Physics and instabilities of low-temperature E × B plasmas for spacecraft propulsion and other applications

In low-temperature $E×B$ plasmas devices, an external magnetic field is placed perpendicular to the discharge current or to the electric field applied between two electrodes. The perpendicular magnetic field increases the electron residence time in the device and the electron collisionality, allowing plasma sustainment by electron impact ionization at low pressures and moderate applied voltage. The perpendicular magnetic field decreases the electron conductivity parallel to the electric field, so an objective of the $E×B$ configuration can also be to generate a large electric field in the quasineutral plasma, which can accelerate ions (application to gridless ion sources such as the Hall thruster). Lowering the electron conductivity with a perpendicular magnetic field is actually not trivial because of the development of instabilities that tend to increase electron transport across the magnetic field. This question is at the center of this tutorial.

In most applications of $E×B$ plasmas, the external magnetic field and the device dimensions are such that electrons are strongly magnetized (Larmor radius much smaller than the device dimensions), while ions are not or are only weakly magnetized. Such plasmas are called “partially magnetized plasmas.” Electron transport along the electric field is, in principle, strongly reduced by the external magnetic field applied perpendicular to the electric field. In most devices, the magnetic field lines intersect a wall (cathode or dielectric surface) so that along the magnetic field line the electrons are electrostatically confined by the wall sheath potential. Electron confinement is therefore a combination of magnetic and electrostatic confinement. Collisionless electrons are trapped in the direction perpendicular to the magnetic field. Only collisions with neutrals can un-trap the electrons from their cyclotron trajectory and allow transport across the magnetic field (“cross field transport”) along the electric field. Electron confinement by the magnetic field in the direction of the applied electric field is generally not as good as predicted by the classical collisional theory because of turbulence and instabilities that lead to cross field transport (“anomalous transport”). Because of their partial magnetization, the physics and instabilities of these plasmas are very distinct from those of other plasmas, e.g., for fusion applications.

The study of instabilities and anomalous transport in $E×B$ plasmas has recently attracted much attention, largely driven by the need to better understand the operation of devices, such as Hall thrusters and magnetron discharges, and to develop predictive models of these devices. Joint efforts by the low-temperature plasma community and the plasma propulsion community “to develop a rigorous understanding of the rich phenomena observed in $E×B$ devices” are described in a recent perspective article by Kaganovich et al.¹ that “presents the current understanding of the physics of these phenomena and state-of the-art computational results, identifies critical questions, and suggests directions for future research”. The present tutorial article can be considered as complementary to the perspective article by Kaganovich et al., but is aimed at non-specialists, with more emphasis on the physics of instabilities in various $E×B$ plasma devices.

In plasmas with perpendicular electric field and magnetic field, there is a charged particle drift in the $E×B$ direction (“ $E×B$ drift”) with a velocity close to $E × B / B 2 = E / B$⁠. This drift plays an important role in regions of large electric fields. Electrons are subject to this drift while ions are accelerated out of the large electric field region before they have time to reach the $E×B$ drift velocity (this is another way to say that the plasma is partially magnetized). The large $E×B$ electron drift and the negligible ion drift in this direction can lead to charge separation and is often responsible for triggering instabilities. The electron pressure gradient $∇ P e$ perpendicular to the magnetic field also creates a drift in the $∇ P e×B$ direction. This is the diamagnetic drift velocity, $∇ P e × B / e n B 2 = ∇ n T e / n B$ where $n$ is the plasma density, $e$ the elementary charge, and $T e$ the electron temperature in eV. $E×B$ and $∇ P e×B$ electron drift can both contribute to the development of instabilities. Instabilities are ubiquitous in low-temperature partially magnetized $E×B$ plasmas and can significantly enhance cross field electron transport. They manifest themselves in a wide range of unstable fluctuations, with frequencies from kilohertz to tens of megahertz and wavelengths from a fraction of millimeter to several centimeters.² 

The electron drift in the $E×B$ direction gives rise to the Hall current. Efficient electron confinement parallel to the electric field requires that the Hall current does not intercept a wall (otherwise the Hall effect would destroy the confinement of electrons). This is possible if the Hall current is closed on itself, as for example, in a cylindrical geometry where the electric field is axial and the magnetic field radial, or the opposite. The Hall current in such geometries is in the azimuthal direction, i.e., is closed on itself. Plasma devices with closed electron drift are called closed-drift devices.³ 

