Transit-time Instability Research Articles

Plasma Dynamics and Electron Transport in a Hall-Thruster-Representative Configuration with Various Propellants: I—Variations with Discharge Voltage and Current Density

The results from a wide-ranging parametric investigation into the behavior of the collisionless partially magnetized plasma discharge of three propellants—xenon, krypton, and argon—are reported in this two-part article. These studies are performed using high-fidelity reduced-order particle-in-cell (PIC) simulations in a 2D configuration that represents an axial–azimuthal cross-section of a Hall thruster. In this part I paper, we discuss the effects of discharge voltage and current density (mass flow rate). Our parametric studies assess the spectra of the resolved instabilities under various plasma conditions. We evaluate the ability of the relevant theories from the literature to explain the variations in the instabilities’ characteristics across the studied plasma parameter space and for various propellants. Moreover, we investigate the changes in the electrons’ cross-magnetic-field transport, as well as the significance of the contribution of different momentum terms to this phenomenon across the analyzed cases. In terms of salient observations, the ion acoustic instability (IAI)-related modes are found to be dominant across the simulation cases, with the ion transit time instability also seen to develop at low current density values. Across the explored parameter space, the instabilities have the main contributions to the electrons’ transport within the plume region. The peak of the electric momentum force term, representing the effect of the instabilities, overall shifts toward the plume as either the current density or the discharge voltage increases. The numerical findings are compared against relevant experimental observations reported in the literature.

Two-dimensional effects on electrostatic instabilities in Hall thrusters. I. Insights from particle-in-cell simulations and two-point power spectral density reconstruction techniques

Using 2D particle-in-cell (PIC) simulations coupled to a fluid description of the gas dynamics, we study the electrostatic instabilities developing in the axial–azimuthal plane of a Hall thruster, during several periods of a low-frequency oscillation (the so-called breathing mode at 10 kHz). As done in experiments, the 2D PIC-MCC (Monte Carlo collision) code is coupled to an electrical circuit in order to partially damp the (otherwise large) discharge current fluctuations at the breathing mode frequency. The different electrostatic higher frequency modes that develop in the plasma are analyzed using a two-point power spectral density reconstruction method, which allows us to generate the dispersion diagrams (in the frequency-wavenumber space) along the axial and azimuthal directions and at different times during the low-frequency breathing mode oscillations. This technique allows us to distinguish between different well-identified instabilities: the electron cyclotron drift instability and its evolution toward an ion acoustic wave and the ion transit time instability. These instabilities are usually considered unidirectional (either axial or azimuthal); however, it is shown here that they exist in both directions. This two-dimensional character is instrumental in understanding where these instabilities grow and how they propagate in the thruster channel and plume. A theoretical discussion of this aspect is proposed in Paper II. The effects of (i) the azimuthal length of the simulation box and (ii) the electron temperature injection at the cathode are also discussed.

Two-dimensional effects on electrostatic instabilities in Hall thrusters. II. Comparison of particle-in-cell simulation results with linear theory dispersion relations

In Paper I, we successfully used an external circuit to significantly damp the Breathing Mode (BM) oscillations in 2D particle-in-cell self-consistent simulations of the axial–azimuthal plane of a Hall thruster. We also introduced the two-point power spectral density reconstruction method (PSD2P) used to analyze electrostatic instabilities and generate dispersion diagrams in azimuthal and axial directions, at various times during the BM period. Here, a 3D Dispersion Relation (DR) for electrostatic modes is calculated by linearizing the continuity/momentum fluid equations for electrons and ions. We show that by taking the appropriate limits, this relation can be simplified to derive the DRs of some well-known E×B instabilities, such as the electron cyclotron drift instability and its evolution to the Ion Acoustic Wave (IAW), and the Ion Transit-Time Instability (ITTI). The PSD2P diagrams demonstrate the importance of considering the 2D nature of the IAW and ITTI, which have been previously considered to be mono-dimensional (azimuthal and axial, respectively). In particular, we show that the IAW grows near the maximum of the magnetic field and due to its axial components propagates toward both the anode and the cathode (in addition to the well-known azimuthal propagation). The resulting wavefront is, therefore, bent. By analogy to the propagation of acoustic waves in gases, it is proposed that the cause of the IAW wavefront bending is the strong electron temperature gradients in the axial direction. We also show that the ITTI has a strong positive growth rate when a small azimuthal component is present. Finally, we observe that the ITTI significantly affects the discharge current.

