Technology for monitoring the surface emission inhomogeneity in plasma electronics devices

In recent years, there has been a surge of interest in studies of secondary electron emission (SEE) due to the wide range of practical applications it offers.¹ The SEE phenomenon plays a pivotal role in developing probe methods for plasma diagnostics,² designing next-generation plasma electronics,³ improving plasma surface treatments,⁴ and other areas.

SEE is especially important in the operation of positively charged particle accelerators. When synchrotron radiation acts on the accelerator walls or residual gas atoms undergo direct ionization, primary electrons are generated within the working volume. These electrons create highly concentrated electron clouds near the walls, causing instabilities in the accelerator.^5–8 

While theoretical and experimental studies devoted to SEE exist,^9–15 there is still a lack of comprehensive understanding of the phenomenon. This can be attributed to the complexity and diversity of plasma processes near cathode surfaces.^13–15 Also, the fact that the near-electrode regions are small and the distribution functions of charged particles are anisotropic makes it challenging to utilize traditional research methods. Emission inhomogeneity leads to the emergence of electric fields near the cathode surfaces, which significantly alters the effective reflection coefficient of slow electrons. These factors, along with possible surface contamination in electronic equipment, sometimes result in researchers obtaining conflicting experimental data.^16,17

Values of the electron reflection coefficient r for standard emission electronics conditions were found and published back in the 20th century.^18–22 However, these values often differ markedly from the results obtained under the operating conditions of electronic equipment. The authors of Refs. 16 and 17 and 23–28 made significant contributions to the development of methods for studying the coefficient of electron reflection from the surfaces of metals with varying degrees of purity.

Recent works (see Refs. 7 and 29–31) have investigated the reflection of electrons with varying energies from copper surfaces. The authors obtained values of r approaching unity for primary electron energies less than 1 eV. Additionally, it was discovered that r ranges from 0.5 to 0.7 at energies of 10–20 eV. The authors believe that the factor contributing to this trend is the purity of the electrode material. This explains not only the emergence of electron clouds in accelerators, but also their impact on the operational modes of Hall-effect thrusters.^32,33

The findings³⁴ of Andronov and Kaganovich raised questions about the results reported in Refs. 7 and 29–31. In support of their arguments, the authors cite Refs. 23–27 and 35–37 reporting reflection coefficient values that do not exceed 0.1. A review (Ref. 38) also showed that earlier theoretical and experimental studies do not contradict the conclusions made by the authors of Ref. 34.

Summarizing the results of experiments on the reflection of slow electrons from copper, gold, and silver surfaces, the authors of Refs. 39 and 40 studied two groups of samples—those that had undergone a plasma treatment with Ar⁺ ions and those that had not. The authors found that the energy dependence of the electron reflection coefficients for these samples at energies above 4 eV showed significant differences. However, at lower energies, the reflection coefficient remained close to unity. Further research is necessary to identify the potential reasons for these conflicting results.

This article presents theoretical and experimental evidence that a significant contribution to the effective reflection coefficient is made by electrons reflected from the potential barrier of spot fields near the surfaces of polycrystalline electrodes.

The authors have developed a method to separately measure the electron reflection coefficients at the surfaces of cathodes and at the potential barrier of spot fields with different work functions. The electric field strength, electron velocities near the cathode surface, and emission parameters of single-crystal and polycrystalline thermionic cathodes were measured for a Knudsen Cs–Ba plasma diode with surface ionization (KDSI).

The resolution of these problems was made possible by utilizing a binary Cs–Ba filler. Barium, with its high heat of adsorption on refractory metals, ensures high cathode emission at low vapor pressures (P_Ва ≈ 10^–3 ÷ 10^–2 Torr). At the same time, barium atoms do not significantly contribute to electron scattering or ion concentration. Under these conditions, cesium serves as the plasma-forming component.

By varying P_Ва through adjustments to the barium thermostat temperature (T_Ba), it is possible to control the degree of compensation, the concentration of charged particles, the electric field strength in the near-electrode layer, and the directed velocity of electrons in the plasma over a wide range, without shifting to the diode's collisional mode.

Let us consider a plasma diode as a one-dimensional model of the near-electrode layer to investigate a strongly nonequilibrium plasma. In the initial stage, the cathode and anode are assumed to be uniform in work function and have zero reflection coefficients.

In the Knudsen mode, plasma is generated on the cathode surface. Ions are produced via surface ionization of the plasma-forming gas (cesium), and electrons are generated through thermionic emission, with the work function of the cathode adjusted by changing the barium vapor pressure. The anode, which absorbs charged particles, is positioned parallel to the cathode at a distance shorter than the mean free path of electrons.