The concept of closed-drift devices has led to important applications in ion sources for space propulsion (Hall thrusters, HT, also called stationary plasma thrusters, SPT,^4–7 and thrusters with anode layer, TAL⁵) and plasma processing (End-Hall ion sources,^3,8,9 closed-drift anode layer ion sources, ALIS,^3,10 or Anode Layer Plasma Accelerators, ALPA^3,11) The originality of these “Hall ion sources” lies in the fact that ions are extracted from the plasma not by a system of biased grids (they are “gridless ion sources”), but by the electric field generated in the quasineutral plasma by the local lowering of electron conductivity due to a magnetic field barrier placed between an emissive cathode and the anode. Magnetron discharges can also be considered as closed-drift ion sources where ions are extracted from the plasma and accelerated in the cathode sheath to sputter atoms from the cathode, with applications to thin film deposition. They can be operated at high power densities in a pulsed regime (HiPIMS, high power impulse magnetron sputtering).^12–15 In magnetron discharges as in Hall thrusters, electrons are heated by the applied dc electric field. They drift in the azimuthal $E×B$ direction, generate the plasma by ionization and slowly drift to the anode due to collisions or instabilities. Penning discharges,^16–19 consisting of two facing cathodes surrounded by a cylindrical anode, with an axial magnetic field, are another type of closed-drift device that has been used in many ion source applications. Hall ion sources, magnetron discharges, and Penning ion sources operate under pressure typically between a fraction of pascals and a few pascals but it is interesting to note that Penning discharges where first implemented as ion gauges^20,21 in the 1930s, operating at much lower pressures. In these conditions, the $E×B$ plasma can become non-neutral (electron density larger than ion density) and specific types of instabilities, called diocotron instabilities^22–26 are likely to develop.

Historical references on the development of $E×B$ devices for electric propulsion can be found in Refs. 3, 4, and 27, for magnetron discharges applications to sputtering and deposition in Refs. 12–15, and for various cross field devices operating over a wide pressure range in Refs. 23 and 28. Recent review or perspective papers on electric propulsion including gridless ion sources can be found in Refs. 29–31.

The question of cross field electron transport or anomalous transport also arises in situations where the plasma simply diffuses across the magnetic field, in the absence of applied voltage perpendicular to the magnetic field. The plasma can be generated by inductive coupling, microwave, of by injection of an electron beam. This is the case of magnetized plasma sources where the plasma is confined by magnetic cusps (applications to plasma processing or gridded ion sources^32–34) or of magnetized cylindrical plasma columns with axial magnetic field. Magnetized plasma columns have been used extensively to study turbulence and anomalous electron transport.^35–43, $E×B$ and $∇ P e×B$ flows combine in these plasmas to ensure a zero net current to the walls. Diffusion is often non-ambipolar and “short-circuit” effects⁴⁴ are possible in the case of metallic walls, i.e., the electron current does not balance locally but globally the ion current and a large fraction of the electron current to the walls is in the parallel direction (along the magnetic field). $E×B$ plasmas have potential applications in mass separation, which requires the imposition and customization of an electric field perpendicular to the axial magnetic field in a plasma column to create an $E×B$ flow.^45,46 Research is ongoing into ways to generate this radial electric field, such as through electrode biasing,^46–48 the use of emissive end-electrodes,⁴⁹ or wave injection.⁵⁰ 

$E×B$ plasma devices, such as Hall ion sources or magnetron discharges, are now currently used successfully in space propulsion and plasmas processing applications. However, despite important research efforts and progress in the last two or three decades, the physics of these devices is still poorly understood and models of these plasmas with predictive capabilities are still lacking. In this article we describe, for non-specialists, the basic physics of low-temperature partially magnetized $E×B$ plasmas, with emphasis on the development of instabilities, and we illustrate with some examples the recent progress in our understanding of these plasmas and their instabilities through experiments, simulation, and theory.

The basic physics and principles of $E×B$ plasma devices are presented in Sec. II. Section III shows how instabilities in $E×B$ plasmas can be described and understood with the combination of theory (dispersion relations) and simulations. Section IV focuses on two important $E×B$ plasma devices, the Hall thruster and the magnetron discharge, with a more detailed discussion of the physics and of the evidence of the presence of instabilities in these devices. Microscopic (kinetic) instabilities in Hall thrusters are studied in Sec. V with the help of a kinetic dispersion relation and particle simulations. The macroscopic instabilities that can be observed in magnetron discharges and under certain Hall thruster conditions (rotating spokes) are described in Sec. VI with particle simulations and comparisons with experiments.