Transit-time resonances enabling amplification and generation of terahertz radiation in periodic graphene p-i-n structures with the Zener–Klein interband tunneling

The Zener–Klein (ZK) interband tunneling in graphene layers (GLs) with the lateral n-i-n and p-i-n junctions results in the specific characteristics that can be used for the rectification, detection, amplification, and generation of the terahertz (THz) signals. The transit-time delay of the tunneling electrons and holes in the depletion regions leads to the phase shift between the THz current and THz voltage causing the negative dynamic conductance in a certain frequency range and resulting in the so-called transit-time (TT) instability. The combination of the ZK tunneling and the TT negative dynamic conductance enables resonant THz the amplification and generation of THz radiation. We propose the THz devices based on periodic cascade GL p-i-n structures exhibiting the TT resonances and evaluate their potential performance. Such structures can serve as THz amplifiers and, being placed in a Fabry–Pérot cavity, or coupled to a THz antenna or using a ring oscillator connection, as THz radiation sources.

The interaction between ion transit-time and electron drift instabilities and their effect on anomalous electron transport in Hall thrusters

Recent simulations and experiments have observed a transition from short to long-wavelength azimuthal instabilities that leads to enhanced electron transport in Hall thrusters. Here we make the hypothesis that this phenomenon stems directly from the interaction between the axial ion transit-time instabilities (ITTI), and the azimuthal electron drift instability (EDI). This interaction is studied using 2D axial-azimuthal self-consistent particle-in-cell simulations which include a 1D neutral dynamics solver. It is found that a short to long-wavelength transition only occurs if the breathing-mode and ITTI are captured in the simulation, and two distinct instability regions can be distinguished depending on the local ion Mach number. Upstream of the ion sonic point the EDI exhibits an ion-acoustic behaviour, and the associated instability-enhanced electron transport is well described by a previously developed model based on kinetic theory. Downstream of the ion sonic point however, the ITTI significantly changes the local plasma parameters, and this modifies the EDI while increasing the electron transport.

Self-pulsing of direct-current discharge in planar and curved geometries

The self-pulsing of direct-current discharges in planar and curved geometries is studied using the two-dimensional axisymmetric fluid model. The simulation results show that in both cases the nature of self-oscillations is the same. They are obtained in the sub-normal mode of the discharge operation, for which the discharge has the negative differential resistance. We demonstrate that the negative differential resistance is due to both the non-linear and non-local dependence of the Townsend ionization coefficient on the electric field. We show that the self-oscillations are due to the ion transit time instability and are not related neither to RC resonance nor to the relation between the negative differential resistance and the ballast resistance as it is often suggested in the literature.

Stability of Space-Charged-Limited Electron Beam Diodes Including Applied- and Self-Magnetic Field Effects

The theoretical analysis of large-area diodes shows that the electron flow in the diode is subject to the transit-time instability (TTI). The TTI can modulate the electron beam energy at the anode and add a transverse beam temperature. Both of these effects can have negative consequences on many diode applications that require a high-quality mono-energetic electron beam. A distributed transmission line model of the diode is developed which allows techniques from microwave engineering to be applied to the unstable diode. The theory suggests that a resistively loaded, periodic structure of slots in the cathode can stabilize the TTI. The results from particle-in-cell (PIC) simulations of a large-area electron-beam diode in 2-D are presented that support this conclusion.

