Trapping and detrapping of electrons in a typical DC glow discharge plasma under double layer condition

Singh, Thangjam Rishikanta; Kommuguri, Sneha Latha; Sinha, Suraj Kumar

doi:10.1063/5.0222639

Observation of a dip in plasma density with the rise of ion plasma waves demonstrates the process of detrapping electrons under the double-layer conditions in a DC glow discharge plasma. This study presents an experimental observation of self-excitation and interplay between electron and ion plasma waves when a high positive DC voltage (⁠ $V_{p} \sim + 100$ V) is applied to a planar probe immersed in plasma. For lower voltages (⁠ $V_{p} \sim + 5$ V), the electron sheath forms on the surface of the probe; however, for sufficiently high applied voltage, plasma could not supply the sufficient number of electrons to shield it from penetrating deep into the plasma. Therefore, the electron-deficient sheath attracts plasma electrons toward the probe, resulting in the excitation of plasma waves and the formation of double layers. Low energy streaming electrons get trapped in the double layers potential step. On ionization of background neutrals, trapped electrons get detrapped. It results in the excitation of ion waves and damping of electron plasma waves. The wavelet analysis of the observed floating potential fluctuations exhibits the interplay between electron and ion plasma waves. The trapping of electrons causes the excitation of electron plasma waves, and detrapping results in the excitation of ion plasma waves as overall electron density dips. It provides new insight into the nonlinear effects of the wave–wave interaction, the onset of Buneman instability, and streaming instability under the double-layer condition.

I. INTRODUCTION

Formation of sheath and excitation of various plasma waves on the application of voltage on a probe immersed in plasma is a well-known phenomenon.^1–6 The excitation of electron plasma waves (EPWs) has been reported in large-amplitude RF pumping.⁷ Further observations reveal that the rise of parametric instability produces saturation states corresponding to pump frequency.⁸ In another experiment, an investigation of collisionless damping of plasma waves in a plasma column was reported,² and the formation of ion-acoustic double layers (DLs) by drifting electrons has also been proposed.^9,10 The DLs are known for non-linearity and the co-existence of electron and ion plasma waves due to the rise of different instabilities. A theoretical solution for the existence of coupled electron and ion plasma waves was obtained.¹¹ Furthermore, the onset of DLs was experimentally observed due to ion trapping in a collisionless plasma column. It explained the significance of Buneman instability and Pierce instability, which triggers the formation of DL.¹² It has also been reported that the acceleration and deceleration of double layers have widespread applications in various fields, such as laser-plasma studies,^13,14 ionospheric and astrophysical plasma, for example, in auroral plasma,^15–18 and dusty plasma.^19–21 The drastic change in global plasma characteristics under DL conditions on perturbation of operating parameters is being studied,^22–24 and our recent work explains that the order–chaos–order transition is a consequence of the interplay between EPWs and IPWs (ion plasma waves). The detailed analysis of fluctuations indicated that the rise and damping of these waves resulted from the trapping and detrapping of electrons.²⁵ The Maxwellian electrons get trapped in the potential well created by ions when the electron's energy is less than the potential energy.²⁶ The trapping splits the distribution into a trapped and non-trapped distribution, leading to localized density and potential structures.^27,28

It raised two questions: what impact do trapping and detrapping have on the global plasma density? How is it associated with nonlinear wave–wave interaction? Therefore, the main aim of the present work is to investigate variation in the plasma density of the system under DL conditions when IPW and/or EPW dominate.

DLs are highly nonlinear potential structures,^17,18,29,30 and attempts at probe measurements were not meaningful since DL vanishes on its intrusion. On the other hand, non-intrusive optical emission spectroscopic measurement (OES) is also not useful because the emission from bulk plasma remains almost unchanged³¹ while floating potential fluctuations (FPF) observed significant change.²² To address this, we decided to make the density measurements using the Langmuir probe in a narrow voltage range. As the characteristic temperature of the glow discharge plasmas is $\sim 1$ eV, Langmuir probe measurements are possible with applied voltage in the range of $- 30$ to $+ 30$ V without affecting the DL condition.

In this study, the current–voltage (I–V) curve was drawn for the probe immersed in the plasma, in the range of 0 to $+ 200$ V. Simultaneously, the variation in the discharge current is recorded to estimate the overall effect of the probe on plasma. Furthermore, the change in FPFs was recorded using an electrostatic probe to understand the time dependence of plasma response to the probe voltage, $V_{p}$ ⁠. Additionally, the variation in plasma density was measured using a Langmuir probe, for different voltages applied to the probe. The I–V characteristic holds information on time-averaged processes, and FPFs provide time-dependent information on waves and instabilities. The FPF recorded for different voltages represents new stationary states in which the recurrence of IPW and EPW varies.³² The time variation in the occurrence of plasma waves of two different stationary states is a consequence of the growth and damping due to density fluctuations. In the experiment, the density fluctuations are induced by changing the probe voltage (⁠ $V_{p}$ ⁠), causing the growth and damping of plasma waves resulting in a new stationary state,²⁵ while the probe measures the variations in plasma density. These experimental observations are analyzed, discussed, and interpreted to understand the governing mechanism, and finally concluded with this investigation.

II. EXPERIMENT

The details of the experimental arrangements are discussed in this section.

A. Experimental setup

The schematic of this experimental setup is shown in Fig. 1. It consists of a cylindrical vacuum chamber made of stainless steel (model SS304 of diameter 30 cm and length 73 cm) and one electrode, namely, the cathode C on which negative bias voltage is applied with respect to the grounded chamber to produce the glow discharge plasma. The cathode is a stainless steel disk with a diameter of 5 cm and 4.2 mm thickness connected to DC voltage $V_{c}$ across a resistor $R_{c}$ of $100 Ω$ ⁠.

FIG. 1.