The typical conditions of low-temperature partially magnetized $E×B$ plasmas devices and the orders of magnitude of the plasma parameters are presented in Subsection II A. The classical, collisional transport theory is briefly recalled in Subsection II B. The geometry and principles of the main closed-drift devices are described in Subsection II C. The Hall effect, which occurs in devices with non-closed-drift, is discussed with two examples in Subsection II D.

Under usual conditions of $E×B$ plasmas applications, the effect of the plasma current on the magnetic field is negligible so we will assume that the magnetic field is purely external, i.e., is generated by a system of coils or magnets and a magnetic circuit. This condition may not be satisfied⁵¹ in the extreme conditions of HiPIMS where very high power densities are injected in the plasma through short pulses of several microseconds.

As said in the introduction, we consider partially magnetized plasmas, which implies that the mean electron Larmor radius $ρ e$ is much smaller than the device dimensions $ρ e≪L$ while the ion Larmor radius is not. For a Maxwellian electron velocity distribution at temperature $T e$⁠, the mean Larmor radius is given by $ρ e= v e ⊥ / ω c e$ where $v e ⊥= e T e / m 1 / 2$ is the electron mean velocity perpendicular to the magnetic field, and $ω c e= e B / m$ is the electron cyclotron angular frequency. Orders of magnitude of magnetic field intensity, charged particles cyclotron frequencies, and Larmor radii are given in Table I. Electron–neutral collisions play an important role in applications of $E×B$ plasmas because they control the electron magnetization and contribute to cross field transport (and also because the plasma is sustained by electron impact ionization). Hall ion sources and magnetron discharges operate at pressures typically between 0.1 and 5 Pa and with magnetic field usually smaller than 100 mT. The plasma dimensions are generally less than 5–10 cm. Table II gives some orders of magnitude of collision frequencies, and free paths of electron and argon ions.

Usual values of plasma densities, plasma frequencies, and electron Debye lengths are given in Table III.

Since the electron trajectory around the magnetic field changes after each electron–neutral collision, leading to transport or diffusion across the magnetic field, an important parameter is the ratio of the electron Larmor frequency over the electron–neutral collision frequency. This ratio, $h= ω c e / ν e n$⁠, is called the Hall parameter and characterizes the electron confinement by the magnetic field. As said in the introduction, plasma fluctuations and instabilities also contribute to un-trapping the electrons from their trajectory around the magnetic field. Their effect can be considered as similar to the effect of collisions, and one can define an effective Hall parameter and an effective collision frequency to describe their effect on cross field transport. It can be deduced from Tables I and II that the Hall parameter associated with electron–neutral collisions, for a magnetic field intensity of 10 mT, and a gas pressure of 0.1 Pa is $h= ω c e / ν e n≈1000$⁠.

Note finally that low-temperature partially magnetized plasmas are far from thermal equilibrium, and that the electron temperature (defined as $2 / 3$ of the thermal energy) is much larger than the ion temperature and the charged particle velocity distribution functions can be very far from Maxwellian.

In this section, the classical, collisional expression of the electron current across a magnetic field is derived from a simplified form of the steady-state, electron momentum equation. We neglect the inertia term and assume that the electron pressure tensor is scalar. We consider a plasma with an electric field is in the $x$ direction, the magnetic field in the perpendicular $z$ direction. We assume that the plasma density gradient is parallel to the electric field, in the $x$ direction (as in Hall thrusters or magnetron discharges). The notations are indicated in Fig. 1.

Along the magnetic field lines (in the $z$ direction), electrons move freely between collisions. The electron conductivity parallel to the magnetic field is, therefore, much larger than the electron conductivity parallel to the electric field. An important consequence is that the magnetic field lines tend to be equipotential. More precisely, from the momentum equation along the magnetic field lines and assuming negligible electron current parallel to the magnetic field, the electric field along these lines must balance the electron pressure gradient. The potential drop along the magnetic field lines is therefore small for low electron temperature or small electron pressure gradient. Note that a magnetic field line should never connect cathode and anode in a magnetized plasma device since this would be equivalent to a short circuit at low pressure.