E × B electron drift instability in Hall thrusters: Particle-in-cell simulations vs. theory

The E × B Electron Drift Instability (E × B EDI), also called Electron Cyclotron Drift Instability, has been observed in recent particle simulations of Hall thrusters and is a possible candidate to explain anomalous electron transport across the magnetic field in these devices. This instability is characterized by the development of an azimuthal wave with wavelength in the mm range and velocity on the order of the ion acoustic velocity, which enhances electron transport across the magnetic field. In this paper, we study the development and convection of the E × B EDI in the acceleration and near plume regions of a Hall thruster using a simplified 2D axial-azimuthal Particle-In-Cell simulation. The simulation is collisionless and the ionization profile is not-self-consistent but rather is given as an input parameter of the model. The aim is to study the development and properties of the instability for different values of the ionization rate (i.e., of the total ion production rate or current) and to compare the results with the theory. An important result is that the wavelength of the simulated azimuthal wave scales as the electron Debye length and that its frequency is on the order of the ion plasma frequency. This is consistent with the theory predicting destruction of electron cyclotron resonance of the E × B EDI in the non-linear regime resulting in the transition to an ion acoustic instability. The simulations also show that for plasma densities smaller than under nominal conditions of Hall thrusters the field fluctuations induced by the E × B EDI are no longer sufficient to significantly enhance electron transport across the magnetic field, and transit time instabilities develop in the axial direction. The conditions and results of the simulations are described in detail in this paper and they can serve as benchmarks for comparisons between different simulation codes. Such benchmarks would be very useful to study the role of numerical noise (numerical noise can also be responsible to the destruction of electron cyclotron resonance) or the influence of the period of the azimuthal domain, as well as to reach a better and consensual understanding of the physics.

Nonlinear ion dynamics in Hall thruster plasma source by ion transit-time instability

High-energy tail formation in an ion energy distribution function (IEDF) is explained in a Hall thruster plasma with the stationary crossed electric and magnetic fields whose discharge current is oscillated at the ion transit-time scale with a frequency of 360 kHz. Among ions in different charge states, singly charged Xe ions (Xe+) have an IEDF that is significantly broadened and shifted toward the high-energy side, which contributes to tail formation in the entire IEDF. Analytical and numerical investigations confirm that the IEDF tail is due to nonlinear ion dynamics in the ion transit-time oscillation.

Coupling of transit time instabilities in electrostatic confinement fusion devices

A model of the behavior of transit time instabilities in an electrostatic confinement fusion reactor is presented in this letter. It is demonstrated that different modes are excited within the spherical cathode of a Farnsworth fusor. Each of these modes is dependent on the fusion products as well as the acceleration voltage applied between the two electrodes and they couple to a resulting oscillation showing non-linear beat phenomena. This type of instability is similar to the transit time instability of electrons between two resonant surfaces but the presence of ions and the occurring fusion reactions alter the physics of this instability considerably. The physics of this plasma instability is examined in detail for typical physical parameter ranges of electrostatic confinement fusion devices.

On the dispersion relation of the transit time instability in inverted fireballs

Recently discovered inverted fireballs are non-linear plasma phenomena, which are formed in hollow grid anodes with high transparency in an existing background plasma. If a sufficiently large potential is applied, accelerated electrons from the bulk start to oscillate through the grid. Experimental investigations have shown that they produce different types of plasma instabilities. One of those oscillations is a transit time instability which originates from strong electron beams that travel through the inverted fireball. This type of instability is similar to vircator reflex oscillations and produces radio frequency waves. Hence, it is suitable to convert DC signals into signals oscillating in the MHz range. This paper analyses the dispersion relation of the transit time instability for three different plasma regimes. The regimes can be divided into a collision less regime, a regime with high collisionality and one in between those former two. It is demonstrated that the plasma properties of the surrounding background plasma have a strong influence on the behavior of the instability itself.

High-Frequency Instabilities in Sheaths and Fireballs

Electron inertia causes a delay between high-frequency currents and electric fields. When the transit time of a bunch of streaming electrons equals one RF period, the electrons can enhance the field and produce oscillations. Here, electron transit-time instabilities have been observed in magnetized sheaths and fireballs. Novel fireballs trapped in a gridded hollow anode have been developed. These produce large-amplitude RF oscillations.

Transit time instabilities in an inverted fireball. I. Basic properties

A new fireball configuration has been developed which produces vircator-like instabilities. Electrons are injected through a transparent anode into a spherical plasma volume. Strong high-frequency oscillations with period corresponding to the electron transit time through the sphere are observed. The frequency is below the electron plasma frequency, hence does not involve plasma eigenmodes. The sphere does not support electromagnetic eigenmodes at the instability frequency. However, the rf oscillations on the gridded anode create electron bunches which reinforce the grid oscillation after one transit time or rf period, which leads to an absolute instability. Various properties of the instability are demonstrated and differences to the sheath-plasma instability are pointed out, one of which is a relatively high conversion efficiency from dc to rf power. Nonlinear effects are described in a companion paper [R. L. Stenzel et al., Phys. Plasmas 18, 012105 (2011)].

