desorbing it (30). The most stable, fully halogenated etch products for Si are SiF4 (boiling point ⫺86⬚C), SiCl4 (boiling point 57.6⬚C), and SiBr4 (boiling point 154⬚C). Due to the ion-assisted desorption, subhalogen etch products may also play a significant role. The absolute reaction rate of F atoms with single crystal Si follows (9): ˚ · min−1 ) = 2.9 × 10−12T 1/2 N exp Rate (A F



−EF RT



where T is the absolute substrate temperature, NF the F-atom number density, EF the measured activation energy of 2.49 kcal ⭈ mol⫺1, and R the gas constant. This equation represents the reaction of an F-saturated Si surface. At a pressure of 10 ˚ ⭈ min⫺1, which mtorr and 25⬚C, the etch rate would be 230 A would produce significant undercut. Cooling the substrate to ˚ ⭈ min⫺1. ⫺100⬚C would reduce this isotropic etch rate to 10 A While substrate cooling is impractical for most manufacturing processes, it is usually necessary to employ He backside cooling of wafers during high-density plasma etching to avoid undercutting. As mentioned previously, the reaction rate of Cl atoms is ˚ ⭈ min⫺1 at 10 mtorr at 25⬚C) with undoped quite slow (앑20 A or p-type Si, but is higher for n⫹ Si. For n ⫽ 5 ⫻ 1018 cm⫺3 Si covered with Cl at steady state (31), ˚ · min−1 ) = 7 × 108 P exp Rate (A Cl



−ECl RT



where ECl has been measured as 6.64 kcal ⭈ mol⫺1 (31). At 10 ˚ ⭈ min⫺1. Similarly, for mtorr and 25⬚C, the etch rate is 100 A Br2 etching (32), ˚ · min−1 ) = 1012P exp Rate (A Br



−EBr RT



PLASMA CHEMISTRY

where EBr is 14.8 kcal ⭈ mol⫺1. Due to this large activation energy, undercutting is much less severe in Br2-based plasma chemistries. HBr (and sometimes also O2) may be added to Cl2 during the entire etch process for selective etching of Si over SiO2, or in many cases this addition is incorporated just prior to reaching the SiO2 layer. ETCHING OF Al AND Al ALLOYS Dry etching of Al interconnect lines is a critical technology for achieving high device densities. Reliable, reproducible etching of Al presents a number of challenges, as follows: 1. Al is always initially covered with a stable native oxide, Al2O3, which prevents etching until it is removed. The first step in the etch process is therefore a breakthrough of this oxide, either by Ar sputter etch or reagents that are strong reducing agents in plasma form. The Lewis acid BCl3 works well for this application through the reaction O + BCl3 → B − OCl + 2 Cl

2.

3.

4.

5.

6.

While CCl4 also is an oxide reducing agent, it is not as effective at obtaining oxygen and water vapor in the reactor as is BCl3 (33–35). This is a very important ability because moisture in the chamber from reactor desorption or resist erosion would react with the Al surface and create a new oxide that would inhibit further etching (36–38). Resist erosion due to Cl attack, which is worsened by the presence of the etch product AlCl3. In most cases, special chlorine-resistant resists are employed to reduce erosion. Toxicity of the Cl2-based gases and their products. These tend to dissolve and become concentrated in the vacuum pump oil, and precautions are necessary in changing and disposing of this oil. Further, oxidation of AlCl3 or of the process oxides SiCl4 and BCl3 produces particulates that need to be filtered from the pump oil. Hygroscopic nature of the etch byproducts AlCl3. The presence of this species on the reactor walls will lead to absorption of moisture if the chamber is opened to air; this is minimized by use of vacuum load locks and keeping the reactor chamber walls at ⱖ35⬚C. Presence of sidewall passivating AlCl3 residues, which can react with moisture in the air upon removal of the wafer from the reactor, corroding the Al line by formation of HCl. Several methods are employed to overcome this, including rinsing in deionized water, in situ removal of resist with an O2 plasma (which also replaces the AlCl3 with Al2O3), and use of an F2-based plasma to form AlF3, followed by rinsing in HNO3 to remove the fluoride and passivate the Al. Difficulty in volatilizing Si and/or Cu additives to the Al (for improved electromigration resistance) (39). In particular, to remove Cu, either very high levels of ion bombardment or elevated substrate temperatures (ⱖ180⬚C) are necessary, requiring special hardening of the resist mask (40).

507

To achieve anisotropy in Al dry etching, it is always necessary to include a sidewall passivant, such as SiCl4, CCl4, CHCl3, or BCl3, usually in concert with a resist mask to produce a sidewall polymer. For example, BCl3 will etch Al isotropically if a dielectric mask is used in place of resist, implicating the latter in the sidewall passivation mechanism. The strong chemical nature of Al etching makes it susceptible to loading effects, and with a small exposed area the etch rate and degree of undercut may be high. ETCHING OF REFRACTORY METALS AND SILICIDES Refractory metal silicides are increasingly replacing doped polysilicon in gate interconnects in MOS integrated circuits. For a number of device applications a polycide (polySi/silicide sandwich) involving MoSi2, WSi2, TaSi2, or TiSi2 layers is employed. In other cases the pure metal is used because it may have a much lower resistivity than the corresponding silicide, and it may be deposited by chemical vapor deposition (CVD), which produces superior step coverage to sputtering. The chlorides of W and Mo have much lower vapor pressures than the corresponding fluorides, and thus the typical chemistries for etching are CF4 /O2, SF6, and so on. For Ti and Ta, the chloride etch products have higher or comparable vapor pressures to their fluoride counterparts. Ion bombardment is necessary in many cases to increase verticality of the pattern transfer, although TiW usually etches anisotropically because of the low volatilities of the tungsten chloride products (41–51). Since the silicides contain more than one element, it is necessary to have product desorption at similar rates to retain good surface morphology. For example, the vapor pressure of SiF4 is several orders of magnitude larger than that of MoF6, and therefore desorption of the latter is generally rate limiting during etching of MoSi2. The desorption rates can be juggled by increased substrate temperature, lowered process pressure, or increased ion bombardment. It is somewhat unusual in Si technology to have a situation in which the rate-limiting step is evaporation of the reaction product, rather than surface reaction kinetics. Often, to improve anisotropy, refractory metal and polycide interconnects are etched with mixed halogen plasma chemistries to achieve high rates (at high F atom concentrations) and yet maintain some anisotropy through formation of chlorinated sidewalls. In general, the metals are less sensitive to undercutting than polysilicon. To avoid differential undercut in polycide structures, multistep etch regimes are often employed in which the refractory metal silicide is removed in a mixed halogen chemistry, followed by Cl2 /O2 etching of the polysilicon. Tungsten and Mo can be etched in BCl3 /Cl2 without undercut, but Ti is susceptible to isotropic sidewalls. ETCHING OF PHOTORESIST AND POLYMERS Removal of photoresist by ‘‘ashing’’ in an O2 plasma to form CO2, CO, and H2O species was the first major application of plasma etching in microelectronics. Typical removal rates are ˚ ⭈ min⫺1, resulting in a 10 min process time for around 1000 A a standard 1 애m thick resist scheme. There is often a strong loading effect because a large number of wafers are cleaned at the same time; and because the process is then diffusion controlled, clearing of the resist begins from the perimeter of

508

PLASMA CHEMISTRY

the wafers and leads to a bullseye pattern during the removal. Eventually, of course, the entire film is cleared. Directional etching for contact hole formation in polyimide dielectric layers, and pattern delineation in organic polymers, is also necessary and is generally carried out in O2 /CF4 plasmas. The absence of ion bombardment during photoresist stripping means there is no ion-induced damage (52–57). Moreover, the addition of a few percent CF4 or SF6 to the O2 flow can substantially increase the resist removal rate. This appears to occur by F removing H from the resist as HF and leaving an alkyl that is very reactive with oxygen. The strip rates are as high at 1 애m ⭈ min⫺1 in O2 /CF4, while elevating the sample temperature to 200⬚C and employing microwave excitation leads to even higher rates, 8 애m ⭈ min⫺1 (52). Etching of resists and polymers is particularly important in bilevel lithography, which is employed to overcome step coverage problems in circuit topology. The scheme typically consists of an initial thick planarizing organic film, followed by a thin dielectric, metal (or semiconductor) layer, usually termed the transfer layer. The stack is completed with a thin imaging resist, which is lithographically patterned. An F2based dry etch is then used to transfer the pattern into the transfer layer, with reasonable selectivity over the underlying planarizing layer. The transfer layer material is then employed as the mask for O2-based etching of the planarizing layer. The advantage of this process is that topology can be planarized using the thick polymer film, but the associated loss of resolution is overcome by using the second, thin imaging resist, which by itself would have insufficient thickness and etch resistance to be suitable. Anisotropic etching of polymers in O2-based plasma chemistries also involves sidewall passivation, generally by redeposition of SiO2 from the mask, crosslinking of the immediate sidewall material, and backsputtered substrate material. Etch-back planarization of SiO2 for defining patterns over high substrate topographies involves equirate etching of resist and oxide, generally in a well-calibrated CF4 /O2 chemistry. LOADING AND ASPECT-RATIO-DEPENDENT ETCHING The loading effect refers to a decrease in etch rate with increasing exposed wafer area, due to a greater consumption of reagents (22). In principle it should be possible to explain these effects by noting the bulk diffusion-limited reagent supply. The practical effect of loading is that an adjustment of etching time must be made for each different amount of material (either wafers or exposed area). Depending on the gas phase mean free path, the effect may be both global and local. The loading effect may be described by several different representations of the same equation: R=

R0 1 + kA

where R0 is the empty chamber etch rate, A is the area of exposed material to be etched, and k is a reactor-dependent constant RA =

(KE /KL ) · G 1 + (KE ρA/KLV )

where RA is the etch rate at area A, G is the rate of production of etchant species, ␳ is the number density of substrate molecules, V is the reactor volume, and KE and KL are rate constants for etching and etchant loss in an empty reactor. For a large wafer load this relationship simplifies to RA =

GV ρA

and the etch rate is inversely dependent on wafer load area A. The loading effect may be reduced by making KL large relative to KE by processes such as high flow rate (rapid pumping) that consume etchant species. Ideally the etching should be independent of feature size, aspect ratio, fraction of exposed material, and mask material. While the loading effect is severe in isotropic etch processes where etch rate is proportional to etchant species, it is generally smaller in anisotropic etching because in the latter case etch rate is mainly controlled by ion bombardment flux. The local or microscopic loading effect occurs when there is sensitivity of etch rate to pattern density, and thus etch rates may change over distances of microns. The reactant concentration varies locally due to consumption by a reactive material and nonconsumption by a nonreactive material. What this means practically when etching Si through to an underlying layer such as SiO2 is that an increased overetch time is required to ensure clearing of Si from the slower etching regions, and thus there is a more severe demand on selectivity. In addition, this overetch solution is not possible for trench etching in single-crystal Si. Another practical result of the loading effect is in etching of Al interconnect lines. In some conditions the Al etch rate will increase as the Al film clears and undercutting will proceed very quickly if the plasma exposure is continued. The aspect ratio dependence of etch rate will become increasingly more of an issue as trench widths decrease while at the same time having even greater depths. In general, the etch rate decreases almost linearly as the aspect ratio increases and is essentially independent of trench diameter opening. A number of mechanisms have been postulated for this effect, including diffusion problems with the supply of reagent to the bottom of the trench, consumption of reactant at the trench sidewalls, and presence of developing electric fields within the trench (4). DAMAGE AND RESIDUES Substrate and oxide damage and the presence of etch residues are all deleterious effects of dry etching (58–63). Typical ion energies range from 50 to 700 eV at fluxes around 1015 ions ⭈ cm⫺2, and this can produce ion bombardment damage. Additionally, particulate formation from the gas phase and sputtering of metallic impurities from reactor surfaces are problems. Since etch anisotropy occurs due to ion bombardment and deposition of polymeric layers, it is inevitable that lattice damage and surface (or sidewall) disruption will be present. A typical Si surface after reactive ion etching (RIE) may con˚ thick, followed sist of a teflonlike polymeric film typically 30 A ˚ Si- and O-rich interfacial polymer, followed by 앑30 A ˚ by a 20 A ˚ of Si-conof heavily damaged Si, followed by up to 250 A taining hydrogen, if the latter was part of the etch chemistry.

PLASMA CHEMISTRY

The organic top film is readily stripped in an O2 plasma, while annealing at 400⬚C is generally sufficient to anneal out lattice disorder. This anneal should also remove the effects of hydrogen passivation of boron dopant impurities. In photoresist stripping there is generally a residue of alkali ions and heavy metals that were present in the resist itself (2). These must be removed by net chemical cleaning to prevent their incorporation into oxide or the Si itself during subsequent processing. Accumulation of charge in the gate area of MOS transistors may seriously degrade device performance and occurs during plasma processing when a wet current is drawn through a wafer containing floating gates (63). This may produce field-induced breakdown of the gate oxide. In general, RIE damage is used to refer to any or all of the following phenomena (58,59): surface residues, especially fluorocarbon films, lattice displacement damage consisting of point defect complexes, hydrogen passivation of dopants, impurity implantation, heavy metal contamination by sputtering of electrode or other reactor materials, mobile ion contamination, surface roughness from micromasking and redeposition, gate oxide breakdown due to change, and postetch corrosion by Cl2 residues, especially on Al. A particular feature concerning the effect of surface residues on morphology of etched Si is the appearance of so-called black silicon after Cl2 trench etching; this consists of grasslike or more isolated surface features resulting from contamination or oxide precipitates that produce micromasking. PLASMA ANALYTICAL TECHNIQUES There are a large number of endpoint detection, plasma diagnostic, and plasma analytical techniques available, and here we will summarize just a few. For detection of the end of an etch process, laser interferometry optical emission spectroscopy and mass spectroscopy are the most common methods. 1. Optical Emission Spectroscopy. Most processing plasmas emit radiation from the infrared (IR) to ultraviolet (UV) regions of the spectrum, and the intensity of these emissions as a function of wavelength can be measured with a spectrometer. Quantitative concentrations of different species giving rise to the peaks can be obtained if the observed intensities are calibrated against those of a small amount of gas such as Ar added for that purpose. This is known as actinometry (64–68). Table 3 lists some common emission lines used for endpoint detection (3,4). 2. Mass Spectrometry. A differentially pumped mass spectrometer unit attached to a reactor can sample constituents of the gas with an electron beam to ionize them and then measure the mass spectrum. One must be aware of the presence of molecule fragmentation. For example, a CF2⫹ ion might originate from the CF2 radical, but could also come from the molecules CHF3 and C2F4. Due to difficulties in assigning peaks due to species with the same mass-to-charge ration (e.g., Si⫹, CCO⫹, C2H4⫹, N2⫹, Fe2⫹) and the fact that the reactive plasma eventually degrades the filament in the mass filter, mass spectrometry is generally not used for endpoint detection.

509

Table 3. Common Optical Emission Lines Used for Endpoint Detection Material Silicon

SiO2 Si3N4

W Al Resist

Etchant Gas

Emitting Species

CF4 /O2 ; SF6 CF4 /O2 ; SF6 Cl2 ; CCl4 CHF3 CF4 /O2 CF4 /O2 CF4 /O2 CF4 /O2 CCl4 ; Cl2 ; BCl3 CCl4 ; Cl2 ; BCl3 O2 O2 O2 O2

F (Etchant) SiF (Product) SiCl (Product) CO (Product) N2 (Product) CN (Product) N (Product) F (Etchant) Al (Product) AlCl (Product) O (Etchant) CO (Product) OH (Product) H (Product)

Wavelength (nm) 704 440; 287 484 337, 387 674 704 391; 261 777; 484, 309 656

777

452–650

394; 396 843 450

3. Laser Reflectance and Interferometry. This monitors changes in film thickness thorugh changes in the reflectance as it is being etched. It therefore gives an accurate measure of etch rates in real time and requires only that the film be partially transparent and that the film and substrate have different optical constants (69,70). The spacing between adjacent maxima in the reflected beam is equal to ␭ /2n, where ␭ is the wavelength of the laser light and n the refractive index of the film. The main drawbacks of this technique are the requirement for a special test area on the wafer and the fact that etch rate is only sampled in this area. 4. Langmuir Probes. A conducting metal probe smaller than the particle mean free path placed directly in the plasma can be used to measure electron density, temperature, and plasma potential from the current-voltage characteristic. An extensive literature exits on probe theory and operation (71–73), but there are numerous pitfalls to their use, including the fact they do not work well in the sheath region of the plasma and they can disturb the electron temperature. Other techniques for plasma diagnosis include optogalvanic spectroscopy, where a change in discharge parameters induced by the absorption of light is monitored (74,75); laserinduced fluorescence, where a tunable dye laser excites species and the emission spectra is measured (77); and ellipsometry. COMPOUND SEMICONDUCTOR ETCHING There are four main etch techniques employed in compound semiconductor etching: 1. Simple Ar⫹ ion milling at energies of 앑500 eV for formation of shallow mesa isolation for field effect transistors (FET) 2. Chemically assisted ion beam milling, where Cl2 gas is injected from a nozzle near the sample and a separate plasma source provides Ar⫹ ion bombardment

510

PLASMA CHEMISTRY

Microwave power (2.45 GHz) Upper magnets (875 Gauss) Gases

Lower magnets (for collimation) Electrode

Substrate bias rf source (13.56 MHz)

ICP RF source (2 MHz) Gases

Electrode

Substrate bias RF source (13.56 MHz) Figure 2. Schematic of Electron cyclotron resonance (top) and inductively coupled plasma (bottom) reactors.

3. RIE, similar to that used in Si technology, which is employed for applications ranging from gate mesa forma˚ , but tion on heterostructure FETs (etch depth 앑300 A high selectivity, ⬎600, required for GaAs over AlGaAs) to through-wafer via creation for power FETs (etch depth 앑100 애m) 4. High-density etch reactors, principally electron cyclotron resonance (ECR) and inductively coupled plasma (ICP), which have ion densities several orders of magnitude greater than RIE systems and produce much higher etch rates at lower ion energies. Schematics are shown in Fig. 2. The typical gas chemistries for III–V semiconductors are listed in Table 4—note that F2-based gases do not create vola-

Table 4. Typical Etch Mixtures for III–V Semiconductors Chemistry (a) Cl2 based Cl2 , SiCl4 , BCl3 , CCl2F2

(b) CH4/H2 based CH4 /H2 , C2H6 /H2 C3H8 /H2 , CH4 /He

(c) I2 based HI, CH3I, C2H5I, I2

(d) Br2 based HBr, CF3Br, Br2

Comments

Typical Rates

Usually have additions of Ar or He. Smooth for GaAs, rough for InP.

˚ ⭈ min⫺1 for 3000 A ˚ ⭈ GaAs; 300 A min⫺1 for InP

Ar often added for stability. Smooth etching of InP. Heavy polymer deposition. High rates at room temperature for InP. Corrosive. No polymer deposition. Corrosive

˚ ⭈ min⫺1 for 300 A InP and InGaAs, lower for GaAs

˚ ⭈ min⫺1 for 5000 A InP and InGaAs; ˚ ⭈ min⫺1 3000 A for GaAs ˚ ⭈ min⫺1 for 600 A ˚ ⭈ GaAs; 400 A min⫺1 for InP

Table 5. Boiling Points of III–V Etch Products Species

Boiling Point (⬚C)

GaCl3 GaBr3 GaI3 (CH3)Ga InCl3 InBr3 InI3 (CH3)3In AlCl3 AlBr3 AII3 (CH3)3Al NCl3 NI3 NF3 NH3 N2 (CH3)3N PCl3 PBr5 PH3 AsCl3 AsBr3 AsH3 AsF3

201 279 Sub 345 55.7 600 ⬎600 210 134 183 263 191 126 ⬍71 Explodes ⫺129 ⫺33 ⫺196 2.9 76 106 ⫺88 130 221 ⫺55 ⫺63

tile etch products, which is an advantage in the sense that dielectric films are easily patterned with high selectivity over the underlying semiconductor. Typical overall etch reactions for Cl2 or CH4 /H2 plasma etching of GaAs follow: GaAs + 6 Cl → GaCl3 + AsCl3 GaAs + 3 CH4 + H2 → (CH3 )3 Ga + AsH3 In practice the Cl2 etch products depend on the chlorine pressure, etch temperature, and ion flux. Chlorine molecules and atoms do not etch GaAs at temperatures below several hundred degrees in the absence of ion bombardment, but desorption of subchlorine and fully chlorinated species occurs at room temperature when ions are present. Indeed virtually all etching of compound semiconductors occurs by ion assistance. Table 5 shows some boiling points for III–V etch products. Note that Cl2 is a good choice for most materials, with the exception of InP, where the InCl3 product is relatively involatile at normal temperatures. To enhance desorption it is necessary either to heat the InP substrate during the etch process or employ a high ion flux to assist removal of the InCl3. At low pressure (⬍20 mtorr) the etching is relatively anisotropic for all III–V semiconductors because of the ion-driven nature of most of the processes (78). Damage in dry etched compound semiconductors consists generally either of ion-induced deep-level compensation, which degrades both the electrical and optical quality of the material, or of stoichiometry changes to the near surface through preferential loss of one of the lattice elements. Damage induced by ions is proportional to the flux-energy product, and therefore in high-density etch systems the ion energy must be kept low to minimize creation of deep-level states (78,79).

PLASMA CHEMISTRY

High-density reactors are particularly useful for patterning of materials with high bond energies, such as GaN, AlN, and InN, which form a particularly attractive alloy system for photonic devices ranging from the red-UV regions of the spectrum. While the etch products for these materials are just as volatile as for more conventional semiconductors such as GaAs, etch rates under RIE conditions are generally factors of 5 to 10 lower because of the low rate of bond breaking. In the much higher ion fluxes available with ECR or ICP reactors, it is easier to break bonds that will allow the etch products to form, and therefore much higher rates are obtained. To achieve smooth etched surface morphology on compound semiconductors, there are several necessary conditions that must be met. First, the native oxide should be removed quickly at the start of the etch process—if it breaks through nonuniformly, this roughness will be replicated in the etched semiconductor surface. Plasma chemistries involving BCl3 or CH4 /H2 readily attack the native oxide. Second, there must be equirate removal of both the group III and group V elements, and this can only be achieved by adjusting the ion/neutral ratio and ion flux and energy, since generally the etch products for these elements have different volatilities. Third, both the mask material and carrier wafer (electrode materials may have a strong influence on etched surface morphology through effects such as micromasking or altering the ion/neutral balance). The same basic plasma chemistries are employed for II–VI compound semiconductors such as ZnSe, SnS, CdS, and HgCdTe. These materials are generally even more susceptible to preferential loss of one of the constituent elements than the III–V materials, but can be etched smoothly and anisotropically in Cl2- or CH4 /H2-based plasma chemistries. For materials such as SiC, which has an extremely high bond strength, high flux reactors produce much higher etch rates than conventional RIE. It appears that fluorine-based plasma chemistries such as NF3 /O2 and SF6 /Ar provide the highest etch rates, with Cl2, Br2, and I2 chemistries being less efficient. Once again, it is necessary to adjust the plasma conditions to avoid preferential loss of C from the near surface. Essentially any material can be etched either in Cl2- or F2-based plasmas, with additional ion bombardment, allowing magnetic, display, or insulators to be patterned for device applications. BIBLIOGRAPHY 1. Applications of Plasma Processes to VLSI Technology, T. Sugano (ed.), New York: Wiley-Interscience, 1985. 2. B. Chapman, Glow Discharge Processes, New York: Wiley, 1980. 3. G. S. Oehrlein, in Handbook of Plasma Processing Technology, S. Rossnagel, J. Cuomo, and R. Westwood (eds.), Park Ridge, NJ: Noyes Publications, 1990. 4. A. J. van Roosmalen, J. A. G. Baggerman, and S. J. H. Broader, Dry Etching for VLSI, New York: Plenum Press, 1991. 5. G. S. Oehrlein and H. L. Williams, J. Appl. Phys., 62: 662, 1987. 6. G. S. Oehrlein et al., J. Electrochem. Soc., 136: 2050, 1989. 7. A. J. von Roosmalen, Vacuum, 34: 429, 1994. 8. L. M. Ephrath and E. J. Petrillo, J. Electrochem. Soc., 129: 2282, 1982. 9. D. L. Flamm, V. M. Donnelly, and J. A. Mucha, J. Appl. Phys., 52: 3633, 1981.

511

10. J. W. Coburn and E. Kay, IBM J. Res. Develop., 23: 33, 1979. 11. D. L. Flamm, V. Donnelly, and D. E. Ibbotson, J. Vac. Sci. Technol., B1: 23, 1983. 12. J. W. Coburn and H. F. Winters, J. Appl. Phys., 50: 189, 1979. 13. H. F. Winters, J. Vac. Sci. Technol., A6: 1997, 1988. 14. C. J. Mogab, A. C. Adams, and D. L. Flamm, J. Appl. Phys., 49: 3796, 1978. 15. K. L. Seaward et al., J. Appl. Phys., 61: 2358, 1987. 16. S. J. Pearton and F. Ren, J. Vac. Sci. Technol., B11: 15, 1993. 17. D. L. Flamm, V. H. Donnelly, and D. E. Ibbotson, Basic principles of plasma etching. In VLSI Electronics Microstructural Science, New York: Academic Press, 1984, pp. 189–251. 18. H. F. Winters and J. W. Coburn, Surface Science Rep., 14: 261, 1992. 19. D. M. Manos and D. L. Flamm, Plasma Etching—An Introduction, Boston: Academic Press, 1989. 20. M. J. Vasile and F. A. Stevie, J. Appl. Phys., 50: 3799, 1982. 21. V. M. Donnelly, in Encyclopedia of Advanced Materials, New York: Pergamon Press, 1994. 22. R. A. Gottscho, C. W. Jurgensen, and D. J. Vitkavage, J. Vac. Sci. Technol., B10: 2133, 1992. 23. M. Sato and Y. Arita, J. Electrochem. Soc., 134: 2856, 1987. 24. R. N. Carlile et al., J. Electrochem. Soc., 135: 2058, 1988. 25. S. Ohki et al., J. Vac. Sci. Technol., B5: 1611, 1987. 26. M. Englehardt and S. Schwarzl, Proc. 2nd Symp., VLSI Science and Technology (Electrochemical Society, Pennington, NJ), ECS Proc., 89-9: 505, 1989. 27. G. R. Powell and A. A. Chambers, Proc. 2nd Symp., VLSI Science and Technology (Electrochemical Society, Pennington, NJ), ECS Proc., 89-9: 498, 1989. 28. Y. H. Lee and M. M. Chen, J. Vac. Sci. Technol., B4: 468, 1986. 29. G. C. Schwartz and P. Schnaible, J. Electrochem. Soc., 130: 1898, 1983. 30. S. Tachi, Proc. ECS, 89-9: 381, 1989. 31. Z. H. Walker and E. Z. Ogryzlo, J. Appl. Phys., 69: 548, 1991. 32. Z. H. Walker and E. A. Ogryzlo, J. Appl. Phys., 69: 2635, 1991. 33. D. L. Smith and R. H. Bruce, J. Electrochem. Soc., 129: 2045, 1982. 34. D. A. Danner, M. Dalvie, and D. W. Hess, J. Electrochem. Soc., 134: 669, 1987. 35. R. H. Bruce and G. P. Malafsky, J. Electrochem. Soc., 130: 1369, 1983. 36. R. A. M. Wolters, Proc. 3rd Symp. Plasma Proc., 82-6: 293, 1983. 37. T. Abraham, J. Electrochem. Soc., 134: 2809, 1987. 38. H. B. Bell, H. M. Andersona, and R. W. Light, J. Electrochem. Soc., 135: 1184, 1988. 39. S. Park, T. N. Rhodin, and L. C. Rathbun, J. Vac. Sci. Technol., A4: 168, 1986. 40. C. K. Hu et al., J. Vac. Sci. Technol., A7: 682, 1989. 41. G. S. Oehrlein et al., J. Electrochem. Soc., 136: 2050, 1989. 42. S. Park, C. Sun, and R. J. Purtell, J. Vac. Sci. Technol., B6: 1570, 1988. 43. F. Fracassi and J. W. Coburn, J. Appl. Phys., 63: 1758, 1988. 44. S. P. Sun and S. P. Murarka, J. Electrochem. Soc., 135: 2353, 1988. 45. F. Y. Robb, J. Electrochem. Soc., 131: 2906, 1984. 46. B. J. Curtis and H. R. Brunner, J. Electrochem. Soc., 136: 1463, 1989. 47. D. S. Fischl, G. W. Rodrigues, and D. W. Hess, J. Electrochem. Soc., 135: 2016, 1989. 48. M. Balooch et al., J. Electrochem. Soc., 135: 2090, 1988.

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49. D. S. Fischl and D. W. Hess, J. Vac. Sci. Technol., B6: 1577, 1988. 50. T. C. Mele et al., J. Electrochem. Soc., 135: 2373, 1988. 51. R. J. Saia and B. Gorowitz, J. Electrochem. Soc., 135: 2975, 1988. 52. B. Robinson and S. A. Shivashankar, Proc. 5th Symp. Plasma Proc., ECS Proc., 85-1: 206, 1985. 53. J. E. Spencer, R. A. Borel, and A. Hoff, J. Electrochem. Soc., 133: 1922, 1986. 54. N. R. Lerner and T. Wydeven, J. Electrochem. Soc., 136: 1426, 1989. 55. V. Vukanovic et al., J. Vac. Sci. Technol., B6: 66, 1988. 56. M. A. Hartney, D. W. Hess, and D. S. Soane, J. Vac. Sci. Technol., B7: 1, 1989. 57. C. W. Jurgensen et al., J. Va. Ci. Technol., A6: 2938, 1988. 58. G. S. Oehrlein et al., Proc. 5th Symp. Plasma Processing, ECS Proc., 85-1: 87, 1985. 59. G. S. Oehrlein, J. G. Clabes, and P. Spirito, J. Electrochem. Soc., 133, 1002 (1986). 60. I. W. H. Connick et al., J. Appl. Phys., 64: 2059, 1988. 61. J. M. Heddleson et al., J. Vac. Sci. Technol., B6: 280, 1988. 62. S. Fujimura and H. Yano, J. Electrochem. Soc., 135: 1195, 1988. 63. K. H. Ryden et al., J. Electrochem. Soc., 134: 3113, 1987. 64. V. M. Donnelly, in Plasma Diagnostics, Vol. 1, O. Auciello and D. L. Flamm (eds.), New York: Academic Press, 1989. 65. J. W. Coburn and M. Chen, J. Appl. Phys., 51: 3134, 1980. 66. R. A. Gottscho and V. M. Donnelly, J. Appl. Phys., 56: 245, 1984. 67. B. J. Curtis and H. J. Brunner, J. Electrochem. Soc., 125: 829, 1978. 68. R. W. B. Pearce and A. G. Gaydon, The Identification of Molecular Spectra, London: Chapman and Hall, 1984. 69. G. S. Selwyn, J. Vac. Sci. Technol., A6: 2041, 1988. 70. P. A. Heiman, J. Electrochem. Soc., 132: 2003, 1985. 71. B. E. Cherrington, Plasma Chem. Plasma Proc., 2: 113, 1982. 72. N. Hershkowitz, in Plasma Diagnostics, Vol. 1, O. Auciello and D. L. Flamm (eds.), New York: Academic Press, 1989. 73. B. Lipschultz et al., J. Vac. Sci. Technol., A4: 1810, 1986. 74. R. Walkup, R. W. Dreyfus, and P. Avouris, Phys. Rev. Lett., 50: 1846, 1983. 75. S. W. Downey, A. Mitchell, and R. A. Gottscho, J. Appl. Phys., 63: 5280, 1988. 76. M. J. Goeckner and J. Goree, J. Vac. Sci. Technol., A7: 977, 1989. 77. S. J. Pearton, C. R. Abernathy, and F. Ren, Topics in Growth and Device Processing of III–V Semiconductors, Singapore: World Scientific, 1996. 78. S. W. Pang, J. Electrochem. Soc., 133: 784, 1986. 79. S. J. Pearton et al., Mat. Chem. Phys., 32: 215, 1992.

S. J. PEARTON University of Florida

PLASMA CHEMISTRY. See PASSIVATION.

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Wiley Encyclopedia of Electrical and Electronics Engineering Plasma Deposition Standard Article Albert A. Adjaottor1 and Efstathios I. Meletis1 1Louisiana State University, Baton Rouge, LA Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5911 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (170K)

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Abstract The sections in this article are Basic Principles of PECVD Equipment Used in PECVD Future Trends About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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512

PLASMA DEPOSITION

ture conditions (⬎850⬚C) limit its suitability. Plasmaenhanced CVD (PECVD) has received substantial attention in recent years because it grows films of metals, semiconductors, and insulators at relatively low temperatures on temperature-sensitive substrates. Low-temperature deposition methods are essential in minimizing defect formation and solid-state diffusion between deposited layers. The fact that insulating films for diffusion masks, interlayer dielectrics, and passivation layers are deposited at low temperatures, so that previous steps are not affected, is extremely important for this industry. The plasma, also known as a glow discharge, generates energetic electrons (1 eV to 10 eV) (1), which interact with and ionize gaseous precursor molecules to form chemically reactive radicals and ions. These active species react either in the gaseous stream in a homogeneous reaction or migrate to the substrate surface where they undergo a heterogeneous reaction to form the film. Reaction byproducts evolve and are carried away in the gas stream. The inherent applicability of the plasma deposition process to uniform deposition over large areas is basically a result of the relative ease by which uniform electric fields are created over large areas. PECVD is currently being used to produce thin films of materials such as amorphous silicon (a-Si), silicon nitride, silicon dioxide, silicon carbide, boron nitride, and diamondlike carbon coatings (DLC). The conductivities of the substrate and the growing film determine the method of plasma creation. Conductive materials are adequately deposited by dc plasma generation. However, dielectrics and semiconductor materials very commonly use RF plasma generation. Other ways of plasma generation, such as by microwave or laser incorporated with magnets are being developed. We begin by reviewing the basic physics and chemistry of nonequilibrium glow discharges.