If the plasma is bounded by walls in the $E×B$⁠, $y$ direction, e.g., dielectric walls, there is an electron flow to one of the walls due to the $E×B$ and $∇ P e×B$ drifts. The electron flow to the wall must be balanced by an ion flow. An electric field is, therefore, generated in the $y$ direction to draw the ions to the wall. This is the Hall effect. This electric field, combined with the magnetic field, will in turn create an $E×B$ electron drift in the $x$ direction that will destroy the electron confinement. The question of Hall effect in low-temperature partially magnetized plasmas has been illustrated analytically in the case of a plasma column in a transverse magnetic field by Ecker and Kanne⁵² and by Kunkel,⁵³ and with particle simulations⁵⁴ in the case of the negative ion source for the neutral beam injector of ITER. This is summarized in Subsection II D. The solution to avoid the Hall effect is to choose a geometry where the $E×B$ electron current is closed on itself. This is possible in a cylindrical geometry where the electric field is axial and the magnetic field is radial or the opposite. Devices with this configuration are called closed-drift devices³ and are described in Subsection II C.

Some classical examples of closed-drift devices are displayed in Fig. 2. In Penning cells and cylindrical magnetrons discharges [Figs. 2(a) and 2(b)], the external magnetic field is axial, and the applied electric field is radial. The concept of Penning discharge [Fig. 2(a)] was introduced by Phillips at the end of the 19th century⁵⁵ and was used for the first time in 1937 by Penning, with the construction of an ionization gauge.²⁰ The different modes of a Penning discharge with a long anode cylinder at low pressure were studied by Knauer⁵⁶ and Schuurmann^16,17 and the diocotron instability that develops in Penning discharge at low pressures was described by Knauer.²² More recent experimental work on Penning discharges and their different regimes can be found in the article by Mamedov et al.⁵⁷ 

Penning discharges are currently used as ion sources¹⁹ (ions can be extracted through an aperture in one of the two cathodes or through the anode) in various applications, e.g., to sputtering and deposition, ion implantation,⁵⁸ isotope separation, production of hydrogen negative ions,⁵⁹ and neutron generators.⁶⁰ The electron vortices associated with the diocotron instability, which develop in Penning discharges and in cylindrical magnetrons at very low pressures, were extensively studied by Kervalishvili and his group^25,61–64 with the help of probe measurements and described with particle simulations in Ref. 26. Cylindrical magnetrons [Fig. 2(b)] consist of two concentric cylindrical electrodes with an axial magnetic field. Early sputtering-deposition applications were demonstrated in cylindrical magnetrons,^14,65 but the planar magnetron configuration [Fig. 2(c)] is now much more commonly used for deposition of metallic or compound thin films in many industrial applications. The planar magnetron can operate in a dc regime in the pressure range 0.2–4 Pa with applied voltages of 300–700 V and a magnetic field of 20–50 mT close to the cathode.¹⁴ Very high power densities can be achieved in a pulsed regime (HiPIMS)^13,15 with repetition frequency in the range 50–5000 Hz and 1%–3% duty cycle,^12–14 applied voltage in the range 500–2000 V and with peak power densities up to several kW/cm². The plasma density is $10 16$– $10 18$ m⁻³ in the dc case and can be larger than $10 18– 10 19$ m⁻³ in the pulsed case. The physics of HiPIMS is extremely complex because of the high plasma density, ionization of the sputtered atoms, instabilities, neutral depletion, etc.

The magnetic field in planar magnetrons is generated by magnets placed behind the cathode as schematically shown in Fig. 2(c). In a Hall ion source [Fig. 2(d)], the objective is to generate an electric field in the quasineutral plasma and to use this field to extract ions from the plasma. The basic idea is, therefore, to place a magnetic barrier perpendicular to the applied electric field between an emissive cathode and the anode, as shown schematically in Fig. 2(d).

Figure 3 shows some practical examples of Hall ion sources. In a Hall thruster^5,7,66 [Fig. 3(a)], the plasma is formed in a channel between two concentric cylinders. The anode is at one end of the channel, and an electron emissive cathode is placed outside the channel. A radial magnetic field with a maximum at the other end of the channel (“exhaust plane”) is generated by coils or magnets and a magnetic circuit. This magnetic field forms a magnetic barrier between cathode and anode. The applied electric field tends to be larger in the region of maximum magnetic field because of the lowering of the electron conductivity due to the magnetic barrier. Ions generated in the channel are extracted by this electric field. The external emissive cathode provides electrons to sustain the discharge as well as to neutralize the extracted ion beam. Several variations around this concept have been proposed and developed. Some of these designs are described in recent review papers.^29,30,67