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Schematic diagram of the experimental setup. Discharge is produced between the vacuum chamber and the cathode. The probe P and the cathode C are connected to the DC power supplies $V_{p}$ and $V_{c}$ with limiting resistors $R_{c}$ and $R_{p}$ ⁠, respectively. Plasma fills the chamber once the discharge is produced between the cathode and the grounded chamber. The DC glow discharge plasma is maintained stable at $V_{c} = - 550$ V. $V_{p}$ is the positive DC voltage source applied on the probe to study its effect, a Langmuir probe (L.P.) and an electrostatic probe (E.P.) are used for plasma density and FPF measurements.

The discharge fills the chamber, and a probe P (tungsten disk with a diameter of 2 cm and 2 mm thickness) is immersed in plasma, with a resistor $R_{p}$ of $100 Ω$ in series, on which the potential $V_{p}$ is applied. The discharge current (⁠ $I_{c}$ ⁠) is measured across $R_{c}$ ⁠, while the probe current, $I_{p}$ ⁠, is measured across $R_{p}$ ⁠. A Langmuir probe (L.P.) and an electrostatic probe (E.P.) are used to measure electron density and record the FPFs. As shown in Fig. 1, the discharge is produced by applying a negative DC voltage $V_{c} = - 550$ V, with a series resistor $R_{c}$ ⁠, between the cathode and the grounded chamber. A positive DC voltage source for the probe, $V_{p}$ is connected in series with resistor $R_{p}$ ⁠. The current $I_{p}$ and $I_{c}$ are measured across the respective resistors $R_{p}$ and $R_{c}$ ⁠. On the monotonic increase in probe voltage, $V_{p}$ in steps of 10 V, from 0 to +200 V, the corresponding changes in current across $R_{c}$ and $R_{p}$ are measured.

B. Effect of probe voltage (⁠ $V_{p}$ ⁠) on discharge current (⁠ $I_{c}$ ⁠)

Figure 2 shows the variation of the discharge current $I_{c}$ measured across $R_{c}$ as the probe voltage $V_{p}$ is increased monotonically by keeping the cathode voltage $V_{c}$ fixed at $V_{c} = - 550$ V. As the applied DC positive bias voltage, $V_{p}$ is increased slowly in a few steps of 10 V from 0 to $+ 200$ V, the current $I_{c}$ was measured. The experiment was repeated for different pressures. At 0.1 mbar, as shown by the solid blue line, the current at $V_{p} = 0$ V is 1.32 mA and, for $V_{p} = 200$ V, the current value is increased to 2.8 mA. In the case of 0.3 mbar at higher pressure, the current is 3.11 mA at 0 V, and when $V_{p}$ is increased to $+ 200$ V, the current is 6.33 mA, shown by the dashed red line. The increase in pressure increases the density of electrons, as does the current. The probe voltage $V_{p}$ and the discharge current $I_{c}$ across $R_{c}$ vary linearly.

FIG. 2.

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The figure shows the variation of increasing probe voltage $V_{p}$ vs the current, $I_{c}$ measured across $R_{c}$ at different pressures 0.1 mbar (solid blue line) and 0.3 mbar (dashed red line), respectively, by keeping the cathode voltage $V_{c}$ fixed at $- 550$ V.

The increase in $I_{c}$ current is minimal at lower applied voltages $V_{p}$ ⁠. As the applied positive DC bias voltage, $V_{p}$ is increased further; it increases the current $I_{c}$ ⁠. The application of voltage $V_{p}$ can be viewed as an overall increase in effective discharge voltage $\approx V_{c} + V_{p}$ ⁠, thus increasing the discharge current, $I_{c}$ ⁠.^1,3,33 However, probe current $I_{p}$ depends on the supply and collection of the number of electrons on the probe, discussed in Sec. II C.

C. Probe current (⁠ $I_{p}$ ⁠) vs probe voltage (⁠ $V_{p}$ ⁠)

The variation of probe voltage $V_{p}$ vs the measured current $I_{p}$ across $R_{p}$ by keeping the cathode voltage fixed at $V_{c} = - 550$ V for different pressures is shown in Fig. 3. It is observed that with the probe voltage $V_{p}$ increases, the current across $I_{p}$ increases exponentially. Figure 3(a) is the plot of the variation of probe voltage, $V_{p}$ ⁠, from 0 to 200 V at 0.1 mbar indicated by the different curves, with the solid blue line from 0 to 40 V and with the dashed blue line from 50 to 200 V. When it is 0 V, the current is $0.3 μ$ A, and at 40 V, the current reaches 1.43 mA. As $V_{p}$ is increased from 50 to 200 V at 0.1 mbar, the current reaches from 1.73 to 7.33 mA, as shown by the dashed blue line. Figure 3(b) shows the current at 0.3 mbar pressure. When $V_{p} = 0$ V, the current is $0.4 μ A$ ⁠, and when it attains 30 V, the current becomes 1.22 mA, shown by the solid red line. As $V_{p}$ is increased from 40 to 200 V, the current increases to 2.85 mA for 40 V and, it reaches up to 9.02 mA for 200 V, which is indicated by the dashed red line.

FIG. 3.

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(a) The variation of probe voltage vs discharge current I_p measured across R_p by keeping fixed cathode voltage $V_{p}$ $V_{c} = - 550$ V at pressure 0.1 mbar showing the rapid exponential increase in probe current $I_{p}$ from 0 to 40 V, indicated by the solid blue line and the slow exponential increase, $I_{p}$ from 50 to 200 V, indicated by the dashed blue line at 0.1 mbar pressure. When the probe voltage $V_{p}$ is increased to 200 V, the current across $R_{p}$ reaches up to 7.33 mA. (b) The variation of probe voltage $V_{p}$ from 0 up to 200 V vs the probe current $I_{p}$ across it at pressure 0.3 mbar, indicated by the solid red line from 0 to 40 V and from 40 to 200 V, shown by the red-blue line, showing the exponential increase in probe current, $I_{p}$ ⁠.