Barium allows for a wide variation in the ratio of the concentrations of ions $n i +$ and electrons $n i −$ produced on the cathode surface (the degree of compensation $γ= n i +/ n i −$⁠). The diode switches from overcompensation $γ>1$ to undercompensation $γ<1$⁠, allowing measurement of the reflection coefficient of slow electrons at various electric field strengths in the cathode layer.

To conduct the study, we made an experimental Cs–Ba diode (see Fig. 1), which consists of a tungsten cathode and a molybdenum anode, both in the form of flat disks with an area of 1 cm². The cathode is heated by an electron gun with a Ta–Nb spiral heater. An optical micropyrometer calibrated against a blackbody cavity and W–Rh micro-thermocouples are used to measure the temperature of the electrodes.

In the experiment, the device was installed inside a cylindrical vacuum chamber. During its operation, a pressure of 10^–8 Torr was maintained by means of an ion getter pump. A cesium thermostat was placed outside the vacuum chamber. Special heaters and an electronic stabilization system ensured that various components of the device had temperatures higher than that of the cesium thermostat. The barium thermostat used in the assembly had elements made of niobium, which enabled operation at temperatures of up to 1500 K to achieve the desired barium vapor pressure P_Ba in the interelectrode gap. The cathode insulator and the anode unit were polished for the device to withstand cesium vapor pressures of up to 5 Torr.

The reflection coefficient was measured in a transverse magnetic field H of about 10⁵ A/m. To generate a magnetic field of this strength, an ARMCO iron magnet with a diameter of 480 mm was used. Two coils with cores made of the same material were attached to the magnet. The values of H were determined using a magnetic induction meter with an accuracy of 5%. Positioning the magnetic system outside the vacuum chamber allowed for precise adjustment of the magnetic field strength vector relative to the electrode planes. The magnetic field was uniform over a region significantly larger than the interelectrode gap of the KDSI.

The magnetic method⁴¹ is based on studying how the electron saturation current I attenuates in a transverse magnetic field H, where electrons move along curvilinear trajectories with the Larmor radius of R (see Fig. 2). When the magnetic field strength increases, the Larmor radius becomes smaller than the interelectrode gap d, and some electrons return to the cathode without reaching the anode. As a result, there is a spatial redistribution of the electron concentration, leading to a change in the potential distribution in the interelectrode gap and the electron trajectories. Under these conditions, electron behavior is described by the Poisson equation and a self-consistent system of stationary kinetic Vlasov equations factoring in an external magnetic field.

In Refs. 42 and 43, a self-consistent solution of this system was achieved and the magnetic curves of the diode ξ(S) were plotted for various values of the degree of compensation γ within the range from 1 to 4 × 10⁴. Here, $ξ= J H/ J 0$⁠, which is the ratio between the values of the electron saturation current in a magnetic field and at a zero magnetic field; $S= e H d 2 m k T c$ is the dimensionless parameter of the magnetic field; H is the magnetic field strength; d is the size of the interelectrode gap; and T_c is the cathode temperature.

In Ref. 44, the authors compared the ion current value obtained by matching the experimental and theoretical magnetic curves of an ideal KDSI with that predicted by the Saha–Langmuir equation. The results show that this ratio increases as the degree of compensation increases, which is an indication of the Schottky anomaly.

Thus, the emission inhomogeneity from the cathode surface strongly affects the electron saturation current in a magnetic field, which made it possible not only to determine r_eff, but also to study its dependence on the electric field strength near the surface.

In the experiment, the magnetic curves were analyzed at anode potentials above 10 V. Under these conditions, the additional reflection of electrons from the potential barrier of the anode spots does not occur, which provides for disregarding the real properties of the anode when determining r_eff. The angular scattering of electrons upon reflection from the surface makes it almost impossible for them to overcome the potential step at the anode and return to the plasma.

It can be seen that for ξ → 1, ξ_r = ξ, and for ξ → 0, $ξ r=ξ/(1– r ¯)$⁠. With an increase in the magnetic field strength, the deviation of the real curves ξ_r(S) from the ideal ones ξ(S) will increase, which will make it possible to measure the value of the reflection coefficient $r ¯$⁠.

Let us now consider the effect of the emission inhomogeneity from the cathode on the magnetic curves in two plasma diode modes—the highly overcompensated mode (γ > 1) and the transition between undercompensation and overcompensation (γ ≈ 1).

In the overcompensated mode, a strong electric field develops near the cathode, which exposes all spots on its surface. The inhomogeneity of the cathode emission results in the dependence of the directed velocity of electrons in the plasma on the area of the cathode from which they are emitted. The magnetic field affects these groups differently.

