Computational study of cathode plasma dynamics in high-power electron beam diodes by particle-in-cell simulations Open Access

High-power electron beam diodes have a wide range of applications, such as radiation effects, high-power microwaves, flash x-ray radiography, and terahertz sources.^1–4 The dynamics of plasma plays a key role in the performance of high-power electron diodes, which can enhance the beam current of the diode owing to the formation of dense plasma near the electrode surface.⁵ However, impedance collapse and even gap closure occur owing to the fast expansion of plasma in high-power diodes.⁶ When high-density cathode plasma forms in a high-power diode and acts as the source of electron emission, the expanding plasma boundary can change the properties of electron emission and the electron beam across the anode-to-cathode gap.⁷ Moreover, the formation of plasma and bombardment of energetic particles also damage the cathode surface, which further limits the lifetime and instability of the operating performance of high-power diodes.⁸ Therefore, a better understanding of the formation and expansion process of cathode plasma is essential for optimizing the performance of high-power diodes.

Previous research has indicated that the discharge of outgoing gas from the electrode surface is the main source of plasma expansion in high-power diodes.⁹ The outgoing gas consisted of contaminants on the electrode surface, such as H₂O and CO₂. Although the species of cathode plasma involve H⁺, C⁺, O⁺, and other ion species, experiments have revealed that H+ dominates the species in the expanding plasma.¹⁰ Experimental results indicate that the expansion velocity of the cathode plasma in high-power diodes ranges from 0.2 to 20 cm/μs in different cases.^11,12 However, when researchers calculate the expanding velocity of the cathode plasma with the working voltage and currents, it shows a time-varying trend, which indicates that the expanding velocity of the cathode plasma is not constant; in contrast, it is time- and space-dependent.¹³ The plasma is not uniform in space. Experiments using a high-speed framing camera or streaked-visible spectroscopy can show images of plasma; however, the images of plasma are diagnosed at time intervals of tens of nanoseconds, which cannot describe the fast process less than tens of nanoseconds, especially for the cathode plasma formation during the initial stage of explosive emission.¹⁴ Computational simulations provide new insights into the plasma dynamics in high-power diodes and magnetically insulated gaps.^15–19 Welch et al. simulated the formation and evolution of cathode and anode plasma in high-power diodes and other devices using a hybrid method;¹⁷ however, a cathode plasma consisting of an electron–ion pair is injected near the electrode at a certain speed, which ignores the discharge process of outgoing gas and is not self-consistent. Xu and Liu explored the formation and expansion process of cathode plasma in high-power diodes using particle-in-cell (PIC) simulations, where the cathode plasma was formed with the discharge of a constant gas layer near the cathode.¹⁵ The process of outgoing gas is also ignored, and the high-power diode is a scaled-down model, which results in an expansion distance of only a few micrometers. Explosive electron emission turns on the cathode surface with the formation of a cathode spot, which is not uniform and results in a non-uniform distribution of temperature. Therefore, the distribution of outgoing gas is also non-uniform and time-dependent, which can influence the discharge process and the formation of cathode plasma consisting of contaminate species.^20–24 However, the material and micro-protrusion of the cathode surface, amplitudes, and rising speed of the applied electric field, and temperature distribution can influence the formation and evolution of the cathode plasma.²⁵ Researchers have performed diagnostic experiments,^26–28 molecular dynamics (MD) simulations,²⁹ and PIC simulations,³⁰ which further emphasized the key point for the outgoing gas during the formation and expansion of cathode plasma; however, the dominant mechanism affecting the plasma dynamics is unclear.

To solve this problem, an outgoing gas model on the cathode surface is developed, which was applied to the cathode dynamics in magnetically insulated transmission lines (MITL) in previous work,¹⁹ in which the heating effects caused by enhanced field emission on the micro-protrusion on the cathode surface are considered to analyze the desorption rate of outgoing gas, which is shown in Sec. II. Section III describes the simulation model for cathode plasma formation. Section IV shows the simulation results of the formation and expansion process of the cathode plasma in the high-power diode, and the effects of different factors on the expansion of the cathode plasma are analyzed and summarized. Section V is devoted to the conclusions.