At pressure 0.1 mbar, the exponential increase in probe current $I_{p}$ from $0.3 μ A$ at $V_{p} = 0$ V to 1.43 mA for $V_{p} = + 40$ V is shown in plot 3.1 (a). Initially, for smaller probe voltages $V_{p}$ (say $V_{p} < + 5$ V), the probe draws the current as the plasma electrons rush to shield the applied positive potential. With further increases in $V_{p}$ ⁠, the supply of electrons from the plasma is not sufficient. Therefore, potential penetrates deeper into the plasma, and thus, an electron sheath forms. Under this condition, electrons from the plasma sheath boundary stream toward the probe. Among the streaming electrons, those with energy lower than the local positive potential, while those that gain sufficient energy ionize the background neutrals. For $N_{2}$ ⁠, electron impact ionization cross section peaks around 30 eV,³³ and therefore, sharp rise in $I_{p}$ is observed $V_{p} = 20 - 50$ V. On further increase in $V_{p}$ up to 200 V, the current $I_{p}$ reaches up to 7.33 mA, as more plasma electrons gain sufficient energy.

Having this understanding, for the two different regions of the I–V characteristics, curve fitting was done using the SciPy algorithm in Python.³⁴ The probe current can be expressed as $I_{p} = I_{1} + I_{2}$ ⁠, where $I_{1} = 0.221 \cdot exp (0.0169 \cdot V_{p}) - 0.028$ ⁠, and $I_{2} = exp (0.0099 \cdot V_{p}) + 0.142$ ⁠. A similar trend was observed at the pressure of 0.3 mbar. The current increases from $0.4 μ A$ at 0 V to a value of 3.13 mA at 50 V, and when probe voltage $V_{p}$ is increased to 200 V, current across $R_{p}$ reaches up to 9.02 mA indicated in the dashed red line. The total probe current is $I_{p} = I_{3} + I_{4}$ ⁠, where $I_{3} = 0.357 \cdot exp (0.405 \cdot V_{p}) - 1.10$ ⁠, and $I_{4} = exp (0.01 \cdot V_{p}) + 1.566$ ⁠.

The first region of the curve, which is expressed by

I_{1}

⁠, shows an exponential increase due to electron impact ionization of background neutrals inside the extended sheath by streaming electrons. However, this results in improved shielding of the probe, and therefore, further increasing applied voltage, the rise in current is almost linear. The expression for probe current at 0.1 mbar is obtained by curve fitting and given as For

V_{p} = 0

–50 V

I_{p} = {I_{1} = 0.221 \cdot exp (0.0169 \cdot V_{p}) - 0.028} .

(1)

For

V_{p} = 50 - 100

V

I_{p} = {I_{2} = exp (0.0099 \cdot V_{p}) + 0.142}

(2)

and the expression of probe current at 0.3 mbar obtained by curve fitting. For

V_{p} = 0

–50 V

I_{p} = {I_{3} = 0.357 \cdot exp (0.405 \cdot V_{p}) - 1.10} .

(3)

For

V_{p} = 50

–100 V at 0.3 mbar,

I_{p} = {I_{4} = exp (0.01 \cdot V_{p}) + 1.566} .

(4)

The variation in the current with voltage results from the increase in the number of electrons pulled by the probe and electron impact ionization of the background neutrals.³³ Figure 4 shows the variation of probe voltage

V_{p}

vs the current

I_{p}

in the semilog scale at pressure 0.1 mbar (solid blue line) and 0.3 mbar (dashed red line). The electron impact ionization of background neutrals is more effective at 0.1 mbar than the pressure at 0.3 mbar. In this figure, the plot for

I_{p}

at 0.1 mbar current varies sharply in the voltage range

0 - 40

V compared to 0.3 mbar. This is the consequence of larger plasma density at a higher pressure as the number of ion-induced secondary electrons emitted from the cathode increases^1,3 and the effective shielding of the applied voltage.

FIG. 4.

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The semilog plot of I–V characteristics of the probe plotted in Fig. 3 pressures 0.1 mbar in solid blue line and 0.3 mbar in dashed red line, respectively.

D. Total discharge current $I_{d} = I_{c} + I_{p}$

In a DC glow discharge plasma, gas ionization occurs at the plasma-cathode sheath (or ion sheath) boundary. The emitted secondary electrons³ from the cathode accelerate sufficient energy inside the cathode sheath to undergo avalanche ionization at the plasma-sheath boundary.¹ However, in the present case, there are two ionization regions. In addition to the cathode sheath region, ionization occurs near the probe P. Here, plasma electrons accelerate toward the probe on the application of $V_{p}$ ⁠. Therefore, the effective discharge current is $I_{d} = I_{c} + I_{p}$ ⁠. Figure 5 shows the variation of probe voltage $V_{p}$ vs the total discharge current $I_{d}$ at 0.1 mbar (shown in solid blue line) and 0.3 mbar (dashed red line) pressures, respectively. The current profile is the same for both pressures. The current at 0.3 mbar pressure is larger since the electron impact ionization frequency increases with pressure.³³

FIG. 5.

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The total discharge current $I_{d}$ ⁠, as the probe voltage $V_{p}$ is varied keeping the cathode voltage $V_{c}$ fixed at $- 550$ V at pressures 0.1 mbar (in solid blue line) and 0.3 mbar (in dashed red line), respectively.

Observing an increase in probe current $I_{p}$ and the total discharge current $I_{d}$ with $V_{p}$ is consistent with the governing discharge processes. However, an interesting question remains: why do FPFs show order–chaos–order transition and the interplay between plasma waves?²⁵ Therefore, we measured plasma density using a Langmuir probe to know how it affects the overall plasma parameters. Langmuir probe measurement details are given below.

E. Langmuir probe measurements

As discussed above, Langmuir probe currents are measured in the narrow range from $- 30$ to $+ 30$ V. In this range of the applied voltages on the Langmuir probe, the disturbances are localized and do not affect the global properties of the plasma. Above this $+ 30$ V, we observe the growth of various instabilities.^3,35–37 In the said range of voltages, the plasma is steady and stable while the plasma parameters are measured.