Here, we only give a brief description of some of these designs to illustrate the different possible $E×B$ configurations. The configuration of a magnetically shielded Hall thruster^68–70 is shown in Fig. 3(b). The magnetic barrier in this thruster is such that the magnetic field in the channel, close to the exhaust plane is parallel to the dielectric walls. Since the magnetic field lines tend to be equipotential (because of the large electron conductivity along these lines), this configuration limits the electron and ion fluxes to the walls and reduces ion erosion in this region. The cylindrical Hall thruster^71–73 [Fig. 3(c)], where the inner cylinder is shorter, is more adapted to small or low-power thrusters since, according to the empirical scaling laws of Hall thrusters, the channel width of the channel between the two dielectric cylinder should decrease with decreasing power. The magnetic field barrier in this thruster is not perpendicular to the thruster axis so the electric field is not perfectly axial, and the divergence of the ion beam is larger than in standard Hall thrusters. The anode layer thruster (or thruster with anode layer, TAL)^74,75 is shown in [Fig. 3(d)]. In this thruster, the anode is very close to the magnetic barrier and the dielectric walls are no longer needed and are replaced by metallic walls. Ions are accelerated in a thin region close to the anode. Ion sources based on the same concept as TAL are also used in plasma processing applications under the name of ALIS^3,10 (anode layer ion sources) or ALPA^11,76 (Anode Layer Plasma Accelerators). Another thruster configuration where the anode is in the magnetic barrier is the wall-less thruster⁷⁷ of [Fig. 3(e)] proposed by Kapulkin et al.⁷⁸ in the 1990s, and by Mazouffre et al.^79–81 later. A variation of a wall-less thruster is based on a standard Hall thruster with a short channel but with a ring anode placed at the channel exit. The ion acceleration in this design takes place outside the channel. The advantage of the configuration of Fig. 3(e) is its simplicity and reduced erosion problems, but the divergence of the ion beam can be an issue. Finally, gridless ion sources are not only used for spacecraft propulsion but also for plasma processing applications. This is the case of the ALIS and ALPA source mentioned above and of the End-Hall ion sources¹⁰ of Fig. 3(f), with a truncated conical anode and a magnetic field barrier along the anode surface. This ion source is used for ion assisted thin film deposition.⁸² Particle simulations of End-Hall ion sources can be found in Refs. 9 and 83.

A final remark on the magnetic field configuration of closed-drift devices is that in all cases, the magnetic field lines intersect the cathode or some dielectric surfaces. Electrons move freely along the magnetic field lines, so, as said in the introduction, the electron confinement is not purely magnetic but is a combination of magnetic confinement in the direction perpendicular to the applied electric field and electrostatic confinement along the magnetic field lines. In configurations where the magnetic field lines end on the cathode (Penning or magnetron discharges), plasma electrons are efficiently confined electrostatically because of the large potential drop in the cathode sheaths. In configurations where the magnetic field lines intersect a dielectric surface (most of the Hall ion sources), low-energy electrons are electrostatically confined by the wall sheath potential, but higher energy electrons can overcome this wall potential and reach the wall. Electron–wall interactions and the resulting secondary electron emission can play an important role in these devices.¹ 

From this brief review, it appears that in closed-drift devices with axial electric field and radial magnetic field (magnetron discharges and Hall ion sources), there can be three generic types of axial profiles of the radial magnetic field, as shown in Fig. 4. The three profiles correspond to (1), magnetron discharges, with a magnetic field decreasing from cathode to anode, (2) standard Hall thrusters, with a magnetic barrier between cathode and anode, and (3) ion sources with anode layers, where the magnetic field increases from cathode to anode. The choice between these three configurations depends on the application and on the objective to be achieved (performance, lifetime, simplicity,…). We will see in the rest of this paper that the magnetic field profile has also important consequences on the nature of the instabilities that develop in these devices and on their consequences on anomalous electron transport.

The Hall effect occurs when the $E×B$ current is not closed on itself but flows to a wall. There is no Hall effect in closed-drift devices like the Hall thruster even though Hall thrusters are sometimes called “Hall effect thrusters.” To illustrate the Hall effect, we consider, as in Kunkel⁵³ (see also Ecker⁵² for the case of a cylindrical column) a plasma slab between two dielectric or floating walls parallel to the $x$ direction and perpendicular to the $y$ direction, an electric field $E 0= E 0 x ̂$ in the $x$ direction and a perpendicular magnetic field $B=B z ̂$ in the $z$ direction. The electric field $E 0$ must adjust so that charged particle losses to the walls are balanced by ionization, as in a positive column plasma. The $E×B$ electron current, in the $−y$ direction, flows to the wall. These conditions are represented schematically in Fig. 5.