The cross section of ionization for H₂O is about $10−20 m2$ for relatively low energy levels. Two-dimensional particle-in-cell simulations were performed to study the plasma formation process on the cathode and anode surfaces of high-power diodes via UNIPIC. Figure 3 shows a schematic simulation model for a simplified high-power planar diode in cylindrical coordinates. The anode-to-cathode (AK) gap of the planar diode was 10 mm, and the radius of the cathode was 11 mm. To reduce the reflective effects of the electromagnetic field in the structure, the input port of the power flow for the electromagnetic field was located in the negative R direction and started at R = 11 mm. The electron emission region was within a radius of 7 mm, which can provide sufficient space for the expansion of the cathode plasma.

In the simulations, the cell sizes in the Z and R directions were 100 and 25 μm, respectively. This resolution ensures 100 grid cells across the AK gap of high power in the Z direction. The time step (Δt) was set to 1.0 × 10⁻¹⁴ s to achieve the Courant condition for grid resolution (Δt < 7.8 × 10⁻¹⁴ s). The time step also satisfies $ωpeΔt∼1$⁠,³³  $ωpe∝ne$⁠, where $ωpe$ and $ne$ are the electron plasma frequency and electron density, respectively. The space–charge-limited (SCL) emission of electrons starts on the cathode surface when the electric field strength reaches a threshold of 10⁷ V/m. The desorption rate of the outgoing gas from the cathode surface, relating to the number of emitted electrons, ranged from 10 to 100. When the outgoing gas is desorbed into the AK gap, a thin layer of gas with a high pressure is formed. The voltage wave applied to the input port was set as a step wave, which could be adjusted to obtain different voltage amplitudes and rising speeds. The voltage amplitude changed from 100 to 1000 kV, which generated a bias electric field varying from 100 kV/cm to 1 MV/cm. The rising speed of the electric field varies from 100 kV·cm⁻¹·ns⁻¹ to 1 MV·cm⁻¹·ns⁻¹, which covers the rising speed of the electric field at pulsed power.

The distribution of the gas pressure on the cathode surface within 0.1 mm is shown in Fig. 6. It can be observed that the gas pressure at the edge of the cathode (R = 7 mm) increases more quickly than at other positions, reaching 1 Torr at 0.4 ns. This was due to the enhancement of electron emission at the edge of the cathode, where the electron emission current and gas release rate were higher. As time passed, the air pressure at other locations gradually increased to ∼2.8 ns, resulting in an overall air pressure of ∼1 Torr.

Figure 7 shows snapshots of the electrons in high-power diodes at different times. This reveals that the electrons are emitted to the AK gap uniformly at 2 ns, as the discharge of the outgoing gas is in the initial stage and the density of plasma is considerably low. However, the current of the diode continued to increase at 2 ns, as the preliminary cathode plasma provided sufficient electrons for emission. As the outgoing gas is discharged substantially, the density of the cathode plasma increases and the electrons aggregate on the cathode surface, as shown in Fig. 7(b). During the process from 12 to 14 ns, electrons near the cathode are far denser than the electrons transported across the AK gap of the diode.

