Theoretical study of particle and energy balance equations in locally bounded plasmas

1.   Introduction

Conservation laws are the most fundamental principles in physics, dictating that physical quantities such as energy, momentum, charge, and others remain conserved before and after physical or chemical reactions [1, 2]. In the plasma reactors used for manufacturing semiconductors and display devices, these conservation laws play a crucial role and are closely connected to determining the characteristics of the plasma, including electron density and electron temperature. In general, the supplied energy for plasma discharge and the total number of plasma particles formed inside the reactor are always conserved. Equations describing these conserved quantities are known as the energy and particle balance equations [3–7]. For the energy balance equation, the total radio frequency (RF) power ${P}_{\mathrm{a}\mathrm{b}\mathrm{s}}$ supplied to create plasma discharge must equal to the total energy lost by ion-electron pairs, and this is expressed as:

Here, ${n}_{0}$ represents the electron density, ${u}_{\mathrm{B}}$ represents the Bohm velocity, and ${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\varepsilon }_{\mathrm{T}}$ respectively show the effective area for particle loss and the total energy lost by ion–electron pairs. Using equation (1), it is possible to determine the electron density based on the ${P}_{\mathrm{a}\mathrm{b}\mathrm{s}}$, ${u}_{\mathrm{B}}$, ${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$, and ${\varepsilon }_{\mathrm{T}}$ [8, 9].

The particle balance equation is based on the premise that the total ionization inside the reactor must equal to the total surface loss inside the reactor, with this condition expressed as:

Here, ${K}_{\mathrm{i}\mathrm{z}}$ represents the volume ionization rate of the gas, ${n}_{\mathrm{g}}$ is the number density of neutral gas, and ${V}_{0}$ denotes the volume of the reactor. Using equation (2), it is possible to determine the electron temperature of the plasma. The energy and particle balance equations provide the simplest approach to determine the electron densities and temperatures inside a plasma reactor, as it is based on the analysis of physical conservation quantities within the reactor.

Despite the importance of predicting both electron temperature and density, applying classical energy and particle balance equations to industrial plasma reactors [10] comes with practical challenges. In plasma etchers used for semiconductor processing, reactor interiors are designed to not fully confine plasma but to allow partial diffusion through components such as plasma confine rings [11–13] and gas baffles [14, 15]. Classical energy and particle balance equations do not include the necessary variables (e.g., open boundary area, ion escape velocity) to effectively model these heterogeneous plasma boundaries. For these reasons, equations (1) and (2) cannot be applied to industrial plasma reactors. To address these challenges, it is essential to develop new equations that incorporate variables associated with heterogeneous plasma boundaries.

In this study, we formulated new energy and particle balance equations for heterogeneous plasma boundaries. Using these equations, we analyzed variations in electron temperature and density based on the open boundary area ($A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$) and ion escape velocity (${u}_{\mathrm{i}}$). At low ion escape velocities (${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}\approx 0.2$) and high open boundary areas ($A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}\approx 0.6$), we found an approximately 38% increase in electron density and an 8% decrease in electron temperature compared to values for a fully bounded reactor. We also analyzed the electric potential equation around the open plasma boundary to determine the ion loss velocity in the energy and particle balance equations. Our analysis revealed that the ion escape velocity through the open plasma boundary exceeds the product of the ion plasma frequency (${\omega }_{\mathrm{p}\mathrm{i}}$) and the electron Debye length (${\lambda }_{\mathrm{D}\mathrm{e}})$. The variables utilized to describe the energy and particle balance equations in this study are summarized in table 1.

Table  1.  Parameter symbols and abbreviations.