Let the electrons coming from the surface area with the work function φ arrive at the area with a different work function φ′ after being rotated by the magnetic field. If φ′ < φ, the electrons will reach the cathode surface regardless of the initial velocity. If φ′ > φ, the electrons encounter a potential barrier which they cannot overcome. The electrons reflected from this barrier go back to the interelectrode space.

Thus, emission inhomogeneity leads to an increase in the effective reflection coefficient, which affects the magnetic curves. To calculate r_eff from magnetic curves, one should use Eq. (2), replacing $r ¯$ with r_eff.

Let us establish a relationship between the effective reflection coefficient and the contrast of the cathode surface using the following notation: $g(φ)dφ$ is the probability that the work function on the cathode surface ranges from φ to $( φ + d φ)$ and r₀ is the surface reflection coefficient.

Let us analyze the behavior of electrons leaving the surface area with the work function φ. In the presence of a transverse magnetic field, some of these electrons return to the cathode surface, approaching the surface area with the work function φ′. We assume that the electrons returned by the magnetic field have a semi-Maxwell velocity distribution. This assumption is valid for large values of the magnetic field strength S, where almost all the electrons return to the cathode. For values of S that are not too large, the distribution function of the returned electrons deviates from the semi-Maxwell distribution. Nevertheless, the distribution function of the returned electrons can be considered as a part of the Maxwell distribution even in such cases.⁴³ 

When deriving Eqs. (8) and (9), it was assumed that the potential barrier at the cathode is parallel to its surface.

It can be seen that if у = 0, r_eff = r₀; if у → ∞, r_eff → 1. Figure 3 shows r_eff(y) functions for several values of r₀. It demonstrates that the effective reflection coefficient can reach values close to unity even at relatively small values of r₀. Thus, the value of r_eff is determined by two parameters (r₀, y), making it impossible to unambiguously derive them from magnetic curves.

It is of interest to plot curves similar to those in Fig. 3 for a zero electric field at the cathode. This mode is witnessed at γ ≈ 1. In this case, a potential equal to the average work function of the cathode φ₀ develops at a certain distance from the cathode.⁴⁵ 

Curves $r eff(y)$ plotted using Eq. (20) for a zero electric field are shown in Fig. 4. It can be seen that the work function inhomogeneity of the cathode does not have such an effect on the value of r_eff as in the case of a strong field. This is due to the fact that at a zero electric field, the surface areas with work functions less than φ₀ are covered by local fields of spots, which makes the contrast of the cathode not as sharp.

The effective reflection coefficient r_eff is the sum of the surface reflection coefficient r₀ and an additional term y due to the average spread of work function values over the cathode surface. The values of r₀ and y can be determined separately by experimentally measuring r_eff for both zero and strong electric fields at the cathode.

Thus, the proposed method enables studying the near-electrode plasma with surface ionization and the emission inhomogeneity from the cathode surface simultaneously.

The multiple reflection of electrons from the cathode was taken into account,⁴⁴ and ξ₀(γ, S) was plotted by solving the self-consistent system of stationary Vlasov equations in the presence of a transverse magnetic field.⁴² 

The effect of real surface properties on the shape of magnetic curves is so strong that ξ_exp(S) intersects several curves ξ₀(S) at different values of γ (Fig. 5). The value of r_eff is found by comparing ξ_exp(S) and ξ₀(S) for a given degree of compensation.

For the undercompensated mode, the values of the function ξ(r_eff, γ, S) were found in a similar way.

Function D was calculated for a number of γ values ranging from γ_h to γ_k, and r_eff was calculated in the range from $r eff h$ to $r eff k$⁠. The values of γ_h, γ_k, $r eff h$⁠, and $r eff k$ changed in the process of adjusting the minimum of D. After performing calculations for γ and r_eff, D(γ, r_eff) was found. One of the calculation results is demonstrated in Fig. 6(a). It shows that D(γ, r_eff) drops steeply to a minimum in the vicinity of the values γ and r_eff that we aimed to find. Similar results were also obtained for other magnetic curves [Figs. 6(b) and 6(c)].

Value δ is the ratio of the average perturbation to the value itself expressed as a percentage. Figure 6(d) shows that at δ ≤ 5% (dashed curves), perturbations have almost no effect on D(γ, r_eff), while values of δ greater than 5% result in some increase in dispersion and a change in the position of the minimum.

Table I presents the calculated and experimental data for γ = 16 and r_eff = 0.5, which corresponds to the dispersion minimum D = 0.0058. For the sake of comparison, the results calculated for γ = 10 and 20 and r_eff= 0 are also given.