BASIC PRINCIPLES OF PECVD

PLASMA DEPOSITION Plasma deposition is a processing technique that has gained popularity in the microelectronics industry in recent years because it operates at relatively low temperatures (⬍250⬚C). Chemical vapor deposition (CVD) has been the primary method of device manufacture, but high operating tempera-

A glow discharge is defined as a partially ionized gas composed of equal volume concentrations of positively and negatively charged species and different concentrations of species in the ground state (2). The processing glow discharge or plasma is produced and driven by external power supplies from dc up to radio frequencies of about 10 GHz and in power up to 30 kW. Figure 1 shows a schematic of the components of a typical apparatus for PECVD (3). There are basically three classes of techniques used to create and sustain the type of plasmas used for thin film growth (4). The first involves applying a high voltage to a metal electrode or set of electrodes within the discharge chamber. The electrodes that are normally in the form of parallel plates are said to have a diode configuration. Devices that use this kind of configuration include dc and RF diodes, magnetrons, and some PECVD systems. The second class of plasma generation technique involves applying electric fields, typically through an insulator. The high electric field is used to help break down the gas and cause ionization. Inductively and capacitively coupled barrel reactors and electron cyclotron resonance (ECR) reactors are examples of devices using this technique. The third class creates and sustains the plasma by injecting large currents of electrons emitted thermionically from a filament or related electron source and then accelerated into the plasma. The accelerated electrons cause ionization and maintain the plasma.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

PLASMA DEPOSITION

513

RF amplifier

Particle Filter

SiH4 B 2H 6

Matching network

Heater

; ; ; ; ;;;

Mass-flow Controller

Substrate

PH3

A

Throttle valve

K

T2 Diffusion pump

T1

Roots blower pump Rotary pump

Matching network

Scrubber

RF amplifier

N2

RF oscillator

Because dielectric and nonconducting materials are mostly used in the microelectronics industry, RF glow discharge is discussed more than dc glow discharge. Potentials Experienced in RF Glow Discharges RF glow discharges used to deposit thin films operate at frequencies between 13.56 MHz and 50 MHz and at pressures between 10 Pa to 300 Pa. The plasma density (i.e., the density of ions and free electrons) is in the range of 108 to 1012 cm⫺3. The degree of ionization is typically ⬍10⫺4 implying that most species in the glow are neutral on the order of approximately 1015 to 1016 cm⫺3. High-energy electrons with energies on the order of 10 to 30 eV generate the reactive species. The average electron energy (1 eV to 10 eV) is considerably higher than the average ion energy (앑0.04 eV) (1,5). The combined density of condensable species, namely, neutral radicals and ions, is small relative to the gas density, and the density of condensable neutrals is much higher than that of charged particles. Therefore, the film is predominantly formed by bonding neutral radicals to the surface of the growing film. The potentials of glow discharges used in PECVD are (1) the plasma potential, (2) the floating potential, and (3) the sheath potential. The plasma potential Vp is the potential of the glow region of the plasma, which is normally considered equipotential. It is the most positive potential in the chamber and is the reference potential for the glow discharge. The floating potential Vf is the potential at which equal fluxes of negatively and positively charged species arrive at an electrically floating surface in contact with the plasma. The resultant potential is represented approximately by the following expression (5): Vp − Vf =

Pressure gauge




mi kTe ln 2e 2.3me



(1)

Figure 1. Typical components of a PECVD apparatus consist of a reactant flow control unit, a plasma reactor, and low and high vacuum pumps (3).

where Te is the electron temperature, e is the unit electron charge, and mi and me are the ion and electron mass, respectively (2,5,6). Most sputtering threshold energies range from 20 to 40 eV (7). Therefore it is desirable to operate the plasma with a (Vp ⫺ Vf ) less than 20 eV to avoid sputtering material off the reactor wall, which may otherwise lead to film contamination. Between the electrode and the glow discharge there is a narrow dark region or sheath (typically 0.01 cm to 1 cm, depending primarily upon pressure, power, and frequency) called the dark space. Within this region, the positive ions are accelerated to the electrode or substrate surface, and secondary electrons emitted from the electrode surface are accelerated out into the glow discharge. The difference between the potential at the surface and the plasma potential determines the maximum energy with which the positive ions bombard the surface and electrons enter the glow discharge. The potential across the sheath is called the sheath potential. The reason for the different potentials within a plasma system becomes obvious when electron and ion mobilities are considered. Imagine applying an RF field between two plates or electrodes positioned within a low-pressure gas. On the first half-cycle of the field, one electrode is negative and attracts positive ions. The other electrode is positive and attracts electrons. Because the mobility of the electrons is considerably greater than that of positive ions and also considering the high frequencies used in RF discharges, the flux of electrons or current is much larger than that of positive ions. This situation depletes electrons in the plasma and results in a positive plasma potential. On the second halfcycle, a large flux of electrons flows to the electrode that previously received the small flux of ions. The plasma potential is nearly uniform throughout the observed glow volume in an RF discharge, although a small electric field directed from the discharge toward the edge of the glow region exists.

514

PLASMA DEPOSITION

There are two distinct regions, low- and high-frequency regimes in the frequency of the input power. The boundary between the two regions is given by the critical frequency f c, which is expressed as (3) fc =

µi E 0 πl

Dissociative Attachment

(2)

where 애i is the ion mobility, E0 is the amplitude of the ac electric field, and l is the electrode spacing. The critical excitation frequency f c is estimated to be between 10 kHz to 100 kHz, below which both ions and electrons respond to the alternating electric field. Beyond f c, ionic species no longer respond as the electrons do. The relatively heavy ions do not respond to the field by significant displacement and therefore are considered immobile. Generation of Active Species As electrons and molecules undergo collisions within the plasma, various excited and metastable states of species are generated. There are two kinds of collision processes generating these species, elastic and inelastic collisions. Elastic collisions between energetic electrons and neutral or ionized species result in little transfer of energy to the atom or ion because of the large mass difference between the species. Inelastic collision processes, on the other hand, lead notably to ionization and generate species in electronically excited states. The ionization process is expressed as e + X → X+ + 2e−

Ionization

the Penning process. Other types of collisional reactions are dissociative attachment and dissociation. The process of dissociative attachment can be represented by the following reaction:

where X is a gaseous species and e is an electron. An additional electron is generated to contribute to maintaining the plasma. Another form of ionization termed dissociative ionization occurs, in which a molecule is dissociated and one of the atoms is ionized. This process requires a little more energy and is expressed as

e + X 2 → X − + X + + e− The process of dissociation is also represented by the following collisional reaction: Dissociation

e + X2 → 2X + e−

The collisional processes in order of increasing energy requirement are (1) excitation, (2) dissociative attachment, (3) dissociation, (4) ionization, and (5) dissociative ionization. The reactive species generated in the plasma have lower energy barriers to physical and chemical reactions than the parent species and consequently react at lower temperatures. Film Deposition in PECVD The process of film deposition in PECVD can be broken down into the following eight primary steps: 1. 2. 3. 4. 5.

Transport of the reactants to the deposition region Generation of condensable reactive species Diffusion of reactant species to the substrate surface Adsorption of the reactants onto the substrate surface Physicochemical reactions leading to the solid film and reaction by-products 6. Desorption of volatile by-products 7. Diffusion of volatile products away from the surface into the main gas stream 8. Transport of by-products away from the desorption region and out of reaction chamber.

Figure 2 (9) shows a schematic of the processes involved in PECVD. The predominant flux impinging onto the substrate

Dissociative Ionization e + X2 → X+ + X + 2e−

;; ; ;;; ; ; ;;; ;;;

Ligand Metal center

The threshold energy for ionizing Ar is 15.8 eV (8). In RF plasmas, the high frequency used is responded to appreciably by the electrons which pick up enough energy from the field to fragment, ionize, and excite the gas molecules. The excitation of an atom or ion, which normally has a very short lifetime, is expressed as:

(7) Transport

e + X → X ∗ + e−

Excitation

The excitation may be rotational, vibrational, or electronic. The threshold energy for excitation of Ar is 11.56 eV (8). In the case of inert gases, some excited atoms may have a longer lifetime of perhaps many milliseconds and during that time may collide with a ground-state atom. This collision that excites or sometimes ionizes that atom is known as the Penning process and is described by: X∗ + Y → X + Y∗

Excitation Ionization

Metal–organic molecule

(1) Transport

∗

+

X +Y→X+Y +e

(18)

Gas-phase + reaction (3) (2) Surface Adsorption reaction

(5) (6) Desorption Nucleation (4) and growth Diffusion +

Substrate

−

where X and Y are arbitrary species. Some species, such as He and Ne, are intentionally added to plasmas to enhance

Figure 2. During PECVD the reactants are transported in the gas phase to the substrate surface, a reaction generates the required material to be deposited, and the byproducts are desorbed and exhausted from the chamber (9).

PLASMA DEPOSITION

surface is inferred to be radicals rather than ions, and the thin film formation process might be controlled either by the generation rate of radicals or by the surface reactions among radicals. As indicated in the preceding section, various collisional processes contribute to the generation of the required condensable reactive species. The rate at which inelastic collisions generate excited species, ions, free radicals, etc., is estimated by a reaction rate equation (10). For example, the rate at which X* is created from the excitation reaction

is given by: ∗

d[X ] = k1 [X ] [e] dt

(3)

where d[X*]/dt is the rate of formation of X*, k1 is the reaction rate coefficient, [X] is the concentration of species X, and [e] is the electron concentration. Because only high-energy electrons participate in inelastic collisions, the rate constant must be defined in terms of the electron velocity and the inelastic collision cross section. The cross section of an electron/particle inelastic collision is proportional to the probability that this inelastic collision occurs and is a function of the electron energy. The rate coefficient ki can be calculated by using the following equation (11):  ki =

∞


2E 1/2

0

me

tion occurs. The most common primary reaction is Eq. (a) and the least probable occurrence will be Eq. (d). The enthalpy of formation ⌬H is calculated from Table 2. It is inferred that the plasma decomposition of NH3 occurs through the following primary step (3): NH3 → NH + H2

( H = 3.9 eV)

(e)

NH3 → NH2 + H

( H = 4.5 eV)

(f)

Secondary reactions between neutral fragments and reactant gas also take place in the plasma, such as (3)

e + X → X ∗ + e−

Excitation

515

σi (E) f (E) dE

(4)

where E is the electron energy, me is the electron mass, ␴i is the collision cross section of reaction i and is a function of E, and f(E) is the electron energy distribution function that gives the fraction of free electrons having a given energy. The integration is carried out over all possible electron energies. The square root term in Eq. (4) is the electron velocity. If an accurate expression for f(E) and the electron collision cross section for the various gas-phase species present are known, the reaction rate coefficients and reaction rates can be calculated theoretically. Unfortunately, such information is generally unavailable for many molecules used in plasma deposition. Dissociation Reactions of Reactants The photolytic decomposition of a gas can provide useful insights to the most probable primary processes in the glow discharge. The reaction steps of representative gases utilized for producing Si3N4, SiO2, a-Si, and BN thin films are summarized in Table 1 (3,12–15). The standard heats of formation of gas molecules and their fragments, shown in Table 2 (3), are useful in predicting the possible electron-impact dissociation reactions. Electron impact dissociation of SiH4 produces primary reaction products as follows:

SiH4 → SiH2 + H2

( H = 2.2 eV)

(a)

SiH4 → SiH3 + H

( H = 4.0 eV)

(b)

SiH4 → Si + 2H2

( H = 4.4 eV)

(c)

SiH4 → SiH + H2 + H

( H = 5.9 eV)

(d)

The differences in the energy requirement for the various dissociation reactions determine the ease with which each reac-

H + SiH4 → SiH3 + H2

( H = −0.5 eV)

(g)

H + Si2 H6 → SiH3 + SiH4

( H = −0.6 eV)

(h)

( H = −0.04 eV)

(i)

H + NH3 → NH2 + N2 + H2

Similarly, radical–molecule reactions also take place (3): SiH2 + SiH4 → Si2 H6

( H = −2.1 eV)

(j)

NH + NH3 → N2 H4

( H = −2.4 eV)

(k)

These reactions are among many that form the basis of the deposition process in PECVD of Si3N4, SiO2, and amorphous Si. Homogeneous and Heterogeneous Processes In RF systems, gas-phase collisions generate condensable reactive free radicals, metastable species and ions, and thus encourage the occurrence of gas-phase (homogeneous) reactions. These reactions lead to powder formation, which is detrimental to film formation. Examples of reactant combinations that lead to homogeneous reactions are silane and oxygen (fast reaction) and silane and nitrous oxide (slow reaction) mixtures, in addition to many hydride-fluoride gas mixtures. There are also the inherent complications of gas-phase reactions between the components for activation and the different gas and plasma kinetics of the gases in the mixture. A combination of electrical power, pressure, and flow rate parameters are chosen to obtain a high deposition rate without any powder formation in the reaction zone. Heterogeneous processes result from interactions taking place between the elements in the plasma and the substrate surface. Following is a summary of the primary processes of interest in plasma deposition (1): • Ion–surface interactions 1. Neutralization and secondary electron emission 2. Sputtering 3. Ion-induced chemistry • Electron–surface interactions 1. Secondary electron emission 2. Electron-induced chemistry • Radical– or atom–surface interactions 1. Film deposition Although vacuum-UV photons and soft X rays in the plasma are sufficiently energetic to break chemical bonds, electron and, particularly, ion bombardments are the most effective methods of promoting surface reactions (16). Ions impinging

516

PLASMA DEPOSITION

Table 1. Photoprocesses of Typical Gases Used in the Production of Si3N4 , SiO2 , a-Si, and BN a Absorption Wavelength (nm) Gas

Edge

SiH4

150

Si2H6

210

PH3 B2H6 NH3

220 200 210

⬍200 180 190

O2 O3 HCl

242 300 200

N2O H2O a b

Maximum

Primary Step

Secondary Step

Ref.

SiH2 ⫹ SiH4 씮 Si2H6 H ⫹ SiH4 씮 SiH3 ⫹ H2 H ⫹ Si2H6 씮 H2 ⫹ Si2H5 H ⫹ Si2H6 씮 SiH3 ⫹ SiH4

12

H ⫹ PH3 씮 PH2 ⫹ H2

14 14 15

140 250 150

SiH2 ⫹ 2H SiH3 ⫹ H SiH2 ⫹ SiH3 ⫹ H SiH3 ⫹ SiH ⫹ 2H Si2H5 ⫹ H H ⫹ PH2 BH3 ⫹ BH3 NH3 ⫹ H NH ⫹ 2H O⫹O O ⫹ O2 H ⫹ Cl

210

180

N2 ⫹ O

180

165

H2 ⫹ O

200

Table 2. Standard Heat of Formation for Various Species of Gas Molecules and Their Fragments a

Ref. 3.

14 14 14 14 14

Ref. 3. M is a nonabsorbing, foreign gas.

on the surface influence the kinetics of network formation and the nature of the resulting film. Radiation damage during PECVD displaces atoms generating vacancies, interstitials, dislocation loops, and stacking faults. Such damage degrades material properties and alters the characteristics of fabricated devices. A fundamental understanding of gas-phase plasma chemistry and physics, along with surface chemistry modified by radiation effects, is needed to define film-growth mechanisms. The complex interactions involved in PECVD

a

NH ⫹ NH3 씮 N2H4 O ⫹ O2 ⫹ M b 씮 O3 ⫹ M b O ⫹ O3 씮 O2 ⫹ 2O or 2O2 H ⫹ HCl 씮 H2 ⫹ Cl Cl ⫹ Cl ⫹ M b 씮 Cl2 ⫹ M b O ⫹ N2 O 씮 N2 ⫹ O2 O ⫹ N2O 씮 2NO O ⫹ H2 씮 OH ⫹ H

13

Species

Standard Heat of formation (eV)

H H2 Si(g) SiH SiH2 SiH3 SiH4(g) Si2H6 N NH(g) NH2(g) NH3(g) N2H4 B(g) BH BH2 BH3(g) P(g) PH PH3(g) NO N2O O

2.27 0 4.69 3.90 2.52 4.00 0.32 0.74 4.92 3.44 1.75 ⫺0.48 0.53 5.78 4.50 2.08 1.11 3.27 2.63 0.06 0.94 0.85 2.59

are outlined in Fig. 3 (17). If the basic or microscopic plasma parameters (neutral species, ion, and electron densities; electron energy distribution; and residence time) are controlled, the gas-phase chemistry can be defined. Many macroscopic plasma variables (gas flow, discharge gas, pumping speed, RF power, frequency, etc.) can be changed to alter the basic plasma conditions. Electrode and chamber materials also alter the chemistry of glow discharges because of chemical reactions on or with the surface. Synergism between these numerous processes results in specific film-growth mechanisms. Ultimately, these factors establish film composition, bonding structure, and thus film properties. EQUIPMENT USED IN PECVD Predicting deposition rates and uniformity require detailed understanding of thermodynamics, kinetics, fluid flow, and mass transport phenomena for the appropriate reactions and reactor design. The plasma deposition process commonly involves using potentially hazardous gases. This requires extreme precautions, such as locating the gases in vented gas cabinets to avoid gas accumulation. The hazard of using silane, the main processing gas in the plasma deposition process, is its rather unpredictable pyrophoricity rather than toxicity (18). However, the pyrophoric tendency of silane can be used with advantage to reduce the toxic hazard of dopant gases, such as diborane and phosphine. It is common practice to acquire the toxic dopant gases at relatively low concentrations (100 ppm to 1%) prediluted in silane, on the premise that the dopant gas will oxidize rapidly in the silane flame which results from a large exposure. Excess flow shut-off valves are also incorporated to close automatically if the gas flow exceeds a preset limit. These shut-off valves are located between the gas tank and the pressure regulator on the tank. Lines are often dedicated to one specific gas and are led to a gas-mixing manifold where the various individual gases are mixed after metering and before entering the reactor. Mass

517

; ; ; ;; ; ; ; ; ; ; ;;

PLASMA DEPOSITION

flow controllers are generally used to gauge and control the gas flow into the plasma reactor because, in contrast to rotameters and needle valves, they are independent of the pressure of the gas at the inlet or the outlet. Different types of reactors for the plasma deposition process (PECVD) are discussed in a later section. Because the process forms particles, large filters are recommended between the reaction chamber and the exhaust pumping system. A variety of devices, such as scrubbers, burn boxes, activated carbon filters, and high velocity ejectors, are used to reduce the hazards of the exhaust gases. Nitrogen at flow rates exceeding the highest process gas flow by at least a factor of 10 (18), is commonly used to force the exhaust effluent from the high-pressure side of the mechanical pump through scrubbing and filtering devices. The potentially hazardous nature of the reaction materials requires placing interlocks to take appropriate action automatically in incidents and emergencies. A combination of electrical power, pressure, and flow rate parameters has to be chosen to obtain a high deposition rate without any powder formation in the reaction chamber. PECVD Reactor Systems PECVD reactors operate mostly on a batch scale, but continuous reactors are being developed to increase throughput. The batch reactors are (1) the radial-flow batch reactor and (2) the tubular reactor. These reactors are designed to be either coldor hot-walled. In the radial-flow batch reactor, the plasma is generated between two parallel, circular electrodes as shown

Kinetics parameters Gas flow Nature of gas Pumping speed

Electrical parameters Power Frequency Geometrical factors

Basic plasma parameters Electron ion densities and fluxes Neutral density Residence time

Growth kinetics Ion energy

Film Properties

Nature of chemical species

Substrate temperature

and mechanisms

Plasma-surface interactions Basic plasma parameters

Surface parameters Nature of surface Material Surface potential

Figure 3. The properties of the films are controlled by complex interactions between the plasma parameters, kinetics of film formation, and plasma-surface interactions (17).

Aluminum electrodes

Substrates

R.F.

Plasma

T.C.

Pyrex cylinder

Gas ring

Pump

SiH4 NH3

Pressure gauge

+Ar or N2

Figure 4. A schematic of the cold-wall radial-flow, plasma-enhanced CVD reactor after Reinberg has the reactants introduced from the periphery of the substrate holder and exhausting in the center (19).

in Fig. 4 (19). This is known as the Reinberg design. The wafers are loaded onto the lower, electrically grounded electrode. The powered electrode in this cold-wall reactor is connected to the high frequency RF power generator through an impedance-matching network. The reactants are introduced from the gas ring, enter the plasma region (i.e., the region between the electrodes) at its outer edge, and flow radially inward toward a pumping port at the center of the lower electrode. Various variations of this design have since been introduced. In the modified Reinberg design developed by Applied Materials, the gas is introduced in the center of the circular electrodes and exhausted at the periphery. Figure 5 (20) shows a further improvement on the radial flow batch reactor in which a perforated upper electrode is incorporated for more uniform gas distribution. This design was developed to cater to gas mixtures of low concentrations of the active components, such as 2% SiH2 in N2. The perforation helped solve the problem of reactant gas depletion before traversing the full cross section of the electrode, thus resulting in nonuniform film formation. The tubular reactor, shown in Fig. 6 (20), consists of a tube of fused quartz within a resistively heated furnace (hot-wall reactor). Vertically oriented rectangular graphite plates carry the wafers in slots, and the wafer surface acts as one electrode. Every other slab is connected to the same RF power terminal as shown schematically in Fig. 6. The glow discharge is generated capacitively between adjacent parallel-plate electrodes. The reactants are directed along the axis of the chamber tube and between the electrodes. Pulsing of the RF power improves deposition uniformity in this reactor. Robotic machinery is used for automatic cassette-to-cassette loading and unloading of wafers, and cantilevers are designed for inserting and withdrawing the electrode units. Contamination by particles has been significantly reduced by these techniques. A continuous-processing, radial, cold-wall reactor by Novellus (21) sequentially indexes each wafer through several deposition stations on a resistance-heated substrate plate

518

PLASMA DEPOSITION

Operational Problems and Solutions

Shielded RF power input

;;;; ;

;;;;;;;; ;;;;;;;; ;;;;;;;; ;;;;;; ;;;;;;;; ;;;;;; ;;;;;;;; ;;;;;; ;;;;;;;; ;;;;;; ;;;;;; ;;;;;; ;;;;;; ;;;;;;;; ;;;; ;;;; ;;;; ;;;; ;;;; ;;;; ;;;; ;;;; ;;;; ;;;; Perforated electrode

Rotating susceptor

Heater

Heater

Rotating shaft

; ; ; ; ;;;;

Out to vacuum pump

Out to vacuum pump

Magnetic rotation drive

Gases in Figure 5. Modified cold-wall radial-flow, plasma-enhanced CVD reactor developed by Applied Materials had the reactants introduced from the center and exhausting from the periphery to improve the film thickness uniformity. A perforated electrode also helped create a uniform plasma (20).

within the reaction chamber. Glow discharge plasma is created selectively above each wafer, so that deposition of the dielectric films occurs only over the wafer area. Initial deposition takes place within 10 s after introducing the wafer into the chamber. The film deposit is formed in seven increments, and a finished wafer leaves the chamber typically every 50 s. Excellent uniformity of film thickness is attained. The system can be operated with dual frequency to improve film characteristics (13.56 MHz-high and 300 kHz to 400 kHz-low).

Figure 6. A schematic of a hot-wall tubular, plasma-enhanced CVD reactor introduced to increase the cassette throughput for film deposition as well as reduce contamination (20).

Most reactors are operated at conditions where the probability of homogeneous (gas phase) and heterogeneous (film forming) reactions are roughly comparable. For a parallel-plate reactor, decreasing the electrode spacing and the radical density reduces the probability of homogeneous reaction. The electrode spacing, however, cannot be arbitrarily reduced. Hence the only practical means for reducing an undesirably high level of homogeneous reactions is through lowering the radical density by reducing the electric input power or the total gas pressure (18). The properties of the end products of a plasma deposition process are generally reproducible for fixed reactor geometry operated under a fixed set of operating conditions. The optimization of these properties is largely heuristic. This makes the translating operating parameters between reactors of different construction, based on the principles of plasma chemistry, generally not possible. The gas purity in the reactor is a function of the purity of the precursor gas, the gas delivery system, and the reactor itself. Not all impurities are equally important. Some, such as the group III and V dopants, affect the electrical behavior of amorphous silicon at ppm levels, whereas others, such as nitrogen, oxygen, and hydrocarbons, have no measurable effects (22,23) on the electrical properties at that level. Of major concern in multilayer plasma deposition processes is the so-called memory effect of gas lines and of the reactor. This memory effect is a cross-contamination phenomenon, which results from the carry-over of gaseous material and its incorporation as impurities in subsequently deposited layers. Some gases are worse than others. Diborane, fluorinated and chlorinated gases, and generally most compounds with low vapor pressure, such as ammonia and metallorganic compounds, are especially notorious in this respect. The problem is minimized by using multichamber reactors, where each chamber is dedicated to depositing a specific material, hot-walled reactors, gas lines dedicated to one gas, and procedural changes, such as sweep runs. In practice, a change of geometry potentially affects a number of important intrinsic parameters, which are not easily affected or corrected by a change in the external variables. Depletion effects, which cause ununiform film thickness and properties, may necessitate moving substrates in the gas stream to ensure the required uniformity of properties. Film material, which accumulates on the surroundings of the deposition zone, has to be regularly removed from any deposition system to avoid the adverse effects of flakes and powder on the quality of the film deposited on the substrates. The probability that film particles dislodge from the surroundings of the reaction zone increases with time. To compound the problem, plasma deposition processes are inherently ‘‘dirty’’ because of the gas-phase reactions, which might occur between radicals during the deposition. Several practical solutions are used to minimize the impact of the powder and flake problem. One is to arrange the substrate surfaces vertically in the deposition chamber or horizontally above a flat counterelectrode. By this configuration, a fraction of the particulate matter generated during the process settles to the bottom of the reactor rather than on the substrate. Some designs offer a demountable counterelectrode to facilitate cleaning by mechanical or wet chemical means between runs. However, if the counterelectrodes are not removable, then the

PLASMA DEPOSITION

Reaction gas (SiH4 etc.) Deposition chamber

Plasma generation gas (O2, N2, Ar2, etc.) ECR ion source

Cooling water

Magnet coil

Plasma flow

; ; ; ; ;; ; ;;;;

reactor has to be etched on a regular basis by an appropriate plasma etching process. Fluorinated gases, such as CF4 and NF3, are commonly used to etch amorphous silicon, silicon dioxide, and silicon nitride deposits. Materials efficiency is defined as the ratio of the weight of the deposited film to the weight of the gas used to form the film. Defined in this way, practical reactor efficiencies rarely exceed 30%. Given the relatively low efficiency of plasma deposition processes and the high cost of the starting materials, the recycling of exhaust gases, after purification is an attractive alternative to the outright disposal of the pump effluents.

519

Substrate

FUTURE TRENDS The prospects for PECVD in the microelectronics industry are along two fronts: (1) development of new reactor designs to improve film quality and properties and (2) film deposition of new dielectric and oxide materials. The need to minimize or reduce radiation damage has led to the development of remote plasma-enhanced CVD reactors that independently control the generation of active species and the reaction chemistry. In this type of reactor, the plasma is confined to a region away from the substrate. This allows selective dissociation of reactants, that is, the plasma dissociates some reactants, which are then transported to the vicinity of the wafer. These active species react with other gaseous reactants that have not been exposed to the plasma, thus forming the desired film deposit. The plasma is confined by striking it in a tube on the sides of the reactor, in one end of a long reaction tube, below the substrate or above the substrate (5). In these plasma reactors, the plasma replaces the thermal energy needed to dissociate parent species by electron–molecule collisions that create a variety of radicals with near-unity reaction probability. New plasma generation techniques, such as those based on pulse dc (24), microwave (25,26) and ECR (27), are emerging for improved film deposition with reduced particle contamination and enhanced film properties. The ECR ion source creates high-density plasma by resonance of microwaves and electrons through a microwave discharge across a powerful magnetic field of 800 G to 1200 G, as shown schematically in Fig. 7 (27). This process allows film deposition at very low gas pressures (13 mPa) and at very low temperature (⬍275⬚C) and yields high rates (20). Microwave multipolar plasmas (MMP) are 2.45 GHz glow discharges confined by multipolar magnetic fields. In MMPs, the plasma excitation and plasma–surface interactions are decoupled. There is no self-bias in MMPs and also the substrate bias and ion bombardment energies are controlled independently. Microwave excitation by distributed electron cyclotron resonance (DECR) is a technologically promising method. Magnetically enhanced plasma deposition is another configuration in which a magnet is added to confine electrons near the electrode, thereby increasing the rate of electron dissociative collisions with molecules. As a result, a glow discharge is maintained at pressures in the 0.1 Pa to 1 Pa range. Polycrystalline silicon (polysilicon) is commercially used as the gate electrode and interconnect material in MOS integrated circuits and as the emitter material in bipolar silicon devices, the active material for thin film transistors (TFTs) and solar energy conversion devices (5). The low-pressure CVD (LPCVD) technique presently being used has two limita-

Sample stage

Shutter

Microwave 2.45 GHz

Figure 7. Schematic representation of the magnetically enhanced electron cyclotron deposition system. This configuration helps reduce the surface damage done to the wafers since the wafers are not in the direct flow of the plasma (24).

tions: (1) deposition rate and film structure are quite sensitive to deposition temperature, and (2) thickness uniformity and deposition rates are affected when relatively high concentrations of dopant species are introduced into the reactor during deposition. PECVD is being studied as an alternative to LPCVD for fabricating polysilicon gate electrodes and also as a low-temperature fabrication technique for TFTs. Silicon epitaxial layers used in silicon device technology are presently deposited by APCVD or reduced (5 kPa to 13 kPa)-pressure CVD at temperatures of 1,050⬚C to 1,200⬚C. PECVD is being considered as a promising low-temperature deposition method because of less stringent vacuum requirements and higher growth rates. PECVD also provides greater flexibility in depositing in situ doped silicon films. Silicides of molybdenum, tungsten, tantalum, and titanium have been successfully deposited by PECVD techniques. Research is focused on producing acceptable film quality. BIBLIOGRAPHY 1. D. W. Hess and D. B. Graves, Plasma enhanced etching and deposition, in D. W. Hess and K. F. Jensen (eds.), Microelectronics Processing: Chemical Engineering Aspects, Washington D.C.: American Chemical Society, 1989, pp. 377–440. 2. B. Chapman, Glow Discharge Processes, New York: Wiley, 1980. 3. M. Hirose, Plasma-deposited films: Kinetics of formation, composition, and microstructure, in J. Mort and F. Jansen (eds.), Plasma Deposited Thin Films, Boca Raton, FL: CRC Press, 1986, pp. 21–43. 4. S. M. Rossnagel, Glow discharge plasmas and sources for etching and deposition, in J. L. Vossen and W. Kern (eds.), Thin Film Processes II, San Diego: Academic Press, 1991, pp. 11–77.