F. Plasma density measurements

With the application of DC voltage on the immersed probe (⁠

V_{p}

⁠), the discharge current

(I_{c})

and the probe current

(I_{p})

increase. In DC glow discharge plasma, the discharge current is associated with plasma density.³ Therefore, diagnosing variations in plasma density is necessary to understand the governing processes. In Fig. 6(I), in the first column, the I–V characteristics of a Langmuir probe when the plasma is maintained stable at

V_{c} = - 550

V and 0.1 mbar for different probe voltages

V_{p}

are shown. The saturation current drawn by the Langmuir probe decreases from

26.8

to

16.4

to

8.0 μ A

⁠, for

V_{p} = 0

⁠, 20, and 40 V, respectively, as shown in Figs. 6I(a)–6I(c). Thus, it was observed that the current decreases as

V_{p}

is increased. The probe with positive bias voltage drains plasma electrons more effectively with the increase in voltage, and consequently, plasma density decreases. The saturation of the Langmuir probe current is not apparent, and therefore, semilog is shown in the second column of Fig. 6(II) for visualization of electron current saturation at 0.1 mbar. Assuming the electron distribution to be Maxwellian, the plasma parameters are determined in the transition region, and the electron current in this region is given by

I_{e} = I_{e, sat} exp [\frac{e (V - ϕ_{p})}{T_{e}}],

(5)

where V is the voltage applied to the electrostatic Langmuir probe (E.P.),

ϕ_{p}

is the plasma potential,

I_{e, sat}

is the electron saturation current, and

T_{e}

is the electron temperature. The graph of voltage, V vs

l n (I_{e})

is plotted from this equation. Taking the inverse slope of any two difference points from this graph provides the value of electron temperature (⁠

T_{e}

⁠) in units of eV. The electron current is found by subtracting the ion current from the total current. The electron density (⁠

n_{e}

⁠) is calculated by inserting the value of

T_{e}

in the equation

n_{e} = \frac{I_{e, sat}}{e A} \sqrt{\frac{2 π m_{e}}{T_{e}}},

(6)

where A is the area of the electrostatic probe (E.P.) of the cylindrical type that has dimensions of 0.5 mm thickness and height 1.70 cm and, thus,

A = 2.708 \times 10^{- 5} m^{2}

⁠,

m_{e}

is the mass of the electron and e is the electronic charge. The I–V characteristics are measured by this electrostatic Langmuir probe (E.P.) corresponding to an increase in probe voltage

V_{p}

⁠. After subtracting the ion saturation part, we have plotted the electron saturation region where the y axis is in the log scale.

FIG. 6.

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In the first column I, the I–V characteristics of the probe at 0.1 mbar and its electron saturation region in the second column II at different probe voltages, $V_{p}$ ⁠.

In Fig. 7(I), the first column shows the I–V characteristics of the Langmuir probe at 0.3 mbar pressure corresponding to the voltage applied on the immersed probe for $V_{p} = 0$ ⁠, 20, and 40 V. Figures 7I(a)–7I(c) shows the saturation current drawn by the Langmuir probe decreases from 26.2, $23.2$ ⁠, to 10.6 $μ A$ ⁠, respectively. For both pressures, the Langmuir probe current decreases as $V_{p}$ is increased, and the reason is the same as discussed above. In Fig. 7(II), the second column shows the electron saturation region in the semilog plot of the probe characteristic at 0.3 mbar for different probe voltages $V_{p}$ ⁠.

FIG. 7.

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The figure represents in the first column I, the I–V characteristics of the probe at 0.3 mbar and its electron saturation region in the second column II at different probe voltage, $V_{p}$ ⁠.

Using these Langmuir probe currents data in Sec. II G, the plasma density variation has been estimated corresponding to an increase in probe voltage, $V_{p}$ ⁠.

G. Variation of plasma density

As discussed above, the plasma density is estimated for each value of the applied voltage on the probe $V_{p}$ ⁠. Figure 8 shows the variation of plasma density as corresponding to $V_{p}$ ⁠. The figure represents the change in electron density, as the probe voltage $V_{p}$ is varied, keeping the cathode voltage $V_{c}$ fixed at $- 550$ V at pressures 0.1 mbar (in solid blue line) and 0.3 mbar (in dashed red line), respectively. For the present experimental condition at 0.1 mbar and 0.3 mbar, the plasma density is measured using the Langmuir probe. It is found that the plasma density goes on reducing from $1.32 \times 10^{12}$ to $3.78 \times 10^{11} m^{- 3}$ for 0.1 mbar when $V_{p}$ is increased from 0 to $+ 40$ V. For 0.3 mbar, on increasing probe voltage $V_{p}$ from 0 to 60 V, the plasma density goes on reducing from $1.3 \times 10^{12}$ to $1.72 \times 10^{11} m^{- 3}$ ⁠. This confirms the experimental evidence of the evacuation of plasma leading to a reduction of plasma density. Figure 8 shows that plasma density dips to a minimum value for both pressures and again recovers marginally and then saturates as $V_{p}$ increases from $0 - 200$ V, while the total discharge current $I_{d}$ increases continuously, as shown in Fig. 5. This an interesting observation that confirms the role of the positively biased immersed probe P, as a sink for electrons. However, as the voltage increases, the rate of loss of electrons also rises; on the other hand, the plasma electrons streaming toward the probe gain sufficient energy to ionize the background neutrals. Initially, the rate of loss of electrons dominates till plasma density attains a minimum value, and thereafter, the rate of ionization is sufficient to recover the loss marginally and finally saturates. The plasma density dip is observed at $V_{p} = 40$ V and $V_{p} = 60$ V for pressures 0.1 mbar (in solid blue line) and 0.3 mbar (in dashed red line), respectively.

FIG. 8.

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The figure represents the change in electron density as the probe voltage $V_{p}$ is varied, keeping the cathode voltage $V_{c}$ fixed at $- 550$ V at pressures 0.1 mbar (in solid blue line) and 0.3 mbar (in dashed red line), respectively.

This suggests that there is competition between the loss rate and ionization rate of electrons, and in Sec. II H, how it influences overall plasma dynamics accordingly, we record and investigate the floating potential fluctuations (FPFs).