Transit time instabilities in an inverted fireball. II. Mode jumping and nonlinearities

A fireball is formed inside a highly transparent spherical grid immersed in a dc discharge plasma. The ambient plasma acts as a cathode and the positively biased grid as an anode. A strong nearly current-free double layer separates the two plasmas. Electrons are accelerated into the fireball, ionize, and establish a discharge plasma with plasma potential near the grid potential. Ions are ejected from the fireball. Since electrons are lost at the same rate as ions, most electrons accelerated into the fireball just pass through it. Thus, the electron distribution contains radially counterstreaming electrons. High-frequency oscillations are excited with rf period given by the electron transit time through the fireball. Since the frequency is well below the electron plasma frequency, no eigenmodes other than a beam space-charge wave exists. The instability is an inertial transit-time instability similar to the sheath-plasma instability or the reflex vircator instability. In contrast to vircators, there is no electron reflection from a space-charge layer but counterstreaming arises from spherical convergence and divergence of electrons. While the basic instability properties have been presented in a companion paper [R. L. Stenzel et al., Phys. Plasmas 18, 012104 (2011)], the present paper focuses on observed mode jumping and nonlinear effects. The former produce frequency jumps and different potential profiles, the latter produce harmonics associated with electron bunching at large amplitudes. In situ probe measurements are presented and interpreted.

Plasmonic noise in silicon nanolayers

We report a microscopic investigation of the spectrum of voltage fluctuations in nanometric n-Si layers. Theory makes use of a Monte Carlo simulator self-consistently coupled with a two-dimensional Poisson solver. We consider layers of variable thickness W in the range of 2–100 nm and variable length L in the range of 10–1000 nm embedded in an external dielectric medium. Calculations are performed at T=300 K for different doping levels and in the presence of an applied voltage of increasing strength. The spectra are found to exhibit peaks centered on the terahertz region. For W≥100 nm and carrier densities of 5×1017 and 5×1018 cm−3, the frequency peaks agree with the value of the three dimensional plasma frequency. For W≤100 nm, the results exhibit a plasma frequency that depends on L, thus implying that the oscillation mode is dispersive. The corresponding frequency covers a wide range of values of 0.2–10 THz and is in agreement with the values of the two-dimensional plasma frequency predicted by existing analytical models. At sufficiently high voltages, the two-dimensional plasma peak is washed out and we observe the onset of a peak in the subterahertz region which is associated with transit time instabilities induced by current saturation conditions.

Plasmonic noise in Si and InGaAs semiconductor nanolayers

By Monte Carlo simulations we investigate the spectrum of voltage fluctuations of Si layers of variable thickness W in the range 2 ÷ 100 nm and variable length L in the range 10÷1000 nm embedded in an external dielectric medium. Calculations are performed at T = 300 K for different doping levels and in the presence of an external bias of increasing strength. For W ≥ 100 nm and carrier densities of 5 × 1017 and 5 × 1018 cm−3 the peaks agree with the value of the three dimensional (3D) plasma frequency. For W ≤ 10 nm the results exhibit a plasma frequency that depends on L, thus implying that the oscillation mode is dispersive. The corresponding frequency covers a wide range of values 0.2 ÷ 10 THz and is in agreement with the values predicted by existing an alytical models. At sufficiently high bias the 2D plasma peak is washed out and we observe the onset of a peak in the sub THz region which, by analogy with the results obtained in n-InGaAs layers, is associated with transit time instabilities induced by current saturation conditions.

Properties of ceramic honeycomb cathodes

A high current electron beam was emitted from slabs of ceramic honeycomb that were placed 2mm in front of the primary electron emitters. Inside the ceramic honeycomb pores, the primary electrons are multiplied many folds by a secondary emission process creating plasma from which the electron beam was emitted. Measurements show that there was no anode-cathode gap closure during the pulse duration and a uniform current density distribution was observed on the anode. Contrary to the measurements of “conventional” large area field emission cathodes, no transit time instability was observed.