520

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5. R. Reif, Plasma-enhanced chemical vapor deposition, in J. L. Vossen and W. Kern (eds.), Thin Film Processes II, San Diego: Academic Press, 1991, pp. 525–564. 6. J. L. Vossen, Glow discharge phenomena in plasma etching and plasma deposition, J. Electrochem. Soc., 126: 319, 1979. 7. A. Sherman, Plasma-assisted chemical vapor deposition processes and their semiconductor applications, Thin Solid Films, 113: 135, 1984. 8. M. Rand, Plasma-promoted deposition of thin inorganic films, J. Vac. Sci. Technol., 16: 420–427, 1979. 9. M. J. Hampden-Smith and T. T. Kodas, Chemical vapor deposition of metals: Part 1. An overview of CVD processes, Chem. Vapor Deposition, 1 (1): 8–23, 1995. 10. H. Y. Kumagai, in McD. Robinson et al. (eds.), Proc. 9th Int. Conf. Chem. Vapor Deposition, Pennington, NJ: Electrochemical Society, Vol. 84-6, 189, 1984. 11. G. Turban, Y. Catherine, and B. Grolleau, Reaction mechanisms of radio frequency glow discharge deposition process in silanehelium, Thin Solid Films, 60: 147–155, 1979. 12. G. C. A. Perkins and F. W. Lampe, J. Amer. Chem. Soc., 102: 3764, 1980. 13. T. L. Pollock et al., Photochemistry of silicon compounds. IV Mercury photosensitization of disilane, J. Amer. Chem. Soc., 95: 1017, 1973. 14. H. Okabe, Photochemistry of Small Molecules, New York: Wiley Interscience, 1978. 15. J. R. MacDonald, V. M. Donnelly, and A. P. Baronauski, ArF laser photodissociation of NH3 at 193 nm: Internal energy distribution of NH2X2B1 and A2A1, and two photon generation of NH A3⌸ and 웁1⌺⫹, Chem. Phys., 43: 271, 1979. 16. U. Gerlach-Meyer, J. W. Coburn, and E. Kay, Ion-enhanced gassurface chemistry: The influence of the mass of the incident ion, Surf. Sci., 103: 177–188, 1981. 17. Y. Catherine, in G. S. Mathad, G. C. Schwartz, and G. Smolinsky (eds.), Plasma Processing, Pennington, NJ: Electrochemical Society, 1985, p. 317. 18. F. Jansen, Plasma deposition processes, in J. Mort and F. Jansen (eds.), Plasma Deposited Thin Films, Boca Raton, FL: CRC Press, 1986, pp. 1–19. 19. W. C. Dautremont-Smith, R. A. Gottscho, and R. J. Schutz, Plasma processing: Mechanisms and applications, in G. E. McGuire (ed.), Semiconductor Materials and Process Technology Handbook, Park Ridge, NJ: Noyes Publications, 1988, p. 191. 20. A. Sherman, Chemical Vapor Deposition for Microelectronics, Park Ridge, NJ: Noyes Publications, 1987, Chaps. 2 and 5. 21. E. R. van de Ven, I.-W. Connick, and A. S. Harrus, Proc. 7th Int. IEEE Multi-Level Interconnection Conf., Santa Clara, CA, 194, June 1990. 22. C. C. Tsai, J. C. Knights, and M. J. Thompson, ‘‘Clean’’ a-Si : H prepared in a UHV system, J. Non-Cryst. Solids, 66: 45, 1984. 23. R. W. Griffith et al., Modifications in optoelectric behavior of plasma-deposited amorphous semiconductor alloys via impurity incorporation, J. Non-Cryst. Solids, 35–36: 391, 1980. 24. K. S. Mogensen et al., Optical emission spectroscopy on pulsed dc plasmas used for TiN deposition, Surf. Coat. Technol., 102 (1–2): 41–49, 1998. 25. F. Galluzzi et al., CVD diamond films as photon detectors, Nuclear Instr. Meth. Phys. Res., Section A, 409 (1–3): 423–425, 1997. 26. T. Osamu and H. Tomofumi, Microwave plasma-enhanced CVD for high rate coating of silicon oxide, Trans. Inst. Metal Finish., 76 Part 1: 16–18, 1998. 27. V. S. Nguyen, Plasma-assisted chemical vapor deposition, in K. K. Schuegraf (ed.), Handbook of Thin-Film Deposition Processes

and Techniques, Park Ridge, NJ: Noyes Publications, 1988, pp. 112–146.

ALBERT A. ADJAOTTOR EFSTATHIOS I. MELETIS Louisiana State University

PLASMA DIAGNOSTICS. See FUSION REACTOR INSTRUMENTATION.

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Wiley Encyclopedia of Electrical and Electronics Engineering Plasma Implantation Standard Article John H. Booske1, Richard J. Matyi1, John R. Conrad1, Kumar Sridharan1 1Department of Engineering Physics, University of Wisconsin, Madison, WI Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5917 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (270K)

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Abstract The sections in this article are Comparison of PSII with Beamline Implantation Physics of PSII Plasma Applications Other Materials and Plasma Systems Commercialization Issues About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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PLASMA IMPLANTATION Plasma source ion implantation (PSII) (1, 2) is an electrical discharge method for modifying the surface properties of materials so as to increase their value for a variety of applications. As such, PSII is one of a class of electrical discharge surface treatment technologies, including cathodic arc vapor deposition (3), magnetron sputter physical deposition (4), plasma-enhanced chemical vapor deposition (5), and ionized physical vapor deposition (6). These electrical discharge surface treatments can be compared in several respects with conventional plating (chemical or electrolytic), but are free of the hazardous chemicals or contaminated fluids typically associated with commercial plating processes (acids, cyanide compounds, contaminated process water, etc.). In electrical discharge surface treatments, ions are formed and transported to the treated object’s surface by placing a voltage bias on the target and immersing it in an ionized gas discharge (plasma) rather than in a liquid chemical or electrolytic bath. Surface modification by PSII has been shown to be effective at increasing wear resistance and corrosion resistance of metals, while advantageously altering the electronic, magnetic, or electromagnetic surface properties of semiconductors, glasses, or other materials. In the PSII process, a plasma is generated in a vacuum chamber and a series of negative voltage pulses (−500 to −100,000 V) are applied to the target immersed in the plasma. As the negative voltage pulse is applied, electrons are repelled and ions in the plasma are attracted to the surface of the target at very high velocities and penetrate the surface of the target material. The implantation of energetic ions into the near surface regions of the target results in chemical and microstructural changes at the surface leading to corresponding changes in the surface properties (mechanical, chemical, electrical, magnetic) of the target. A significant advantage of the PSII process is its non–line-of-sight nature, which makes it highly effective for the implantation of three-dimensional targets. In addition to implantation, the PSII process can be used in combination with thin film deposition techniques to realize superior mixing at the filmtarget interface, thereby ensuring good adhesion. Following its invention (1) and initial demonstration (2) by Conrad, PSII has grown to a world-wide activity that is under development in approximately 120 laboratories. During its expansion, it has been referred to by a variety of names, such as plasma immersion ion implantation (PIII), plasma implantation (PI), and plasma doping (PLAD), in addition to the acronym PSII used in this article. The proceedings from the Workshops on Plasma-Based Ion Implantation (7–9) and other publications (10, 11) provide extensive reviews of the scope of plasma and materials issues related to the science and technology of the PSII process.

COMPARISON OF PSII WITH BEAMLINE IMPLANTATION The critical attributes of PSII that make it desirable as a materials modification tool can be best appreciated by com-

parison with conventional beamline ion implant processes. Conventional ion implantation (see Fig. 1) is a line-of-sight process in which ions are extracted from an ion source, accelerated as a directed beam to high energy, and then raster-scanned across the target. The acceleration voltage is high enough to bury the ions below the target’s surface. Depending on the application, the accelerated ion energies can range from a few kilo-electron-volts (keV) to several mega-electron-volts (MeV). Ion implantation was first developed as a means of doping the semiconductor elements of integrated circuits. Because of the speed, accuracy, cleanliness, and controllability of the process, it has become the standard for this type of work. In the early 1970s, it was found that ion implantation of metal surfaces could improve their wear, friction, and corrosion properties. Ion implantation of specific tools is now preferred over other types of coating technologies because the ion-implanted layer does not delaminate, does not require high processing temperatures to produce, and does not add more material on the surface (which would change the dimensions of critical components). Unfortunately, conventional beamline implanters are large, complicated, and expensive instruments. Because of the capital costs involved with setting up an ion implant capability and the significant operating expenses that it incurs, ion implantation is now used regularly only to implant specific tools and equipment with a high value-added potential (e.g., score dies for aluminum can pop-tops and artificial knee and hip joints). Studies have shown that a considerably greater number of applications would benefit from ion implantation, but that the expense of the process prevents it from becoming cost effective for those applications. PSII is essentially a modified method for ion implantation (see Fig. 1). As mentioned above, in PSII, the target is placed directly in the plasma source and pulsed to a high negative potential relative to the chamber walls. Pulses typically last for tens of microseconds at a duty cycle that may range from about 0.1% to a few percent. No ion beam is extracted and none needs to be manipulated or focused. PSII thus achieves ion implantation in a compact bell jar environment without the need for the complex beam optics and sample manipulation necessary for a conventional implanter. The high-voltage negative pulses applied to the target in PSII attract positive ions in the plasma which naturally tend to strike all parts of the target at normal incidence. Fortuitously, this normal incidence is indeed the optimal angle for ion implantation, because the physical process of sputtering occurs simultaneously with ion implantation. Sputtering is the removal of material from a surface due to transfer of momentum from an atom or ion in the gas phase to an atom (or several atoms) in a surface. It is characterized by a parameter known as the sputter yield, which is defined by the ratio of the mean number of emitted atoms per incident ion on the surface (12). The sputter yield depends on a number of factors, with the most important being (a) the structure and composition of the target, (b) the mass and energy of the incident atoms or ions, and (c) the experimental geometry. Thus, for a given accelerating voltage and ion species, a nonplanar target will present a range of incident angles if it is fixed in a beamline implanter. At

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright © 2007 John Wiley & Sons, Inc.

2

Plasma Implantation

Figure 1. Comparison of (upper) conventional beamline and (lower) plasma source ion implantation.

the off-normal angles of incidence to the surface, the rate of material removal by sputtering can be high enough to equal the delivered dose to the substrate—in other words, the surface is eroded as quickly as ions are implanted into it. The dependence of the retained dose on the incident angle θ (with normal incidence given by θ = 0) can be expressed as (13)

Since these last two operations must be performed within the vacuum environment of the implanter, they result in a significant increase in the complexity and expense in the modification of nonplanar surfaces by ion implant. With PSII, however, the near-normal trajectory of the incident ions greatly reduces the retained dose problem, thus allowing higher doses to be delivered to complex workpieces in shorter times.

where

PHYSICS OF PSII PLASMA

D= the retained dose N= the target atomic density Rp = the projected ion range S= the sputter yield

An understanding of the physics of the plasma discharge during PSII is useful for designing and optimizing the process. A rather sophisticated understanding of the plasma physics has been developed and the interested reader is encouraged to consult Refs. 14 through 15–27 and the references cited therein. Here, a simplified description of the basic phenomenology will suffice to impart an appreciation for the most crucial factors for process adoption and implementation. The process starts with immersion of the target object in a relatively low-density plasma discharge, with densities typically between 108 cm−3 and 1011 cm−3 . Lower-density plasmas would be associated with unacceptably long process times (because of a slow average supply rate for ions)

In a conventional beamline implanter, the only reliable solution to the “retained dose problem” is to ensure that the incident ion beam maintains an approximate normal incidence to the surface. This is accomplished by (a) rastering the beam across the surface being implanted, (b) partially masking the surface to prevent the ions from striking the target at incidence angles greater than approximately 30◦ off-normal, and/or (c) rotating or otherwise manipulating the sample to maintain a near normal incidence.

Plasma Implantation

and higher-density plasmas are difficult to sustain in large volumes and are more susceptible to arcing during the application of the high-voltage pulse. The treatment then consists of the repetitive application of negative voltage pulses to the target object, until the desired number (or dose) of ions have been implanted. At the beginning of each negative voltage pulse, electrons are repelled from the region immediately surrounding the target toward the walls of the vacuum chamber, which are usually held at ground potential. Since the ions are more massive and move more slowly than the electrons, this initially results in an “ion matrix sheath” or “cathode fall” region surrounding the target which is populated almost exclusively by ions, the electrons having been expelled. Almost all of the applied voltage difference occurs across this region, as indicated in Fig. 2. During this transient ion matrix phase of the sheath evolution, it is possible to determine the extent of the sheath in various geometries from Poisson’s equation

where φ is the potential, ni and ne are the ion and electron densities, and α = 0, 1, and 2 for planar, cylindrical, and spherical geometries, respectively. If it is assumed that, before the potential is applied to the substrate, the plasma has a uniform density n0 = ni = ne , then the thickness of the ion matrix sheath s can be expressed in a normalized form for a planar geometry by

where φ˜ 0 = eφ0 /Te , s˜ ≡ s/λD , and the Debye length λD =
kTe /4πn0 e2 (with Te being the electron temperature and k equal to the Boltzmann constant). Similar expressions can be derived for cylindrical and spherical geometries. Following the ion matrix phase (typically several microseconds), positive ions begin to accelerate within and across this sheath, attracted to the negative potential at the target. The rate at which the ions are drawn from the edge of this sheath region to the target increases with increasing applied voltage and decreasing sheath thickness is described by the Child–Langmuir relation (27):

where j is the Child–Langmuir space charge limited current, q is the ion charge, m is the ion mass, and n is the plasma density. The rate of ion transport across the sheath is typically faster than the rate at which ions flow to the sheath edge from the surrounding plasma. Hence, at the edge of the sheath—as within the sheath itself—the ion and electron densities are markedly reduced (as they are drawn in toward and repelled away from the target, respectively), and the effective sheath edge expands outward. The expansion rate for the sheath can be calculated by using the expression from Lieberman (19) that equates the current density to the sheath expansion by

3

where vd is the ion drift velocity. If it is assumed that the drift velocity of an ion is zero until the sheath reaches it, then the two expressions given above can be combined to form a differential equation for the sheath edge position (18):

This expression can be integrated (18) to yield

where ωpi is the ion plasma frequency and s0 is the sheath thickness at t = 0. This sheath expansion continues until the inward flux rate of ions across the sheath decreases to equal the rate at which ions flow from the surrounding plasma to the sheath edge. At this point, the sheath boundary becomes stationary, assuming that the source for the surrounding plasma discharge can globally replenish the plasma ions as quickly as they are “consumed” by implantation into the target. In the event that the plasma replenishment rate is relatively slow (e.g., low-density plasma discharges), the sheath boundary will continue to expand until the negative voltage pulse is terminated or the sheath edge reaches the vacuum chamber walls (i.e., complete consumption of the plasma ions into the target surface). Knowing the spatial extent of the plasma sheath at the end of each voltage pulse is important for selection of spacing between multiple targets for batch processing (28). If the targets are spaced too closely, then the sheath boundaries will overlap in the regions between the targets before the end of each pulse. This would result in reduced implantation dosage on those interior regions of the target surfaces. Careful selection of multiple target spacing based on an ability to predict maximum sheath thickness for each pulse has been shown to yield good uniformity of implantation in batch processing (29). Limitations of the PSII Process When compared directly with beamline implantation, PSII has a number of potential problems that need to be addressed. First, the PSII implant has an inherent energy inhomogeneity because the ions that are first to be accelerated across the sheath experience a different spatial potential distribution than do the ions accelerated at the end of the voltage pulse. In addition, the complex nature of the plasma means that a variety of ionized species are likely to be implanted during a given voltage pulse. These factors combine to make an accurate theoretical prediction of the profile of the implanted ion much more difficult than is the case with monoenergetic beamline implantation of a particular species. Comparisons of beamline versus PSII implant profiles consistently show both qualitative and quantitative differences (30, 31). In addition to knowing the implant profile, determination of the exact dose delivered to the substrate during PSII implant is somewhat complicated (32, 33). The simple expedient of monitoring the current delivered to the substrate is not a reliable indicator of ion dose, since the current consists of two contributions, namely, the (positive) ion flux

4

Plasma Implantation

Figure 2. Time evolution of plasma sheath during a PSII pulse.

into the target and the (negative) flux of secondary electrons ejected by the target due to ion irradiation. While it may appear possible to measure the secondary electron current by collecting them as they impinge on the chamber wall, the fact is that the secondary electrons will (as they hit the wall) generate tertiary electrons, which then can produce quaternary electrons, and so on. While an alternative scheme for monitoring the dose during plasma implant has been proposed in Ref. 33, accurate dosimetry remains a significant issue in PSII. Secondary electron emission poses other problems for PSII. First, it represents a significant electrical inefficiency, since the secondary current can be many times larger than the ion current. This increases the size (and cost) of the high voltage modulator that will be required to attain a particular dose in a specified time. More seriously, when the secondary electrons hit the internal walls of the vacuum chamber, they create bremsstrahlung X rays. This X-ray production may require personnel exclusion or additional shielding for operation at high applied voltages (in excess of 20 kV), depending on the thickness and the materials used in the construction of the vacuum chamber (34). Finally, the secondary electrons sheath the vacuum chamber walls, desorbing gases and contaminants and thus requiring active cooling of the chamber in high dose, high voltage applications. Uniformity of the implant over large areas is an obvious concern for PSII (as well as for other implant technologies) for semiconductor applications in particular, since silicon wafer diameters are expected to increase to 300 mm in the near future. The uniformity of a plasma implant over a large area would be expected to be a function of all of the factors that influence the spatial properties of the plasma, including chamber geometry, the electric field generated by the voltage pulse, and the characteristics of the gas used to generate the plasma. By careful consideration of chamber design and the specifics of the implant process, doping uniformity of better than 2% across a six-inch diameter wafer has been demonstrated (35). Contamination due to the unintentional incorporation of impurities during implantation is, like uniformity, a perennial concern in all implant processes, but especially so when PSII is used for semiconductor doping applications. Since there is no mass analysis of the implanted

species in PSII (unlike that in beamline implant processes), any impurity that exists or can be inserted into the process gas can appear as a contaminant. Thus contamination can arise either from the starting impurities present in the gas supply or from unintentional sputtering from chamber components onto the implant target. The latter source is of particular concern to PSII, since everything that is biased during the voltage pulse is subject to implantation and thus can become a potential sputter source of impurities. Fortunately, like uniformity, concerns about contamination have largely been mitigated by studies that have shown very low levels of unintentional impurity incorporation in welldesigned implant systems (35, 36). A final concern with PSII is the possibility that the plasma environment (which is obviously in contact with a wafer being processed) may cause undesired changes to the substrate through either etching or charging. For shallow p-type implants into silicon (discussed below), BF3 gas is typically used to supply the implant species. In the plasma environment above the wafer substrate, BF3 typically decomposes to BF+ 2 and F+ , with the ionized fluorine being a very effective silicon etchant. Thus early attempts at PSII doping from a BF3 plasma were complicated by the etching of the silicon simultaneously with implantation (37, 38). Fortunately, by altering conditions such as the gas pressure and plasma power, it is possible to reduce the etch rate to essentially zero under PSII conditions (38). In a similar vein, the rich mix of ionized atoms and radiation that is characteristic of a plasma environment could be a source of damage to sensitive structures such as metal–oxide–semiconductor (MOS) transistors. Extensive studies using antennae test devices have confirmed that radiation and charging damage are minimal in the PSII environment, presumably because of the rapid neutralization of charge imbalance by the ionized species in the plasma (39). APPLICATIONS Materials Systems Many studies have demonstrated the effectiveness of the PSII process for improving the surface hardness, wear, and corrosion characteristics of engineering materials. When

Plasma Implantation

Figure 3. Weak-track profile for PSII nitrogen-implanted S-1 tool steel.

5

ment mode. Examples of coatings deposited using this technique include W, Pt, Cr, TiN, TaN, CrN, and Cr–Mo alloy films. The implant characteristics of PSII result in ionenhanced mixing at the substrate-film interface, thus leading to superior adhesion of the deposited layer (50, 51). PSII nitrogen ion implanted metallic films have been shown to have superior diffusion barrier characteristics. For example, nitrogen ion implanted Ti and Ta films has been shown to exhibit superior high temperature diffusion barrier characteristics between Cu and Si, a materials system of considerable relevance in microelectronics applications (52, 53). Finally, in materials where the implanted species can diffuse, PSII at elevated temperatures can produce a substantially thicker diffusion zone supporting the implanted layer, leading to major increases in hardness and load-bearing capacity (discussed in Ref. 10). Many other examples and detailed studies of the materials science of metallurgical applications of PSII have been conducted at many institutions throughout the world as shown in Refs. 7 through 9. Diamond-like Carbon (DLC) Films

Figure 4. Microhardness profiles for untreated and PSII nitrogen-implanted A-2 tool steel.

performed with gaseous implant species (e.g., nitrogen, oxygen, or carbon-containing vapors), these unique surface properties are obtained primarily through the formation of nitrides, oxides, and carbides. For example, PSII nitrogen ion implantation of American Iron and Steel Institute (AISI) S-1 tool steel (40) at a target bias of −50 kV and a dose of 3 ×1021 atoms per square meter results in an implanted layer approximately 100 nm thick. The corresponding increase in wear resistance as measured with a pin-on-disk wear tester is shown in Fig. 3. Under similar conditions, implantation of an AISI A-2 tool steel (41) increases hardness by more than 50% (Fig. 4), while the friction coefficient drops by a factor of two. In field tests of A-2 tool-steel score dies used for stamping tabs in aluminum cans, PSII nitrogen implantation improved service life from 8.5 million to 14 million hits (39). Corrosion resistance of aluminum and bearing steel alloys has been substantially improved by nitrogen PSII (42, 43), and the orthopedic alloy Ti-6Al-4V (used for artificial replacement joints) has shown a significant increase in wear resistance after nitrogen ion implantation (44). PSII of Ni-Ti shape memory alloy has been investigated for the improvement biocompatibility and corrosion resistance in biomedical applications (45, 46). Carbon ion implantation using methane precursor gas has been shown to increase the surface hardness of stainless steels (47, 48) and wear and galling resistance of AISI 52100 bearing steel (49). It has been noted that for an austenitic stainless steel, depending on dose and dose rate, the near-surface structure could be altered from crystalline to amorphous or a combination of the two phases (47). An enormous range of adherent, high-value-added coatings and surface treatments can be realized by combining PSII with plasma or physical vapor coating deposition treatments for an “ion beam enhanced deposition” treat-

PSII has been used for depositing diamond-like carbon (DLC) films. DLC films are being considered for a variety of applications because of their high hardness and low friction, wear resistance, biocompatibility, chemical inertness, and optical transparency. DLC films have been traditionally synthesized by ion beam processes and high temperature CVD methods (54, 55). PSII is a near room temperature, non-line-of-sight method for depositing DLC films [56]. A significant advantage of PSII in regards to DLC films is the ability to implant carbon at higher energies (>10 keV) to form a carbide seed layer followed by deposition of DLC at lower energies (1 to 5 keV). Commonly, methane has been used as precursor gas for carbon ion implantation and acetylene has been used for DLC deposition. This approach has been shown to enhance the adhesion of DLC films to a variety of metallic substrates (57, 58), and has been trademarked by General Motors (IoncladTM ). The hardness and elastic modulus of DLC films synthesized by PSII range from 5 to 25 GPa and 50 to 200 GPa, respectively depending on the process parameters and the hydrogen content of the films. Film thicknesses are usually limited to about 3µm due to limitations imposed by film stresses. DLC films elementally modified by with silicon and fluorine can be produced by PSII using appropriate precursor gases (e.g., Si-DLC with hexamethyl-disiloxane and F-DLC with tetrafluoro-ethane). Si-DLC films have a higher hardness, lower surface energy, and greater high temperature stability compared to unmodified DLC films, whereas the F-DLC films are softer. Elementally modified DLC films synthesized by PSII are similar to those that have been successfully produced using the CVD technology (59). Finally, DLC films produced by PSII have been examined for nanoscience and technology, particularly in the study of friction and tribology on a near-atomic level by the conformal deposition of these films on AFM tips (60). DLC films produced by PSII could have potentially wide range of applications in tools and components, MEMs devices and computer hard disks (61).

6

Plasma Implantation

Semiconductor Systems PSII has a number of unique and important applications to the processing of semiconductor materials; excellent reviews regarding the application of PSII to semiconductors have been published in Refs. 62–67. As mentioned before, during each voltage pulse the ions are accelerated across the ion sheath and implanted into the surface. It is important to note that this process can proceed at arbitrarily low accelerating potentials, implying that the PSII implantation process has no limit at low energies. In contrast, a decrease in accelerating voltage reduces the efficiency of beamline implanters as the accelerating voltage is decreased, thus reducing the dose that can be delivered in a given process time. PSII has an added advantage in a semiconductor manufacturing environment in that the space requirement of a typical implant system (i.e., the “footprint” of the installation) is much smaller than that of a typical beamline implanter. Given the escalating costs of typical Class 1 cleanroom space, a decrease in footprint offers the prospect of significant cost-of-ownership advantages for a PSII-based system. A third advantage of PSII for semiconductor applications is that the implantation can be performed (in principle) into substrates of arbitrary sizes. There has been great interest, for instance, in using PSII for the implant step in the fabrication of thin film transistors (TFT) in flat panel displays (FPD). As mentioned above, beamline implanters typically require complex and expensive target rotation and manipulation mechanisms in order to achieve uniform implantation over large areas. Since PSII is a plasma process, it shares many of the characteristics of these processes as they are used in the semiconductor industry. This means, for instance, that a PSII implanter can be integrated along with other processing stations into a “cluster tool” environment. Also, it is conceivable that the implant capabilities of PSII could be combined with other plasma processes such as sputter deposition or reactive ion etching to achieve new process capabilities. PSII has long been recognized for its potential for shallow junction doping, where its combination of high ion flux at low energies makes it ideally suited. This is especially true in the case of p-doping (impurity doping that generates acceptor sites or “hole” carriers) of silicon by boron, where the low mass of the boron ion results in a relatively long projected ion range. Boron implantation thus requires very low energies in order to achieve the shallow junction depth anticipated for advanced device geome˚ 300 Afor ˚ tries [viz. 100 Ato 0.07 µm process according to the National Technology Roadmap for Semiconductor Processes (62)]. Numerous studies (described in Refs. 63–69) have therefore been conducted on the formation of shallow junctions by PSII doping. For instance, 0.5 kV PSII implantation from a BF3 plasma yielded a junction depth ˚ of less than 400 Aafter a 10 s anneal at 950◦ C (69). In preamorphized samples, transient-enhanced diffusion and dopant trapping was observed at low temperature (550◦ C) anneals (69). High-resolution transmission electron microscopy and x-ray diffraction analyses of the as-implanted silicon has shown the expected formation of a thin (5 nm)

amorphous layer in silicon implanted from a BF3 plasma at 3.5 kV. Dislocations or other extended defects are not observed (70, 71). Other variants on the PSII process show promise for semiconductor materials processing. For instance, instead of implanting boron directly into the silicon and then activating the implant by an anneal process, it is possible to sputter deposit boron onto the silicon surface and then use ion-assisted mixing to incorporate the boron into the silicon lattice (72). An improvement in properties of polysilicon TFT transistors following PSII implant has been reported (73). PSII has also been used to synthesize TiN diffusion barriers (74). PSII has been shown to be useful for the fabrication of silicon-on-insulator structures via the SIMOX (separation by implantation of oxygen) process (75, 76). SIMOX materials are typically synthesized by performing a high dose silicon implant (>1017 cm−2 ) to form a subsurface layer that is supersaturated with oxygen; a subsequent hightemperature anneal converts this layer to a buried oxide. Cheung and coworkers have shown that PSII can effectively form a high-quality SIMOX structure by performing an oxygen implant at 60 kV followed by an anneal at 1270◦ C (75). Moreover, it was claimed (75) that the high ion doses that can be achieved by PSII could make SIMOX fabrication by this method more cost effective than conventional beamline implant synthesis. While one of the main advantages of PSII over conventional beamline implantation is the natural near-normal incidence of the accelerated ions, it has been shown (77) that conformal doping of deep silicon trenches can be performed by this method. For trenches with aspect rations as great as 12:1, very good doping uniformity was observed. Since the uniformity depended primarily on the implant bias while the pressure had relatively little effect, the conformal doping was attributed primarily to collisional scattering of the incident ions as they crossed the plasma sheath (62).

OTHER MATERIALS AND PLASMA SYSTEMS While metallurgical and semiconductor applications of PSII have garnered the greatest attention, other materials can be treated as well by use of this process. One of the more intriguing applications has been in the implantation of polymer films to improve their wetting characteristics. The exposure of polymer surfaces to a plasma in order to improve their wetability is an established practice; however, generally the contact angle of a liquid placed on the surface increases with time (minutes to hours) after the plasma treatment is completed. If polymer surfaces are implanted with oxygen by PSII, the degradation of the wetability can be significantly retarded, or, in some cases, eliminated (78). Modification of glass surfaces to induce changes in optical or magnetic properties has also been investigated. For example, it has been demonstrated that adding oxygen into iron-doped magnesium aluminosilicate glasses results in near-surface precipitation of nano-scale magnetic spinel domains of potential interest for high storage-density mag-

Plasma Implantation

netic recording media (79). Similar possibilities exist for modifying optical transparency characteristics of specialized window glass by formation of buried TiN layers. This technology has been demonstrated using conventional ion implantation (80), and is considered a reasonable candidate for PSII technology as well. While the original PSII concept utilized low pressure, weakly ionized gas discharges, other approaches have been investigated (81, 82). In particular, cathodic arcs have been successfully employed as a source of metal ions for PSII processes. By pulsing a cathodic arc in synchronization with the PSII pulse bias it is possible to perform pure metal ion implantation without deposition. Although the directional nature of the cathodic arc may obviate the advantage of treating the entire workpiece simultaneously, excellent conformality can be achieved on surfaces facing the arc. This is due to the fact that cathodic arcs can produce very dense plasma (typically 1012 cm−3 ) which will result in a subcentimeter sheath thickness. COMMERCIALIZATION ISSUES Cost effectiveness for high-volume, low-cost-per-part commodities favors large-batch processing, which, in turn, implies large processing chambers and corresponding vacuum systems (28). If the capital costs are amortized, then the annual operating expenses become dominated by personnel costs, and are therefore competitive with other processing technologies. However, the potentially large equipment investment may mean that the time for return on investment is significantly longer than that of competing coating technologies, such as electrolytic plating or vacuum arc deposition. This, of course, depends on the rate of treated parts (or total treated surface) that the market for that product is expected to consume in a year. Therefore, the best opportunities for commercial adoption include large-volume-consumption-rate commodities whose customers are willing to pay premium prices for a high-value-added process. The most obvious candidate, therefore, is plasma doping or related PSII treatments for next generation (ultra-large-scale integrated) microelectronic devices. Other possibilities include niche applications where the wear stress on the material surface is so severe as to cause adhesion failure of more conventionally deposited coatings. A semiempirical model for the cost of a commercial PSII system has concluded that such a facility should be able to treat a surface area of 104 m2 /year at a cost of $0.01 per cm2 (83). Opportunities exist for commercial adoption in the automotive or fine-finish architectural coatings markets, especially as restrictions on wet chemical plating processes arise from environmental regulations on disposal of hazardous or polluted fluids conventionally used in that industry (84, 85). BIBLIOGRAPHY General J. R. Conrad Method and apparatus for plasma source ion implantation, United States Patent 4,764,394, August 16, 1988.

7

J. R. Conrad et al. Plasma source ion implantation technique for surface modification of materials, J. Appl. Phys., 62: 4591–4596, 1987. P. C. Johnson The cathodic arc plasma deposition of thin films, in Thin Film Processes II, in J. L. Vossen and W. Kern (eds.), New York: Academic Press, 1991. M. Konuma Film Deposition by Plasma Techniques, New York: Springer-Verlag, 1992. G. Lukowski D. E. Ibbotson D. W. Hess (eds.) Characterization of Plasma-Enhanced CVD Processes, Mater. Res. Soc. Symp. Proc., 1990, p. 165. W. Wang et al. Magnetic field enhanced argon plasma for ionized magnetron sputtering of copper, Appl. Phys. Lett., 71 (12): 1622–1624, 1997. J. R. Conrad K. Sridharan (eds.) Proc. 1st Int. Workshop PlasmaBased Ion Implantation, New York: Amer. Vacuum Soc. Amer. Inst. Phys., 1994. G. Collins (ed.) Proc. 2nd Int. Workshop Plasma-Based Ion Implantation, in Surf. Coatings Technol., 1996. Proc. 3rd Int. Workshop Plasma-Based Ion Implantation, in W. M¨oller (ed.), Surf Coatings Technol., 1997. J. V. Mantese et al. Plasma-immersion ion implantation, MRS Bulletin, August, 52–56, 1996. P. K. Chu et al. Plasma immersion ion implantation—a fledgling technique for semiconductor processing, Mat. Sci. Engineer, R 17: 207–277, 1996.

Sputtering in Ion Implantation G. Carter J. S. Colligon Ion Bombardment of Solids, Amsterdam, The Netherlands: Elsevier, 1968. F. A. Smidt B. D. Sartwell Nucl. Instrum. Methods, B 6: 70, 1985.

Plasma Science in PSII M. Hong G. A. Emmert Two-dimensional fluid simulation of expanding plasma sheaths, J. Appl. Phys., 78 (12): 6967–6973, 1995. G. A. Emmert Model for expanding sheaths and surface charging at dielectric surfaces during plasma source ion implantation, J. Vac. Sci. Technol., B 12 (2): 880–883, 1994. B. Wood Displacement current and multiple pulse effects in plasma source ion implantation, J. Appl. Phys., 73 (10): 4770–4778, 1993. G. A. Emmert M. A. Henry Numerical simulation of plasma sheath expansion, with applications to plasma-source ion implantation, J. Appl. Phys., 71 (1): 113–117, 1992. J. T. Scheuer M. Shamim J. R. Conrad Model of plasma source ion implantation in planar, cylindrical, and spherical geometries, J. Appl. Phys., 67 (3): 1241–1245, 1990. M. A. Leiberman Model of plasma immersion ion implantation, J. Appl. Phys., 66 (7): 2926–2929, 1989. M. Shamim J. T. Scheuer J. R. Conrad Measurements of spatial and temporal sheath evolution for spherical and cylindrical geometries in plasma source ion implantation, J. Appl. Phys., 69 (5): 2904–2908, 1991. M. Shamim et al. Measurement of electron emission due to energetic ion bombardment in plasma source ion implantation, J. Appl. Phys., 70 (9): 4756–4759, 1991.

8

Plasma Implantation

S. M. Malik et al. Sheath dynamics and dose analysis for planar targets in plasma source ion implantation, Plasma Sources Sci. Technol., 2: 81–85, 1993. M. J. Goeckner et al. Laser-induced fluorescence measurement of the dynamics of a pulsed planar sheath, Phys. Plasmas, 1 (4): 1064–1074, 1994. W. N. G. Hitchon E. R. Keiter Kinetic simulation of a timedependent two-dimensional plasma, J. Computational Phys., 112: 226–233, 1994. R. A. Stewart M. A. Lieberman Model of plasma immersion ion implantation for voltage pulses with finite rise and fall times, J. Appl. Phys., 70 (7): 3481–3487, 1991. T. E. Sheridan Effect of target size on dose uniformity in plasmabased ion implantation, J. Appl. Phys., 81: 7153–7157, 1997. D. K. Cheng Field and Wave Electromagnetics. 2nd ed., Reading, MA: Addison-Wesley, 1989. T. S. Ebert et al. A cost model of commercial plasma source ion implantation for corrosion protection of multiple objects via batch processing, accepted for publication in Surface and Coatings Technology, 1997. J. R. Conrad et al. Plasma source ion implantation dose uniformity of a 2×2 array of spherical targets, J. Appl. Phys., 65: 1707, 1989.