H. Floating potential fluctuations (FPFs)

The FPFs are the collective response of electron loss and generation due to the ionization of background neutrals. It is recorded by the electrostatic probe shown in Fig. 1. The FPFs are recorded for corresponding to the change in probe voltage $V_{p}$ using a digital storage oscilloscope (Lecroy Wavejet-354A 500 MHz) having the sampling rate of 2 $GS / sec$ with a high impedance passive probe of 10 $M Ω$ with an attenuation factor of 10:1. These FPFs are analyzed using continuous wavelet transform to simultaneously obtain the localized information in both the time and frequency domains.

I. Continuous wavelet transform (CWT) analysis

The wavelet transforms of a continuous signal

x (t)

with respect to the wavelet function

ψ (t)

are defined as

C (a, b) = \frac{1}{\sqrt{| a |}} \int_{- \infty}^{+ \infty} x (t) ψ^{⋆} (\frac{t - b}{a}) d t,

(7)

where a is the scaling parameter and b is the location parameter, respectively, providing both the translated and dilated functions of the mother wavelet

ψ (\frac{t - b}{a})

and the signal

x (t)

⁠. The scaling parameter a is related to frequency, F by

F = \frac{F_{c}}{a \cdot Δ}

⁠, where

F_{c}

represents the center frequency of the mother wavelet and

Δ

is the sampling interval. This is particularly useful for analyzing signals such as noisy, transient, and intermittent signals.^38,39

Analysis of the observed FPFs by CWT confirms the presence of low-frequency ion plasma waves (IPWs) and high-frequency electron plasma waves (EPWs) as shown in Fig. 9. For 0.1 mbar, Fig. 9(a) shows the fluctuations change with $V_{p}$ and corresponding wavelet spectra in 3D are shown in Fig. 9(b). In the wavelet spectrum, the y axis scale is inversely proportional to the frequency of FPFs (⁠ $y \propto 1 / f$ ⁠). The observed high-frequency FPFs $V_{p} = 0$ V appear like background noise, which becomes more periodic for $V_{p} = 20$ V shown in Figs. 9(II-a) and 9(II-b). On the further increase in $V_{p} = 40$ V, Figs. 9(III-a) and 9(III-b) show that fluctuations are intermittent and have a low frequency, and it corresponds to IPW (as the first kind of high-frequency and the second kind of low-frequency waves correspond to EPW and IPW).²⁵ With further increasing $V_{p} = 40$ V, the IPW becomes more periodic, and for this voltage, a multi-layered globular protuberance appears on the surface of the immersed probe. It is referred to as DL. The frequency of IPW increases with increasing $V_{p}$ ⁠, and it completely dampens, and EPW rises again at $V_{p} = 120$ V as shown in Figs. 9(VI-a) and 9(VI-b). The change in fluctuations and the respective wavelet spectra for 0.3 mbar is shown in Fig. 10. The FPFs for $V_{p} = 0$ V are shown in Fig. 10(I-a) the occurrence of high-frequency EPW in the CWT spectrum in Fig. 10(I-b). On the increase in voltage $V_{p} = 20$ V, the IPW arises. Further increasing the probe voltage $V_{p}$ to 80 V, the IPW dampens, and EPW rises again. The rise of IPWs is the effect of applied DC voltage $V_{p}$ ⁠, when electrons have already been depleted. Under this condition, on increasing $V_{p}$ ⁠, sheath voltage increases, resulting in a higher frequency of IPW. The IPW generates a potential well, which further traps electrons. On the other hand, an increase in $V_{p}$ also pulls more and more plasma electrons to be trapped. When a sufficiently large number of electrons are trapped, these IPW dampens, as the applied potential has the least effect on ions, and these trapped electrons give rise to EPW. Figure 9 shows the transition from IPW to EPW for pressure 0.1 mbar is very slow; however, for 0.3 mbar shown in Fig. 10, the transition from IPW to EPW is achieved at lower $V_{p}$ and very rapid as compared to the lower pressure of 0.1 mbar since the ionization rate is very high.

FIG. 9.

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The first column labeled as (a) represents the FPFs at different probe voltage $V_{p}$ by keeping the cathode voltage $V_{c}$ fixed at $- 550$ V at pressures 0.1 mbar, the second column labeled as (b) represents the corresponding CWT spectrum by Mexican hat wavelet. The x axis represents the position of the fluctuations over time. The y axis shows the scale, which is inversely proportional to the frequency of the fluctuations. The color at each x–y point shows the magnitude of the wavelet coefficient shown on the z axis.

FIG. 10.

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The first column labeled as (a) represents the FPF at different probe voltage $V_{p}$ by keeping the cathode voltage $V_{c}$ fixed at $- 550$ V at pressures 0.3 mbar, the second column labeled as (b) represents the corresponding CWT spectrum by Mexican hat wavelet in 3D. The x axis represents the position of the fluctuations over time. The y axis shows the scale, which is inversely proportional to the frequency of the fluctuations. The color at each x–y point shows the magnitude of the wavelet coefficient shown on the z axis.

The increase in applied voltage $V_{p}$ on the immersed probe causes variations in electron density, shown in Fig. 8, and wavelet spectra show the interplay between EPW and IPW, in Fig. 9, are correlated and provide the understanding of the governing mechanism. The main observation is the dip in plasma density that is associated with IPW.

III. DISCUSSIONS

A. Effects observed at cathode region

When we increase the cathode voltage, more current flows. Upon utilizing it for a very long time, the size of the cathode region becomes larger. The effect of ionization is more pronounced in the cathode sheath region. The ions' density decreases due to charge exchange collisions as they flow toward the cathode but are accelerated when they enter the sheath. The electron emission from the cathode is proportional to temperature. The ions produced in the cathode sheath could not travel to the anode because of the polarity of the cathode sheath field. The current at the cathode is the sum of ion current and secondary electrons emitted from the cathode.^1,3,33

Analysis of cathode region: The current density carried by the electrons in the glow plasma at a distance r is denoted by $J (r) = e n (r) μ e E$ ⁠. This current is flowing at the edge of the cathode sheath region because of the continuity of the current. The current carried only by the ions is given by $J_{i} (r) = ens (r) μ B$ ⁠, where $μ B = \sqrt{\frac{e T_{e}}{M}}$ is the Bohm velocity. This current density is very much less than the electron drift velocity given by $v_{d} = μ e E$ ⁠. The power input is equal to the power output due to power balance, so $P_{input} = 2 π \int_{0}^{R} j E_{r} d r$ and the power output is $P_{output} = 2 π R Γ_{r} e W_{T}$ ⁠; $Γ_{r} = - D_{a} \frac{d n}{d r}$ is the radial particle flux and $W_{T}$ is the total energy gained for every electron–ion pair generated.^3,33

The diffusion equation is given by

- \nabla (D_{a} \nabla n) = v_{i z} n,

(8)

where

D_{a}

is the ambipolar diffusion coefficient and

v_{i z}

is the ionization rate. In terms of cylindrical coordinates, the above equation is written as

- \frac{d^{2} n}{d r^{2}} = \frac{1}{r} \frac{d n}{d r} + \frac{v_{i z}}{D_{a}} n .