The distribution of the ions in the cathode plasma is shown in Fig. 8. The snapshots at 2 and 8 ns indicate that the ions exist in a small region near the cathode surface before 8 ns, which possesses a thickness of ∼100 μm. This phenomenon is caused by insufficient discharge during this period, which may produce low-density plasma. Another reason for the narrow region of ions is the accelerating effects caused by the electric field, which pulls the ions to the cathode and enhances the emitting current of the electrons. Although the gas density reaches a peak of 1 Torr at ∼2 ns, the plasma density is relatively low owing to the low ionization degree of the gas. The expansion speed of the plasma increased after 8 ns, and the thickness of the ion layer in Fig. 8(c) at 12 ns was about 2 times of that in Fig. 8(b) at 8 ns. When the electrons collide substantially with the neutral atoms, the density of the cathode plasma increases, and the cathode plasma near the emission edge of the cathode expands further into the anode-to-cathode gap. This phenomenon is caused by the enhanced current density at the emitting edge of the cathode, causing a higher temperature and a larger number of molecules of outgoing gas into the gap; therefore, a much denser plasma expands faster. As shown in Fig. 8(d), the average thickness of the ion boundary layer is ∼0.3 mm. Therefore, the expansion velocity of the cathode plasma before 14 ns was ∼2.2 cm/μs, which agrees with the simulation and experimental results.

Figures 9 and 10 show the contours of the electron and ion densities for plasma in a high-power diode. The images in Figs. 9(a) and 10(a) illustrate that low-density plasma formed near the emission edge of the cathode surface, which was ∼10¹³ cm⁻³. The electron density across the AK gap of the diode was nearly uniform and in the range of 10¹¹–10¹² cm⁻³. At 8 ns in Figs. 9(b) and 10(b), a thin layer of plasma with a higher density in the range of 10¹³–10¹⁴ cm⁻³ is formed near the cathode surface. As the number of particles increased and particle merging occurred after 10 ns, the density of the plasma increased sharply. The peak plasma density at 12 ns was up to 3 × 10¹⁵ cm⁻³, which is approximately 10 times that at 8 ns. The ion density near the emission edge (6–7 mm) was much higher than that in other regions owing to enhanced electron emission, resulting in a higher desorption speed of neutral gas. Emitted electrons with higher emission currents collide and ionize with neutral gas during the plasma formation process, resulting in a higher plasma density. Therefore, the plasma gradients near the emission edge should be much higher, resulting in a faster plasma expansion speed. This phenomenon causes the boundary of the plasma near the edge to extend further into the AK gap compared with other regions.

The distributions of the electron and ion relativistic velocities are shown in Fig. 11. The velocity on the y axis is relativistic velocity. The results indicate that the axial velocity of the electrons is much higher than the radial velocity, which is attributed to the accelerating effects of the electric field. The maximum axial velocity of an electron is as high as 5 × 10⁸ m/s, which is approximately nine times the peak radial velocity. Figure 11(a) reveals that most electrons possess a low radial velocity, which ranges from −5 × 10⁶ to 5 × 10⁶ m/s. As the outgoing gas is discharged substantially, more electrons with low velocity are generated near the cathode surface at 12 ns, as shown in Fig. 11(b). The radial velocity of the electrons shown in Figs. 11(a) and 11(b) displays a larger positive velocity, which expands away from the axis. As higher emission current and plasma density occur near the emission edge on the cathode surface (Figs. 7 and 9), higher electron velocities are formed. Moreover, the bunching region of the electrons appears to possess a negative radial velocity. This phenomenon can be explained by the increasing current and resulting pinching effects of the electron beam. The ion velocity is at the level of 1 × 10⁴ m/s, and the axial velocity is almost equal in the positive and negative directions owing to field-screen effects. A small portion of ions possess a larger radial velocity toward the axis than that away from the axis because of the diffusion of high-density plasma near the emission edge.

Based on the velocity information of the cathode plasma, the energy spectra of the electrons and ions at different times were analyzed and are shown in Fig. 12. The results indicate that the majority of the electrons and ions are located in the energy range below 10 eV. The maximum energy of the electron can reach 340 keV, which is only a small portion of the total electrons in the AK gap of the diode, as shown in Fig. 11. The proportion of high-energy electrons and ions was even lower than that of electrons.