Comparison With Beamline Implant E. Jones et al. Source/drain profile engineering with plasma implantation, 96 Proc. 11th Conf. Ion Impl. Tech., 1997, pp. 745–748. S. B. Felch et al. Formation of deep sub-micron buried channel pMOSFETs with plasma doping, 96 Proc. 11th Conf. Ion Impl. Tech., 1997, p. 753.

Complicating Factors in PSII J. Shao et al. Dose-time relation in BF3 plasma immersion ion implantation, J. Vac. Sci. Technol., A 13 (2): 332–334, 1995. E. C. Jones N. W. Cheung Plasma doping dosimetry, IEEE Trans. Plasma Sci., 25 (1): 42–52, 1997. B. P. Wood et al. Design of a large-scale plasma source ion implantation experiment, Mat. Res. Soc. Symp. Proc., 279: 345–350, 1993. D. L. Chapek Optimization of a plasma doping technique for ultrashallow junction formation in silicon, Ph.D Dissertation, Univ. Wisconsin-Madison, 1995. S. Qin C. Chan An evaluation of contamination from plasma immersion ion implantation on silicon device characteristics, J. Electron. Mater., 23: 337, 1994. E. C. Jones et al. Anomalous behavior of shallow BF3 plasma immersion ion implantation, J. Vac. Sci. Tech., B 12: 956, 1994. J. Shao et al. J. Vac. Sci. Tech., A 13: 332, 1995. T. Sheng S. B. Felch C. B. Cooper III Characteristics of a plasma doping system for semiconductor device fabrication, J. Vac. Sci. Tech., B 12: 969, 1994.

Materials Applications

A. M. Redsten et al. Nitrogen plasma source ion implantation of AISI S1 tool steel, J. Materials Process. Technol., 30: 253, 1992. J. R. Conrad et al. Surface modification of materials by plasma source ion implantation, Proc. ASM-TMS Plasma Laser Process. Mater., 1991, pp. 141–149. J. H. Booske et al. Nitrogen plasma source ion implantation for corrosion protection of aluminum 6061-T4, J. Mater. Res., 12 (5): 1356–1366, 1997. W. Wang et al. Modification of bearing steel surface by nitrogen plasma source ion implantation for corrosion protection, J. Mater. Res., 1997, submitted. N. C. Horswill K. Sridharan J. R. Conrad A fretting wear study of a nitrogen implanted titanium alloy, J. Mater. Sci. Lett., 14: 1349–1351, 1995. L. Tan, R.A. Dodd, and W.C. Crone,“Corrosion and Wear-Corrosion Behavior of NiTi Modified by Plasma Source Ion Implantation”, Biomaterials, 24 (2003)pp. 3931–3939. P. K. Chu,“ Bioactivity of Plasma Implanted Biomaterials”, Nuclear Instruments and Methods in Physics Research B, vol. 242, no. 1–2, (2006)pp. 1–7. J. Chen J. R. Conrad R. A. Dodd Dose and dose rate effects on the structure of methane plasma source ion implanted 304 stainless steel, Mater. Sci. Eng., A 161: 97, 1993. J. Chen et al. Structure and wear properties of carbon implanted 304 stainless steel using plasma source ion implantation, Surf. Coat. Technol., 53: 267, 1992. K. Sridharan, E.H. Wilson, D.F. Lawrence, and J.R. Jacobs, “Application of Hydrocarbon Plasmas for Modifying Near-Surface Characteristics of Bearing Steel” Applied Surface Science, 222, (2004)pp. 208–214. I. G. Brown et al. Plasma synthesis of metallic and composite thin films with atomically mixed substrate banding, Nucl. Inst. Methods B, 80/81: 1281–1287, 1993. B. P. Wood K. C. Walter T. N. Taylor Plasma source ion implantation to increase the adherence of subsequently deposited coatings, inA. Hobayashi andN. M. Ghoniem (eds.), Advances in Plasma Science, Osaka: Institute Applied Plasma Science, 1997. Erik H. Wilson, Ph.D. Thesis, Energetic ion based ion deposition and implantation of metal nitride diffusion barriers for microelectronics applications by plasma source ion implantation. Department of Engineering Physics, University of Wisconsin, Madison, (2001). M. Kumar et al.“ Effect of Plasma Immersion Ion Implantation on Thermal Stability of Diffusion Barriers” Surface Coatings and Technology, vol. 186,No. 1–2, (2004)pp. 77–81.

Diamond-Like Carbon (DLC) Films B. Bhushan“ Chemical, mechanical, and tribological characterization of ultra-thin and hard amorphous carbon coatings as thin as 3.5nm: recent developments”. Diamond and Related Materials, vol.8, (1999)pp. 1985–2015. B. K. Gupta, B. Bhushan“ Micromechanical properties of amorphous carbon coatings deposited by different deposition techniques”, Wear, 190, (1995)pp. 110–122. Non-Semiconductor Applications of Plasma Immersion Ion Implantation & Deposition”; K. Sridharan, S. Anders, M. Nastasi, K. C. Walter, A. Anders, O. R. Monteiro, and W. Ensinger, Handbook of Plasma Immersion Ion Implantation & Deposition, publisher John Wiley & Sons, New York, September (2000),pp. 553–636.

Plasma Implantation G. W. Malaczynski et al. Surface Enhancement by shallow carbon implantation for improved adhesion of diamond–like coatings, J. Vac. Sci. Technol. B, 17(2), (1999)pp. 813–817. “Advances in PSII Techniques for Surface Modification”; K. C. Walter, M. Nastasi, N. P. Baker, C. P. Munson, W. K. Scarborough, J. T. Scheuer, B. P. Wood, J. R. Conrad, K. Sridharan, S. Malik, R. A. Breun, Surface and Coatings Technology, 103/104, (1998)p. 205. M. Grischke et al.Application-oriented Modifications of Deposition Processes for diamond-like Carbon-based Coatings, Surface Coatings and Technology, 74–75 (1995)pp. 739–745. “ Atomic Scale Friction and its Connections to Fracture Mechanics”; R. W. Carpick, E. E. Flater, K. Sridharan, D. F. Ogletree, and M. Salmeron, JoM,October (2004)p. 48–0. R. PetersM.S. Thesis, Characterization of thin amorphous hydrogenated carbon films produced by plasma source ion implantation as protective coatings on computer hard disks,Department of Materials Science and Engineering, University of Wisconsin, Madison (1991).

Semiconductor Applications E. C. Jones B. P. Linder N. W. Cheung Plasma immersion ion implantation for electronic materials, Jpn. J. Appl. Phys., 35: 1027–1036, 1996. R. Burger W. Howard (eds.) Semiconductor Industry Association technology working group report. Santa Clara: SIA, 1993. S. B. Felch et al. Proc. Meas. & Char. Ultra-Shallow Doping Profiles Semicond., 3: 25.1, 1995. E. Ishida S. B. Felch J. Martens Proc. Meas. & Char. Ultra-Shallow Doping Profiles Semicond., 3: 37.1, 1995. S. B. Felch et al. Studies of ultra-shallow p+ -n junction formation using plasma doping, Proc. Ion Impl. Tech., 94: 981, 1995. S. Qin et al. Plasma immersion ion implantation doping using a microwave multiplier bucket plasma, IEEE Trans. Elec. Dev., 39: 2354, 1992. S. Qin C. Chan N. E. McGruer Energy distribution of boron ions during plasma immersion ion implantation, Plasma Source Sci. & Tech., 1: 1, 1992. C. M. Osborn et al. Proc. 11th Int. Conf. Ion Implant. Tech. 96, 1997, pp. 607–610. D. L. Chapek et al. Structural characterization of plasma-doped silicon by high resolution x-ray diffraction, J. Vac. Sci. Tech., B 12 (2): 951–955, 1995. R. J. Matyi et al. Boron doping of silicon by plasma source ion implantation, Proc. 3rd Int. Conf. Plasma-Based Ion Implantation, Surf. Coat. Technol., 93 (2–3): 247–253, 1997. H. L. Liu et al. Recoil implantation of boron into silicon for ultrashallow junction formation: Modeling, fabrication, and characterization, J. Vac. Sci. Technol., B 16 (1): 415–419, 1998. J. D. Bernstein et al. Hydrogenation of polycrystalline silicon thin film transistors by plasma ion implantation, IEEE Electron Dev. Lett., 16 (10): 421–423, 1995. W. Wang et al. TiN prepared by plasma source ion implantation of nitrogen into Ti as a diffusion barrier for Si/Cu metallization, J. Mater. Res., 13 (3): 1998. J. B. Liu et al. Formation of buried oxide in silicon using separation by plasma implantation of oxygen, Appl. Phys. Lett., 67 (16): 2361, 1995. L. Zhang et al. Low energy separation by implantation of oxygen structures via plasma source ion implantation, Appl. Phys. Lett., 65 (8): 1–3, 1994.

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C. Yu N. W. Cheung Trench doping conformality by plasma immersion ion implantation (PIII), IEEE Electron. Dev. Lett., 15: 196, 1994.

Other Materials and Applications J. W. Lee et al. Investigation of ion bombarded polymer surfaces using SIMS, XPS and AFM, Nucl. Instr. Meth., B 121: 474, 1997. L. Zhang et al. Plasma-immersed oxygen ion implantation of irondoped glass for nonmetallic magnetic hard disks, J. Vac. Sci. Tech., B 12 (6): 3342–3346, 1994. G. Was et al. Formation of buried TiN in glass by ion implantation to reduce solar load and reflectivity, J. Appl. Phys., 80: 2768, 1996. A. Anders et al. Metal plasma immersion ion implantation using vacuum-arc plasma sources, J. Vac. Sci. Tech., B 12 (2): 815–820, 1994. C. P. Munson et al. Recent advances in plasma source ion implantation at Los Alamos National Laboratory, Surf. Coat. Technol., 84 (1–3): 528–536, 1996. D. J. Rej R. B. Alexander Cost estimates for commercial plasma source ion implantation, J. Vac. Sci. Tech., B 12 (4): 2380–2387, 1994. J. R. Conrad K. Sridharan M. M. Shamim Plasma source ion implantation: An environmentally acceptable alterantive to wet chemical plating, Int. J. Environmentally Conscious Des. Manuf., 2 (1): 61–66, 1993. A. Chen et al. Chromium plating pollution source reduction by plasma source ion implantation, Surf. Coat. Technol., 82: 305–310, 1996.

JOHN H. BOOSKE RICHARD J. MATYI JOHN R. CONRAD KUMAR SRIDHARAN Department of Engineering Physics, University of Wisconsin, Madison, WI

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Wiley Encyclopedia of Electrical and Electronics Engineering Plasma Switches Standard Article Frank Hegeler1 1Commonwealth Technologies, Inc./Naval Research Laboratory, Washington, DC Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5920.pub2 Article Online Posting Date: August 17, 2007 Abstract | Full Text: HTML PDF (1176K)

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Abstract The sections in this article are General Switch Parameters Gas Spark Gaps Surface Discharge Swtches Diffuse Discharge Opening Switch Plasma Opening Switches Plasma Flow Switches Thyratrons Tacitrons Crossatrons Pseudospark Switches

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PLASMA SWITCHES Switches are common components in electric circuits that are familiar to our everyday experience. In applications for which intense bursts of energy are required, switching becomes a technical challenge. It is in these cases that plasma switches are typically employed. Reliable plasma switches are essential components in pulsed power systems. Holdoff voltages of these switches range from kilovolts (kV) to megavolts (MV), and conduction currents vary from amperes to mega-amperes (MA). A simple multiplication of the current and voltage rating indicates that some of the largest devices transfer power in excess of terawatts (TW). For comparison, a large electric power plant generates on the order of 1 gigawatt (GW) of continuous power. Pulsed power technology is used in many applications such as lasers, radar, X-ray generators, particle accelerators, pollution control, material surface treatments, and nuclear weapons effects simulators. A pulsed power system consists of an energy store, which is charged slowly at low power (i.e., charging time constants between milliseconds and minutes) and then is quickly discharged within nanoseconds (ns) to milliseconds (ms) into a load (Fig. 1). The switch accomplishes the rapid transition from the charging to the discharging stage. Switches are categorized into two major groups: opening and closing switches. Capacitive energy storage systems need closing switches (Sc ), whereas inductive energy storage systems require opening switches (So ), as shown in Fig. 2. The capacitor is charged with a source voltage Vs through a charging resistor; Rcharge , [Fig. 2(a)]. Power amplification is given by the ratio of load current to charging current. For inductive energy storage systems, the increase of the inductor voltage during the rapid opening of the switch, So , results in power amplification at the load, Zload [Fig. 2(b)]. Reliable and repetitive closing switches have been developed successfully, and thus the capacitive energy storage system is being used extensively. Fast, reliable, and repetitive opening switches present a greater technological challenge. While there is little physical impediment to the operation of a closing switch, an opening switch must work against inductance (manifested as a tendency to increase voltage in order to maintain current) that tends to oppose the opening of the switch. Recently, research has been directed toward opening switches because the energy storage density of typical inductors is 2 to 3 orders of magnitude greater than that of capacitors. Table 1 gives a list of the most common opening and closing switches.

Figure 1. Pulsed power systems are charged slowly and discharged rapidly, which achieves high output power amplitudes.

Figure 2. (a) Capacitive energy storage system with a closing switch and (b) inductive energy storage system with an opening switch.

This article describes plasma switches, such as plasma arc switches (gas spark gaps and surface discharge switches), diffuse discharge switches, and low-pressure switches (plasma opening switches, plasma flow switches, thyratrons, tacitrons, crossatrons, pseudospark switches, and ignitrons). GENERAL SWITCH PARAMETERS The following parameters are used to characterize most switches. Hold-off Voltage. Maximum blocking voltage of switch in the “open” state. Exceeding the hold-off voltage causes an electrical breakdown in the switch, in most cases, owing to field emission at the electrodes and/or ionization of a background gas and rapid formation of a conducting plasma (see article on Conduction and breakdown in gases Conduction and breakdown in gases). Peak Conduction Current. Maximum switch current in the “closed” state. The switch impedance in the open state, the load impedance, and hold-off voltage limits the peak charge voltage, and hence the peak current in a capacitive energy store system. The switch impedance in the closed state limits peak current in both the capacitive and inductive energy storage systems. Current Rise (dI/dt). Rate at which the conduction current can be applied without device damage, or in some cases, the value of current rise limited by switch or circuit configuration. Voltage Rise (dV/dt). Rate at which a voltage can be applied during the open state without causing switch closure. (applies to opening switch) Forward Voltage Drop. Voltage drop of the switch during the on state. It is found by multiplying the switch impedance in the closed state with the conduction current. Closing Delay Time. Time between the triggering of a switch and the beginning of the conduction state. It often relates to a plasma drift time for closing the electrode gap or a gas ionization time. Opening Delay Time. Time between triggering a control grid of the switch and the termination of the conduction current.

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright © 2007 John Wiley & Sons, Inc.

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Plasma Switches Table 1. Some of the Most Common Opening and Closing Switches in Pulsed Power Systems

Closing Switches Crossatron switches Exploding foil dielectric switches Ignitrons Krytrons Mechanical switches Photoconductive switches Pseudospark switches Spark gaps Solid-state switches Surface-discharge switches Tacitrons Thyratrons Vacuum switches

Opening Switches Chemically exploding switches Crossatron switches Diffuse discharge opening switches Electrically exploding fuses and foils Mechanical switches Plasma (erosion) opening switches Plasma flow switches Solid-state switches Vacuum switches

Recovery Time. Time between the end of the conduction current and the point at which a voltage with a certain dV/dt can be applied without breakdown. In most cases, it is the recombination time of the gap plasma. Pulse Repetition Rate. The rate at which the switch can be “closed” and/or “opened” without degradation of switch characteristics. The usual units are pulses per second (pps) or hertz (Hz). Jitter. Variation in the opening or closing delay times. Switch Lifetime. Number of “shots” (switching operations) or amount of time during which the switch operates within its normal specification. Electrode or grid erosion, mechanical failure, and insulator degradation, are typical factors that limit switch lifetime.

GAS SPARK GAPS Gas sparks gaps are plasma arc switches that are widely employed in pulsed power technology. They are closing switches that are conceptually simple and easily manufactured. Voltage and current parameters are scalable, with hold-off voltages varying from kilovolts to gavolts and conduction currents ranging from amperes to megaamperes. They operate to the right of the so-called Paschen minimum (see article on Conduction and Breakdown in Gases), at pressures of 105 Pa to 106 Pa. Dry air, and sulfur hexafluoride (SF6 ) are the most common insulating gases in spark gaps. The hold-off voltage of a spark gap depends on the electrode geometry and material, gap spacing, gas pressure, gas species, and the applied pulse duration. Rough surfaces on electrodes tend to lower the breakdown strength due to microscopic field enhancements (i.e., the electric field at a tip of the rough surface is significantly higher than the average surface field). To decrease macroscopic field enhancements, the radius of the electrodes should be larger than the gap distance. At high electric fields, materials with a low work function tend to give a lower breakdown voltage due to a reduced field emission threshold. The product of electrode gap spacing and gas pressure is roughly a constant function of the electric breakdown field strength (see Conduction and Breakdown in Gases); at constant electrode gap spacing, the hold-off voltage increases with gas pressure and vice versa.

The insulating gas in the spark gap has a notable effect on the breakdown strength. With SF6 , an electronegative gas with excellent arc-quenching capacity, the breakdown strength is 2.5 to 2.8 times that of air; a 10% SF6 −90% N2 mixture has twice the strength of pure N2 . The dc breakdown field in dry air at atmospheric pressure (105 Pa) is about 30 kV/cm for parallel plate electrodes. In addition, the hold-off voltage can be larger than that calculated for a Townsend dc breakdown (see article on CONDUCTION AND BREAKDOWN IN GASES) if the applied pulse duration is less than the ionization time. The spark gap closes if the applied voltage exceeds the self-breakdown voltage of the gap or if the gap is triggered. Ionization of the gas and the formation of an arc discharge in the plasma close the switch. Triggering of gas-filled gaps is generally achieved by one of the following methods: field distortion, arc initiation (trigatron), laser triggering, electron-beam triggering, UV irradiation, or any other means of initiating ionization. For good switching performance, the gap voltage should be above 60% of the selfbreakdown voltage by choosing the appropriate gap spacing and gas pressure. Voltage trigger generators should have a low internal impedance, fast voltage rise times, and an open-circuit voltage that is several times greater than the self-breakdown of the igniter gap. A field distortion spark gap is presented in Fig. 3. The trigger electrode is located on an equipotential line in the gap, thus, maximizing the hold-off voltage. A trigger pulse initiates breakdown between one electrode and the trigger plate, causing practically the full gap voltage to appear across the other half of the gap, initiating a Townsend breakdown. The trigger electrode may be located halfway into the electrode gap, or closer to the cathode. Although trigger voltage requirements are reduced with shorter trigger electrode–cathode distances, closing delay times are often increased. Figure 4(a) shows the trigatron arrangement. An arc discharge is formed between the trigger electrode and one of the main electrodes, which initiates breakdown both by providing seed electrons for ionization and UV light for gas volume photoionization. Reliable triggering can be achieved with lower trigger voltage amplitudes compared to the field distortion spark gaps, but the trigatron has a lower hold-off voltage due to field enhancements at the trigger pin–cathode interface. Lasers can be used to achieve very accurate triggering of megavolt spark gaps as shown

Plasma Switches

Figure 3. A field distortion spark gap with interchangeable anode and cathode. The electrode gap holds off the applied voltage between the anode and cathode until a breakdown is initiated by energizing the trigger plate, which distorts the electric field in the gap.

Figure 4. Configuration of (a) a trigatron and (b) a laser-beamtriggered spark gap.

in Fig. 4(b). When the anode is irradiated by energetic photons, the gap is closed by streamer propagation (see article on Conduction and Breakdown in Gases), which is 10 to 100 times faster than breakdowns obtained from Townsend avalanche formation. Rail-gap switches (i.e., electrodes are much longer than they are wide) are usually UV laser triggered along the axis of the electrode rail, and its switch inductance is lower due to a parallel multichannel discharge. Spark gaps have low conduction losses with forward voltage drops ranging from 100 V to a few kV, depending on the configuration. In small, high-pressure, coaxial gaps, closing times of less than 1 ns have been achieved; for large gaps, delay times of 10 ns to 100 ns are common. The recovery time in sealed spark gaps is a few milliseconds. Increasing the gas pressure improves the recovery rate, However, gas flushing is required for extended repetition rate operation, which provides cooling of the switch electrodes and gas as well as removes debris from the electrode gap. Repetition rates of up to 100 Hz are typical; higher rates of up to 10 kHz can be obtained with proper electrode configuration, cooling, and gas flushing. Jitter ranges from as low as 25 ps for UV-irradiated gaps to as low as 1 ns for trigatrons and field distortion spark gaps. The switch lifetime is mainly limited by electrode erosion, which is affected by the conduction current rise time and amplitude, pulse duration, repetition rate, electrode material, gas type, electrode temperature, and other factors that contribute to ero-

3

sion. Charge transfers of more than 100 C per shot have been demonstrated with a total switch lifetime in excess of a megacoulomb; smaller spark gaps typically conduct total charges of tens of kilocoulombs over the lifetime of the switch. Copper, brass, stainless steel, aluminum, tungsten, graphite, molybdenum, and alloys (for example elkonite, which is a mixture of tungsten and copper) are commonly used as electrode material, with tungsten having one of the highest erosion resistances. For further information on electric field enhancements, spark gaps, and gas breakdowns see Refs. 1, 6. Applications. The spark gap is the most common pulsed power closing switch. It is used in capacitive pulse generators, as a peaking gap to sharpen the voltage rise times, and in prototype pulsed power systems.

SURFACE DISCHARGE SWTCHES The surface discharge switch also falls in the category of plasma arc switches. It is a simple high-current closing switch used with low-impedance loads. A schematic of this switch is shown in Fig. 5. It consists of two electrodes on a dielectric surface and a trigger electrode in the electrode gap, which is generally on or above the insulator surface. The position of the anode and cathode is determined by the polarity of the applied high voltage (HV), and either gas or vacuum surrounds the switch. Typical gap dimensions are 1 cm to 40 cm in width (anode-to-cathode separation) and 1 cm to 50 cm in length. In a gas environment, the surface breakdown proceeds in several stages. First, after a voltage pulse has been applied to the trigger electrode, the capacitance of the electrode gap is charged. The trigger pulse should have a fast voltage rise of about 1012 V/s to assure multiple arc channel formation, and a high open-circuit voltage is essential, similar to trigger requirements for spark gaps. Electrons are then field emitted from the cathode-dielectric-gas (or vacuum) triple point and accelerated by the applied electric field. They ionize the gas in the electrode gap, and plasma spreads with a speed of 105 cm/s to 108 cm/s, depending on the gas type and pressure, towards the anode. An exponential rise of the breakdown current is seen in the third stage, which is characteristic of a Townsend-like breakdown, and in the final stage, the conduction current flows through the plasma (switch closure). For vacuum surroundings, some of the field-emitted electrons, with energies between 30 eV to 2 kV, hit the dielectric surface and generate secondary electrons. A positive surface charge is generated, which leads to a saturated surface current on the order of tens of milliamperes. Gas, which had been absorbed on the insulator surface, is released by electron impact and drifts away from the surface at its thermal velocity. A gas layer with a thickness of less than 1 mm and pressures on the order of 104 Pa is formed above the dielectric and is ionized by high-energy electrons. In vacuum, plasma velocities of 106 cm/s to 2 × 107 cm/s have been reported. In both gas and vacuum surroundings, the discharge arc is located on the dielectric surface due to geometric field enhancements (i.e., the relative permittivity εr of the dielectric is larger

4

Plasma Switches

Figure 6. Electrode configuration of a diffuse discharge switch. The switch is closed as long as the applied electron beam or photons ionize the gas between the electrodes.

Applications. Surface discharge switches can be used for capacitive, megaampere, pulsed power systems with low-impedance loads. DIFFUSE DISCHARGE OPENING SWITCH Figure 5. Cross section of a surface discharge switch with trigger: (a) trigger electrode on surface, and (b) trigger rail above dielectric surface.

than that of the surrounding medium). A surface discharge switch without a trigger electrode usually operates as a peaking gap. The desired hold-off switch voltage determines the minimum thickness of the dielectric with its bulk breakdown strength. For fast conductmion current rise times, a high switch capacitance with a low inductance is necessary; thus, the dielectric should be thin and multiple, parallel arc channels should form during switch closure. Gap length, gas type and pressure, insulator properties, surface roughness, and electrode configuration determine the hold-off voltage of surface discharge switches. Single-gap switches usually withstand less than 100 kV. Conduction currents depend on the number of simultaneous arc channels formed during a discharge. Currents of up to a few megaamperes have been achieved for large gaps, with tens of arc channels generated by trigger voltage rises of more than 1012 V/s. Pulse durations of a few microseconds to milliseconds, maximum current rises of 1011 A/s to 1012 A/s, closing delay times of 100 ns, and jitters of a few nanoseconds are typical for surface discharge switches. Repetition rates of 0.1 Hz for megaampere currents to 100 Hz for kiloampere conduction currents with gas-purge velocities of 10 m/s are possible. Recovery times vary from a few to tens of milliseconds in gases and hundreds of microseconds to a few milliseconds for the vacuum case. Usually, the dielectric material determines the lifetime of the switch. In high current applications, some areas of the dielectric surface are metallized with electrode material. Therefore, erosion of the dielectric is necessary to clean the insulator surface. However, material loss will eventually cause dielectric bulk breakdowns. Lifetimes range from 103 shots to 106 shots, depending on the conduction current amplitude, pulse width, cooling, and other factors that impact electrode and dielectric erosion. Reference 7 provides more information on surface discharge switches.

The operating principle of an externally controlled diffuse discharge opening switch is given in Fig. 6. It consists of an electrode gap and a high-energy electron beam, a UV source, or laser radiation that ionizes the gas mixture in the electrode gap during the closing state of the switch. Plasma between the electrodes conducts the current and the electric field strength to gas density ratio (E/N) or electric field strength to gas pressure ratio (E/p) is kept in a range in which ionization by discharge electrons is negligible. When the external ionization source is turned off, electron attachment and recombination processes in the gas mixture cause the conductivity to decrease and the switch opens. To increase switch performance, the gas should have a large electron mobility and small attachment and recombination rate coefficients during the conduction state (at low E/N ratios). In the opening stage at high field strengths, a low electron drift velocity, large electron attachment and recombination coefficients, high breakdown strength, and selfhealing gas mixtures (gas-mixture compositions that do not change in time) are desired. Typically, methane (CH4 ) and/or argon (Ar) are used as buffer gases, which do not have electron attachment at low E/N ratios, and are mixed with an electron attacher gas such as freon (C2 F6 ) or perfluoropropane (C3 F8 ). Optically enhanced attachment by laser radiation can also be used to decrease switch opening times because certain gases have a substantially larger electron attachment cross section in their excited state. Pulsed hold-off voltages of up to 300 kV, with peak conduction currents of up to 10 kA, and conduction current densities of 10 A/cm2 to 30 A/cm2 have been reported. Electrode gap distances of a few millimeters to tens of centimeters, electrode diameters of 2 cm to 30 cm, and gas pressures of 105 Pa to 106 Pa are common switch parameters. Conduction times are limited by the external ionization source and range from 0.5 µs to 5 µs. Operation as a closing switch is possible, but short conduction times restrict its application. Opening times of 50 ns to 3 µs have been measured, depending on the gas type, gas pressure, and the use of optically enhanced electron attachment

Plasma Switches

5

Figure 8. Circuit schematic of an inductive pulsed power system with a plasma opening switch. After the switch Sc closes, the generator current IG flows through the plasma. The load current IL is initially zero (IG = IP ).

Figure 7. Simple coaxial geometry plasma opening switch. Plasma streams through the mesh and electrically shorts the inner and outer conductors.

methods, and repetition rates of up to 1 kHz are possible. Further information on of diffuse discharge switches is given in Refs. 48 and 9. Applications. The diffuse discharge switch was studied as an opening switch for inductive pulsed power systems in the 1970s and 1980s, but little information exists on practical applications. PLASMA OPENING SWITCHES




η J⊥ J × E + v⊥ − µ0 ne e



Imax = 2πrl and

Plasma opening switches (POS) and plasma erosion opening switches (PEOS) have been studied extensively as repetitive opening switches for compact, energy-dense, inductive storage generators since the 1970s. Figures 7 and 8 give the coaxial geometry of a simple POS and a schematic of a pulsed power system with a POS, respectively. A plasma source, which consists of a flashboard, plasma gun, or laser-illuminated target, is triggered and emits plasma that drifts through a mesh and electrically shorts the inner and outer conductors. Initially, the generator current IG charges the circuit inductance without passing through the load; thus the plasma current IP equals IG . The fast opening of the switch occurs due to a “plasma thinning” mechanism that locally decreases the plasma density near the inner conductor, and a load current begins to flow. The plasma opening mechanisms are not fully understood. It is postulated that magnetic pressure, erosion, or combinations of both mechanisms generate a plasmafree region near the inner electrode. The magnetic pressure, B2 /2µ0 , is derived from the Poynting vector, S = E × B/µ0 , S=

the magnetic field, J⊥ is the current density perpendicular to the magnetic field, ne is the plasma electron density, and e is the electron charge. This magnetic pressure displaces the plasma on the generator side, resulting in a lowdensity region that generates an axial current density [see Fig. 9(a)]. The radial J × B force [see Fig. 9(b)] or plasma erosion opens the plasma gap, and load current begins to flow. Based on the erosion model, the PEOS opens if the amplitude of the plasma current Ip reaches the maximum allowed current Imax due to the space-charge limit[ see Fig. 10(a)

 B2 2µ0

(1)

where η is the plasma resistivity, µ0 is the permeability of free space, J is the current density, B is the magnetic flux density, v⊥ is the electron velocity perpendicular to

Ii = Ie

mi Zeni vd Zme



Zme mi

(2)

(3)

where r is the radius of the inner conductor, l is the length of the plasma channel, mi is ion mass, me is the electron mass, Z is the ion charge state, ni is the plasma ion density, vd is the ion drift speed, Ii is the ion current, and Ie is the electron current. For C2−+ plasmas, the Ie /Ii ratio is about 100. When the generator current exceeds Imax , ions are removed from the plasma faster than they are replaced, and the ion current is determined by



Ii = 2πrlZeni

vd +

dD dt



(4)

where D is the plasma gap spacing. In the enhanced erosion phase [see Fig. 10(b)], a magnetic field is generated by the initial load current, which significantly increases the plasma gap opening rate. The final magnetic insulation phase occurs when the plasma gap is larger than an electron gyroradius, as shown in Fig. 10(c). Typical dimensions of plasma opening switches are an inner electrode radius of 3 mm to 10 cm, an outer electrode radius of 2 cm to 20 cm, and a plasma channel length of 2 cm to 15 cm. Equation (2) indicates that the maximum conduction current increases with switch size. The POS permits switch powers of megawatts to tens of terawatts, with hold-off voltages ranging from kilovolts to a few megavolts, conduction currents of kiloamperes to megaamperes,

6

Plasma Switches

load current rises of 3 × 1013 A/s have been measured, indicating a factor-of-10 enhancement in current rise by the plasma flow switch. The PFS is a single-shot device due to the total destruction of the wire array and barrier foil. See Refs. 10, 14, and 15 for additional information. Application. The PFS has been used to study high-power radiation loads (Z pinch by metal foil implosion).