(9)

The solution of the above equation is Bessel's function

n = n_{0} J_{0} (β r)

⁠, which is the standard boundary condition obtained when we assume at the chamber wall there is no density,

J_{0}

is the zero-order Bessel's function.

β = \sqrt{\frac{v_{i} z}{D_{a}}} = \frac{χ_{01}}{R}

and

χ_{01} = 2.405

⁠. The ionization rate is dependent upon temperature, and its value is given by

v_{i z} = cPexp (\frac{- U_{i z}}{T_{e}}) = \sqrt{\frac{cpexp (\frac{- U_{i z}}{T_{e}})}{D_{a}}} = \frac{χ_{01}}{R}

⁠.

B. The wall of the vacuum chamber

Since the chamber is grounded, it acts as a sink, as the electrons are lost to the chamber walls and the boundary. Near the vicinity of the probe, P, some interesting effects are observed. Due to electron emission from the cathode, the temperature is increased near the probe region, and with increasing temperature, sheath ionization occurs, thereby creating an ion current at the probe. As the electrons enter the sheath, they are reflected or deflected. The probe also repels some of the random flux of electrons and reduces the current density. The secondary electrons ejected when ions strike the cathode result in the generation of fast electrons in the cathode sheath, and stream toward the probe is accelerated with high energy, giving input power to the probe.^3,15,33

In a gas discharge, the dominant mechanism of generating charges is due to the ionization of atoms and molecules by electron impact. The rate of generation of charges is given by

{(\frac{d n_{e}}{d t})}_{i} = ν_{i} n_{e} = k_{i} N n_{e}

⁠, where

ν_{i}

denotes the ionization frequency, which indicates the number of ionization events occurred by an electron per second,

N

is the total number of electrons,

k_{i}

represents the reaction rate constant. Under certain conditions, where ionization occurs in such a way that

ν_{i} = constant

and the removal of electrons has been ignored. In this case, the electrons are found growing exponentially, developing an avalanche condition, which is mathematically represented by

n_{e} = n_{e} (0) exp (ν_{i} t) .

(10)

The avalanche produced by an electron drifts along the direction of knocked-out electrons in an electric field. The number of electrons in the avalanche that knocked out

N_{0}

electrons and grew toward the probe at a distance x is given by

\frac{d N}{d x} = α N, N (x) = N_{0} exp (α x),

(11)

where

α

represents the ionization coefficient, and it is related to ionization frequency,

ν_{i}

as

α = \frac{ν_{i}}{v_{d}}

⁠, and

v_{d}

denotes the drift velocity. At the probe, the collected electron current is given by

I = e N_{0} exp (α d),

(12)

d represents the separation distance between the probe and the cathode.^1,3,33

During the course of the drift, the electrons are also attached in the electric field and multiplication of electrons takes place. The number of attachment events as the electrons drift along the field is called the attachment coefficient (a) denoted by

a = \frac{v_{a}}{v_{d}}

⁠, and

v_{a}

represents the attachment frequency. The avalanche equation, including the term effective ionization coefficient,

α_{eff}

⁠, where

α_{eff} = α - a

⁠, now becomes

\frac{d N_{e}}{d x} = (α - a) N_{e}; N_{e} \propto exp (α - a) x .

(13)

The equation that describes the overall ionization kinetics of the gas, that is, the total number of electrons,

N_{e}

in the discharge volume is given by

\frac{d N_{e}}{d t} = (ν_{i} - ν_{a} - ν_{d}) N_{e},

(14)

where

ν_{d}

represents the frequency of diffusion losses of electrons. Then, the value of

N_{e}

is given by

N_{e} = N_{e 0} exp (ν_{i} - ν_{a} - ν_{d}) t = N_{e 0} exp (\frac{t}{Θ})

⁠. The symbol

Θ

is the avalanche time constant, and

N_{e 0}

represents the number of seed electrons to start the avalanche.

C. Evacuation of plasma: The plasma density dip

In 1967, Alfvén and Carlqvist gave a qualitative explanation for the current interruption in plasma, suggesting that there is a local decrease in plasma density because of current-driven instability. In 1972, Carlqvist suggested a local evacuation of plasma leads to DL formation. Furthermore, he reported the decreasing density in plasma violated the quasi-neutrality condition of plasma and suggested this may be due to a current interruption. The current-carrying electrons are trapped at lower current densities. The current interruption may be scattered by wave fields or seized by a coherent large-amplitude wave. The previously trapped electrons heat the incoming streaming electrons. The DLs produced enhance the thermalization of background plasma from wave–particle interaction, and wave–wave interaction, which disperse and scatter the plasma electrons. It has been experimentally verified that the electron density within a DL is significantly lower as compared to the surrounding plasma (Lindberg and Torven, 1979).^15,40,41

Generally, DL's role is to accelerate the charged particles to higher energies, and the current in a DL is mainly provided by the drifted electron current. In the region where there is a dip in the electron density, the electrons must be accelerated. In this way, the density of electrons

n_{e}

follows the local density of ions

n_{i}

⁠. Subsequently, the electric field, E, is generated, which is given by

E \equiv - \frac{\partial ϕ}{\partial x} = - \frac{m_{e} v_{e}}{e} \frac{\partial v_{e}}{\partial x} = \frac{\partial}{\partial x} (\frac{m_{e} i_{e}^{2}}{2 e^{3}} \frac{1}{n^{2}}) .