The mean velocity parallel to the electric field $, E x$⁠, averaged in the $z$ direction is $v e x ¯=− μ e 0 1 + h 2( 1 + h 2 α) E 0$⁠. This is to be compared to the mean velocity in the absence of Hall effect, given by Eq. (2), $v e x ¯=− μ e 0 1 + h 2 E 0$⁠. Since the ratio of ion to electron mobility without magnetic field, $μ i / μ e 0$ can be on the order of 10^–2 or less, we have generally $( 1 + h 2 α)≫1$⁠. The electron mobility parallel to the electric field $E x$ in the presence of Hall effect is therefore considerably larger than without Hall effect and, in the limit $μ i / μ e 0→0$⁠, $α→1$⁠, and we have $v e x ¯→− μ e 0 E 0$⁠. We can conclude that the Hall effect tends to destroy the electron confinement perpendicular to the magnetic field.

The simple example above corresponds to an axially uniform plasma column between parallel walls in the presence of a perpendicular magnetic field.

We consider now the more complex example of a negative ion source used for the neutral beam injection system of ITER.⁸⁴ The design of this source is schematically represented in Fig. 6. The source consists of a driver where electrons are heated by inductive coupling, a diffusion region with a magnetic filter, and an extraction region. The magnetic field is perpendicular to the plane of Fig. 6. The plasma is produced in hydrogen or deuterium at 0.3 Pa. The negative ions of hydrogen or deuterium are generated in the plasma volume by attachment on vibrationally excited states, and on the surface of the plasma grid covered with a cesium layer. Hydrogen or deuterium atoms hitting the caesiated grid are converted into negative ions. Negative ions are extracted through a system of biased grids. Negative ions can be destroyed by fast electrons coming from the driver. The objective of the magnetic filter in the diffusion region is to increase the residence time of electrons and increase their energy losses through collisions with neutrals. A second objective is also to lower the density of electrons close to the extraction region and to limit their extraction through the grids (in addition to the magnetic filter, a magnetic field is also generated by magnets near and parallel to each grid hole to limit this flux).

The plasma density gradient from the driver to the magnetic filter due to the large electron pressure in the driver creates a charged particle flux toward the extraction grids. The grid voltage is positive with respect to the grounded walls to enhance the flux of negative ions to the extraction system. In this configuration, there is an axial flux of charged particles perpendicular to the magnetic field. We therefore expect to have a combination of diamagnetic drift and $E×B$ drift in the $y$ direction of Fig. 6. These drifts are not closed and are perpendicular to the rectangular chamber walls, so we expect a Hall effect to occur. This Hall effect is discussed in Refs. 85–88, on the basis of PIC-MCC simulations. We present here a few results from Ref. 85. Figure 7 shows the 2D distributions of the electron density and plasma potential in the ion source. The plasma potential in the driver is much larger than in the diffusion chamber because electrons are heated in the driver and lose energy in the filter.

We observe on the potential and plasma density distributions an asymmetry, which can be attributed to the Hall effect. There is a potential difference (Hall voltage) between the sheath edges on the top and bottom walls. The plasma density is higher on the lower side of the chamber (while the electron temperature is lower on the lower side, see Ref. 89).

Figure 8 displays the intensity and streamlines of the electron and ion current densities in the chamber. We can clearly see the combination of the different electron drifts in the $y$ direction at the driver exit (diamagnetic and $E×B$ drift; a drift due to the magnetic field gradient, proportional to $∇B×B$ is also present). The more important plasma asymmetry predicted by the model at the driver exit has been confirmed by experiments.⁹⁰ Compared with a closed-drift system, the Hall effect enhances the electron flow through the magnetic filter, and no instability is observed in these conditions.

If the walls are removed in the $y$ direction and periodic conditions are used instead (see Ref. 91), the drift is closed on itself and electron confinement in the axial direction is much more efficient. In these conditions, instabilities are present,⁹¹ which is not the case in the bounded conditions of Figs. 7 and 8. Finally, we note, by comparing Figs. 8(a) and 8(b), that the plasma diffusion in this source is clearly non-ambipolar except in the magnetic field free driver. Non-magnetized ions simply go down the potential while the streamlines of the electron current density are much more complex. This feature is typical of partially magnetized plasmas.