THYRATRONS Figure 9. Magnetic pressure opening mechanism of a plasma opening switch. (a) Plasma displacement by magnetic pressure, and (b) gap opening. As the gap opens, current flows through the load.

very low forward voltages, and opening times of 10 ns to 100 ns. The closing stage conduction time is limited to the range of 50 ns to 1 µs, depending on the applied current rise time and plasma characteristics, and the switch jitter relies on the reproducibility of the plasma parameters. Repetition rates are seldom published since they are often not limited by the POS, but rather by the recharging time of the generator or other factors. See Refs. 8 and 10–13 for additional information. Application. Plasma opening switches are used in inductive energy storage systems with low impedance loads, which produce high-current electron beams or high-power radiation (Z pinch). PLASMA FLOW SWITCHES The plasma flow switch (PFS) technique employs a vacuum discharge through a plasma armature that stores magnetic energy for several microseconds and rapidly transfers current and energy to a load when the armature exits the inductive store structure. A schematic of the PFS operation is presented in Fig. 11. Prior to switching, the PFS consists of a vacuum electrode gap, a wire array, barrier foil, and a load, as shown in Fig. 11(a). With the application of a megaampere current, a plasma armature is created by electrical explosion of the wire array and its impact on a plastic barrier foil. The total mass of the plasma ranges between 50 mg and 200 mg, and the plasma armature is accelerated by J × B forces to velocities of up to 70 km/s [see Fig. 11(b)]. Current flows through the load when the plasma armature reaches the plasma dump as shown in Fig. 11(c). The electrode gap is on the order of 3 cm, and plasma is accelerated over a length of 6 cm. Conduction times of 3 µs to 4 µs are typical for these plasma accelerating lengths. A cylindrical design of the PFS decreases the circuit inductance compared to a planar design, which makes short current rise times possible. The electrode gap is reduced by the plasma dump, which decreases the circuit resistance during load current conduction. The hold-off voltage is on the order of 100 kV,. The advantage of the PFS over the spark gap is the capability for, peak conduction currents of a few megaamperes, producing switching powers of up to 1 TW. Opening times of hundreds of nanoseconds and

The thyratron is a low-pressure plasma-closing switch that is generally filled with hydrogen. A schematic of a simple single-gap hydrogen thyratron is given in Fig. 12. In the opened stage of the thyratron, prior to triggering, the control grid usually has the same potential as the cathode. It shields the anode so that very little or no electric field from the anode penetrates to the cathode. A grid baffle prohibits a direct path from the cathode to the anode, which prevents spurious triggering. The auxiliary grid limits the cathode current and maintains a low-amplitude dc current discharge during the opening stage of the switch. The use of both the baffle and the auxiliary grid increases the holdoff voltage of the thyratron. To reduce the trigger voltage requirements, the distance between the control grid and the cathode times the pressure of the hydrogen gas is close to the Paschen minimum. The hydrogen gas pressure, controlled by a hydride reservoir, is on the order of 70 Pa. In hot-cathode thyratrons, the metal oxide or tungsten cathode is heated to 700◦ to 1200◦ C, depending on the cathode material, providing thermionic electron emission. Current densities of 30 A/cm2 for metal oxide and 100 A/cm2 to 150 A/cm2 for impregnated tungsten cathodes are typical for modern thyratron cathodes with microsecond pulses. The hold-off voltage of the switch (40 kV to 50 kV) is limited by electrical breakdown in vacuum between the control grid and the anode. Gas breakdown during the opening stage of the switch is unlikely since the control grid–anode distance (top gap) is less than one electron mean free path. To close the switch, the control grid is triggered with a voltage of about 1 kV above the cathode potential; i.e., the trigger voltage depends on the gas pressure and the control grid–cathode gap (bottom gap). A discharge is formed between the control grid and the cathode, hydrogen gas is partially ionized in the bottom gap, and the resulting plasma drifts into the top gap. High electric fields in the top gap cause a rapid ionization of the gas, which forms a highly conductive plasma. The potential of the control grid changes to that of the anode, and a secondary ionization of the hydrogen gas takes place in the bottom gap. The plasma shorts both the top and the bottom gaps, and the voltage drop during this closing stage is roughly 100 V. In this stage, the switch current density in the anode gap is larger than the emitted cathode current density since the high electric field pulls the emitted electrons through the auxiliary grid towards the anode. This grid area must be large enough to ensure that the current density does not exceed approximately 1.5 kA/cm2 ; otherwise, plasma instabilities may occur, which could significantly increase the closing resistance and thereby the voltage drop of the switch. Once the top gap is filled with plasma, the switch

Plasma Switches

7

Figure 10. Erosion opening mechanism of a plasma opening switch or plasma erosion opening switch. (a) Initial erosion phase, (b) enhanced erosion phase, and (c) magnetic insulation phase.

Figure 11. Schematic of the plasma flow switch opening process. (a) Initial stage, (b) plasma flow stage, and (c) opening stage. Plasma drifts down the vacuum gap, which allows current to flow through the load.

cannot be turned off until the switch current decreases to zero and sufficient time (up to a few tens of microseconds) is given for the recombination of the conducting plasma. The control grid potential strongly affects the recovery time. A negative grid bias of 50 V accelerates the plasma recombination process by a factor of 5 compared to the case with no grid bias. The thyratron conducts peak currents of tens of kiloamperes for a few tens of microseconds. The advantage of this switch is its high repetition rate. At high currents (kA), the thyratron is capable of operating at a few hundred Hz, while at lower currents (tens of amps) the repetition rate is as high as 100 kHz. High current rises of 1011 A/s to 1012 A/s have been achieved. The cathode predominantly limits the life of the switch, usually 10,000 h for metal oxide and 30,000 h for impregnated tungsten cathodes. The main disadvantage is the delay time, that is, time between the triggering of the control grid and the closing of the thyratron, due to the plasma that must form a conduction path between the cathode and the anode. Delay times of 20 ns to 30 ns are typical among modern, hot-cathode thyratrons with an associated jitter of 1 ns to 10 ns. The back-lighted thyratron (BLT) is a different class of thyratrons. The cathode of the BLT is radiated by intense UV light, which generates photoelectron emission. These electrons drift into the anode–cathode gap and initiate a glow discharge, from which current densities of up to 10 kA/cm2 over a 1 cm2 cathode area have been reported. This design does not require a cathode heater or grids, that is,

Figure 12. Simplified cross section of a modern, single-stage, high-power, hot-cathode, hydrogen thyratron. The small gap between the anode and the control grid makes ionization of the hydrogen gas unlikely. Plasma is formed always in the auxiliary grid/cathode gap.

the switch is triggered by laser light. Advantages of this switch are cold cathodes, higher peak switching currents, lower voltage drops, and a simpler and smaller switch design. Disadvantages are longer delay times (200 ns to 300 ns) and the need for a laser–UV trigger system. The design of BLTs is similar to that of pseudospark switches. More

8

Plasma Switches

facts on thyratrons can be found in Refs. 2 and 16–19. Application. Thyratrons are used in pulsed, high-power gas lasers and radar modulators. Low power thyratrons have been phased out and replaced with thyristors (a solidstate semiconductor device).

TACITRONS The tacitron is a hydrogen gas triode that can be used as an opening or closing switch. While the construction of his device is very similar to that of a hydrogen thyratron, the tacitron has smaller grid apertures, with a diameter of 0.5 mm and a lower gas pressure of 0.7 Pa to 4 Pa. The initial ionization in the tube is limited to the control-grid–anode region, so that complete grid control of the tube is possible. Tacitrons components include an anode, control grid, hot cathode, and heated reservoir (usually titanium or tantalum hydride) for gas-pressure control. Russian hydrogen tacitrons have peak ratings of a few megawatts, with holdoff voltages of up to 12 kV, conduction currents of a few hundred amperes, and peak pulse repetition rates up to 200 kHz. The grid-driven commutation causes the tacitron to absorb more energy than an externally commutated thyratron of equivalent power, since the tacitron must interrupt the full conduction (or discharge) current while the thyratron ceases to conduct near zero current. Thus, the design of the tacitron control grid is more massive and includes cooling. The typical flow rate for cooling water is 5 L/min for the control grid and 10 L/min for the anode. The recovery time is faster than an equivalent thyratron, no more than a few hundred nanoseconds, but the forward voltage drop during the closing stage is typically a few tens of volts higher than a thyratron. Since the 1970s, cesium (Cs) and cesium–barium (Ba) mixtures have been investigated for use in tacitrons instead of hydrogen. Cesium has the lowest first ionization potential of any element (only 3.89 eV versus 13.6 eV for hydrogen, 10.5 eV for mercury, and 12.1 eV for xenon), and multistep ionization takes place through a resonant state (metastable states) of the Cs atom. The use of cesium reduces the voltage drop by more than one order of magnitude compared to hydrogen tacitrons and allows the possibility of using this plasma switch as a low-loss dc inverter in high-temperature or high-radiation environments. The disadvantage of cesium is its mass; cesium ions have less mobility than hydrogen ions, causing Ca–Ba tacitrons to have larger turn-on and turn-off times. Russian Cs tacitron prototypes had low forward voltage drops of 1.75 V to 2.75 V, hold-off voltages of 50 V, and modulation frequencies of up to 1 kHz. The major limitation of the Cs vapor tacitron is its low emission current density of <1 A/cm2 . By adding small amounts of Ba vapor (<2%), the emission current of a Cs–Ba vapor tacitron is increased by more than one order of magnitude. At the present time, the Cs–Ba tacitron is still very much an experimental device. Figure 13 shows a simplified cross section of a Cs–Ba tacitron. It does not require any water cooling since peak power ratings are on the order of kilowatts. Cs and Ba vapor pressures are controlled by heating a pure source

Figure 13. Cross section of a planar, high-current, demountable Cs–Ba tacitron. The Cs plasma has a low ionization energy, which results in a low voltage drop during the closing stage.

rather than storing them in a chemically bound form as with the metal hydride reservoirs used in commercial hydrogen tubes. The cathode (emitter) is heated to temperatures of 1100◦ to 1300◦ C and the 1 mm thick molybdenum grid with a transparency of 35% is placed between the molybdenum cathode and molybdenum anode (collector). It has complete grid control of the tube conduction like the hydrogen tacitron, but a very low forward voltage drop of 1.3 V to 3 V. Reverse voltages or hold-off voltages of 100 V to 200 V have been demonstrated in laboratory devices that lack grid baffles. Peak conduction current densities range from 10 A/cm2 to 30 A/cm2 and current rise times are on the order of 10 µs. The modulation frequency depends on the conduction current and plasma density—higher plasma densities require higher current densities in order to deplete ions from the plasma during conduction. Ion depletion reduces the turn-off time and thereby increases the maximum attainable modulation frequency. Modulation frequencies of 10 kHz have been obtained with an average current density of 5 A/cm2 ; at lower frequencies the current density can be increased. The control grid is triggered with a voltage of 10 V to 50 V and a current density of 2 A/cm2 . The grid turn-on delay time is 10 µs, the turn-off delay varies from 0.1 µs to 3 µs, and current rises of 107 A/s have been measured. See Refs. 20–22 for additonal information. Application. Hydrogen tacitrons are used in cyclotron accelerators, copper vapor lasers, and microsecond pulse generators. CROSSATRONS Cross-field tubes work to the left of the Paschen minimum breakdown region at low hydrogen pressures (2 Pa to 7 Pa); consequently field emission or vacuum breakdown

Plasma Switches

9

Figure 14. Cross section of a simple, grid-free cross-field tube. Electrons drift in cycloidal orbits, which increases the ionization probability of the fill gas.

phenomena limit the hold-off voltage. A cross section of a simple, grid-free cross-field tube is shown in Fig. 14. The electrode spacing d is less than one electron mean free path, and thus a glow discharge cannot be sustained. With the application of a magnetic field normal to the electric field, electrons travel in cycloidal E × B orbits around the cylindrical anode. This increases the collisional probability for ionization of hydrogen due to the greater electron path length; plasma is formed in the electrode gap, which drifts toward the anode, and the switch closes. Removal of the magnetic field will result in recombination of the conducting plasma. Therefore, the cross-field tube is an opening switch as well as a closing switch. The crossatron is a cross-field tube with additional grids. Figure 15 shows a crossatron modular switch configuration. The cylindrical symmetric crossatron uses rare-earth (SmCo) permanent magnets attached outside of the switch to generate the E × B electron drift for efficient ionization of the helium or hydrogen background gas. The cathode material is typically molybdenum or chrome. The stainlesssteel source grid is continuously biased at 500 V to 700 V to sustain the plasma between the cathode and the source grid. This decreases the turn-on jitter of the switch and maintains a low-power discharge (about 4 W) to the cathode during the stand-by mode and between high-voltage pulses. The control grid (CG) is pulsed above the 200 V plasma potential, usually at 600 V to 5 kV. Plasma streams through the 50% to 80% transparent grid, with apertures of 0.5 mm to 2 mm towards the anode, and closes the switch. Lowering the control-grid potential to the cathode voltage turns off the switch; the plasma recombines in the CG–anode region, and the full anode voltage appears across the CG–anode vacuum gap. Crossatrons typically range from 6 cm to 15 cm in diameter and 7 cm to 30 cm in length. The cathode does not require any heating, and the gas pressure in the switch is controlled by an internal Zr–Al reservoir. Holdoff voltages up to 120 kV have been achieved, with peak conduction currents of 2.5 kA, time average currents of 5 A, current rises of 5 × 1010 A/s, and forward voltage drops of 200 V to 500 V. Other parameters are modulation frequencies of 1 kHz to 1 MHz, opening delay times of 100 ns to 300 ns, closing delay times of 50 ns to 150 ns, a recovery time of less than 1 µs, a switching jitter of 1 ns to 10 ns,

Figure 15. Crossatron modular switch configuration with source grid (SG) and control grid (CG). The source grid is used to sustain the plasma, whereas plasma streams towards the anode only if the control grid is triggered at a voltage above the plasma potential.

and a switch lifetime of greater than 2 × 1010 shots. See Refs. 8 and 23 for additional information. Applications. Crossatrons have been used in groundbased radar modulators and lasers, and they have been proposed as inverter switches in mobile power conditioning systems. PSEUDOSPARK SWITCHES The pseudospark switch (PSS) is a low-pressure discharge device that operates in a glow discharge mode. Figure 16 gives a schematic cross section of a pseudospark discharge closing switch. It includes a hollow cathode and anode, trigger, and preionization electrodes (shown for a glowdischarge trigger configuration), and a blocking grid (optional). Characteristic features of the PSS are two parallel plane electrodes with a central hole, separated by a dielectric insulating ring. Switch closure is initiated by triggering (1) a surface flashover in the cathode hole, in the hollow cathode, or behind the cathode cage, (2) a glow discharge in the back of the hollow cathode, or (3) UV photons on the back side of the cathode. Hydrogen or deuterium gas at pressures between 10 Pa and 100 Pa is ionized and plasma streams through the cathode hole and electrically shorts the electrode gap. In the glow-discharge trigger configuration, the preionization grid is dc biased at − 2 kV, with a current of 100 µA, whereas the trigger electrode receives a 3 kV to 5 kV positive voltage pulse. The surface flashover trigger configuration is similar to a surface discharge switch with a shorted load and without trigger electrode (compare to Fig. 5). Typical electrode gaps are 3 mm to 5 mm and

10

Plasma Switches

are possible, but electrode erosion increases significantly when conduction currents exceed 50 kA, resulting in lifetimes of 105 shots to 106 shots for gigawatt devices and 109 shots for hundreds of megawatt switches. More details on pseudospark switches are given in Refs. 2 and 24–27. Applications. Pseudospark switches are used in pulsed gas discharge lasers, X-ray microscope systems, and in pulsed power systems for CERN’s Large Hadron Collider (LHC) project.

IGNITRONS

Figure 16. Pseudospark switch configuration. Plasma is generated inside the hollow cathode, drifts into the cathode–anode gap, and closes the switch.

electrode holes are 3 mm in diameter. The PSS opens only after the conduction current amplitude decreases to zero and the plasma in the electrode gap recombines. Short recovery times are achieved by applying a 300 V pulse to the blocking electrode after cessation of conduction, which increases the ion mobility and thus decreases the recombination time of the plasma. Hold-off voltages are limited to 35 kV for a single electrode gap, with peak conduction current amplitudes ranging from hundreds of amperes to hundreds of kiloamperes. A multielectrode gap PSS has higher hold-off voltages, up to 70 kV for a three-electrode PSS. Trigger electrodes are often mounted behind the hollow cathode to protect them from erosion by the main discharge. Disadvantages of this setup are a switch jitter of tens of nanoseconds and closing delay times of 100 ns to 500 ns. Conduction pulse lengths are on the order of a few microseconds, with plasma densities of 5 × 1016 cm−3 to 1017 cm−3 generated in the conduction gap. Pulse length and conduction current increase with larger electrode apertures since more plasma drifts in the electrode gap region; on the other hand, the rate of current rise decreases. Repetition rates depend upon the conduction current amplitude and 1 Hz, 1 kHz, and 5 kHz have been achieved for 5 GW, 200 MW, and tens of megawatts switching powers, respectively. The switch lifetime is limited by electrode erosion around the cathode aperture. Maximum charge transfers of several 105 C over the lifetime of the switch

Although the ignitron can be categorized as a liquid-metal cathode vacuum switch, it is described briefly since the final conduction phase occurs through plasma. A schematic of an ignitron is given in Fig. 17. A simple ignitron consists of a liquid cathode, mercury (Hg), which, in principle, allows massive coulomb transfer without destructive effects. It is a low-pressure device, and vacuum surrounds the 10 cm to 20 cm wide electrode gap during the opening stage. Switch closure is achieved by applying a trigger pulse of 2 kV to 4 kV to the ignitor. An arc is formed between the ignitor and the cathode, some mercury is vaporized and plasma is emitted, pressure increases in the electrode gap, and a glow discharge initiates, which further develops into an arc between the anode and cathode. The switch opens at zero current, the plasma recombines, and mercury condenses at the walls of the ignitron and returns to the cathode pool. A control grid may be added [see Fig. 17(b)] to achieve precision in closure time. During opening, the grid is held at zero (or negative) potential relative to the cathode. Thereupon, the switch will close only after the ignitor has been triggered and a positive pulse has been applied to the control grid. A gradient grid is placed less than one mean free path from the control grid, thereby preventing breakdown prior to triggering the ignitor. The shield grid helps prevent spurious breakdowns and decreases the voltage requirements of the control grid. While the splash baffle blocks mercury from being sprayed directly onto the molybdenum anode and insulators, it may reduce the hold-off voltage of the switch due to field enhancement. Maximum hold-off voltages are on the order of 50 kV, in addition to conduction current amplitudes of 1 kA to 500 kA and time average currents of approximately 10 A. The ignitron has a low forward voltage drop of 10 V to 100 V, closing delay times

Figure 17. Basic ignitron configurations: (a) simple ignitron and (b) ignitron with (1) gradient grid, (2) control grid, (3) shield grid, and (4) splash baffle. These particular ignitrons are unidirectional switches.

Plasma Switches

between 50 ns to 300 ns, and jitters as low as 10 ns. Simple ignitrons are operated at repetition rates of less than 1 Hz, while gridded ignitrons operate up to a few 100 Hz. The ignitor usually limits the lifetime. Metallic sputtering may shorten the trigger gap, or erosion at the ignitor tip may increase the trigger gap. Lifetimes of 108 shots at 1 kA and 1000 shots at hundreds of kiloamperes have been obtained with microsecond pulses. References 2 and 28, 29 provide additional data on ignitrons. Application. The ignitron is used as a reliable, almost maintenance-free, closing switch in low-repetitionrate pulse power system with kiloampere load currents. We presented an overview of a variety of plasma switches that are used in pulsed power applications. Future trends in this area include more compact switches with longer lifetimes. Research is continuing in this field, and the latest advances are found in journals and conference proceedings that are provided in the reading list. The following encyclopedia articles provide additional background information: CATHODES, CONDUCTION AND BREAKDOWN IN GASES, FIELD EMISSION, FIELD IONIZATION, GASEOUS INSULATION, VACUUM INSULATION, AND VACUUM SWITCHES.

BIBLIOGRAPHY 1. R. E. Friedrich, (ed.) Application of Power Circuit Breakers, New York: IEEE, 1975. 2. G. Schaefer, A. Guenther, M. Kristiansen (eds.), Gas Discharge Closing Switches, New York: Plenum, 1990. 3. S. C. Brown, Basic Data of Plasma Physics, New York: Technology Press of MIT and Wiley, 1959. 4. M. A. Gundersen, Gas-phase pulsed power switches, IEEE Trans. Plasma Sci., 19: 1123–1131, 1991. 5. L. Liebing, Motion and Structure of a plasma produced in a rail spark gap, Phys. of Fluids, 6: 1035–1036, 1963. 6. R. S. Taylor and K. E. Leopold, UV radiation-triggered rail-gap switches, Review Sci. Instrum., 5: 52–63, 1984. 7. R. E. Reinovsky, Surface-discharge switches for highperformance closing applications, IEEE Trans. Plasma Sci., 32: 1765–1777, 2004. 8. A. Guenther, M. Kristiansen, T. Martin, (eds.) Opening Switches, New York: Plenum, 1987. 9. K. H. Schoenbach, G. Schaefer, M. Kristiansen, H. Krompholz, H. C. Harjes, and D. Skaggs, An electron-beam controlled diffuse discharge switch, J. Appl. Phys., 57: 1618–1622, 1985. 10. G. Cooperstein, P. F. Ottinger (eds.), Special issue on fast opening vacuum switches, IEEE Trans. Plasma Sci., PS–15: 629–780, 1987. 11. B. V. Weber, R. J. Commisso, P. J. Goodrich, J. M. Grossmann, D. D. Hinshelwood, J. C. Kellogg, and P. F. Ottinger, Investigation of plasma opening switch conduction and opening mechanisms, IEEE Trans. Plasma Sci., 19: 757–766, 1991. 12. M. E. Savage, D. B. Seidel, and C. W. Mendel Jr., Design of a command-triggered plasma opening switch for terawatt applications, IEEE Trans. Plasma Sci., 28: 1533–1539, 2000. 13. J. W. Schumer, S. B. Swanekamp, P. F. Ottinger, R. J. Commisso, B. V. Weber, D. N. Smithe, and L. D. Ludeking, MHDto-PIC transition for modeling of conduction and opening in a plasma opening switch, IEEE Trans. Plasma Sci., 29: 479–493, 2001.

11

14. R. L. Bowers, A. E. Greene, D. L. Peterson, N. F. Roderick, R. R. Bartsch, J. C. Cochrane, and H. Kruse, Initiation and assembly of the plasma in a plasma flow switch, IEEE Trans. Plasma Sci., 24: 510–522, 1996. 15. R. L. Bowers, J. H. Brownell, A. E. Greene, D. L. Peterson, N. F. Roderick, and P. J. Turchi, Effects of Plasma Surface Layers on the Efficiency of Plasma Flow Switching, IEEE Trans. Plasma Sci., 26: 1420–1436, 1998. 16. W. J. Sarjeant R. E. Dollinger, High-Power Electronics, Blue Ridge Summit: TAB, 1989. 17. B. M. Penetrante and E. E. Kunhardt, Kinetics of hydrogen thyratron plasmas during the conduction phase, J. Appl. Phys., 59: 3383–3396, 1986. 18. W. Hartmann, V. Dominic, G. F. Kirtman, and M. A. Gundersen, An analysis of the anomalous high-current cathode emission in pseudospark and back-of-the-cathode lighted thyratron switches, J. Appl. Phys., 65: 4388–4395, 1989. 19. G. Kirkman-Amemiya and M. A. Gundersen, High current back lighted thyratron switch, Appl. Phys. Lett., 60: 316–318, 1992. 20. E. O. Johnson, J. Olmstead, W. M. Webster, The tacitron, a low noise thyratron capable of current interruption by grid action, Proc. IRE, 42: 1350–1362, 1954. 21. C. Murray, B. Wernsman, M. S. El-Genk, and V. Kaibyshev, Ignition and extinguishing characteristics of Cs-Ba tacitron, J. Appl. Phys., 72: 4556–4565, 1992. 22. B. Wernsman and M. S. El-Genk, Experimental investigation of current modulation in a planar Cs-Ba tacitron, IEEE Trans. Plasma Sci., 23: 196–203, 1995. 23. D. M. Goebel, Cold-cathode, pulsed-power plasma discharge switch, Rev. Sci. Instrum., 67: 3136–3148, 1996. 24. K. Frank M. A. Gundersen (eds.), Special issue on physics and application of pseudospark discharges, IEEE Trans. Plasma Sci., 23: 209–392, 1995. 25. M. Gundersen and G. Schaefer, Physics and Applications of Pseudosparks, NATO ASI Series, Plenum Press, New York, 1990. 26. K. Frank, O. Almen, P. Bickel, J. Christiansen, A. Gortler, W. Hartmann, C. Kozlik, A. Linsenmeyer, H. Loscher, F. Peter, A. Schwandner, and R. Stark, Pseudospark switches for high repetition rates and high current applications, Proc. IEEE, 80: 958–970, 1992. 27. Y. D. Korolev and K. Frank, Discharge formation processes and glow-to-arc transition in pseudospark switch, IEEE Trans. Plasma Sci., 27: 1525–1537, 1999. 28. J. Slepian and A. H. Toepfer, Cathode spot fixation and mercury pool temperatures in an ignitron, J. Appl. Phys., 9: 483–484, 1938. 29. D. L. Loree, M. G. Giesselmann, M. Kristiansen, A. P. Shulski, and R. Kihara, Recent advances in high-power ignitron development, IEEE Trans. Electron Devices, 38: 720–725, 1991.

12

Plasma Switches

Reading List J. D. Cobine, Gaseous Conductors: Theory and Engineering Applications, New York: Dover, 1958. Yu. D. Korolev G. A. Mesyats, Physics of Pulsed Breakdown in Gases, Yekaterinburg: URO-Press, 1998. L. Latham, (ed.) High Voltage Vacuum Insulation, New York: Academic, 1995. T. H. Martin A. H. Guenther (eds.), J. C. Martin on Pulsed Power, New York: Plenum, 1996. IEEE Conference Record of the Power Modulator Symposium (biannual, even years). Proceedings of the IEEE International Pulsed Power Conference (biannual, odd years). Proceedings of the International Conference on High Power Particle Beams (biannual, even years). Special Issue on Pulsed Power Science and Technology, IEEE Trans. Plasma Sci. (biannual, even years). Special Issue on Vacuum Discharge Plasmas, IEEE Trans. Plasma Sci. (biannual, odd years).

FRANK HEGELER Commonwealth Technologies, Inc./Naval Research Laboratory, Washington, DC

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Wiley Encyclopedia of Electrical and Electronics Engineering Plasma Turbulence Standard Article W. Horton1 and G. Hu2 1The University of Texas, Austin, TX 2Globalstar LP, San Jose, CA Copyright © 1999 by John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5921 Article Online Posting Date: December 27, 1999 Abstract | Full Text: HTML PDF (278K)

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Abstract The sections in this article are Current-Driven Turbulence Spatial-Gradient-Driven Turbulence in Magnetized Plasma Acknowledgments About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

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PLASMA TURBULENCE

527

PLASMA TURBULENCE Plasma occurs in states of turbulence under a wide range of conditions including space and astrophysical plasmas as well as those produced in laboratory confinement devices. The strength of the turbulence increases as the plasma is driven J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

528

PLASMA TURBULENCE

further away from thermodynamic equilibrium. While there are many ways to drive the plasma away from equilibrium with particle beams, laser beams, and radio-frequency waves, the universally occurring departures from equilibrium considered here are (1) the presence of plasma currents and (2) the existence of spatial gradients. While both driving forces may exist simultaneously, it is sufficient to consider the effects independently here. In the section entitled ‘‘Current-Driven Turbulence’’ we describe the types of plasma turbulence that occur in the uniform plasma in which the role of magnetic fields is negligible on the fast ion acoustic wave turbulence and the Buneman instability driven by a current density j ⫽ ⫺enu (A/m2) in a plasma. Here n is the plasma density and u the relative drift velocity of the electrons with respect to the ions. The electron charge is ⫺e and mass is me. The plasma turbulence provides the mechanism for the exchange of energy and momentum between the electrons and ions. The turbulence determines the current–voltage relationship through the anomalous resistivity ␩ ⫽ me␯eff /ne2 where ␯eff is the rate of change of the electron momentum meu, measured relative to the ion rest frame, due to the turbulent electric fields. The energy density in the turbulence is W (J/m3). The dependence of ␯eff on u/ve for both regimes is shown in Fig. 1. Experiments have confirmed the general features of the small fraction turbulence level W/neTe producing a substantial anomalous resistivity ␯eff /웆pe ⱗ 0.2(me /mi)1/3 and a fast turbulent heating of the electrons. Once the electron temperature Te increases to the value such that the associated thermal velocity ve ⫽ (kBTe /me)1/2 ⬎

0.020

Te Ti

= 50

0.015

νeff ω pe

Te Ti

= 30

0.010 u νeff –3 ω pe ≈ 1.2 × 10 ve

Te Ti

0.005

0

0

0.5

1.0

1.5

u ve

Figure 1. The dependence of the effective electron collision frequency ␯eff on the drift velocity of the electrons with respect to the ions. The regime of u/ve ⱗ 0.5 is dominated by ion acoustic turbulence while the regime of u/ve ⲏ 1 is that of the two-stream instability.

2u, the turbulence enters a weaker regime where ␯eff 앓 웆pe(u/ve)(Te /Ti) ⫻ 10⫺3. The dependence of ␯eff on the drift velocity u makes Ohm’s law, E ⫽ ␩j ⫽ (me␯eff /ne2)j, nonlinear. There is substantial plasma heating and the production of a fast ion tail on the ion distribution in the ion acoustic turbulence phase of the experiments. In the section entitled ‘‘Spatial Gradient-Driven Turbulence in Magnetized Plasma’’ the case of plasma turbulence produced by the spatial gradients from its confinement. The confinement or trapping of a plasma is produced both in the laboratory and in space/astrophysics by magnetic fields that cause the charged particles to gyrate with radius ␳a ⫽ mava /qaB around the local magnetic field B(x). The confinement along B (parallel to B) occurs either due to the large increase of 兩B兩 giving rise to the mirror effect as in Earth’s magnetosphere or due to the field lines forming closed, nested toroidal surfaces as in solar current loops and the laboratory tokamak device. From over 30 years of laboratory research in the tokamak confinement studies of plasma, there is a detailed understanding of the intrinsic, irreducible plasma turbulence that develops from spatial gradients. This turbulence is generically called drift wave turbulence and is driven by the cross-field gradients of the plasma density ⵜn ⫽ ⫺(n/Ln)eˆx and temperature ⵜT ⫽ ⫺(T/LT)eˆx. While there are many detailed forms known for the turbulence depending on the plasma parameters, there are essentially three generic results for the plasma cross-field diffusivities D (m2 /s) of particles and ␹ (m2 /s) for thermal diffusivity. The three functionally distinct forms are: (1) the Bohm diffusivity DB ⫽ 움B(Te /eB), (2) the gyro-Bohm diffusivity from drift waves Ddw ⫽ 움dw(␳i /LT)(Te /eB), where ␳i ⫽ (miTi)1/2 /eB is the thermal ion gyroradius, and the (3) collisional turbulence diffusivities Dr ⫽ 움rg␯e␳e2(Ls2 /LTRc) where ␯e is electron–ion collision frequency associated with resistivity E储 ⫽ ␩j储 along the magnetic field. The Bohm diffusivity varies as T/B and is documented in Taroni et al. (1) and Erba et al. (2), for the tokamak with a low coefficient 움B 앑
. The drift wave transport varies as T3/2 /B2L and is documented in Horton (3) with a coefficient 움dw 앑 0.3 (4). The resistive interchange mode diffusivity var2 ies as ␹rg 앑 n/T1/2 e B and has a coefficient of order unity (5). The coefficients 움B, 움dw, and 움rg are weak functions of many detailed plasma parameters such Te /Ti, LT /R, 웁 ⫽ 2애0p/B2, and more. Just as in neutral fluid turbulence, there are many degrees of freedom excited in plasma turbulence and there is great difficulty in determining the details of these formulas either through theory or numerical simulations. Nonetheless, the years of experience with the tokamak program have led to rather firm general conclusions about the turbulent diffusivities. The study of the boundary layers between different types of plasmas in the magnetosphere have also shed light on the limits of the collisionless plasma transport rates. Now we present some details of these two fundamental forms of plasma turbulence.

CURRENT-DRIVEN TURBULENCE Large currents naturally occur in a plasma due to its low resistivity. Since the typical current system in a plasma has a long L/R decay time, the plasma electric field from E(t) ⫽ ␩(t)j varies rapidly with the level of the plasma turbulence. One of the well-documented settings of current-driven turbu-

PLASMA TURBULENCE

lence is the ionosphere (Ref. 6, pp. 167–182 and 397–419). In this weakly ionized plasma, large currents are driven by the magnetospheric coupling to the solar wind dynamo. Both the Buneman (7) two-stream instability at high drift velocities and the E ⫻ B gradient instability at low drift velocities (8,9) are well-documented sources of plasma turbulence. Here we summarize the first considerations for determining the plasma turbulence and resistivity.

1.0 S(k) S0 0.5

Two-Stream Instability In the initial phase of a plasma carrying a high current, the electron drift velocity relative to the ions u ⬅ ⫺j/ene exceeds the electron thermal velocity ve ⫽ (kBTe /me)1/2. These conditions produce a strong, unstable electrostatic wave with an intermediate phase velocity that grows until it traps most of the electrons. The trapping mechanism thermalizes the electron distribution to the new, high temperature Te 앓 4(mi /me)1/3meu2 which may be described as a turbulent heating with the effective collision frequency ␯eff 앓 0.2웆pe(me /mi)1/3 as shown in the laboratory by Hamberger and Jancarik (10). At the end of the turbulent heating of the electrons, the plasma current is still present, but now the plasma instabilities are kinetic with the ion acoustic wave turbulence driven by the positive slope on the drifting electron velocity distribution function.

529

u 0.0 kλ D 0.5 1.0 1.5 kλ D

Figure 2. Spectral distribution of wave vectors for current driven turbulence in unmagnetized plasma. The k vectors form a cone in the direction of the electron drift velocity u. The shape of the cone is determined by the balance of the linear growth rate and the nonlinear scattering of the waves (12).