The results of the numerical experiment clearly demonstrate that the method is valid.

Figure 7 shows the results of measuring r_eff and γ on single-crystal and polycrystalline surfaces. A strong electric field near the cathode is created in a highly overcompensated regime, while a zero electric field is observed upon transition from the overcompensated mode to the undercompensated one (γ = 0.405).⁴⁶ Notably, polycrystalline tungsten [Fig. 7(a)] exhibits rather large values of the effective reflection coefficient in the highly overcompensated mode. The value of r_eff ceases to change at γ above 8, and at γ < 8, r_eff stonically decreases with a decrease in γ and reaches zero at γ of about 10^–2.

On a single-crystal cathode [Fig. 7(b)], a noticeable decrease in r_eff is observed only upon transition to the undercompensated mode, i.e., at γ < 0.5. Under these conditions, a potential barrier (virtual cathode) emerges near the surface, which grows with a decrease in γ.⁴⁷ 

In the highly undercompensated mode, the inhomogeneous potential distribution near the cathode surface is covered by a potential barrier. The electrons returned by the magnetic field to the cathode always overcome this barrier due to its uniformity, and the reflection of electrons from the cathode surface does not lead to their moving back to the plasma as they are scattered around the angles simultaneously with the electron reflection. This explains the decrease in r_eff to zero.

Let us consider the technique of deriving r₀ and y from r_eff(y) shown in Figs. 8 and 9. The values of r₀ and y can be found by comparing the experimental and theoretical values of r_eff in two limiting cases (γ ≈ 0.5 and γ ≈ 100). Δφ is determined from the known values of y and the cathode temperature using Eq. (17). Table II shows the values of r₀, y, and Δφ for polycrystalline tungsten cathode and the (110) crystal face of tungsten.

As the table shows, the values of y and Δφ are notably smaller for single-crystal tungsten compared to polycrystalline tungsten, and the surface reflection coefficient r₀ is higher by a factor of 1.5. The homogeneity of the single-crystal cathode was evaluated using the Schottky anomaly in the vacuum mode, with a direct dependence $lg(j/ j 0)=f ( V )$ (here, j₀ is the saturation current density for γ = 0.405) used as a criterion.

By knowing the values of γ, cathode temperature, and electron current, we can calculate the electric field E near the cathode. Figure 10 shows the calculated E(γ) and experimental r_eff(γ) functions for a polycrystalline surface. It can be seen that r_eff changes greatly in the range of electric field strengths corresponding to the Schottky anomaly, and r_eff reaches saturation at E = 1.6 × 10³ V/cm, which corresponds to a spot size of 10^–4 cm and agrees with the existing data on the structure of polycrystalline cathodes.^45,48

Academic literature on the topic provides almost no experimental data on measuring the effective electron reflection coefficient r_eff for the cathode materials of interest here and for electrons at thermal energies. Measuring r_eff directly from an inhomogeneous emitting surface is difficult and complicated by the need to evaluate the contribution of electrons reflected not only from the surface but also from the potential barrier of the spots. Traditional methods for measuring r_eff are usually utilized under conditions that are not representative of those in which thermionic cathodes operate in plasma devices.

A magnetic technique has been developed for measuring the effective reflection coefficient of slow electrons, r_eff, from polycrystalline and single-crystal surfaces. The experiments were conducted at different values of electric field strengths near the cathode, which enabled the determination of the reflection coefficient directly from the surface (r₀) and from the potential barrier created by the electric fields of spots with different work functions. The results show that surfaces with significant emission inhomogeneity exhibit the maximum values of r_eff, and the reflection of electrons from the electric fields of spots can significantly exceed the reflection from the surface. This observation explains the high values of the effective reflection coefficient (r_eff ≈ 1) reported by other researchers.

The developed technology addresses several important applied issues:

• - monitoring the emission inhomogeneity of thermal cathode surfaces;

• - identifying the most effective methods for mitigating the formation of electron clouds under operating conditions in charged particle accelerators.

These electron clouds may not be caused by surface contamination but by the formation of local electric fields. Developing methods to suppress these fields will enable stable operation of accelerators across a wide range of operating parameters.

The authors have no conflicts to disclose.

A. S. Mustafaev: Conceptualization (equal); Investigation (supporting); Methodology (equal); Supervision (lead). A. Y. Grabovskiy: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead). V. S. Sukhomlinov: Conceptualization (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Validation (lead). E. V. Shtoda: Data curation (lead); Formal analysis (supporting); Investigation (equal); Validation (supporting).

The data that support the findings of this study are available within the article.