The basic way to study the existence of instabilities in a physical system is to look at how perturbations around a given steady state grow in time or are damped. This is done by linearizing the equations characterizing the steady state regime. In plasmas, instabilities are described by linearizing transport equation coupled with field equations. For some instabilities, a fluid approach of the transport equations is sufficient (macroscopic instabilities) but in some cases, a kinetic description is required (microscopic, or kinetic instabilities). The linearization of the equations gives the dispersion relations, i.e., the frequency and growth rate of the perturbations as a function of wavenumber, and are extremely useful to understand the mechanisms responsible for the development of the instabilities. However, this approach can describe only the linear stage of the development of the instability and cannot predict how the instability saturates. Numerical simulations based on fluid or kinetic (e.g., Particle-In-Cell simulations) transport equations must be used to study the evolution from the linear stage to the saturated stage of the instability but the interpretation of numerical results is not always straightforward. The best approach to fully understand the physics of the instabilities is, therefore, to combine the derivation of the dispersion relations by linearization of the equations, with numerical simulations.

$E×B$ plasma devices are complex physical systems, and it should be kept in mind that before trying to develop fully predictive engineering models of these devices, it is important to build simplified or reduced models at the right level of description and approximation. As stated in Ref. 92, “Every good model starts from a question. The modeler should always choose the correct level of detail to answer the question.” This is clearly true for the derivation of analytical dispersion relations but also applies to numerical simulations.

This section is organized as follows. The principles of the derivation of fluid dispersion relations in the context of $E×B$ plasmas are described in Subsection III A. The derivation of kinetic dispersion relations is only briefly discussed in Subsection III B. The kinetic dispersion relation of an $E×B$ plasma in the conditions of a Hall thruster, leading to the electron cyclotron drift instability will be illustrated in Sec. V. In Subsection III C, we recall the basic principles of Particle-In-Cell Monte Carlo Collisions (PIC-MCC) simulations that are often used to study $E×B$ plasmas and their instabilities. Finally, Subsection III D shows some examples of instabilities predicted by PIC-MCC simulations and that can be understood on the basis of fluid dispersion relations.

In this section, we describe some aspects of the derivation of the fluid dispersion relations of a low-temperature partially magnetized $E×B$ plasmas based on Refs. 2 and 93–95. We use the notations of Fig. 1, where $E$ is in the $x$ direction, $B$ in the $z$ direction, and $E×B$ in the $−y$ direction. The plasma density gradient is parallel to the electric field, in the $x$ direction.

We look for solutions of Eqs. (11)–(14) above, of the form $n= n 0+ n ̃exp(−iωt+iky)$⁠, $ϕ= ϕ 0+ ϕ ̃exp(−iωt+iky)$ where $ω= ω R+iγ$ is the complex wave frequency and $k$ the wavenumber in the $y$ direction (i.e., $k≡ k y$⁠). $ω R$⁠, the real part of $ω$⁠, is the wave angular frequency and $γ$⁠, the imaginary part, is the growth rate. The perturbation grows when $γ>0$⁠. In general, the equilibrium and perturbed density, $n 0$ and $n ̃,$ are function of the axial $(x)$ coordinate, and the full nonlocal theory should be used. For the long-wavelength perturbations, $( k L n ) − 1<1,$ the local theory can be used, assuming the density gradients constant.

For $ω 0=0$⁠, the solution of Eq. (21) is purely real. This corresponds to the stable, anti-drift mode.^2,36,96 The $− ω 0$ term in the denominator, corresponding to the $E×B$ flow, is a possible destabilizing term, since, for example, when $ω 0≫ω$ and $ω * / ω 0 > 0$⁠, $ω 2$ becomes negative and Eq. (21) has a positive imaginary root (⁠ $γ>0$⁠) and the instability grows. This instability, which occurs due to the destabilization of the anti-drift mode by the $E×B$ electron flow, is called the gradient-drift instability or the collisionless Simon–Hoh instability or the modified Simon–Hoh instability.^35,36 In the collisionless case, the instability is reactive, i.e., the dispersion relation is real, and the complex roots come in complex conjugate pairs. This is an example of negative energy mode in flowing plasmas and fluids when the instability is caused by the coupling of positive and negative energy modes (a plasma wave is said to have negative energy if energy is removed from the plasma when the wave is excited; energy is added to the plasma in the case of a positive energy mode). Negative energy modes may become unstable solely due to dissipation. As discussed by Sakawa and Joshi,³⁶ the dissipative Simon–Hoh instability was studied by Simon,⁹⁷ Hoh,⁹⁸ and Thomassen⁹⁹ in a weakly ionized, inhomogeneous, collisional, magnetized plasma column and was found to be unstable when the density gradient and electric field were in the same direction (i.e., $ω * / ω 0 > 0$⁠, as in the collisionless case). An important point in the Simon–Hoh instability is that the $E×B$ electron drift is much larger than the ion $E×B$ drift. This difference in electron and ion drift can be due to the weak magnetization of ions or to collisions.