Ion Acoustic Turbulence Driven by the Plasma Current After the electron temperature rises to the level where the drift parameter u/ve ⬍ 1 the uniform plasma has unstable ion acoustic waves that provide an anomalous resistivity ␩ and thermal diffusivity ␹. There is also a condition that Te /Ti Ⰷ 1 for the ion acoustic turbulence to be strong. When the large Te /Ti condition is not satisfied, the nonuniformity or gradient drift velocity must be present for the system to be unstable. This is the usual regime of laboratory plasma confinement experiments and is the topic of the section entitled ‘‘SpatialGradient-Driven Turbulence in Magnetized Plasma.’’ When the ion acoustic turbulence is unstable, the system exhibits the effective collision frequency that rises linearly with u/ve as shown in Fig. 1 for u/ve ⱗ
. The higher u/ve region is the transition to the Buneman two-stream turbulence. The turbulence exists as a cone-shaped spectrum of wave vectors pointing in the direction of the electron drift velocity u measured with respect to the ion rest frame. An ion acoustic spectrum is shown in Fig. 2. The waves receive energy and momentum from the electrons and deposit energy and momentum in the ions. The response of the ions depends on their mass and collision frequency with the neutrals. For collisionless, light (hydrogenic) ions, the turbulence produces a fast ion tail on the ion velocity distribution. Ion energies beyond 5Te are produced and the density nt of the ion tail varies roughly as nt /ne 앓 (me /mi)1/4. In plasmas with a mixture of ion species, the hydrogenic component is easily accelerated to form the energetic ion tail and thus shuts off the linear growth mechanism (11). Thus, the turbulence tends to appear in bursts. The transport coefficients are given in Horton et al. (12). For complex structured signals ␾(x, t), typical of what is meant by the term turbulence, the standard measure of the coherence of the signals is the two-point, two-time correlation

function C12 (x, t1 , x2 , t2 ) =
φ(x1 , t1 )φ(x2 , t2 )

(1)

where the average 具 典 is taken over a time period that contains many oscillations of the associated field. The correlation functions for the plasma electric potential ␾ and the fluctuations of the electron density 웃ne ⫽ ne(x, t) ⫺ 具ne典 are the principal structure functions of use in plasma turbulence research. The dependence of C12 on r ⫽ x2 ⫺ x1 and ␶ ⫽ t2 ⫺ t1 is strong, decaying to small values for large r, ␶ ; and the dependence on R ⫽ (r1 ⫹ r2)/2 and t ⫽ (t1 ⫹ t2)/2 is weak for turbulent systems. In the physics idealization of homogeneous, stationary turbulence C12 씮 C12(r, ␶) independent of R and t. It is evident that C12(r, ␶) is maximum at r ⫽ 0, ␶ ⫽ 0 where C12 씮 具␾2典 and falls to low values for large 兩r兩 or 兩␶兩. These properties provide the definition of the correlation distances ⌬xc, ⌬yc, ⌬zc and correlation time ␶c by choosing a critical point for the fall-off of C12 /C12(0) usually taken as the 1/e-point of the envelope. For the ion acoustic turbulence the correlation function of the electrostatic potential turbulence is shown in Fig. 3 from the experiment of Stenzel (12a). In Fig. 3(a) the decay of the two-point correlation function of separation ⌬z of the twospace points along the direction of the current j ⫽ ⫺enu driving the turbulence is shown for all other separations (⌬x ⫽ ⌬y ⫽ ␶ ⫽ 0) equal to zero. In Fig. 3(b) the constant level contours of C12 of the correlation function for separations in ⌬z along the current and in an orthogonal direction ⌬y are shown. Note that the central maximum (approaching 40 units) occurs at the ⌬y ⫽ ⌬z ⫽ 0 point and the correlation length (⌬yc ⬵ 6 mm) perpendicular to the current is longer than the correlation parallel to the current ⌬z ⫽ 4 mm. This anisotropy of the turbulence is a characteristic of plasmas due

530

PLASMA TURBULENCE

1.0 0.8 I (f )

Correlation function

C12 (∆y, ∆z, 0)

0.6 0.4 0.2

Distance along j0 ∆z

0

(a) 4 0

2

0

0

0 10

+

30 –

5

0

–

20

400 600 800 Frequency f (kHz)

1000

–10

+

–15 + –10

–2

5

2 0

–3 –4

200

Figure 4. Frequency spectrum of ion acoustic turbulence showing the broadband potential fluctuation spectrum with short correlation time ␶c ⫽ 1/⌬웆 ⬵ 0.5 애s. The spectral distribution decreases slightly faster than 1/웆 between the low-frequency cut-off due to nonlinear effects and a high-frequency limit due to landau damping on the thermal ions (courtesy of Stenzel).

C12 = const.

0

0

0

–5

0

0 –5 0

2

0

–2 0 2 Distance along j0 ∆z (mm)

4

(b) Figure 3. Characterization of the turbulent electrostatic potential by the two-point correlation function C12 in Eq. (1): (a) Decay of the correlation function with separation of the two points along an axial line parallel to the current; (b) Oscillatory decay of the correlations for two points in the y–z plane where the yˆ direction is perpendicular to the current and zˆ is the parallel direction (courtesy of Stenzel).

to the preferred directions of unstable wave propagation associated the geometry and the magnetic fields in the plasma systems. For broadband turbulence there is a simple rule for estimating the correlation time ␶c and correlation lengths ⌬rc from the spectral width of the power spectrum I(k웆) defined by the Fourier transform of the correlation function C12(r, ␶). The correlation time is given by ␶c ⫽ 1/⌬웆 and the correlation length ⌬z ⫽ 1/⌬kz where ⌬웆 ⫽ 웆max ⫺ 웆min, ⌬kz ⫽ kmax ⫺ kmin are the widths of the highest levels of the fluctuation power spectrum I(k, 웆). As an example, the ion acoustic turbulence in the Stenzel experiment is shown in Fig. 4 where a spectrum consistent with turbulence theory has been reported. The spectral width from the 1/e point is approximately 400 kHz, consistent with the directly measured ␶c ⫽ 3 애s correlation time. Physically, the correlation time ␶c is the maximum time interval over which the field, in this case the electrostatic potential, maintains a given structure. In the next correlation time the complexion, or structure, of the field is qualitatively different. SPATIAL-GRADIENT-DRIVEN TURBULENCE IN MAGNETIZED PLASMA In nonuniform, magnetized plasmas the ion acoustic waves are modified into two branches with different parallel phases

velocities due to the presence of the diamagnetic currents ja ⫽ eanavda required from the ja ⫻ B ⫽ ⵱pa force balance. Here, each charged particle species is designated by the subscript a. In a sense, the relevant drift velocities for driving the plasma turbulence change to the small diamagnetic drift velocities vda ⫽ Ta /eaBLpa ⫽ (␳a /Lpa)vTa, where L⫺1 pa ⫽ ⫺⭸x ln pa(x). Even for small values of ␳a /Lpa, these diamagnetic currents drive low-frequency (웆 Ⰶ eaB/ma) waves with k almost parallel to B ⫻ ⵱pa unstable. These waves are called drift waves, and their effect is to produce a cross-field transport of particle energy and momentum. A direct spectroscopic measurement of the turbulent transport of injected impurity ions is given in Ref. 13. Drift Waves in the Laboratory The collisional drift waves with growth rates determined by resistivity ␩ and thermal diffusivity ␹e were the first drift waves to be discovered and thoroughly investigated (14). The identification was made in low-temperature steady-state plasmas produced by thermal (contact) ionization of Alkali elements (principally cesium and potassium) in long cylindrical devices with closely spaced Helmholtz coils. Correlations between the observed potential-density waves with the properties predicted by the linear dispersion relation and the singlewave finite amplitude formulas (15) were used to establish that the radially localized, 10 kHz rotating wave structures were the drift waves. The dimensionless density n˜ /n and potential e␾˜ /Te waves are approximately equal amplitude sinusoidal oscillation with n˜ leading ␾˜ by 30⬚ to 45⬚ in phase. Figure 5 shows the drift wave potential and density isolines. Vortex dynamics has also been observed in the plasmas produced in these devices called Q-machines. Here Q stands for quiet. In the experiments of Pe´cseli et al. (16,17), externally excited vortices of like signs were shown to coalesce into one vortex. Vortices of opposite signs were reported to interact with each other forming a dipole vortex pair. A variety of drift-type instabilities relevant to toroidal magnetic fusion devices, including the trapped electron modes by Prager et al. (18), the trapped ion instability by Slough et al. (19), and the collisionless curvature-driven trapped parti-

PLASMA TURBULENCE

N(x) N>(x) VE

δ n2

δ n1

+ Γ–

B

φ1

z

y

φ2

Γ+

ψδ n,φ

x Figure 5. A segment of a drift wave fluctuation showing the variation of the electrostatic potential and the density perpendicular to the magnetic field at a given instant of time. The contours of ␾ in the plane perpendicular to B ⫽ Bz are the stream lines of the E ⫻ B particle motion. The potential and the density variation are out of phase by ␺웃n,␾ , producing the net downward flux.

cle mode by Scarmozzino et al. (20) have been produced and identified in the Columbia linear machine (CLM). The drift wave driven by the radial ion temperature gradient in a collisionless cylindrical plasma was demonstrated in the modified Columbia linear machine by Sen, Chen, and Mauel (21) by using biased wire screens to create a Ti储(r) gradient sufficient to excite an m ⫽ 2, 10 kHz (in the plasma frame) drift wave oscillation. The toroidal ITG mode driven by the magnetic curvature was also produced and identified by Chen and Sen (22) in the same machine. In the CLM experiments the plasma is in a steady state like the Q-machine experiments except that the plasma temperatures are an order of magnitude higher (Ti ⲏ Te 앑 6 eV) and the density is lower (N 앑 109 cm⫺3), giving the collisionless condition (␯ ⬍ 웆k) for waves with angular frequency 웆k in the working gas of hydrogen. There are approximately 15 ion gyroradii in the plasma radius. Drift waves were found in the transient plasmas produced in the multipole confinement devices that were both linear and toroidal devices with strongly varying B fields from parallel conductors carrying large currents from external power supplies. The multipole plasmas of hydrogen, helium, and argon were produced by microwave frequency heating. The theory for the drift waves in the multipole takes into account the localization of the unstable oscillations to regions of unfavorable gradient B and curvature particle drifts and the shear in the helical B(x) field (22a). These experiments provided further evidence for the universal appearance of drift waves in confinement geometries. The correlation of drift wave theory with the multipole and spherator experiments are described in Section 3.3 of the Horton (3) review article. The main result to be noted here is that the experiments show that increasing the magnetic shear reduces the fluctuation amplitudes (23). The multipole devices are unique in being able to continuously vary the magnetic shear parameter strength from zero to of order unity. Even with the strongest magnetic shear, however, the fluctuations were not eliminated. The magnetic shear plays a central role in the linear and nonlinear theory of the cross-field transport consistent with the role of shear on the fluctuations measured in these experiments. In recent theory and experiments for tokamak con-

531

finement devices the combined roles of Er ⫻ B sheared flows and magnetic shear are known to produce enhanced confinement regimes (24,25). The improved confinement occurs over narrow radial regions giving rise to new confinement regimes with internal transport barriers (26–28). Table 1 gives the high-performance fusion plasma parameters achieved with internal transport barrier in the JT60U machine (26). The principal tools available for understanding these changes in transport are the dependence of drift wave turbulence on the system parameters, especially the magnetic shear in B(x) and mass flow shear in the hydrodynamic flow velocity u(x). In tokamaks the identification of drift waves in the core plasma came from the microwave scattering experiments (29) and infrared CO2 laser scattering experiments (30,31). These measured fluctuations were explained in the context of drift waves existing at the mixing-length level of saturation, taking into account the response of the trapped electrons in the drift-wave dissipation. Subsequently, many experiments around the world have observed the universal appearance of a broad band of drift wave fluctuations with 웆/2앟 앓 50 kHz to 500 kHz at k⬜ ⫽ 1 cm⫺1 to 15 cm⫺1 in toroidal confinement devices for both the tokamak and helical-stellarator systems. Many fluctuation and transport studies in toroidal confinement facilities around the world, including TFTR, Alcator, TEXT, TORE SUPRA ATF, Heliotron, JFT2M, and ASDEX, were undertaken in the 1980s and 1990s that have confirmed these initial findings of drift wave turbulence and the associated radial turbulent transport. A basic physics research program on plasma fluctuations and anomalous transport was carried out from 1982 to 1994 in the TEXT tokamak at The University of Texas at Austin. This experiment provided the most complete correlated data sets of core fluctuations from five k⬜ values (2, 4.5, 7, 9, 12 cm⫺1) from far-infrared (FIR) laser scattering, complex probe arrays for edge turbulence, the heavy ion beam probe (HIBP) for measurements of the radial electric field Er, and the space–time localized fluctuating potential ␾˜ in addition to the usual complement of spectrometers, interferometers, bolometers, and magnetic coils for determining the state of the plasma. A review of the FIR and HIBP data, as well as other diagnostics, leading to the conclusion that the drift waves are present and responsible for the transport, is given in Bravenec et al. (32). A review of the diagnostics on TEXT and other tokamaks is given by Gentle (33). Table 2 gives the plasma parameters and the drift wave turbulence characteristics for TEXT and the approximately twice larger TFTR tokamaks. There is substantial supporting evidence in the high-level edge fluctuations data for resistive-interchange turbulence giving ␹rg. Table 1. JT-60U High-␤p Experiment 17110 with Internal Transport Barrier: Phase I R/a B␾ Ip PNBI nD(0) Ti(0)/T e(0) nD␶ETi v ␾ (0)

3.1 m/0.7 m 4.4 T 2 MA 27 MW 4.1 ⫻ 1019 m⫺3 38 keV/12 keV 1.1 ⫻ 1021 m⫺3 ⭈ s ⭈ keV ⫺100 km/s

532

PLASMA TURBULENCE

Table 2. Plasma Drift Wave Parameters Parameter Magnetic field Major/minor radii Electron temperature Density n e and Gradient length L n Drift velocity v d k scattering experiment 웆 scattering experiment n˜e /ne

Physics of Plasma Transport and the Propagation of Disturbances

TFTR

TEXT

4.8 T 2.45 m/0.8 m 6 keV 4 ⫻ 1013 cm⫺3 20 cm 3 ⫻ 105 cm/s 1–20 cm⫺1 10–500 kHz 5 ⫻ 10⫺3 to 0.02

2T 1.0 m/0.27 m 500 eV 3 ⫻ 1013 cm⫺3 10 cm 1 ⫻ 105 cm/s 1.5–15 cm⫺1 10–1000 kHz 0.01–0.1

Other drift turbulence is observed due to parallel shear flow, impurity drift modes, and recombination ionization. In the core plasma the dissipative trapped electron mode dominates in TEXT (32). In contrast to the ohmic heated TEXT experiments, for auxiliary heated plasmas where the Ohmic heating is a fraction of the total input power, the ion-temperaturegradient-driven drift wave is the dominant driving mechanism for the drift wave turbulence. The TEXT experiments were ohmic discharges with toroidal magnetic fields B ⫽ 2 T, plasma current I ⫽ 200 kA to 400 kA, and inductive loop voltage E l ⫽ 2V from the iron core transformer. The discharges produced hydrogen plasmas with Te ⱗ 2 keV and Ti ⱗ 1 keV from the collisional resistivity. In contrast, the major fusion confinement experiments have high-power neutral beam injection (NBI) systems providing injected powers Pb ⫽ 5 MW to 30 MW producing Te ⱕ 8 keV and Ti ⱕ 40 keV. Thus, the nature of the instability driving the plasma turbulence switches from the electron temperature gradient T⬘e (r) parameterized by ηe =

∂r ln Te ∂r ln ne

(2)

to the ion temperature gradient T⬘i (r) parameterized by ηi =

∂r ln Ti ∂r ln ni

(3)

in going from ohmic discharges in TEXT and ALCATOR to the high-power NBI heated discharges in TFTR, JET, DIIID, and JT60U. The energy-momentum deposition profiles and magnitudes from the auxiliary heating become key control parameters in the later experiments. The latest results for the two major fusion laboratories with billion-dollar tokamaks are found at www.jet.uk and www.jt60.naka.jaeri.go.ip. Currently, the JET machine has the record of 13 MW of peak fusion power with a ratio of fusion power to net input power Q of 60%. The conclusion of the TEXT experiment is that the drift wave fluctuations account for both the particle and thermal energy transport to the edge of the large Ohmic heated tokamak. At the extreme edge the plasma is in contact with a metallic diaphram where collisional and radiative losses dominate the transport and cooling of the plasma. The same conclusion that the small-scale drift wave turbulence determines the radial transport properties of the plasmas appears to apply to the largest (I ⲏ 2 MA) tokamaks. There is some indication that the very core of the large machines may enter a quieter regime with lower transport levels (33a).

Now we analyze the ion motion in the E ⫻ B convection. For the small, localized excess of ion charge shown in Fig. 5 the vE =

E ×B cE B2

(4)

convection rotates plasma clockwise around the potential maximum ␾ ⬎ 0, which is also the density and electron pressure maximum in the adiabatic response. Now, if the ambient plasma is uniform (⭸xna ⫽ ⭸xTi ⫽ 0) across the convection zone, then the cell rotates without plasma transport. When the plasma has an x-gradient of density (pressure), however, there is a rapid transport of the structure along the symmetry direction yˆ with a small diffusive transport across an x ⫽ const surface. The speed of the localized structure in Fig. 5 along the symmetry direction is approximately the electron diamagnetic drift speed vde ⬅ cTe /eBLn, where L⫺1 n ⫽ ⫺⭸r ln N. The analytical description of the net convective flux particle and thermal fluxes across a given surface S is given by a = qa =

1 S




3 2S

S

a = −D11 nav E · da


S

dne dTe − D12 dx dx

a = −D21 na Tav E · da

dne dTe − ne D22 dx dx

(5) (6)

In the absence of the phase shift 웃n ⫺ ␾ in Fig. 5, the transport vanishes. In the proper set of flux-driving gradient variables the transport matrix has Onsager symmetry (34). The Onsager symmetries guarantee the correct thermodynamic properties for the system. For the positive potential structure in Fig. 5, the clockwise E ⫻ B rotation brings higher density N⬎ (and higher pressure N⬎Te) plasma to the right and lower density N⬍ (pressure) to the left, resulting in a shift of the maximum density and potential, linked through the electron response by 웃ne ⬵ ne(e␾ /Te), to the right. The speed of the translation is proportional to the gradient of the density L⫺1 n ⫽ ⫺⭸r ln N and inversely proportional to the strength of the magnetic field B. The speed also increases with electron temperature Te since the potential fluctuation e␾ scales up with Te. For a negative potential structure the E ⫻ B rotation is counterclockwise, but the structure moves to the right with same speed (in the limit of small e␾ /Te) since now lower-density plasma is brought to the right, shifting the minimum in that direction. Now, the ion density at this location builds up in the time 웃t equal to that of the original electron maximum 웃ne ⫽ N(e␾ /Te) when the condition δni = −

eφ δtcφ ∂N =N Bδy ∂x Te

(7)

is satisfied. In the last step we use quasineutrality taking 웃ni ⫽ 웃ne ⫽ N(e␾ /Te), which is valid for fluctuations that are large compared to the Debye length. During the time 웃t the convection moves the maximum of the structure to the right by 웃y ⫽ vde웃t, where vde =

cTe ∂N δy =− δt eBN ∂x

(8)

PLASMA TURBULENCE

Drift Wave Diffusivities and the Ion Inertial Scale Length It is conventional in the study of drift waves and transport to introduce gradient scale lengths and reference diffusivities. Thus, the length Ln is defined as the density gradient scale length through the relation 1/Ln ⫽ ⫺⭸x ln N. The temperature gradient scale length LT is defined similarly. The space–time scales of the waves lead to two different dimensional scalings for the plasma diffusivities. The reference diffusivities are the Bohm diffusivity DB =

Te eB

(9)

and the drift wave diffusivity Ddw =


ρ 
T  s

e

Ln

eB

(10)

also commonly called the gyro–Bohm diffusivity in reference to the factor ␳s /Ln Ⰶ 1. Here ␳s ⫽ (miTe)1/2 /eB is effective gyroradius parameters for hot electrons Te ⲏ Te. Clearly, the scaling of the Bohm and gyro–Bohm diffusivities are markedly different with DB 앜 Te /B independent of the system size 2 while Ddw ⫽ T3/2 e /B L, decreasing with the system size. There is a long history of confinement scaling studies that have correlated the thermal and/or particle confinement with either the Bohm or the drift wave scaling laws. The issue is still actively debated as to which transport scaling is to occur under given confinement conditions (36). In short, the Bohm [Eq. (9)] scaling arises from mesoscale drift wave structures ⌬x ⫽ (␳s LT)1/2 and thus is expected near marginal stability (37–40). When the convective cells size reduces to ⌬x ⫽ ␳s the drift wave diffusivity [Eq. (10)], more commonly called gyroBohm, applies. A compendium of thermal diffusivity formulas collected from the literature on drift wave turbulent transport is given by Connor and Wilson (41). Equations (9) and (10) must be multiplied by dimensionless functions of the system parameters to explain transport in a particular device. Currently, large-scale particle simulations are used to address this issue from first principle calculations (42,43). In defining the dimensionless gyroradius parameter ␳ , it * is usual to replace the space–time varying length Ln with the relatively constant value a of the plasma minor radius. Thus, a key issue is the scaling of plasma confinement systems with ρs ρ∗ ≡ a

port. Power balance in the JET discharge up to 7 MA of plasma current is obtained with ␹e ⫽ 움eq2(a/Lp)DB and ␹i ⫽ 움i␹e ⫹ ␹ineo with 움e ⫽ 2.1 ⫻ 10⫺4 and 움i ⫽ 3.0 (1,2). Here 1/q(r) ⫽ RB␪ /rBT gives the local pitch of the helical magnetic side line. The fluctuation measurement at wavenumbers k⬜ ⱗ 1 cm⫺1 require the techniques of reflectometry (46,47) and the indirect method of beam emission spectroscopy as in the Durst et al. (48) experiment. Let us close by showing the evidence from the FIR scattering experiment in the core of the TEXT experiment (49,50). Figure 6 shows the peak of the frequency of the electron density fluctuations inferred from the dynamical scattering factor S(k⬜, 웆) versus the wavenumber k⬜ from the scattering geometry. The frequency of the spectrum in the lab frame follows 웆 ⫽ 웆k(␩i) ⫹ k␪vE to a good approximation where vE ⫽ ⫺cEr /B is the Lagrangian velocity of the plasma (ions and electrons) relative to the laboratory frame of reference and 웆k(␩i) is the drift wave frequency in the plasma frame. The plasma rest frame frequency 웆k is shown by dotted lines for the range of ␩i ⫽ ⭸ ln Ti /⭸ ln ni values in this experiment. The corresponding laboratory frame frequencies are given by the dashed lines. To find the transformation velocity vE, and thus the Doppler shift, the HIBP diagnostic is used to determine the radial electric field Er(r). Quasilinear theory (QT) predicts the spectrum of density fluctuations  2   δne (k k, ω)  S k )δ(ω − k · u − ωk ) k, ω) = 0  S(k   I(k VT ne where V and T are the sample volume and time interval. Renormalized turbulence theory (RNT) gives a Lorentzian spec500

400

ω k+ k θVE 300

ω /2 π (kHz)

The x-displacement of the plasma during this motion is ␰x ⫽ vx웃t ⫽ ⫺웃t웃␾ /B웃y. When this displacement becomes comparable to 웃x the motion is nonlinear, leading to the formation of nonlinear vortex structures. Locally, the plasma is mixed over the length 웃x in one rotation period when the amplitude ␰x ⫽ t 兰 dt⬘vEx ⫽ 웃x. The nonlinear problem is treated in Chaos and Structures in Nonlinear Plasmas by Horton and Ichikawa (35).

200

100

ω k ( η i = 1) 0

(11)

(44,45). Drift wave theory is able to account for confinement scaling either as DB or ␳ DB. Transport dependent on ␳ de* * pends on the average mass mi of the working gas ions since 1/2 ␳* ⫽ (miTe) /eB. The Perkins et al. (45), Petty et al. (36), and Erba et al. (2) studies present evidence for the Bohm-like scaling of trans-

533

ω k ( η i = 3) –100

0

2

4

6

8

10

12

14

–1

kθ (cm ) Figure 6. The measured frequency spectrum obtained from far-infrared laser scattering from the electron density fluctuations from the drift waves in the TEXT Tokamak. (Courtesy of Brower).

534

PLASMA TURBULENCE

tral distribution k, ω) = Srn (k

k )νk I(k (ω − ωk )2 + νk2

(12)

with the turbulent decorrelation rate k · v E )2 1/2  νk 
(k

ck2⊥ φ˜ B

(13)

experiment in Fig. 4. Parameters for such a regime are ␩i ⫽ 3, S ⫽ Ln /Ls ⫽ 0.1, and ␶ ⫽ Te /Ti ⫽ 1 as in the three-dimensional (3-D) FLR fluid simulation (4,58a) and the gyrofluid simulation (58b,59,59a). The 3-D-slab model simulations (4,58a) show that the ion thermal diffusivity is χislab = 0.3


L 1/2 s

Ln

ρs cTe (η − ηc ) Ln eB i

(14)

The agreement of the peak of the FIR measurement of S(k, 웆) in Fig. 6 and the theoretical formulas for the drift waves is the direct signature of the drift wave turbulence. The main features of the measured fluctuations are explained by drift waves and the drift wave turbulence. Comparing the fractional fluctuation level n˜ /n from the early smaller machines ATC (29) with the intermediate size TEXT and the large TFTR machine shows that the core fractional fluctuation level scaling is consistent with (␳s /a)움 with an exponent between
 and unity. In a review article it is important to point out parallels with other areas of physics. The closest and most important parallel to plasma drift waves is the analogy with the Rossby waves and vortices in geophysical atmospheric and oceanographic disturbances with periods long compared to the rotational period of the planet. Hasegawa and Mima (51,52) and Hasegawa et al. (53) develop the limit in which the two models become isomorphic. The correspondence is due to the Coriolis force having the same mathematical form as the Lorentz force. The analogy was also recognized by Petviashvili (54) which led to the first rotating parabolic water tank experiments by Antipov et al. (55,56) in Kurchatov and by Antonova et al. (57) in Tiblisi. This aspect of the drift-wave–Rossby problem is found in the Horton and Hasegawa (58) article in the special issue of CHAOS devoted to such geophysical vortex structures. Further development of the theory from the plasma physics perspective is given in Chaos and Structures in Nonlinear Plasmas by Horton and Ichikawa (35).

for the magnetic–shear parameter S ⫽ Ln /Ls ⬎ S1 ⫽ 0.05 and ␩c ⱕ ␩i ⱗ 4. For small shear S ⬍ S1 the scaling S⫺1/2 is too strong and the parameterization with S is given as exp (⫺S/S0) in (58a) and as q/(S ⫹ S0) in Kotschereuther et al. (58c) and Waltz et al. (59,59a). When the turbulence level is high we expect S0 independent of ␳*: For low turbulence levels, S0 is proportional to ␳*. Near the critical gradient the exponent on ␩i ⫺ ␩c in Eq. (14) becomes unity. The supporting nonlinear bifurcation analysis is found in Hamaguchi and Horton (58a). For the regime of ␩i Ⰷ 1 the density gradient parameter Ln drops out of the system and the parameterization of the 3-D simulations gives ␹i ⫽ 0.8 (␳s /LT)(cTi /eB) g(q, s, Ti /Te). This is called the flat density regime where ␩i ⲏ 3 to 4. In full toroidal models there are additional parameters. The instability in the toroidal system changes character from that in the cylindrical system. It is the linear eigenmode problem (59b) that determines how the plasma adjusts through the self-consistent field dynamics to the gradient driving mechanisms and the sheared toroidal magnetic field to obtain the fastest release of stored thermal energy contained in the pressure gradient. In the toroidal regime the ion thermal diffusivity formula is the formula given by (59c):

Scaling Laws of the Ion Temperature Gradient Turbulent Transport

with c1 of order unity. Owing to the factor of q the confinement is inversely proportional to the product of self-field Bp 앒 Ip /a and the external toroidal field B ⬵ B␾ . The peak of the growth rate driving the turbulence is at ky ⫽ ␳⫺1 s [1/(1 ⫹ ␩i)]1/2. Near this wavenumber the phase velocity 웆k /ky is changing from the electron diamagnetic direction for smaller ky␳s to the ion diamagnetic direction at larger ky␳s. Thus, there is a critical wavenumber where the phase of the wave is stationary in the plasma rest frame and the mode grows exponentially at the interchange growth time ␶ ⫽ (RLTi)1/2 /cs, which is on the order of a few microseconds. In the absence of the ITG fluctuations (␩i ⬍ ␩c) the calculation of the ion thermal flux in the torus is still a complicated problem due to the geometry and complex ion orbits. The collisional thermal flux for low collisionality ␯*i ⬅ (␯iqR/vTi)(R/r)3/2 ⱕ 1 is from the random walk of the banana orbits formed by the guiding center drifts of the trapped ions. Most large tokamaks (I ⬎ 1 MA) are in this low collisionality regime. The collisional thermal flux qi ⫽ ⫺ni␹ineo dTi /dr (which is only one diagonal element of a large transport matrix) is given by Chang and Hinton (60) and Hirshman and Sigmar (60a). In the small ⑀ ⫽ r/R limit the ion thermal diffusivity

The most important instability with respect to the limits on the ion thermal confinement for nuclear fusion is the ion temperature gradient (ITG) instability. This drift wave instability goes by both the name ITG and the eta-i (␩i ⬅ Ln /LTi) mode due to the key dimensionless parameter ␩i that measures the strength of the ion temperature gradient. For a specific toroidal machine with major radius R, it is advantageous to use the gradient parameter 애i ⫽ R/LTi since, unlike Ln, the major radius R of the machine is fixed. In certain regimes there is a well-defined critical value ␩i,crit, starting at the minimum of , above which there is a strong drift wave instability producing anomalous ion thermal flux qi. First studies of the ITG turbulent transport were naturally concerned with regimes where the mode growth per wave pemax riod 웂max occurs approxik /웆k is substantial (ⱖ 0.1). The 웂k mately at the wavenumbers ky␳i 앓 (1 ⫹ ␩i)⫺1/2 Ⰶ 1. In this regime the mixing scale length for an isotropic fluctuation spectrum (␭x 앓 ␭y) of wavelengths ␭ gives ␹i 앑 ␭⬜2 웂max 앓 (␳i /LTi)(cTe /eB) g(␩i, s, q, ⑀n) at the fluctuation levels n˜ /n ⫽ e␾˜ /Te 앓 ␳s /LTi as known from theory and associated with the

χiHCT = cT

qρs cTe [2 n (ηi − ηc )]1/2 sLn eB

(15)

PLASMA TURBULENCE

from collisions is 2 χineo = νi ρiθ (0.66 + 1.88(r/R)1/2 )(r/R)1/2

535

tinuity equation give, in dimensional form, (16)

where the 0.66 arises from the pitch angle scattering and the 1.88 arises from energy scattering (60b). Here ␳i␪ ⫽ q␳i / ⑀ and the post-factor of (r/R)1/2 takes into account that only the trapped and barely passing ions, whose fractional density increases as (r/R)1/2, contribute to the ␹ineo in the banana regime. At larger collisionality (R/r)3/2 ⬎ ␯ i ⬎ 1 the ion detrapping * collision occurs before the banana orbit is formed. The ion thermal diffusivity from collisions in this regime is ␹ineo ⫽ 2.6(vi /qR)(q␳i)2. The theoretical structure of the full transport problem treating the collective drift wave fluctuations and the collisional interactions on equal footing is given in a series of works by Sugama and Horton (60c,d) for electrostatic turbulence and by Sugama et al. (34) for electromagnetic turbulence. Both transport processes arise from the Coulomb interactions of the charged particles. The collision operator takes into account the thermal fluctuation levels at Debye length space scales k␭De ⲏ 1, while the drift wave takes into account the collective Coulomb interactions on space scales covering many Debye lengths k␭De Ⰶ 1. The general procedure of predicting the Ti(r) profiles from the energy–momentum deposition profiles for the evolution of the core plasma is carried out with transport codes. An example is shown in Fig. 7 from the IFS/JAERI collaborative analysis of an important discharge (60e). The normalized gradient scale lengths computed for the JT60-U discharge 17110 is shown in Fig. 7. The principal system and plasma parameters for the discharges are given in Table 1. Recent deuterium discharges in JT60-U have reached sufficient plasma confinement in the presence of the drift wave turbulence to achieve conditions showing that the equivalent deuterium–tritium (D–T) plasma would have achieved fusion power breakeven. The D–T fuel is roughly 200 times more reactive than the deuterium–deuterium (D–D) fusion fuels at these ion temperatures. Extrapolations of the JET and JT60U parameters to a 20 MA tokamak predict a fusion power sufficient for ignition. Resistive Drift Wave and Interchange Turbulence The collisional drift wave is a paradigm for anomalous transport that has been extensively investigated with many different modelings. A particularly simple two-dimensional (2D) model, called the Hasegawa–Wakatani (61) model with an adiabaticity parameter 움, has been investigated by Wakatani and Hasegawa (5), Krommes and Hu (61a), and Hu et al. (61b). To understand the origin of the simple 움-model and to appreciate its limits, we briefly present the 3D resistive drift model. For finite resistivity ␩ ⫽ me␯e /nee2 the parallel current carried by the electrons in Eq. (11) yields j储 ⫽ ⫺(ne e2 /me␯e)ⵜ储(␾ ⫺ Te /e ln n) using the isothermal approximation 웃pe ⫽ Te웃ne. The collisional drift wave equation follows from the divergence of the current ⵱ ⭈ j ⫽ 0 with the polarization current balancing j储 through ⵱ ⭈ jp ⫽ ⫺⵱储 j储 ⫽ ␩⫺1ⵜ储2(␾ ⫺ (Te /e) ln n) and the electron continuity equation. The rotational part of the plasma momentum for the vorticity ⵱2␸ is equivalent to the current closure equation. The vorticity equation and the electron con-

mi nc d 2 B ∇ ϕ = 0 ∇  j + zˆ · ∇ pe × ∇ B0 dt c 1 cTe n0 d (n + n1 ) = ∇  j + zˆ · ∇ dt 0 e eB0


n

(17) 1

n0

−

eϕ Te



× ∇

(18)

where ⵱⍀ is the effective g force used to relate the curvature and gradient-B effects to the classical Rayleigh–Taylor instability. The computation of ⍀(r) for the average curvature of the magnetic field line is extensively used in stellarator/heliotron research (62). The derivatives on the left side of Eqs. (17) and (18) are the E ⫻ B convective derivatives defined by df /dt ⫽ ⭸t f ⫹ vE ⭈ ⵱f. The model equations (17) and (18) have a conserved potential vorticity ␨ given by ζ =

mi c 2 2 n ∇ ϕ − ln n0 − 1 − eB2 ⊥ n0

(19)

The potential vorticity includes the influences of the density variations and the nonuniformity of the magnetic field. From the conservation of the potential vorticity ds/dt ⫽ 0 many of the complex vortical structures of the flow can be understood. This 3-D model in Eqs. (17) and (18) has resistive drift waves driven by the density gradient (⭸xn0)2 through the charge separation from finite k⬜2 ␳s2 and has resistive interchange-driven modes from 웆 웆D ⬎ 0, where 웆D ⫽ * (ck␪T/eB)(d⍀/dr) is the averaged grad-B/curvature drift frequency. The linear eigenmodes are of two types: localized to the rational surfaces where k储 ⫽ 0 and global modes. Wakatani et al. (63) carried out 3-D simulations of this system, and the results for one case are shown in Fig. 8. The density fluctuations are large and give a qualitative explanation of the similar tokamak and stellarator/helical device edge density fluctuations. The electric potential, shown in the left column in Fig. 6, has the important property of developing an m ⫽ 0/n ⫽ 0 component with a well-defined circular null surface. This ␸0,0(r, t) ⫽ 0 surface partially blocks the turbulent losses from the core of the cylindrical model. For stellarators the m ⫽ 1, n ⫽ 1 rational surface is near the edge of the plasma and the dominant modes in this simulation are the m ⫽ 3/n ⫽ 2 and m ⫽ 2/n ⫽ 1 fluctuations and the m ⫽ 0/n ⫽ 0 background profile for v␪ ⫽ ⫺cEr /B. These simulations with ␯e /웆ce ⫽ 1.4 ⫻ 10⫺4 are too collisional to apply to the edge of tokamaks with I ⬎ 1 MA confinement devices (where ␯e /웆ce ⱗ 10⫺6). Wakatani et al. (63) investigations of the model include an externally imposed electric field Er(r) exceeding the strength of self-consistently field generated from the m ⫽ 0/n ⫽ 0 modes. The Er ⬍ 0 field suppresses the turbulence during the growth phases, but produces only a weak reduction of the flux in the saturated state. The collisionality dependence of the particle flux is shown to increase with ␯e for ␯ /웆ce ⬍ 10⫺3 and for ␯ /웆ce ⬎ 10⫺3. then to increase as ␯1/3 e In the widely investigated reduced 2-D model of the Hasegawa–Wakatani equations the operator ⵜ储 씮 ik储 or ⫺i/L c, where k储 is the relevant mean parallel wavenumber and L c is the connection length to the divertor end plates in the scrapeoff layer (open field lines) modeling the new parameter 움 appears.