(15)

Since the plasma is quasi-neutral, we assume

n_{e} = n_{i} = n

and

i_{e}

denotes the electron current. This electric field acts on the ions and, in turn, produces an outward-driving force given by

F_{E} = - \frac{\partial}{\partial x} (m_{e} n_{e} v_{e}^{2})

⁠. The electric field enhances the acceleration of ions, causing further reduction of density and producing local regions with extremely low densities, which favors the formation of DL. The generated driving force

F_{E}

should be greater than the force produced due to the ion-pressure gradient for evacuating the plasma and concludes that to excite the Buneman instability, which is a critical condition for DL formation, the electron velocity must exceed a certain critical value. The charge neutrality condition suddenly breaks down after the DL formation has taken place.^15,28

D. Particle trapping and detrapping

Particle trapping is one of the fundamental phenomena in plasma physics. For a particle moving with velocity v, trapping is only possible if it has lower energy as compared to wave potential denoted by the expression, $q ϕ > \frac{1}{2} m {(v - v_{ϕ})}^{2}$ ⁠, where $v_{ϕ}$ denotes the speed of the wave, and the particles moving at wave speed are known as resonant particles.^2,42 Even though there are small amplitudes, the resonant particles are easily trapped, and more particles in a wide range of distribution functions can also be trapped when the wave's amplitude becomes larger. This can be seen in the CWT spectrum shown in Figs. 9(III-b), 9(IV-b), and 9(V-b) for $V_{p} = 40$ ⁠, 50, and 60 V. The density variation for 0.1 mbar pressure, as shown in Fig. 8 (in solid blue line), the plasma density is minimum for $V_{p} = 40$ V; however, the I–V curve shown in Fig. 3(I-a) that current increases exponentially. This suggests that the current increases due to ionization and this ionization leads to detrapping electrons in DL. The detrapping causes an increase in local ion density and, thus, rise of IPW. However, with a further rise in electron density fluctuations with increasing probe voltage corresponding to $V_{p} = 50$ V and $V_{p} = 60$ V, electrons are again get trapped in the ion potential well, and the amplitude rises as shown in Figs. 9(IV-b) and 9(V-b). Similarly, for 0.3 mbar pressure, as shown in Fig. 8 (in dashed red line), the plasma density is minimum for $V_{p} = 60$ V. The density increases again for $V_{p} = 80$ V due to the trapping of electrons, which leads to the rise of EPW. These EPW trap electrons. The corresponding CWT spectra in Fig. 10(IV-b) and Fig. 10(VI-b) show that the amplitude of EPW becomes larger. In the phase space plots, the trapped particles are identified by closed orbits. The trapped particles possess a finite lifetime, and they suddenly lose their energy and become detrapped. The detrapping occurs when the streaming electrons in the extended anode sheath occur intense ionization with background neutrals. It has also been reported that the number of trapped particles is proportional to wave amplitude and inversely proportional to the speed of the wave. To determine whether a particle is trapped or not, it depends on the energy acquired in overcoming the potential, $ϕ_{c}$ ⁠, and hence, the trapping condition becomes $q ϕ > \frac{1}{2} m {(v - v_{ϕ})}^{2} + q ϕ_{c}$ ⁠. The detrapping process involves the particles' energy loss and the interaction of trapped particles and several streaming particles in its path. In this process, the kinetic energy of the trapped particles is distributed to other particles through various means, such as the excitation of waves and collisions, where there is the transfer of energy from particles to waves.⁴³

For a wave traveling in the x-direction at the speed,

ω / k

⁠, the particle observes a constant wave profile. Thus, the equation of motion of an electron is

m \ddot{x} = - e E (x) = e \frac{d ϕ (x)}{d x},

(16)

where

ϕ (x)

denotes the electrostatic potential. Integrating the above equation, we get

\frac{1}{2} m {\dot{x}}^{2} - e ϕ (x) = E_{0}

(17)

with

E_{0}

⁠, a constant equal to the electron's total energy. This equation shows that if

E_{0} > - e ϕ (x)

⁠, the kinetic energy is positive for all x, resulting in untrapped electrons. However, all electrons with energy values

E_{0}

below wave peaks are trapped and able to oscillate back and forth in wave troughs between the points at which

E_{0} = - e ϕ (x)

⁠. Thus, we conclude that only those resonant electrons moving at the speed

v \approx ω / k

are trapped in wave troughs. The trajectories of trapped electrons display an oscillatory motion. Suppose the electric field for the most strongly trapped electrons is denoted by

E (x) = \tilde{E} s inkx ≅ \tilde{E} k x

⁠, then from the equation of motion

m \ddot{x} = - m ω^{2} x = - e \tilde{E} k x,

(18)

where

ω^{2} = \frac{e \tilde{E} k}{m} \equiv ω_{B}^{2}

⁠, where

ω_{B}

is known as bounce frequency. The wave energy decays for

t < \frac{2 π}{ω_{B}},

and releases some of its trapped particles by oscillating at a frequency of the order of

ω_{B}

⁠, which increases its frequency for an increase in t. On the other hand, for

t ≫ \frac{2 π}{ω_{B}},

the wave energy becomes a constant value, which is very much less than its initial value.^2,42

E. Wave instabilities

The onset of DL is associated with various waves and the evolution of different instabilities.^2,17,41,42 In a current-carrying DL plasma, at a certain critical condition, the drift velocity of electrons

(v_{d}) \geq 1.3 \times v_{t e}

⁠, where

v_{t e} = {(\frac{K T_{e}}{m_{e}})}^{1 / 2}

is the electron thermal velocity, and

T_{e} ≫ T_{i}

⁠.^27,44 This resulted in triggering the Buneman instability, which gives the exact threshold condition for the formation of the double layer.¹⁷ Therefore, the onset of streaming instability depends on the current primarily on the drift velocity of electrons. Streaming instabilities are observed in plasma where one particle species moves with a net velocity relative to another. Since electrons move with a velocity

v_{0}

relative to the ions considered, ions are stationary and form a neutralizing background. The dispersion relation for streaming instabilities is given by