The expansion velocity of the plasma can influence the equivalent AK gap of high-power diodes, which enhances the current density and even causes gap closure of the device. Therefore, the intrinsic mechanism and related rules of plasma expansion velocity are essential. To investigate the effects of different operating conditions on the plasma velocity of high-power diodes, a series of simulations was conducted and analyzed. The expansion velocity of the cathode plasma depending on the desorption rate of the outgoing gas is shown in Fig. 13, which shows a positive correlation, that is, $vp=0.38rd0.5(cm/μs)$⁠, where rd is the desorption rate of gas molecules per electron. As a higher desorption rate of the outgoing gas can release more molecules to the vacuum of the AK gap, a high-pressure gas layer will be formed on the cathode surface. Therefore, high-density cathode plasma occurs more quickly and expands more rapidly owing to the high plasma concentration gradients. As cathodes with different materials result in different desorption rates of outgoing gas owing to material and microstructure properties, the experiments of plasma velocity for high-power diodes with different cathode materials were compared with the simulation results. The reported plasma velocity agrees well with the simulation results, and the fitting curve shows an empirical relationship between the plasma velocity and desorption rate of the outgoing gas. The reported research shows a maximum desorption rate of gas at about 100 molecules per electron.

Moreover, the dependence of the plasma velocity on the total volume of the outgoing gas is shown in Fig. 13, which also shows a positive correlation with $vp=0.24nML0.5(cm/μs)$⁠, where n_ML is the number of monolayers on cathode surface. This phenomenon is assumed to be caused by the higher plasma concentration resulting from the discharge of high-pressure gas near the cathode surface. The discharge of high-pressure gas becomes more intensive than that of low-pressure gas because of the high collision frequency of electrons with neutral atoms. However, the high desorption rate of outgoing gas from the cathode surface of high-power diodes can also release a large amount of gas into the AK gap; thus, the expansion velocity is increased. Figure 13(d) shows the dependence of the plasma velocity on the increasing speed of applied electric field on the AK gap of diode. The physical quantity dE/dt corresponds to the typical explosive electron emission threshold and emission point start time, which influence the formation of cathode plasma greatly. The plasma velocity is expressed as $vp=2lg(dE/[MV/cm]dt/[1/ns]0.5 (cm/μs)$⁠. Results indicate that higher increasing speed of electric field causes higher expansion velocity of cathode plasma, which can be explained by the faster explosive emission and the higher corresponding desorption rate of outgoing gas.

A higher increase in the diode voltage per unit time can provide more energy to the electrons and the process of impact ionization, resulting in a faster discharge of outgoing gas and a higher density of cathode plasma. This is consistent with the trend shown in Fig. 14, which implies an empirical expression of the expansion velocity relating to the rise speed of the diode voltage, $vp=0.2rdnMLlg(dE/[MV/cm]dt/[ns])0.5(cm/μs)$⁠, where E = V/d is the average electric field applied on the diode, which equals the ratio of voltage to the distance of anode-to-cathode gap. However, the maximum diode voltage shows no clear influence on the expansion velocity of the cathode plasma. In these situations, the rising speed of the diode voltage is fixed and equal. The reason for this phenomenon may be that the formation process of the cathode plasma greatly influences the expansion of the plasma, while the previous process occurs mainly in the rising stage of the diode voltage. Therefore, the amplitude of diode voltage slightly affected the expansion velocity of the cathode plasma. In pulsed power applications, the rise time of high-power pulses ranges from tens to hundreds of nanoseconds and the distance of AK gap ranges from mm to cm, while the applied electric field on AK gap amplitude ranges from hundreds of kilovolt per centimeter to several megavolt per centimeter. Therefore, $lg(dE/[MV/cm]dt/[ns]$ ranges from 0.001 to 1.