Symbols	Descriptions	Dimensions
${A}_{\mathrm{L}}$	Bounded reactor wall area	m2
$A'_{\mathrm{L}}$	Unbounded reactor wall area	m2
${A}_{\mathrm{T}}$	${A}_{\mathrm{L}}+A'_{\mathrm{L}}$	m2
${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$	$h{\cdot A}_{\mathrm{L}}$	m2
$A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$	${h'}\cdot A'_{\mathrm{L}}$	m2
${A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$	Total effective wall loss area: $h{\cdot A}_{\mathrm{L}}+{h'}\cdot A'_{\mathrm{L}}$	m2
${d}_{\mathrm{e}\mathrm{f}\mathrm{f}}$	Effective plasma size: ${V}_{0}/{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$	m
${d}_{\mathrm{i}\mathrm{z}}$	$u_{\mathrm{i}\mathrm{z}}/\nu_{\mathrm{i}\mathrm{z}0}$	m
$e$	Elementary charge	C
${E}_{0}$	Electric field at an open boundary	V·m−1
${E}_{\mathrm{s}}$	Electric field at a sheath edge	V·m−1
$h$	${n}_{\mathrm{s}}$/${n}_{0}$	-
${h'}$	$n'_{\mathrm{i}}$/${n}_{0}$	-
${K}_{\mathrm{e}\mathrm{l}}$	Elastic rate constant	m3·s−1
${K}_{\mathrm{e}\mathrm{x}}$	Excitation rate constant	m3·s−1
${K}_{\mathrm{i}\mathrm{z}}$	Ionization rate constant: $K_{\mathrm{i}\mathrm{z}0}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\varepsilon_{\mathrm{i}\mathrm{z}}/T_{\mathrm{e}}\right)$	m3·s−1
${K}_{\mathrm{i}\mathrm{z}0}$	Preexponential factor	m3·s−1
${m}_{\mathrm{e}}$	Electron mass	kg
${M}_{\mathrm{i}}$	Ion mass	kg
$n\left(x,y,z\right)$	Spatial electron density (x, y, z)	m−3
${n}_{0}$	Electron density at a bulk plasma	m−3
${n}_{\mathrm{e}}$	Electron density	m−3
${n}_{\mathrm{e}}\left(\mathrm{F}\right)$	Electron density (fully bounded plasma)	m−3
${n}_{\mathrm{e}}\left(\mathrm{L}\right)$	Electron density (locally bounded plasma)	m−3
$n'_{\mathrm{e}}$	Electron density at an open boundary	m−3
${n}_{\mathrm{g}}$	Number density of neutral gas	m−3
${n}_{\mathrm{i}}$	Ion density	m−3
$n'_{\mathrm{i}}$	Ion density at an open boundary	m−3
${n}_{\mathrm{s}}$	Electron density at a sheath edge	m−3
${N}_{A}$	$A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$	-
${N}_{u}$	${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$	-
$P_0$	Gas pressure	Torr
${P}_{\mathrm{a}\mathrm{b}\mathrm{s}}$	Total energy loss: ${P}_{\mathrm{c}}+{P}_{\mathrm{e}}+{P}_{\mathrm{i}}$	J·s−1
${P}_{\mathrm{c}}$	Volume energy loss in a plasma	J·s−1
${P}_{\mathrm{e}}$	Kinetic energy loss of electrons at a reactor wall	J·s−1
${P}_{\mathrm{i}}$	Kinetic energy loss of ions at a reactor wall	J·s−1
${T}_{\mathrm{e}}$	Electron temperature	V
${T}_{\mathrm{e}}\left(\mathrm{F}\right)$	Electron temperature (fully bounded plasma)	V
${T}_{\mathrm{e}}\left(\mathrm{L}\right)$	Electron temperature (locally bounded plasma)	V
${T}_{\mathrm{i}}$	Ion temperature	V
${T}_{\mathrm{n}}$	Neutral gas temperature	V
${u}_{\mathrm{B}}$	Bohm velocity: ${\left(e{T}_{\mathrm{e}}/{M}_{\mathrm{i}}\right)}^{{1}/{2}}$	m·s−1
${u}_{\mathrm{i}}$	Ion escape velocity at an open boundary	m·s−1
${u}_{\mathrm{i}\mathrm{b}}$	Initial ion velocity at a bulk plasma	m·s−1
${u}_{\mathrm{i}\mathrm{z}}$	${\left(e{\varepsilon}_{\mathrm{iz}}/{M}_{\mathrm{i}}\right)}^{1/2}$	m·s−1
${V}_{0}$	Reactor volume	m3
${V}_{\mathrm{s}}$	Sheath voltage	V
$\alpha$	${V}_{0}{{n}_{\mathrm{g}}K}_{\mathrm{i}\mathrm{z}0}$	m3·s−1
$\beta$	$e/{M}_{\mathrm{i}}$	C·kg−1
$\gamma$	${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}\left(1+{N}_{u}{N}_{\mathrm{A}}\right)$	m2
${\varGamma }_{\mathrm{L},\mathrm{e}}$	Electron flux to a bounded reactor wall	s−1·m−2
$\varGamma'_{\mathrm{L},\mathrm{e}}$	Electron flux to an unbounded reactor wall	s−1·m−2
${\varGamma }_{\mathrm{L},\mathrm{i}}$	Ion flux to a bounded reactor wall: ${n}_{\mathrm{s}}{u}_{\mathrm{B}}$	s−1·m−2
$\varGamma '_{\mathrm{L},\mathrm{i}}$	Ion flux to an unbounded reactor wall: $n'_{\mathrm{i}}{u}_{\mathrm{i}}$	s−1·m−2
${\varepsilon}_{0}$	Permittivity of vacuum	F·m−1
${\varepsilon }_{\mathrm{c}}$	Collisional energy loss per electron–ion pair creation	V
${\varepsilon }_{\mathrm{e}}$	Electron kinetic energy loss at a bounded wall	V
${\varepsilon }_{\mathrm{e}\mathrm{x}}$	Excitation energy of neutral gas	V
${\varepsilon }_{\mathrm{i}}$	Ion kinetic energy loss at a bounded wall	V
${\varepsilon}_{\mathrm{i}\mathrm{z}}$	Ionization energy of neutral gas	V
${\varepsilon }_{\mathrm{T}}$	Total energy lost per electron–ion pair	V
$\varepsilon' _{\mathrm{e}}$	Electron kinetic energy loss at an unbounded wall	V
$\varepsilon' _{\mathrm{i}}$	Ion kinetic energy loss at an unbounded wall	V
$\theta$	$A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$	-
${\lambda }_{\mathrm{D}\mathrm{e}}$	Electron Debye length: ${\left({\varepsilon }_{0}{T}_{\mathrm{e}}/en'_{\mathrm{e}}\right)}^{1/2}$	m
${\lambda }_{\mathrm{i}}$	Ionization mean free path	m
$\mu$	$\beta \cdot{\left(\gamma /\alpha \right)}^{2}$	V−1
${\nu }_{\mathrm{i}\mathrm{z}}$	Ionization frequency: ${{n}_{\mathrm{g}}K}_{\mathrm{i}\mathrm{z}0}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\varepsilon}_{\mathrm{i}\mathrm{z}}/{T}_{\mathrm{e}}\right)$	s−1
${\nu }_{\mathrm{i}\mathrm{z}0}$	Preexponential factor: ${{n}_{\mathrm{g}}K}_{\mathrm{i}\mathrm{z}0}$	s−1
$\varPhi$	Electric potential	V
$\psi$	$\mathrm{i}\sqrt{2}{(d}_{\mathrm{i}\mathrm{z}}/{d}_{\mathrm{e}\mathrm{f}\mathrm{f}})\left(1+{N}_{\mathrm{u}}{N}_{\mathrm{A}}\right)$	-
${\omega }_{\mathrm{p}\mathrm{i}}$	Ion plasma frequency: ${\left({e}^{2}n'_{\mathrm{i}}/{\varepsilon }_{0}{M}_{\mathrm{i}}\right)}^{1/2}$	s−1