Typically, for unmagnetized ions the ion azimuthal drift is neglected. In the case of weak magnetization $k ρ i≫1$⁠, the ion drift can be estimated as¹⁰⁰  $v 0 i= v 0 I 0 k ρ i 2exp − k ρ i 2$ where $I 0$ is the modified Bessel function (see Sakawa et al.¹⁰⁰ for the derivation of this expression, which is valid only in a slab geometry and does not include centrifugal effects).

These expressions are valid when the plasma current is neglected, i.e., it does not modify the magnetic field and $∇×B→0$⁠. These expressions also show that for a uniform magnetic field, electron temperature gradients and generally electron temperature dynamics are not coupled to the electron density directly (unless the electron temperature changes the density via ionization). For a nonuniform magnetic field, the compressibility of the electron diamagnetic flow in Eq. (29) may involve temperature evolution directly. In that case, the dispersion relation (25) becomes more complex and involve the equilibrium temperature gradients, as described in Frias et al.⁹³ Among other effects, the temperature gradients affect the condition for the instability (27) and may result in the appearance of unstable modes in the direction opposite to the $E×B$ drift.⁹³ 

Because of the condition (29), $∇ · ( n v pe)=0$ in the case of a uniform magnetic field, the diamagnetic drift does not result in the frequency Doppler shift in the electron dynamics, as it occurs due to the $E×B drift,ω→ω− ω 0$⁠, and does not result in additional terms in the Simon–Hoh dispersion relation (24), contrary to the statement in Ref. 102. We note also that the condition $v p e> v T e$⁠, obtained in Ref. 102, is equivalent to the condition $ρ e> L n$⁠, which is not possible in the fluid theory but requires a full kinetic treatment.

The derivation of the dispersion relations (21) and (25) was made neglecting the inertia terms in the electron momentum equation. Also, the approach based on the standard momentum equation assuming scalar pressure cannot describe situations where the wavelength of the instability becomes close to the electron Larmor radius (i.e., when $k 2 ρ e 2≫1)$ and a kinetic approach is necessary.

For very large values of $k ρ e$⁠, the general form of Eq. (30) gives the ion sound mode $ω=k c s$ or $ω= k c s / 1 + k 2 λ D e 2$ for finite Debye lengths.

Note that Eq. (30) can be written in dimensionless units where the frequencies are normalized by $ω L H$⁠, the lengths are normalized by $ρ e$⁠, and the velocities are normalized by $c s$⁠. If we write $Ω = ω / ω L H$⁠, $K=k ρ e$⁠, the roots $Ω(K)$ of Eq. (30) depend on three parameters: $V 0= v 0 / c s$⁠, $V *= v * / c s$⁠, and $N e n= ν e n / ω L H$⁠. If finite Debye length is taken into account, another parameter is $λ D e / ρ e$ (or, equivalently, $ω c e / ω p e$⁠).

We show in Figs. 9 and 10 some examples of solutions $Ω(K)$ of the dispersion relation (30) for different values of $V 0$ and $V *$⁠, with $ν e n / ω L H=6.23$ and $λ D e / ρ e= ω c e / ω p e=0.16$ and a uniform magnetic field. In Fig. 9, the dimensionless diamagnetic drift velocity $V *$ is fixed and the $E×B$ drift velocity $V 0$ is varied. In Fig. 10, $V 0$ is fixed and $V *$ is varied. The corresponding numerical values of $E 0$⁠, $L n,$ and other parameters are given in the figure captions, for a particular case corresponding to argon, a magnetic field intensity of 50 mT, and an electron temperature $T e$ = 5 eV.