1 - \frac{ω_{p e}^{2}}{{(ω - k \cdot v_{0})}^{2}} = 0,

(19)

where

ω_{p e}

represents the plasma frequency, k is the wave vector. Due to these instabilities, the electrons drifting toward the extended anode sheath are trapped, leading to the rise of EPWs. These streaming electrons gain sufficient energy and cause ionization with background neutrals. Buneman instability arises when the drift velocity of an electron is greater than the electron's thermal velocity.^17,44 In this condition, the DLs are observed in current-carrying plasma. The Buneman instability, which is a current-driven instability, plays a crucial role in triggering the formation of DL as it provides the required critical current density and localization of electron space charges. The dispersion relation of Buneman instability^26,42 is given by

1 - \frac{ω_{p e}^{2}}{{(ω - k \cdot v_{0})}^{2}} - \frac{ω_{p i}^{2}}{ω^{2}} = 0,

(20)

where

ω_{p e}^{2}

and

ω_{p i}^{2}

are, respectively, the electrons and ion plasma frequencies, and a velocity

v_{0}

for electrons moving relative to the ions. As proposed by Smith and Goertz, the mechanism for triggering the formation of DL may be due to pondermotive force (⁠

F_{p}

⁠), when acts on charged particle species

α

disturb the motion of the electron and it is given by

F_{p} \propto - \frac{1}{m_{α}} \frac{\partial}{\partial x} ⟨ \frac{E^{2}}{8 π} ⟩

⁠, where the term

⟨ \dots ⟩

indicates averaging over the wave period.⁴¹ The reflected electrons bounce back and forth between the walls of the wave potential, thus increasing the electric field with the growing wave field. The larger the amplitude, the reflected electrons oscillate in the wave field. These reflected electrons extract the monotonically increasing streaming electrons from the distribution function, and they lose their capability of further feeding the instability, thereby causing the stabilization of the instability.²⁶

IV. SUMMARY

On the subsequent increase in applied positive DC voltage to a metal probe immersed in plasma, the plasma electrons could not supply sufficient electrons to shield the applied voltage applied on the probe, which resulted in extending the sheath deeper into the plasma, causing self-excitation of EPW, forming DL and low energy electrons get trapped in DL potential. On intense ionization with background neutrals, the trapped electrons get detrapped leading to the damping of EPW and the rise of IPWs. The rise of IPWs is associated with detrapping of electrons, with a dip in plasma density. The wavelet analysis reveals the interplay between EPWs and IPWs. Overall, the sole process occurring in a discharge chamber depends on the density of electrons, their production and removal, heating of the gas, etc. Higher pressures result in discharge contraction and higher current. This observation of producing AC signals on the application of DC voltage is similar to that of the Gunn-like effect in semiconductor devices. However, in the case of plasma, it results in the generation of plasma waves.

V. CONCLUSIONS

Based on the experimental observation, we conclude that the application of positive DC voltage on a probe immersed in plasma results in the excitation of plasma waves, which is similar to the GUNN EFFECT in semiconductor devices. The excitation of high-frequency EPWs results from the trapping of electrons in the extended sheath region of the immersed probe when a sufficiently high voltage is applied. The trapping also causes the formation of DL. The immersed probe acts as a sink for electrons, and these accelerating electrons toward the probe cause the ionization of background neutrals. Ionization causes detrapping of the trapped electrons, leading to a dip in plasma density. Under these conditions, the applied positive voltage excites low-frequency IPW. The competition between the rate of loss of electrons and the ionization of background neutrals causes the interplay between EPW and IPW. At lower pressure, the impact of DLs is more significant than at higher pressure because we observe important processes like ionization and excitation more appreciably. The DL formation occurs when there is a local dip in plasma density than in its surrounding regions. The electron temperature increases after the formation of DL, corresponding to an increase in the applied probe voltage.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Thangjam Rishikanta Singh: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Sneha Latha Kommuguri: Methodology (equal). Suraj Kumar Sinha: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

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

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© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Trapping and detrapping of electrons in a typical DC glow discharge plasma under double layer condition

I. INTRODUCTION

II. EXPERIMENT

A. Experimental setup

B. Effect of probe voltage (⁠ $V_{p}$ ⁠) on discharge current (⁠ $I_{c}$ ⁠)

C. Probe current (⁠ $I_{p}$ ⁠) vs probe voltage (⁠ $V_{p}$ ⁠)

D. Total discharge current $I_{d} = I_{c} + I_{p}$

E. Langmuir probe measurements

F. Plasma density measurements

G. Variation of plasma density

H. Floating potential fluctuations (FPFs)

I. Continuous wavelet transform (CWT) analysis

III. DISCUSSIONS

A. Effects observed at cathode region

B. The wall of the vacuum chamber

C. Evacuation of plasma: The plasma density dip

D. Particle trapping and detrapping

E. Wave instabilities

IV. SUMMARY

V. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Trapping and detrapping of electrons in a typical DC glow discharge plasma under double layer condition

I. INTRODUCTION

II. EXPERIMENT

A. Experimental setup

B. Effect of probe voltage (⁠Vp⁠) on discharge current (⁠Ic⁠)

C. Probe current (⁠Ip⁠) vs probe voltage (⁠Vp⁠)

D. Total discharge current Id=Ic+Ip

E. Langmuir probe measurements

F. Plasma density measurements

G. Variation of plasma density

H. Floating potential fluctuations (FPFs)

I. Continuous wavelet transform (CWT) analysis

III. DISCUSSIONS

A. Effects observed at cathode region

B. The wall of the vacuum chamber

C. Evacuation of plasma: The plasma density dip

D. Particle trapping and detrapping

E. Wave instabilities

IV. SUMMARY

V. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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B. Effect of probe voltage (⁠ $V_{p}$ ⁠) on discharge current (⁠ $I_{c}$ ⁠)

C. Probe current (⁠ $I_{p}$ ⁠) vs probe voltage (⁠ $V_{p}$ ⁠)

D. Total discharge current $I_{d} = I_{c} + I_{p}$