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2.   Heterogeneous plasma boundary in the process reactor

To integrate the concepts of bounded and unbounded reactor walls into particle and energy balance equations, it is essential to define their respective physical roles. Bounded walls serve to confine plasma physically. These reactor walls can be categorized based on their electrical properties as: (1) electrically grounded walls, (2) floating walls, and (3) powered walls. In plasma reactors used for semiconductor production, reactor walls are coated with dielectric films such as Y₂O₃, YOF, YF₃ [16, 17], which block direct current from the plasma. Electrons and ions generated from bulk plasma transport their kinetic energy to the reactor wall, where it dissipates on the surface of the reactor wall. This plasma energy loss varies substantially depending on factors such as plasma potential, direct current (DC) self-bias, discharge gas pressure, and sheath characteristics. Under DC discharge conditions, ion and electron energy losses on the reactor-wall surfaces are determined by the sheath voltage (${V}_{\mathrm{s}}$) and electron temperature (${T}_{\mathrm{e}}$), respectively, and are known as ${\varepsilon }_{\mathrm{i}}={V}_{\mathrm{s}}$+(1/2)${T}_{\mathrm{e}}$ and $\varepsilon_{\mathrm{e}}=2T_{\mathrm{e}}$ [3]. Generally, the kinetic energy loss characteristics of plasma can be influenced by factors such as discharge gas temperature and gas flow dynamics within the reactor. However, in typical plasma etchers, the discharge pressure is very low (a few millitorr). Under these conditions, the neutral gas temperature (${T}_{\mathrm{n}}$​) is significantly lower than either ion temperature (${T}_{\mathrm{i}}$​) or electron temperature (${T}_{\mathrm{e}}$​), making its influence on plasma kinetic energy negligible. Furthermore, within this range of discharge pressure, the flow dynamics inside the reactor can effectively be disregarded.