Transit-time instability in Hall thrusters

Longitudinal waves characterized by a phase velocity of the order of the velocity of ions have been recurrently observed in Hall thruster experiments and simulations. The origin of this so-called ion transit-time instability is investigated with a simple one-dimensional fluid model of a Hall thruster discharge in which cold ions are accelerated between two electrodes within a quasineutral plasma. A short-wave asymptotics applied to linearized equations shows that plasma perturbations in such a device consist of quasineutral ion acoustic waves superimposed on a background standing wave generated by discharge current oscillations. Under adequate circumstances and, in particular, at high ionization levels, acoustic waves are amplified as they propagate, inducing strong perturbation of the ion density and velocity. Responding to the subsequent perturbation of the column resistivity, the discharge current generates a standing wave, the reflection of which sustains the generation of acoustic waves at the inlet boundary. A calculation of the frequency and growth rate of this resonance mechanism for a supersonic ion flow is proposed, which illustrates the influence of the ionization degree on their onset and the approximate scaling of the frequency with the ion transit time. Consistent with experimental reports, the traveling wave can be observed on plasma density and velocity perturbations, while the plasma potential ostensibly oscillates in phase along the discharge.

Emission of an intense large area electron beam from a slab of porous dielectric

Inserting a thick slab of porous dielectric (e.g., ceramic honeycomb) in front of the emitting surface of a large-area planar diode improves the electron beam emission uniformity, decreases the beam current rise and fall times, and maintains a more constant diode impedance. Particle-in-cell simulations of the first few nanoseconds of diode operation show that initially numerous secondary electrons and ions load the ceramic honeycomb. The electrons and ions were confined within the ceramic pores, redistributing the electric field by reducing it within the ceramic pores and increasing it on the cathode surface (by a factor of 2–3). After the initial stage, plasma fills the ceramic pores and the space between the cathode and the ceramic. A space-charge-limited electron beam was then emitted from the ceramic honeycomb. No surface plasma was detected outside the pores inside the diode vacuum. The introduction of dielectric into the diode solves two additional problems associated with large-area planar diodes: (1) Space-charge-limited flow in large-area planar diodes is susceptible to the transit time instability. Feeding the instability are growing transverse electromagnetic (TEM) waves that propagate along the anode-cathode gap. By inserting the slab of honeycomb ceramic in front of the emitting cathode these TEM waves are suppressed. (2) A planar diode emits an electron beam with an enhanced current density at the edges (edge effect). The ceramic slab can be easily machined and contoured so as to reduce this effect. The insertion of ceramic honeycomb into the diode had little affect onthe postshot diode pressure. However, deposition of gamma alumina on the ceramic reduced the postshot diode pressure by 80%. This enables the diode to be repetitively pulsed (rep-rated mode). The modified diode was fielded on Electra, a high-power, rep-rated, electron-beam pumped KrF laser. It operated for 50 000 shots at 1Hz and 8000 shots at 5Hz with little or no degradation in the pulse shape and with undetectable loss of cathode material.

Initiation and prevention of the transit time instability in large area diodes

Optimum pumping of krypton fluoride (KrF) lasers requires electron beams that are stable against the transit time instability and spatially uniform. Eliminating the enhanced current density (beam halo) along the edges of a large area homogenous electron beam has been achieved by surrounding the flat diode emitting area with a protruding metallic frame. However, modification to eliminate the beam halo can trigger the transit time instability in an otherwise stable diode. Feeding the instability are growing transverse electromagnetic (TEM) waves that propagate along the anode–cathode (AK) gap reflecting back and forth from the protruding frame. Suppressing these TEM waves was achieved by inserting a slab of honeycomb ceramic in front of the emitting cathode. Electron emission was improved and the presence of the dielectric prevented TEM waves from being generated in the cathode–anode gap. Only surface waves can exist and propagate along the AK gap when the ceramic slab is inserted in the diode. These waves do not couple to the electron beam and cannot support the instability. In such a diode no beam halo and no instability were observed.