The above results show a clear dependence of the plasma velocity on the gas desorption rate, total volume of outgoing gas, and rising speed of the diode voltage. To better analyze the intrinsic mechanism of the cathode plasma velocity, a synthesis expression is proposed to fit the simulation results and predict the plasma velocity. The plasma velocity showed a linear dependence on the synthesized physical quantity, which is shown as $vp=2lg(dE/[MV/cm]dt/[ns]0.5(cm/μs)$⁠. The results indicated that the gas desorption rate and total amount of outgoing gas were the dominant factors influencing the expansion velocity of the cathode plasma. The large difference in the gas desorption rate and the total amount of outgoing gas on the cathode with different materials can generate different plasma velocities by an order of magnitude. For example, the plasma velocities of the cathodes made of carbon fiber and velvet were 2 and 6.5 cm/μs in the experiments,^6,27 respectively. This discrepancy can be attributed to the difference in the gas desorption rate and volume, by which the velvet cathode possesses a gas desorption rate of 98 molecules/electron.²⁴ The carbon fiber cathode exhibited a gas desorption rate of about 10 molecules/electron. Therefore, the dependence of expansion velocity of cathode plasma for carbon fiber and velvet on $rdnMLlg(dE/[MV/cm]dt/[ns]$ can be calculated and shown in Fig. 12. As discussed in above contents for pulsed power system, the desorption rate of outgoing gas on cathode surface of electron beam diode ranges from several to hundreds of molecular per electron, while the number of monolayers for contaminates on cathode surface ranges from several to tens, and the voltage shows $lg(dE/[MV/cm]dt/[ns]$ ranging from 0.001 to 1. Therefore, the synthesized variable of gas desorption rate, gas volume, and voltage shown as $rdnMLlg(dE/[MV/cm]dt/[ns]$ ranges from zero to thousands. The empirical expression of plasma velocity depending on operating parameters concluded in this work will be helpful for analyzing the plasma dynamics in high-power electron beam diodes.

The dynamics and evolution of cathode plasma in high-power diodes were investigated via particle-in-cell simulations. The results indicate that the gas pressure near the emission edge of the cathode surface grows much faster than that in other regions owing to the enhanced electron emission and gas desorption. Therefore, faster formation and expansion of plasma occurs near the region of enhanced electron emission, which possesses a plasma density as high as 10¹⁵ cm⁻³. The average velocity of plasma expansion in the high-power diode with a gas desorption rate of 38 molecules per electron was ∼2 cm/μs. Most electrons and ions are located in the low-energy range, which generates a cathode plasma temperature of approximately several electron volts. Simulations for high-power diodes with different operating conditions reveal that the velocity of plasma expansion grows linearly with the square of desorption rate, total volume of outgoing gas, and the logarithm of the rising speed of the diode voltage, respectively. The amplitude of diode voltage did not show a clear correlation with plasma velocity. Finally, the dependence of the plasma velocity on a normalized variable including the gas desorption rate, total gas volume, and rising speed of the diode voltage is summarized in this work. The expansion velocity of the cathode plasma in the experiments reported in references was compared with the prediction model of the fitting curve, which showed good agreement. This work provides new insights into the dynamics of cathode plasma in high-power diodes and may be helpful for engineering design.

This work was supported by the National Natural Science Foundation of China (52007152 and 12275222) and the Special Foundation of the State Key Laboratory of Intense Pulsed Radiation Simulation and Effect (SKLIPR2005).

The authors have no conflicts to disclose.

Wei Luo and Yu Gu contributed equally to this paper.

Wei Luo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yu Gu: Formal analysis (equal); Validation (equal); Writing – original draft (equal). Jianwei Zhang: Formal analysis (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Lanpeng Qiang: Data curation (equal); Visualization (equal); Writing – original draft (equal). Li He: Methodology (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). Baoshan Tang: Resources (equal); Software (equal); Visualization (equal). Quanzhen Wan: Investigation (equal); Methodology (equal); Visualization (equal). Kequan Wu: Investigation (equal); Methodology (equal). Yuyao Guo: Data curation (equal); Formal analysis (equal); Methodology (equal). Shilin Xing: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal). Yongdong Li: Methodology (equal); Supervision (equal); Writing – review & editing (equal). Pengfei Zhang: Funding acquisition (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.