536

PLASMA TURBULENCE

6.0

0.6 t4 = 6.00

5.0

t4 = 6.00 t1 = 5.55

0.4 t3 = 5.75 Lne R

ne (r) (1019 m–3)

4.0 3.0 t2 = 5.60

0.2

2.0 1.0 t1 = 5.55 0

0

0.5

0

1.0

0

r (m)

0.5

1.0

r (m) (a)

50.0 t4 = 6.00

t3 = 5.75

t1 = 5.55 LTi R

Ti (r) (keV)

30.0

t2 = 5.60 20.0

10.0

0 0

t4 = 6.00

0.4

40.0

0.2

t1 = 5.55 0 0.5

1.0

0

r (m)

0.5

1.0

r (m) (b)

Figure 7. Evolution of the temperature and density profiles for the JT-60U experiment during the formation of the internal transport barrier at r/a ⫽ 0.7. (a) Smoothed density profiles used to compute the gradient scale lengths at four times during the growth of the core pressure from the internal transport barrier. The inset gives ⑀ ⫽ Lne /R. (b) Ion temperature profiles and the inset shows ⑀ ⫽ LTi /R.

The space–time units are changed to the local scales of ␳s and Ln /cs in these 2-D studies. The standard form of the Hasegawa–Wakatani 2-D model is then d (∇ 2 φ) = α(φ − n) + µ∇ 4 φ dt dn ∂φ = −κ + α(φ − n) + D∇ 2 n dt ∂y

(20) (21)

where the viscosity 애 and D are taken small, but finite to absorb all fluctuation energy reaching the smallest resolved space scales in the simulation system. The system’s strong turbulence features a small 움 where 움/웆 앑 1, where k, 웆, 웂 are taken at the peak of the energy spectrum. Here the over-

bars on k, 웆, 웂 denote a mean value near the peak of the energy spectrum Ek. One can show that k 앒 움1/3, 웂 앒 움1/3, and Ek 앓 웂 2 /k3 앓 1/움1/3 (61a). In the large 움 limit the density n 씮 ␾(1 ⫹ O (1/움)) approaches the adiabatic limit and a weaker turbulence appears with Ek 앓 웂웆/k3 앒 1/움 since 웂 앑 1/움, and k ⫽ 움0 and 웆 ⫽ 움0 independent of 움. These 움-scalings in the small-움 and large-움 regimes have been verified by direct numerical simulation and the statistical closure method. The analytical closure method used here is a realizable Markovian closure (RMC). Figure 9 shows the good agreement between the direct numerical simulation (DNS) and the RMC closure calculation. The new parameter 움 ⫽ k储2Te /me␯e웆0 measures the parallel electron diffusion in a characteristic wave period (1/웆0) and determines the properties of the waves. For 움 Ⰷ 1 the elec-

PLASMA TURBULENCE

Time = 3.0

Time = 3.0

Time = 4.5

Time = 4.5

Time = 6.0

Time = 6.0

537

trons tend to the Boltzmann distribution n˜ ⫽ e␾ /Te and the Hasegawa–Mima equation is recovered. For intermediate values of 움 ⲏ 1 the Terry-Horton equation (64) applies. Plasma turbulence appears as ubiquitous as plasma itself. Space, solar, and astrophysical plasmas show many varied forms of plasma turbulence ranging from large-scale magnetohydrodynamic (MHD) turbulence (65) to the smallest Debye length scale Langmuir turbulence. For example, the energy released during solar flares heats and accelerates the plasma in the solar corona. Electrons on the open coronal magnetic field lines caused by large-scale MHD reconnection events stream at relativistic speeds into the interplanetary plasma. The electron beams drive Langmuir turbulence creating intermittent bursts of radio noise known as type-III radio sources. The peculiar intermittency of Langmuir turbulence called nonlinear wave collapse was first described by Zakharov (66). The phenomenon of nonlinear wave collapse is reviewed by Robinson (67) in a wide range of laboratory and space physics settings. The general theoretical analysis of wave turbulence for plasmas and neutral fluids is given in Zakharov, L’vov, and Falkovich in Kolmogorov Spectra of Turbulence (68). ACKNOWLEDGMENTS

Potential contour (a)

The work was supported by the Department of Energy under Grant No. DE-FG03-96ER-54346.

Density contour (b)

Figure 8. The evolution of (a) the electrostatic potential and (b) the density contours for the 3-D turbulence given by the resistive drift wave model with average unfavorable toroidal curvature. The time values are in units of (Courtesy of Wakatani).

25 RMC closure code Pseudospectral DNS code 20

15 Ek

α

–1/3

10

5

α 0 0.01

0.1

1

–1

10

α Figure 9. The power law decrease of the turbulent kinetic energy with adiabaticity parameter 움 in the Hasegawa–Wakatani equations. In the limit of large 움 the drift wave model reduces to the Terry– Horton model applicable at higher plasma temperatures.

BIBLIOGRAPHY 1. A. Taroni et al., Global and local energy confinement properties of simple transport coefficients of the Bohm type, Plasma Phys. Contr. Fusion, 36: 1629–1646, 1994. 2. M. Erba et al., Extension of a Bohm model for L-mode electron heat transport to ion heat transport and to the ohmic regime, Plasma Phys. Control. Fusion, 37: 1249–1261, 1995. 3. W. Horton, Nonlinear drift waves and transport in magnetized plasma, Phys. Rep., 192: 1, 1990. 4. W. Horton, R. Estes, and D. Biskamp, Fluid simulation of ion pressure gradient driven drift modes, Plasmas Phys., 22: 663– 678, 1980. 5. M. Wakatani and A. Hasegawa, A collisional drift wave description of plasma edge turbulence, Phys. Fluids, 27: 611, 1984. 6. M. C. Kelley and R. A. Heelis, The Earth’s Ionosphere, Plasma Physics and Electrodynamics, San Diego, CA: Academic Press, 1989, Vol. 43. 7. O. Buneman, Instability, turbulence, and conductivity in a current-carrying plasma, Phys. Rev. Lett., 10: 285, 1963. 8. M. J. Keskinen, S. L. Ossakow, and P. K. Chaturvedi, J. Geophys. Res., 85: 3485, 1980. 9. R. N. Sudan, Geophys. Res. Lett., 88: 4853, 1983. 10. S. M. Hamberger and J. Jancarik, Experimental studies of electrostatic fluctuations in a turbulently heated plasma, Phys. Fluids, 15: 825, 1972. 11. R. E. Slusher et al., Study of current-driven ion-acoustic instability using CO2-laser scattering, Phys. Rev. Lett., 36: 674, 1976. 12. W. Horton, D.-I. Choi, and R. A. Koch, Ion acoustic heating from renormalized turbulence theory, Phys. Rev. A, 14: 424, 1976. 12a. R. L. Stenzel, Experiment on current-driven three-dimensional ion sound turbulence, Phys. Fluids, 21: 93, 99, 1978. 13. W. Horton and W. Rowan, Impurity transport studies in the Texas Experimental Tokamak (TEXT), Phys. Plasmas, 1: 901– 908, 1994.

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17. H. L. Pe´cseli, J. Juul Rasmussen, and K. Thomsen, Nonlinear interaction of convective cells in plasmas, Plasma Phys. Contr. Fusion, 27: 837–846, 1985. 18. S. C. Prager, A. K. Sen, and T. C. Marshall, Dissipative trappedelectron instability in cylindrical geometry, Phys. Rev. Lett., 33: 692–695, 1974. 19. J. Slough, G. A. Navratil, and A. K. Sen, Production and observation of the dissipative trapped ion instability, Phys. Rev. Lett., 47: 1057, 1982. 20. R. Scarmozzino, A. K. Sen, and G. A. Navratil, Production and identification of a collisionless, curvature-driven, trapped-particle instability, Phys. Rev. Lett., 57: 1729–1732, 1986. 21. A. K. Sen, J. Chen, and M. Mauel, Production and identification of the ion-temperature-gradient instability, Phys. Rev. Lett., 66: 429–432, 1991. 22. J. Chen and A. K. Sen, Experimental study of a mixed slab and toroidal ion-temperature-gradient mode, Phys. Plasmas, 2: 3063, 1995. 22a. T. Ohkawa and M. Yoshikawa, Experimental observation and theoretical interpretation of low-frequency interchange instability in ‘‘average minimum-B’’ configurations, Phys. Rev. Lett., 19: 1374, 1987. 23. M. Okabayashi and V. Arunasalam, Study of drift-wave turbulence by microwave scattering in a toroidal plasma, Nucl. Fusion, 17: 497–513, 1977. 24. E. J. Synakowski et al., Phys. Plasmas, 4: 1736–1744, 1997. 25. K. H. Burrell, Effects of E ⫻ B velocity shear and magnetic shear on turbulence and transport in magnetic confinement devices, Phys. Plasmas, 4: 1499–1518, 1997. 26. Y. Koide et al., Internal transport barrier on q ⫽ 3 surface and poloidal plasma spin-up in JT-60U high-웁p discharges, Phys. Rev. Lett., 72: 3662–3665, 1994. 27. F. M. Levinton et al., Improved confinement with reversed magnetic shear in TFTR, Phys. Rev. Lett., 75 (24): 4417–4420, 1995.

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28. E. J. Strait et al., Enhanced confinement and stability in DIII-D discharges with reversed magnetic shear, Phys. Rev. Lett., 75 (24): 4421, 1995.

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51. A. Hasegawa and K. Mima, Stationary spectrum of strong turbulence in magnetized plasma, Phys. Rev. Lett., 39: 205, 1977.

30. C. M. Surko and R. E. Slusher, Study of the density fluctuations in the adiabatic toroidal compressor scattering Tokamak using CO2 laser, Phys. Rev. Lett., 36: 1747–1750, 1976.

52. A. Hasegawa and K. Mima, Pseudo-three-dimensional turbulence in magnetized nonuniform plasma, Phys. Fluids, 21: 87–92, 1978.

31. C. M. Surko and R. E. Slusher, Study of density fluctuations in the Alcator Tokamak using CO2 laser scattering, Phys. Rev. Lett., 40: 400–403, 1978. 32. R. V. Bravenec et al., Core turbulence and transport studies on the Texas Experimental Tokamak, Phys. Fluids B, 4: 2127– 2135, 1992.

53. A. Hasegawa, C. G. MacLennan, and Y. Kodama, Nonlinear behavior and turbulence spectra of drift waves and Rossby waves, Phys. Fluids, 22: 2122–2129, 1979. 54. V. I. Petviashvili, Self-focusing of an electrostatic drift wave, Fiz. Plazmy, 3: 270 (Sov. J. Plasma Phys., 3: 150), 1977. 55. S. V. Antipov et al., Rossby soliton in the laboratory, Zh. Eskp. Teor. Fiz., 82: 145–160, 1982 (Sov. Phys. JETP, 55: 85, 1982).

33. K. W. Gentle, Diagnostics of magnetically confined high-temperature plasma, Rev. Mod. Phys., 67: 80, 1995.

56. S. V. Antipov et al., Rossby autosoliton and laboratory model of Jupiter’s Great Red Spot, Sov. Phys. JETP, 62: 1097, 1985.

33a. E. Mazzucato et al., Turbulent fluctuation in TFTR configurations with reversed magnetic shear, Phys. Rev. Lett., 77: 3145, 1996.

57. R. A. Antonova et al., Drift solitons in a shallow rotating fluid, Pisma Zh. Eksp. Teor. Fiz., 37: 545, 1983 (JETP Lett., 37: 651, 1983).

POLYNOMIALS

539

58. W. Horton and A. Hasegawa, Quasi-two-dimensional dynamics of plasmas and fluids, CHAOS, 4: 227–251, 1994. 58a. S. Hamaguchi and W. Horton, Phys. Fluids, B2: 1833, 1990. 58b. G. W. Hammett, W. Dorland, and F. W. Perkins, Phys. Fluids, B4: 2052, 1992. 58c. M. Kotschereuther et al., Phys. Plasmas, 2: 2381, 1995. 59. R. E. Waltz, G. D. Kerbel, and J. Milovich, Toroidal gyro-Landau fluid model turbulence simulations in a nonlinear ballooning mode representation with radial modes, Phys. Plasmas, 1: 2229, 1994. 59a. R. E. Waltz et al., Phys. Plasmas, 2: 2408, 1994. 59b. G. Rewoldt and W. M. Tang, Phys. Fluids, B2: 318, 1990.

pendence of the turbulent ion conductivity, Plasma Phys. Contr. Fusion, 39: 1461–1477, 1997. C. C. Petty et al., Nondimensional transport scaling in DIII-D: Bohm versus gyro-Bohm resolved, Phys. Plasmas, 2: 2342–2348, 1995.

59c. W. Horton, D.-I. Choi, and W. M. Tang, Phys. Fluids, 24: 1077, 1981. 60. C. S. Chang and F. L. Hinton, Effect of impurity particles on the finite-aspect ratio neoclassical ion thermal conductivity in a tokamak, Phys. Fluids, 29: 3314–3316, 1986. 60a. S. P. Hirshman and D. J. Sigmar, Neoclassical transport of impurities in tokamak plasmas, Nucl. Fusion, 21: 1079, 1981. 60b. C. Bolton and A. A. Ware, Phys. Fluids, 26: 459, 1983.

PLASTIC PACKAGINGS. See ENCAPSULATION MATERIALS

60c. H. Sugama and W. Horton, Phys. Plasmas, 2: 2989, 1995. 60d. H. Sugama and W. Horton, Phys. Plasmas, 4: 405, 1997. 60e. W. Horton et al., Plasma Phys. Control. Fusion, 39: 83, 1997. 61. A. Hasegawa and M. Wakatani, Self-organization of electrostatic turbulence in a cylindrical plasma, Phys. Rev. Lett., 59: 1581– 1584, 1987. 61a. J. A. Krommes and G. Hu, Phys. Plasmas, 1: 3211, 1994. 61b. G. Hu, J. A. Krommes, and J. C. Bowman, Phys. Plasmas, 1: 3211, 1996. 62. B. A. Carreras, L. Garcia, and P. H. Diamond, Theory of resistive pressure-gradient driven turbulence, Phys. Fluids, 30: 1338, 1987. 63. M. Wakatani et al., Resistive drift wave and interchange turbulence in a cylindrical plasma with magnetic and velocity shear, Phys. Fluids B, 4: 1754–1765, 1992. 64. P. W. Terry and W. Horton, Drift wave turbulence in a low-order k space, Phys. Fluids, 26: 106–112, 1983. 65. D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge Univ. Press, 1997, Chaps. 7–10. 66. V. E. Zakharov, Zh. Eksp. Teor. Fiz., 62: 1745, 1972 (Soviet Phys. JETP, 35: 908, 1972). 67. P. A. Robinson, Nonlinear wave collapse and strong turbulence, Rev. Mod. Phys., 507–574, 1997. 68. V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectral of Turbulence, New York: Springer-Verlag, 1992. Reading List H. L. Berk, D. D. Ryutov, and Yu. A. Tsidulko, Temperature-gradient instability induced by conducting end walls, Phys. Fluids B, 3: 1346–1354, 1991. J. W. Connor and H. R. Wilson, Plasma Phys. Controlled Fusion, 36: 719, 1994. A. Hasegawa and M. Wakatani, Phys. Rev. Lett., 50: 682, 1983. W. Horton, M. Wakatani, and A. J. Wootton, Ion temperature gradient driven turbulent transport, American Institute Physics Conf. Proc. No. 284, New York, 1994. B. B. Kadomtsev, Plasma Turbulence, New York: Academic Press, 1965, p. 68. B. B. Kadomtsev and O. P. Pogutse, Trapped particles in toroidal magnetic systems, Nucl. Fusion, 11: 67–92, 1971. M. Ottaviani, W. Horton, and M. Erba, The long wavelength behaviour of ion-temperature-gradient-driven turbulence and the radial de-

W. HORTON The University of Texas

G. HU Globalstar LP

AND PROCESSES.

PLATFORMS FOR MOBILE ROBOTS. See MOBILE ROBOTS.

PLLS. See PHASE LOCKED LOOPS. PN MODELS. See PETRI NETS. POLARIMETRY, RADAR. See RADAR POLARIMETRY. POLLUTION, FROM POWER PLANTS. See AIR POLLUTION CONTROL.

POLYCRYSTALLINE THIN FILM SOLAR CELLS. See TERNARY SEMICONDUCTORS.

POLYNOMIAL EQUATIONS. See POLYNOMIALS.

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Wiley Encyclopedia of Electrical and Electronics Engineering Thermal Spray Coatings Standard Article P. Fauchais1 and A. Vardelle 1SPCTS, UMR 6638, Univ. of Limoges, France Copyright © 2007 John Wiley & Sons, Inc. All rights reserved. DOI: 10.1002/047134608X.W5923.pub2 Article Online Posting Date: November 16, 2007 Abstract | Full Text: HTML PDF (2466K)

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Abstract Thermal spraying is a group of processes in which finely divided metallic or nonmetallic surfacing materials are deposited in a molten or semi-molten condition on a prepared substrate to form a spray deposit. The surfacing material may be in the form of powder, rod, cord, or wire, and the different spraying processes are generally categorized into two groups according to the method of heat generation: combustion or electrical. This article is restricted to electrical processes: plasmas jets (nontransferred direct current arcs or induction discharges), wire arcs, and transferred arcs. Nowadays, thermal spray is part of engineering technology for wear protection, thermal insulation or barriers, oxidation protection, corrosion or oxidation resistance, abradable seals, electrically conductive or insulating materials, dimensional restoration or manufacture, free-standing shapes, and medical implants. Keywords: plasma spraying; beads; plasma torch; pass; nozzle shroud; arc spraying About Wiley InterScience | About Wiley | Privacy | Terms & Conditions Copyright © 1999-2008John Wiley & Sons, Inc. All Rights Reserved.

file:///N|/000000/0WILEY%20ENCYCLOPEDIA%20OF%20ELE...ICS%20ENGINEERING/45.%20Plasma%20Science/W5923.htm16.06.2008 0:23:21

40

THERMAL SPRAY COATINGS

THERMAL SPRAY COATINGS Thermal spraying is a group of processes in which finely divided metallic or nonmetallic surfacing materials are deposited in a molten or semimolten condition on a prepared substrate to form a spray deposit (1). The surfacing material may be in the form of powder, rod, cord, or wire, and the different spraying processes are generally categorized into two groups according to the method of heat generation: combustion or electrical. This article is restricted to electrical processes: plasmas jets (nontransferred direct current arcs or induction discharges), wire arcs, and transferred arcs. Plasma jets utilize powdered materials whereas in the wire arc, the arc is struck between two consumable, automatically advanced wires. In these two processes, the substrate (e.g., metal, glass, ceramic, plastic, composite) is kept far below its melting temperature. This is not the case with the transferred arc process where the metallic substrate and the powdered surfacing material are molten, and the resulting coating is welded to its substrate. Now thermal spray is part of engineering technology for wear protection, thermal insulation or barriers, oxidation protection, corrosion or oxidation resistance, abradable seals, electrically conductive or insulating materials, dimensional restoration or manufacture, free-standing shapes, and medical implants (1–3). This market represented about $1.2 billion US in 1995 (with 40% to 60% in plasma technology, depending on the sources) and is poised for substantial growth (4). This article presents the spraying processes with coating formation, plasma jet spraying, arc spraying, sprayed materials, and applications. the plasma transferred arc process with the plasma torch, the heat flux control for substrate and powders, and the coatings and their applications. COATING FORMATION Key Parameters The heat source (plasma or arc) is independent of the substrate which can be cooled independently and whose surface temperature can be as low as 50⬚ to 70⬚C, for example, with cryogenic cooling for plasma spraying, and typically below 150⬚ to 200⬚C with compressed air cooling. The plasma jet flow, or the propelling compressed gas for the wire arc, entrains the surrounding atmosphere. Therefore, spraying in air may result in surface oxidation of the hot particles in flight and the flattened particles and coatings, when substrate and coating are kept at temperatures over 200⬚ to 500⬚C, depending on the materials. The only way to reduce oxidation drastically is to spray in a controlled atmosphere or preferably in a soft vacuum (앑20 kPa to 30 kPa) which increases the cost of the industrial spray equipment by at least an order of magnitude. The use of nozzle shields to spray in air reduces oxidation problems, but they are not as effective as spraying in a soft vacuum. The surfacing material must hit on the substrate or the previously deposited layers in a molten or plastic state. When it is in a plastic state, its velocity must be high, typically over 300 m/s to 350 m/s for particles in the 30 애m range. Thus, only materials that melt at temperatures at least 300 K below their vaporization or decomposition temperatures can be J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

THERMAL SPRAY COATINGS

sprayed. The particles strike the surface, flatten, and form thin platelets (splats) that conform and adhere to the irregularities of the prepared surface and to each other with occasional diffusion or alloying. The splats cool down below their plastic state in a very short time (앑5 애s to 30 애s), and the next particles impinge on the same location in time of the order of tenths to hundreds of ms, that is, after the splats have solidified and cooled well below their melting point. Thus, the coating is built up, particle by particle, into a lamellar structure as shown in Fig. 1. Except for oxide inclusions (for metals and alloys sprayed in air), the coating is generally porous and micro- and macrocracked, as explained later. Thus, its properties differ from those of the bulk material. For example, its Young’s modulus is one-third to one-quarter of that of the bulk material in most cases. Substrate Preparation Coating adhesion is directly related to the cleanliness and the roughness of the substrate surface (1,2,4,5). Preparations of the surface is the most critical step in a thermal spraying operation. The first step in the preparation is to remove all surface contaminants, such as scale, oil, and paint, by different techniques: scraping, wire brushing, machining, vapor degreasing, vapor blasting, acid pickling, or oven backing. These must to be adapted to the substrate material. After contaminants have been removed, the cleanliness must be maintained until spraying is completed: no finger prints, clean fixtures and materials, and no airborne debris especially during spraying. Thus, the air or gas flow close to the substrate must be carefully controlled, especially when the substrate is kept hot (T ⬎ 400⬚C) during spraying, to minimize, as much as possible, recirculation of submicron-sized debris or recondensed vapors. After cleaning, the substrate surface has to be roughened to promote adhesion. In most cases this is achieved by grit blasting. Blasting parameters have to be controlled carefully to obtain reproducible surfaces, to reduce the embedded grit debris as much as possible, and to limit the induced compressive stresses (5) in a thin layer (tens of mm) below the gritblasted substrate surface. Higher surface roughness, as measured by the root-mean-square average of surface roughness (often termed Ra) requires bigger grits which induce higher compressive stresses and more embedded debris. The embedded debris, which reduces coating adhesion proportionally to

Figure 1. Bilayer coating with a metal bond coat and a ceramic coat.

41

the quantity, can be avoided by blasting with high pressure (100 MPa) water jets (6). The size of the peaks has to be adjusted to the size of the particles or more precisely to that of the resulting splats. If the peak height is 100 애m (Ra 앑 12 애m) splats with mean diameters below 90 애m will have poor or even no mechanical adhesion. Roughening of composite or plastic-based materials is usually achieved by chemical methods. Splat Formation Splat formation has been extensively studied in the past decade, and work is continuing to get a better understanding of the involved phenomena. For example, in dc plasma spraying the short flattening (앑1 애s) and solidification (앑10 애s) times and solidification starting before flattening is completed make experiments and modeling very complex. The main conclusions of these studies are the following (7): Substrates Orthogonal to Particle Trajectory. For a given particle material on smooth surfaces (Ra ⬍ 0.4 애m), there is a critical temperature Tc for the substrate preheating temperature Tp above which the splat has a disk shape and excellent contact with the substrate (thermal contact resistance Rth ⬍ 10⫺7 m2 ⭈ K/W) corresponding to at least 70% to 80% of perfect contact. Below Tc, the splat is extensively fingered with Rth ⬎ 10⫺6 m2 ⭈ K/W and has about 20% of good contact with the substrate. There is actually no clear explanation for Tc, which is not linked to the particle melting temperature. For example, for Al2O3 or ZrO2, Tc 앑 200⬚C. However, this phenomenon is linked to the fact that if the actual temperature of the substrate Tp ⬍ Tc, solidification starts before flattening is completed (7). In general, splat adhesion to the substrate increases with substrate temperature (over Tc), provided that the metallic substrate is not too oxidized. The oxidation of metals or alloys depends on the original oxide layer thickness and composition. On oxidized surfaces, splats are ‘‘fingered’’ when the substrate temperature Tp is below Tc. For Tp ⬍ Tc, splats are very easily pulled off by the diamond tip of a perthometer even at very low load (⬍0.1 N). The splat cooling rate, which controls their microstructure (size of columns or grains and phases), is strongly linked to Rth and splat thickness e. Very fast cooling rates (up to 106 –108 K/s for dc plasma-sprayed splats) may result in metastable phases or even amorphous structures. This is less likely when e and/or Rth increases. e depends on the particle impact velocity. For example, with 30 애m alumina particles hit at v ⫽ 250 m/s, e 앑 1 애m (dc plasma spraying) and 3 애m for v 앑 50 m/s (RF plasma spraying). The tensile stresses induced upon splat cooling (quenching stresses) can be very high, and they are partly relaxed by creep in metals and greatly relaxed by microcracking in ceramics (for more details about the other relaxation possibilities see Ref. 8). For alumina splats sprayed on a substrate at Tp ⬎ Tc, most of the splat surface exhibits microcracks. When Tp ⬍ Tc, only a small part, that is, in areas where the contact is good (⬍20%), exhibits these cracks. As the peaks and valleys perturb the liquid particle flow on rough surfaces, even when Tp ⬎ Tc, the splats are more extensively fingered and thicker. Their cooling rate is greater than that obtained on smooth surfaces, and the microcrack network with alumina, for example, is less dense. When Tp ⬍

42

THERMAL SPRAY COATINGS

Tc, the splats are extensively fingered, and their cooling rate is much slower (one order of magnitude) than when Tp ⬎ Tc. On Inclined Substrates. For Tp ⬎ Tc and smooth substrates, splats have an elliptical shape, and their thickness increases in the direction of the liquid flow (due to substrate inclination). However, when this inclination is increased, the splats become fingered, and areas of poor contact appear. They can be distinguished easily by atomic force microscopy because crystalline growth in these areas is parallel to the substrate (9). The splats are more easily detached by the tip of a perthometer, especially when the substrate inclination is greater than 30⬚. For T ⬍ Tp on smooth surfaces, the splats are hardly collected on inclined surfaces. Thus if possible the torch and the coated surface have to be orthogonal.

(i.e., by machining or grinding in conjunction with bond coating and/or grit blasting), or by annealing the coating and substrate when a certain coating thickness has been deposited. Coatings are not generally fully dense depending on the way they were sprayed, especially when sprayed in air, and porosity can affect the properties of thermally sprayed coatings significantly. The porosities generally observed are chamber-type pores a few micrometers in size because of shadow effects or poor accommodation of the gap between two splats, thin gaps between lamellae (앑0.1 애m), and cracks. Thus, measured Young’s modulii of metallic air plasma spray (APS) deposits are generally around one-third of bulk values and in ceramics they can be one-fifth (12). This heterogeneous structure also results in hardness values lower than those for equivalent materials in either cast or wrought form.

Bead and Pass Generation

PLASMA SPRAYING

When translating the torch relative to the substrate at a speed vt, a bead is obtained (7). As a first approximation, its shape is Gaussian. Its thickness eb depends on vt, powder mass flow rate mp⬚ , deposition efficiency ␳d, and the substrate stand-off distance. eb increases when mp⬚ or ␳d increase and vt decreases. In fact, to coat simple (flat or cylindrical) or complex structures, additional handling equipment is required, for example, torch translation at vt and cylindrical substrate rotation. In most complex structures, the torch is mounted on the arm of a computer-operated robot and the part to be coated is also moved by another unit. Such devices allow keeping the pass thickness constant and the torch orthogonal to the part of substrate it faces. The spray pattern results in overlapping beads. The thickness increase ep of a pass depends on eb and the rate of overlapping. Each pass thickness ep varies generally from a few 애m up to 100 애m. The thinner the pass, the easier the coating and substrate cooling and thus the control of substrate and coating mean temperature Tc between successive passes. The properties of the coatings are strongly linked to the substrate preheating temperature and to the control of Tc. The adhesion/cohesion of coatings increases with preheating temperature Tp, provided that the substrate (if metallic) is not too oxidized. This means that, depending on the substrate material, Tp is limited to temperatures between 300⬚ and 600⬚C in air. These temperatures are too low to improve adhesion/cohesion by diffusion, a phenomenon which is observed only when spraying in soft vacuum and when the substrate is hot enough (for example, 앑950⬚C for superalloy sprayed on superalloys). As is expected from the study of splats collected on inclined substrates, the adhesion/cohesion of coatings diminishes drastically when spraying on inclined substrates (10). Thus care has to be taken to keep the torch and the coated substrate orthogonal to each other during spraying. When Tp and/or Tc increase, residual stresses increase also: quenching stress (due to the improvement of the contact between splats and substrate or previously deposited layers) and expansion mismatch stress. Too high a residual stress can result in coating peeling off, especially for thick coatings (11). These stresses result in microcracks within splats in ceramics and macrocracks parallel or perpendicular to the substrate (see Fig. 1). This can be handled in certain cases by using a bond coating, whose expansion coefficient is between that of the substrate and the coating, by macroroughening

Plasma Torches The following three types of plasma torches are used: 1. Dc Plasma Torch with a Stick Type Cathode. These torches represent about 95% of the torches used in industry. The plasma torch consists of a stick-type cathode (made of tungsten doped with 2 wt% of thoria to lower the electron extraction potential) and a watercooled (at a pressure of 1.5 to 2 MPa) anode nozzle (Fig. 2) that has the same axis of symmetry. The arc strikes between the tip of the conical cathode and the anode. Because of the arc restrike mode at the anode, the arcroot lifetime at the same location is usually below 100 애s (13). It drastically reduces anode erosion but induces voltage fluctuation and thus power fluctuation at frequencies in the range 3 to 20 kHz. The anode nozzle shape is usually very simple, a cone followed by a cylindrical duct. The nozzle may also have a divergent section to produce supersonic jets. In most cases, it is made of OFHP copper sometimes with a tungsten insert. The method of injecting the plasma forming gas plays an important role in developing the plasma jet at the nozzle exit (13). The most traditional method is to inject it as a vortex (creating a cold boundary layer at the anode wall and pushing the arc root further downstream), longitudinally along the anode axis or along the cathode conical tip which creates extra gas stabilization of the arc column. Within the arc column, temperatures in the range 10,000 K to 18,000 K are achieved, and the plasma-forming gas is dissociated (if molecular) and ionized. Downstream of the arc root, the ions, electrons, and atoms recombine releasing energy which maintains a plasma jet as it exits from the nozzle. This high velocity jet (from 1000 m/s to 2700 m/s for subsonic jets) creates vortical rings at the nozzle exit which coalesce and create large eddies that result in an engulfment process. Cold surrounding gas bubbles entrained by the large eddies mix with the plasma and cool it down while they are heated up, so that their density becomes close to that of the jet. At the nozzle exit, where temperatures of 13,000 K to 14,000 K are achieved, the plasma’s specific mass is about one-fortieth of that of the cold gas. The most common plasma gases are mixtures of a pri-

THERMAL SPRAY COATINGS

Powder injection Working gases inlet

Anode

Water outlet

Cathode

Electrical insulator Water inlet

(a) mm 10 8 6 4 2

Mean temperature + root mean square 9kK 0

20 13kK 30 13kK

10

40

50

60 mm

9kK

2 4 6 8 10

Mean temperature – root mean square

mm

(b) m/s Gas velocity 2000 1500 1000 500 Plasma jet centerline mm 10

20

30

40

50

60

(c)

Figure 2. (a) Scheme of a dc plasma torch. (b) The upper and lower parts of the figure show the variation in the temperature fields of an Ar-H2 plasma jet resulting from the arc root movement on the anode wall. (c) Evolution of plasma jet velocity along the jet center line (nozzle i.d. 7 mm, Ar 45 slm, H2 15 slm, I ⫽ 600 A, V ⫽ 62 V, rth ⫽ 52%).

mary heavy gas Ar or N2 with a secondary gas either hydrogen or helium which improves the gas thermal conductivity and thus the heat transfer (14). Helium increases plasma viscosity at T ⬎ 10,000 K and delays mixing with the surrounding atmosphere. Ternary mixtures (Ar–He–H2) are sometimes used. Typical power levels are in the range of 40 kW, but some plasma torches operate at 80 kW especially in producing supersonic jets. Typical nozzle internal diameters are in the range 6 mm to 8 mm but generally the working parameters for a given nozzle are rather narrow. For example, with a 6 mm i.d. nozzle and a total gas flow rate of 40 slm (Ar ⫹ 25 vol% H2), the arc current can vary between 400 A and 700 A resulting in voltages between 80 and

43

65 V. Such torches are used to spray up to 6 kg/h to 8 kg/h of powder which is injected orthogonally to the jet axis inside (downstream of the arc root) or outside of the nozzle. 2. Torches with Three Cathodes. Recently torches with three stick-type cathodes set at 120⬚ around the torch axis have become available. Their main advantage is the possibility of injecting the particles axially (between the cathodes). Thus, particles are accelerated and heated by the hot plasma column, and the heat and momentum transfer are far more efficient than with conventional torches. 3. High-Power Torches. Dc vortex plasma torches using a button type cathode and high gas flow rates have been developed to spray relatively large quantities of powder (up to 20 kg/h to 25 kg/h). Figure 3 shows a schematic diagram of such a plasma torch (15). The nozzle internal diameter is about 8 mm and typical working conditions are 200 slm to 300 slm N2, 100 slm to 150 slm H2, 500 A, and 400 V. The plasma jet maximum temperature is below 9000 K and has a rather flat distribution. Supersonic flows, if necessary, are achieved more easily (16) than with the stick-type cathodes where temperatures are as high as 14,000 K. Because of the higher plasma density, the particle acceleration is also more efficient than with the torches previously described. 4. RF Plasma Torches. A typical commercial torch is shown in Fig. 4. The ceramic chamber internal diameter is about 50 mm for a 50 kW torch, thus resulting in much lower flow velocities (about 100 m/s). The wall is protected by a strong vortex in which a few vol% of H2 can be introduced. The plasma-forming gas is mainly argon to limit constricting it and requiring more dissipated energy. Because of the inductively coupled power, some sort of skin effect is observed, and the maximum temperature (below 9000 K) is off-axis. The central part of the plasma is heated mainly by conduction and convection. The gas flow may recirculate, especially at the top of the plasma. That is why particles are injected axially with a water-cooled probe at the level of the first loop of the coil. Larger particles (up to 150 애m) can be sprayed because of the lower gas velocity. This offers some advantages for easily oxidized materials, such as titanium–aluminum alloys, used to make the matrix of a carbon–carbon fiber reinforced material. Such torches

Plasma gas

Powder or wire

Shock diamond Plasma jet Extended arc Cooling water in Dc power (–)

Cooling water out Dc power (+)

Figure 3. Schematic diagram of a dc vortex plasma torch (from Ref. 15).

44

THERMAL SPRAY COATINGS Powder + carrier gas Gas distributor head Sheath gas Central gas

Torch cooling water in Intermediate tube

Torch cooling water in

Torch body

Powder injection probe

the complexity of these systems for industrial production (see Fig. 5) (18). 6. Nozzle Shrouds. To avoid the cost of CA chambers, water-cooled nozzle shields are located at the torch nozzle exit. Similar to Laval nozzles, they are especially designed to eliminate low pressures close to the nozzle shield wall to avoid a countercurrent flow that entrains air within the plasma jet. Although these devices consume 10 to 30% of torch energy, they reduce particle oxidation during their flight and are used commercially to spray superalloys (19). Plasma Particle Heat and Momentum Transfer

Plasma confinement tube Induction coil

Exit nozzle

Figure 4. Schematic diagram of an RF plasma torch (from Ref. 17).

operate at pressures ranging from 10 kPa to 200 kPa (17). 5. Controlled Atmosphere or Soft Vacuum. Controlled atmosphere (CA) or soft vacuum spraying (VPS) are used to confine hazardous materials (toxic beryllium or pyrophoric magnesium, titanium) and restrict forming oxides that would occur in air spraying. A controlled-atmosphere (CA) chamber includes a tight box, watercooled if below a few m3, the plasma gun and its mechanical equipment, a vacuum pumping system, and an inert gas backfill manifold. The chamber is pumped down to a low pressure (a few pascals) and then backfilled with high-purity, dry argon to maintain oxygen levels below 30 ppm. When the plasma jet flows in an argon atmosphere, it is longer and fatter than in air because there is no dissociation and ionization of the entrained gas, both of which consume a lot of energy. These CA chambers are used to spray carbides, borides, and similar materials. Thick coatings (up to a few cm) are achieved when the coating and substrate are cooled by liquid argon droplets during spraying. When the pressure of the chamber is maintained at 10 kPa to 30 kPa during spraying, long plasma jets (up to 60 cm) at lower temperatures (앑4000 K to 5000 K) are obtained. The optimum spraying conditions, especially the chamber pressure, vary with the chemistry and particle size of each spray material. VPS leads to closer control of the composition and morphology of sprayed coatings, characterized by greater structural homogeneity, very low level of oxides, improved hardness, thickness capability, and deposition efficiency, and much better adhesion because of diffusion (for example a superalloy substrate can be kept at 950⬚C during spraying). However, the capital cost of such equipment increases considerably (by one to two orders of magnitude) depending on

Many models (20) and measurements (21) have been devoted to the study of plasma particle interactions. The heat and momentum transfer to a particle depend on its residence time within the hot zone of the plasma and the plasma-forming gas composition. The heat and momentum transfer is by far more complex (20) than in flames because of the high temperature gradients around the particle, its evaporation, and the Knudsen effect. This last noncontinuum effect is observed in plasmas at atmospheric pressure because the gas species mean free path ᐉ is already a few micrometers which is not very small compared to particles whose mean size is often around 20 애m to 30 애m. Of course, this effect is enhanced in VPS where ᐉ increases when the pressure chamber decreases in size. In spite of long jets, for example, at 10 kPa, melting particles, whose size is more than 40 애m and melting temperature more than 1500⬚C, is almost impossible. In fact, the difficulty in melting a particle increases as the factor Q/ 兹␳ where ␳ is its specific mass and Q ⫽ (m ⭈ Cp ⭈ ⌬T ⫹ mLm) where m is the mass of the particle, Cp its specific heat at constant pressure and Lm its latent heat of fusion. Q/ 兹␳ for alumina is almost three times the value for molybdenum in spite of the higher melting temperature of the latter. When the particle has low thermal conductivity, compared to the mean thermal conductivity of the plasma, heat propagation occurs and the particle surface starts to evaporate when its center is still cold. This is the case for zirconia particles in an Ar ⫹ 25 vol% H2 plasma. Thus the stand-off distance has to be long enough

Traverse axis

Three-axis robotic gun manipulator

Up-down axis

Pitch

T

s ran

lat

Sting rotation

ion

Two-axis part manipulator (“sting”)

Figure 5. Schematic diagram of a soft vacuum plasma spraying chamber and its utilities (from Ref. 18).

THERMAL SPRAY COATINGS

for the particle to be fully molten. In general, better coatings are obtained with dense particles than with porous particles, such as agglomerated particles because thermal conductivity decreases for porous particles. The key parameter for achieving fully molten particles is the method of injection into the plasma jet. With most dc torches, particles are injected radically to the plasma jet with the help of a carrier gas. However, it is a complex phenomenon (13). Particles have a size distribution and with the carrier gas they get a velocity distribution within the injector. Moreover, the particles collide with the injector wall, especially the finer ones, and with each other. Thus, at the injector exit they exhibit divergent trajectories, especially for those whose diameter ⬍ 20 애m. The optimum trajectory depends on the force balance between that imparted by the carrier gas and that resulting from the drag of the plasma. It is difficult to achieve an optimum trajectory for all of the particles and injection, in spite of its simplicity, has to be carefully checked. The position of the injector tip, injector tilting, and carrier gas flow rate may also perturb the plasma (22). Finally, the stand-off distance d is a compromise between good particle melting, which is achieved for dc plasma torches at distances between 60 mm and 70 mm downstream of the nozzle exit, and good control of substrate and coating heating by the plasma heat flux, which varies as exp(⫺d/d0). Process Control Developing and using efficient controls for thermal spraying processes is absolutely necessary to produce coatings more consistently day after day. Most of the conventional systems control the arc current and voltage and the gas flow rates. For example, when the voltage decreases with time because of electrode erosion, a slightly higher percentage of H2 is introduced to keep it constant. In addition to the energy parameters, the spray pattern and the carrier gas flow rate are also controlled very often by computerized systems which adapt all of these parameters to preset values for a given coating. Recently (7,23) more sophisticated systems have been developed for simultaneously monitoring a combination of particle parameters: velocity, trajectory, and mean surface temperature. These systems are adapted to a production environment and take advantage of hot particle radiation which makes it possible for them to react to the process and correct any drift detected. In the author’s opinion, the most important operating parameters to control are the particle trajectories and the coating surface temperature. Different optical and noncontact techniques have also been developed for continuously measuring coating thickness. However, establishing a control strategy for a thermal spray process as a whole is very complex because the process involves many variables that concurrently influence the properties of the deposited coatings. A lot of work remains to be done in this field. Auxiliary Equipment In addition to the cleaning and grit blasting systems, power supply, pressurized deionized water cooling system, and torch and robot control panel, plasma spraying requires a specially designed, soundproof spray booth (a supersonic plasma gun can generate a noise up to 140 db), that is ventilated and allows keeping particles air borne (mini-

45

mum diameter, gas flow rate, and velocity) and a water wash efficient cleaning system; an exhaust system that has ventilation equipment and adapted powder collectors which comply with safety regulatory requirements. The torch and robot control equipment are located in a room close to the booth from which the operator observes and controls the operation through a window (with UV filters) or with cameras. ARC SPRAYING In this process, illustrated in Fig. 6, two consumable wire electrodes automatically advance at a controlled rate to meet at a point and form an atomizing gas stream. A potential difference of 18 to 40 volts applied across the wires initiates an arc that melts the tips of the wires. The molten metal is atomized and propelled onto a prepared substrate by a stream of compressed gas, which is air in most cases. The design of the atomizing nozzle is very important (24). The arc source is at a constant potential, and usually arc current is limited to 400 A. The wire control unit is composed of two reel or coil holders, insulated from each other. Thus, this process is limited to relatively ductile, electrically conductive wire about 1.5 mm to 1.8 mm in diameter. In most instances this process is less expensive to operate than other processes discussed in this article, and the power consumption is low. Only fully molten particles are propelled to the substrate which is heated almost entirely by the impinging molten droplets (no flame touches the substrate). In general this process has high deposition rates, from 20 kg/h to a maximum of 55 kg/h, and rather high bond strengths (often ⬎50 MPa). SPRAYED MATERIALS As already noted, any material whose melting temperature is at least 300 K lower than its vaporization or decomposition Compressed air Air control Air cleaner AC power

Wire reels

Arc spray gun

Dc power supply (a)

1 Wire

5 Air

1 Wire

Atomizing nozzle 2 3 4 3 2 Contact tips Wire guides (b)

Figure 6. (a) Arc spray components. (b) Schematic cross section of an arc spray gun (from Ref. 1).

46

THERMAL SPRAY COATINGS

temperature can be sprayed. Sprayable materials include metals and alloys (including self fluxing alloys), oxides, cermets (such as carbides or graphite cladded with Ni and Co), carbides and borides (sprayable only in a controlled atmosphere), and composites (e.g., superalloys agglomerated with polyimide). The powders are prepared by different techniques, such as (2,25): gas or water atomization for metals and alloys (almost spherical particles) fusing or sintering for oxides, carbides, and cermets (blocky and irregular shapes) spray drying, a very versatile technique (nearly spherical shapes), but the mechanical resistance of powders is poor and they have to be sintered to reinforce them cladding where a core of one material is coated with a porous or dense layer of another material many other methods, such as the sol-gel technique, mechanofusion and so forth All of these techniques allow preparing a wide variety of alloys, composites, and cermets tailored to the application and make spraying a unique technique for producing coatings suited to a wide range of service conditions. For example, spraying allows producing metallic coatings reinforced with homogeneously dispersed ceramic particles (in the micrometer range). The key parameters for a ‘‘good’’ powder are its flowability, low porosity, and size distribution. In most cases, particles range in size between 10 애m and 45 애m (below 10 애m powder flowability is very poor) for ceramics. For metals and alloys, in general, the size range is between 45 애m and 90 애m. Of course, wider distributions (for example from 10 애m to 106 애m) can be used to produce porous coatings. In wire arc spraying the choice is more restricted because of the need for an electrically conductive and ductile wire.

APPLICATIONS Choosing a thermal spray material for an application is more complex than selecting a wrought or cast material for the same application because coating properties are not as predictable as those of conventional materials. However, now many applications are well established and new ones are being developed that allow pairing optimum base material and surface coating properties to obtain a combination of characteristics that would not be possible with homogenous materials. Because the aircraft and aerospace industries have provided an ideal proving ground for testing and integrating a few coating concepts, this technology has advanced steadily to the point where it has increased the credibility and reliability of coatings. The success in manufacturing aircraft components has led to applications in other markets, such as the paper, printing, steel, metal processing, textile, synthetic fibres, chemical, oil and gas, automotive, plastic, glass, pumps, pneumatic and hydraulic systems, mechanical engineering, shaped part, turbine, nuclear, electronic, and electrical industries (1–4,26). The most common applications today are the following:

• Minimization of wear, caused by fretting, sliding, impact, abrasion, erosion and other service conditions. Hard metal coatings, such as Cr-Ni-B-X alloys where X represents Nb, Ta Ti, self-fluxing alloys of Fe, Ni, Co, high carbon steels, NiCr alloys, and Mo, are used to improve the wear properties of low alloy steel and nonferrous metal surfaces. Al2O3 and Al2O3-TiO2 coatings are used in the textile industry. Cr2O3 coatings are used in pumps and printing presses. Cr2O3-SiO2 coatings are used to absorb energy from impact loading. The carbide cermets are very popular for landing gear components, valves in oil drilling equipment, ‘‘Yankee’’ dryers in paper machines (coated on site), for large rollers in sheet steel production, grinding wheel mold rings, and cam followers. • Wear resistant coatings often exhibit good resistance to heat and chemical attack. When using ceramic coatings this is achieved by using VPS bond coats. • Corrosion protection coatings are used to protect iron and steel in a range of corrosive environments. The particular environment dictates the appropriate alloy selection. Sealed austenitic stainless steel, aluminum bronze, and Co and Ni base alloys are often used below 200⬚C. The sealant, for example, is Teflon or epoxy (below 150⬚C) or Cu or Zn deposited by electrolysis for higher temperature applications (⬍400⬚C). Zn and Al provide the broadest atmospheric protection and represent the biggest successful commercial application of wire arc spraying. Spraying is used to protect structural steel work, parts subjected to sulfurous atmospheres, offshore assemblies, pipelines, and ships. • Oxidation protection: Ni and Co base, Ni, Cr and Co alloys and superalloy coatings are the most popular protection. Oxidation protection requires fully dense coatings and the superalloys are sprayed at high temperature (앑950⬚C) under low vacuum. • Liquid metal protection: Extrusion dies are coated with NiCr doped with Al2O3 particles or with CrO3, Cr2O3 ⫹ TiC. MgO ⫹ ZrSiO4 coatings are used for protection from liquid copper, stabilized zirconia against liquid zirconia, and so forth. • Thermal barrier coatings are used in diesel engine power assemblies (piston dome heads and valves), gas turbines, and aircraft engines (not yet on turbine blades). The material used is ZrO2 stabilized with CaO or MgO for Ts ⬍ 800⬚C and with Y2O3 or CeO2 for higher temperatures. • Heat transfer improvement can be achieved by using coatings with better thermal characteristics, such as Cu or Al or BeO for ceramics when electrical insulation is mandatory. • Electrical properties: Electrically insulating surfaces (mostly oxides and especially Al2O3) are used for dielectrics (for example, ozonizers), high temperature strain gages, oxygen sensors (ionic conduction of zirconia), and insulation of induction heating coils. Electrical contacts are produced with Al or Cu coatings. Resistors are made of NiCr, superalloys doped with Al2O3 particles. • Self-lubricating coatings containing free carbon or MoSi2 are plasma sprayed.

THERMAL SPRAY COATINGS

• Electromagnetic applications: Electromagnetic or radio frequency interference which can damage electronic components or interfere with low level signals can be reduced by shielding. This is achieved by a metallic coating that is wire-arc sprayed on the inside surface of an instrumentation cabinet, which is usually made of plastic or a composite material. Microwave integrated circuits (MIC) are made of plasmasprayed Mg-Mn ferrite. Al, Ta and Nb capacitor electrodes are plasma-sprayed in air. Sputtering sources are also plasma-sprayed. • Dimensional buildup for salvage of worn or overmachined parts. All thermal spraying processes are used for this salvage work and very often NiCr or NiAl are used. • Clearance control: Abradable coatings wear preferentially when contact is made with the mating part, thus making possible to establish the necessary clearances automatically. High technology abradable coatings consists of two powders blended or agglomerated, for example, a metal matrix and a nonmetallic filler, such as graphite, polyester polymide, boron nitride, or a friable mineral. • Free standing shapes: Parts are fabricated from hard-tomachine materials by building up a coating on a removable form. This process is less costly than conventional processing methods, such as sintering or hot isostatic pressing. Ceramic tubes 1 m in diameter, 10 m long and with wall thickness of a few mm can be produced. Ceramic membranes, rocket nozzles, and ceramic crucibles are manufactured. • Mold making is an important application of wire-arc spraying. A prototype part is coated with a mold release and then sprayed with tin-zinc, or NiAl, or similar suitable material to form the mold face. Backup is provided by a filled epoxy cast onto the mold face. • Nuclear applications: In addition to the usual mechanical applications (however, with short lifetime elements, no Co, for example), coatings are used in Tokamak reactors and in magnetic fusion devices. • Prostheses (orthopedic and dental): Alumina is used on the parts where friction occurs, and hydroxyapatite (of a composition similar to that of bone) is used to ensure a reliable contact between the implant surface and the living bone tissue to replace cement.

The Plasma Torch Figure 7 shows a typical plasma torch used for PTA. It consists of a stick type thoriated tungsten cathode (5 mm to 8 mm in diameter) with a conical tip, a nozzle which can be used as an anode to form a pilot blown arc, a powder injector at 30⬚, and a shroud gas (very often Ar ⫹ 5 to 10 vol% H2, except when working with metals sensitive to H2) that is injected around the plasma. The arc is transferred to the substrate, which forms the anode, where a molten pool is created. The aim of the pilot arc is to allow better control of the transferred arc current. For a given anode–cathode distance, nozzle dimensions, and plasma-forming gas flow rate, there is a minimum transferred arc current to sustain it as a cathode jet-dominated (CJD) arc (27). The pilot arc, usually operating at a low power level, ionizes the gap between anode and cathode and heats the substrate allowing a CJD arc with much lower currents than those necessary without it. The control of heat transfer and size of the molten pool created at the arc root together with the plasma velocity close to the pool are the key parameters for achieving good coatings. In fact (27) when a CJD arc is obtained, the heat flux is distributed over a larger area and thus it can be one order of magnitude less than that obtained with an anode jet-dominated arc, thus limiting drastically the depth of the molten pool and its evaporation. CJD arcs are obtained when the anode is directly heated by the convective flux of the plasma. This convective effect is increased when the nozzle constricts the arc column, but it increases the plasma flow velocity and may result in turbulence of the molten pool, even blowing it out. That is why, in most cases, the nozzle diameter is larger than higher (a few millimeters) whereas the anode–cathode distance is below 10 mm. However, recently PTA torches with an important anode striction have been designed to achieve a higher density of energy, thus limiting the size of the heat-affected zone (28). Such arcs work with pure argon, as the plasma-forming gas, at a low flow rate (for example 3 slm for 200 A). Sometimes, the plasma gas is pure helium which reduces its momentum at the anode pool but increases the gas temperature at the anode. Three different types of torches are used depending on their melting capacity as shown in Table 1. Increasing the transferred arc current requires higher plasma-forming gas

• Decoration: By changing the metal oxidation or the TiO2 stoichiometry, attractive colors are achieved. This list is far from exhaustive but shows the wide applications of plasma sprayed coatings.

PLASMA TRANSFERRED ARCS The plasma transferred arc (PTA) process adds the capability of substrate surface heating and melting to plasma-arc spray. The advantages of this direct heating are metallurgical bonding, high density coatings, high deposition rates (up to 24 kg/ h), and high thickness per pass (up to 14 mm).

47

Figure 7. Schematic diagram of the PTA process.

48

THERMAL SPRAY COATINGS

Table 1. Characteristics of Available PTA Torches ⬍150 ⬍200 6

Transferred arc current, A Pilot arc current, A Melting capacity, kg/h

⬍200 ⬍300 10

⬍250 ⬍500 24

flow rates and powder carrier gas to achieve a higher powder carrying capability. The Deposition Parameters Plasma-forming gas flow and transferred arc current affect the quality of the deposit. When too high turbulence is generated (29), more pores and oxides are created. The evaporation of certain alloys also increases. Both of these two parameters must be adjusted to the powder flow rate. The powders have to be injected so that the arc strikes mainly on the powders and not directly on the substrate. When increasing the powder feed rate, the dissipated powder must also be increased. A good size range for the particles is ⫺180 애m ⫹ 53 애m. Complete melting is not achieved beyond 200 애m, and below 53 애m the dilution is drastically increased. Particle injection as close as possible to the tip of the cathode is also important. For a 10 mm anode to cathode distance, residence time is about 1 ms to 2 ms, insufficient time to melt particles completely but allowing them to melt partially and stick to the substrate. Torch speed is a key parameter in reducing the dilution rate, that is, allowing melting of the base material to the smallest possible depth. This is achieved by adjusting the ratio of heat impact/torch speed. The torch speed v for given arc conditions can be calculated (28) by v = dp

a 5p2

where d is the diameter of the anode spot, a is the base material diffusivity, and p is the depth of the molten base material. It can be readily seen that p is reduced if v is increased. However, too high speed causes back fusion and also pores, and if v is too low, the plasma arc is directed on top of the molten pool and increases the porosity. Finally the speed has to be adjusted so that the arc root is in front of the molten pool. The speed and the thickness of the pass also control the coating microstructure. A higher speed and thinner pass result in a finer microstructure. Oscillation width is also an important parameter: If the energy input is too low for a given width, the molten pool is smaller than the width of the layer, and the layer has to be remelted at the next oscillations of the torch past the coated surface. This may cause incomplete melting between the base material and the deposited layer. If the energy is too high, the entire width of the layer is molten, and its protection against oxidation is no longer assured. In general, an oscillation width of 30 mm is a maximum. The oscillation must also be adjusted to give a satisfactory overlap. Small ‘‘ripples’’ are obtained with a feed in the surfacing direction shorter than 3 mm at each double oscillation of the torch. The overlapping also depends on the wettability of the surfacing material. To conclude, when deposition parameters are well adjusted, PTA results in low dilution of base material (10% for single pass coatings), a smooth surface finish, limited splat and clean pro-

cessing, fine microstructures, automated processing, a broad range of coating thickness in one pass (from 0.5 mm to 14 mm), and high deposition speed. The drawbacks are a limited range of alloys, compared to other spray techniques, and expensive, bulky equipment. Deposited Materials and Applications The materials used in PTA spraying are mainly hard alloys that have good corrosion resistance at high temperatures. Most of them are Ni, Co, and Fe alloys, heat resistant steels, and Mn-hardened steels. However, recently new chromium base alloys (Cr-Ni-X such as Cr-Ni-Si-B with a chromium-rich primary crystal and eutectic) have been developed successfully. To increase the wear resistance without affecting other properties, composites are used that consist of a blend of matrix alloy and harder particles. For example, fine tungsten carbide (WC) particles are used against erosion or abrasion. Because their density is higher than that of the matrix, they sink in the coating during deposition without going through a liquid stage. Another route to improve wear resistance is to change the alloy system and rely on the precipitation of hard particles, for example, NiCrMo alloys. In all cases, however, the choice of the base material-deposited alloy couple is very important for avoiding cracks (not too high an expansion mismatch and reasonable quantity of hard phases). Applications can be divided into two groups: Reclamation of a worn, corroded, or overmachined part. By properly choosing the deposited alloy, the restored part can have wear and/or corrosion resistance higher than the initial part. Reinforcement of a new part when special resistance to corrosion, oxidation, or wear is required. This is the case, for example, with valves in the automotive industry, guide rollers in rolling mills, rollers in continuous casting machines, teeth of brown coal dryers, cutting and stabilizer pads of thermal drilling equipment, roller-crushers in cermet plants, and heat exchangers in corrosive environment. FINISHING TREATMENTS Thermal spray coatings or PTA deposits have common features: a sandpaper-like surface finish of a coating and a wavy shape for PTA deposits; and for coatings a porous structure (2 to 17 vol% depending on the process and sprayed material). In many cases, coatings or deposits must have a high surface finish and, when exposed to corrosive environment, the coatings must be sealed (PTA deposits are generally fully dense). Sealing The pores (or cracks) of coatings must be filled to avoid infiltration by fluids or corrosive media that would contribute to premature failure. Sealing materials are petroleum-based wax, phenolic resins, polyester resins, vinyl, aluminum silicate, silicone, and electroplated metals. Most resins are limited to temperatures ranging from 90⬚ to 250⬚C, but silicone can reach 400⬚C. For electrodeposited metals, the maximum operating temperature depends principally on the coating material.

THERMIONIC CONVERSION

Surface Finishing Machining can be used for most of the metals and alloys. Depending on their hardness, either high speed steel or carbide tools are used. Grinding can be used for metals and alloys and is mandatory for ceramics. In the latter case, care must be taken to avoid crack formation, and wet grinding is generally better than dry grinding. Polishing, burnishing, grinding, honing, lapping, and superfinishing are used for surface finish. A roughness of Ra ⬍ 0.1 애m can be achieved with metal and alloys surfaces, but the best value for ceramics is a Ra 앑 0.4 애m to 0.5 애m. BIBLIOGRAPHY 1. Thermal Spraying, Miami, FL: Amer. Welding Soc., 1985. 2. L. Pawlowski, The Science and Engineering of Thermal Spray Coatings, New York: Wiley, 1995. 3. P. Fauchais and M. Vardelle, Plasma spraying: Present and future, Pure Appl. Chem., 66: 1247–1258, 1994. 4. F. N. Longo, Industrial guide-markets, materials and applications for thermal-sprayed coatings, J. Therm. Spray Technol., 1 (2): 143, 1992.

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7. P. Fauchais et al., Formation of plasma sprayed oxide coatings, in H. Y. Sohn, J. W. Evans, and D. Apelian (eds.), The Julian Szekely Memorial Symposium on Materials Processing, T.M.S., 1997, pp. 571–592. 8. S. Kuroda, T. Dendo, and S. Kitahara, Quenching stress in plasma sprayed coatings and its correlation with the deposit microstructure, J. Therm. Spray Technol., 4 (1): 75–84, 1995. 9. L. Bianchi et al., Microstructural investigation of plasma-sprayed ceramic splats, Thin Solid Films, 299: 125–135, 1997. 10. M. F. Smith, R. A. Neiser, and R. C. Dyhuizen, An investigation of the effects of droplet impact angle in thermal spray deposition, in C. C. Berndt and S. Sampath (eds.), Thermal Spray: Industrial Applications, OH: ASM Int., 1994, pp. 603–608. 11. T. W. Clyne and S. C. Gill, Residual stresses in thermal spray coatings and their effect on interfacial adhesion: A review of recent work, J. Therm. Spray Technol., 5 (4): 401–418, 1996. 12. S. Kuroda, Properties and characterization of thermal sprayed coatings. A review of recent research progress, Proc. ITSC ’98, Nice, France, 1998 (to be published). 13. P. Fauchais and A. Vardelle, Thermal plasmas, IEEE Trans., 25 (6): 1258–1280, 1998. 14. M. I. Boulos, P. Fauchais, and E. Pfender, Thermal Plasmas: Fundamentals and Applications, New York: Plenum, 1994, Vol. 1. 15. T. Morishita, Coatings by 250 kW Plazjet system, in S. BlumSandmeier et al. (eds.), 2nd Plasma Technik-Symposium, Wohlen, CH: Plasma Technik, 1: 137–142, 1991. 16. B. Pateyron et al., Sound velocity in different reacting thermal plasma system, Plasma Chem. Plasma Proc., 16 (1): 39–57, 1996. 17. J. Jurewicz and M. I. Boulos, High energy density induction plasma system for materials processing, in C. C. Berndt and T. F. Bernecki (eds.), Thermal Spray Coatings: Research, Design and Applications, OH: ASM Int., 1993, pp. 89–95. 18. P. J. Meyer and D. Hawley, (Electro-Plasma Inc.), LPPS production systems, in T. F. Bernecki (ed.), Thermal Spray Coatings:

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Properties, Processes and Applications, OH: ASM Int., 1991, pp. 29–38. P. Sahoo and R. Raghuraman, Gator-Gard applied blond coats for TBCs, in C. C. Berndt and T. F. Bernecki (eds.), Thermal Spray Coatings: Research, Design and Applications, OH: ASM Int., 1993, pp. 369–374. M. Boulos et al., Fundamentals of plasma particle momentum and heat transfer, in R. Suryanarayanan (ed.), Thermal Spraying, Singapore: World Scientific, 1993, pp. 3–60. P. Fauchais et al., Diagnostics of thermal spraying plasma jets, J. Therm. Spray Technol., 1 (2): 117–128, 1992. B. Dussoubs et al., Injection and trajectories of particles in dc plasma spray processes, Proc. 12th Int. Symp. Plasma Chem., Univ. of Minnesota, Minneapolis, 1995, Vol. 3, pp. 1343–1348. C. Moreau, Towards a better control of thermal spray process, Proc. ITSC ’98, (ed.) C. Coddet (Pub.) ASM Int., OH, USA 2, 1998, pp. 1681–1692. X. Wang et al., Effect of shrouded CO2 gas atomization on coating properties in wire arc spray, in C. C. Berndt and S. Sampath (eds.), Thermal Spray Science and Technology, OH: ASM Int., 1995, pp. 31–38. E. J. Kubel, Jr., Powder dictate thermal, Adv. Mater. Process., 12: 24–32, 1990. J. H. Clare and D. E. Crawmer, Thermal spray coatings, Surf. Clean., Finish. Coat., 5: 361–374, 1990. E. Pfender, Heat transfer in thermal plasmas, in The Julian Szekely Memorial Symposium on Materials Processing, H. Y. Sohn, J. W. Evans, and D. Apelian (eds.), T.M.S., 1997, pp. 475–508. A. Proner, J. P. Dacquet, and R. Romanet, The plasma high energy, a new hard facing technique, in P. Fauchais (ed.), Progress in Plasma Processing of Materials, New York: Begell House, 1997, pp. 787–794. M. Ducos, PTA reclamation, in Electrical processes in surface finishing (in French) ed. (Dopee 85) (Pub.) Electra 38 A. F. Roosevelt 77210 Avon, France, 1989, 147–158.

P. FAUCHAIS A. VARDELLE University of Limoges