Unbounded reactor walls do not confine plasma physically. Electrons and ions escape to the outside of the reactor through open areas. In this case, the kinetic energy transferred from plasma can be determined by ion velocities and electron temperatures around bounded areas. Examples of widely applied unbounded reactor walls in plasma reactors include the plasma confinement ring (figure 1(a)) and the gas baffle (figure 1(b)). The plasma confinement ring consists of 4 quartz rings, with the spacing between them varying according to process pressure. These structures confine plasma to localized regions, allowing chemical by-products generated during wafer processing to be exhausted through the spaces between the confinement rings. The gas baffle is located between the reactor process zone and the gas pumping system, controlling pressure gradients and ensuring stable plasma confinement within the reactor process zone.

Figure  1.  Example of locally bounded plasma wall: (a) plasma confinement rings, (b) gas baffle slits in an industrial plasma reactor. Plasma is locally confined by slit structures, and heterogeneous plasma boundaries are formed.

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In a plasma reactor system with fully bounded walls (figure 2(a)), the inner reactor represents the area where plasma processes occur, assuming conditions where the process pressure is sufficiently low (“ionization mean free path (${\lambda }_{\mathrm{i}}$) $\gg$ reactor scale” [18–20]), ensuring the formation of uniform plasma. Under these conditions, the Bohm criterion is valid at the sheath edge, and ions enter the sheath with the Bohm velocity ${u}_{\mathrm{B}}$. The electron temperature and density can then be calculated using the particle and energy balance equations as indicated in equations (1) and (2).

Figure  2.  Schematic description of (a) fully bounded plasma boundary, (b) locally bounded plasma boundary. In the case of a fully bounded plasma boundary, the ion velocity in the plasma-sheath boundary is determined by the Bohm velocity ${u}_{\mathrm{B}}$. However, in the case of a reactor having a locally bounded plasma boundary, the ion diffusing to the unbounded area is not limited to the Bohm velocity.

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In a plasma reactor system featuring an unbounded wall (figure 2(b)), the outer reactor defines a space where plasma diffuses through the unbounded wall (open area) of the inner reactor. Ions passing through the unbounded area do not encounter the presheath-sheath structure and are not constrained by the Bohm criterion. Mathematically, this unbounded area can be represented as a local Gaussian surface [21] through which electron and ion fluxes pass. The volume ionization formed within the Gaussian surface remains in equilibrium with the plasma loss through it. Consequently, the electron and ion fluxes passing through the Gaussian surface determine the global electron temperature and electron density within the reactor. Even if the reactor wall is fully open, it remains possible to solve energy and particle balance equations using the electron and ion fluxes passing through the Gaussian surface. To determine the global electron density and electron temperature under locally bounded plasma conditions, it is necessary to consider the plasma loss characteristics at heterogeneous boundaries separately, and equations (1) and (2) should be modified accordingly. The development of an analytical model for this is described in subsequent sections.

3.   Particle balance equation in the locally bounded plasma

It can be generally assumed that even for locally bounded plasma, the plasma volume ionization within the reactor always remains in balance with the plasma wall loss. Denoting ion fluxes reaching the bounded wall area ${A}_{\mathrm{L}}$ and the unbounded wall area $A'_{\mathrm{L}}$ as ${\varGamma }_{\mathrm{L},\mathrm{i}}$ and $\varGamma '_{\mathrm{L},\mathrm{i}}$ respectively, we obtain the following particle balance equation:

Here, ${\nu }_{\mathrm{i}\mathrm{z}}$ and $n\left(x,y,z\right)$ represent the plasma volume ionization frequency and spatial electron density, respectively. For ion fluxes reaching the reactor’s boundaries, we define the ion density at the bounded wall’s sheath edge as ${n}_{\mathrm{s}}$ and the ion density at the open boundary as $n'_{\mathrm{i}}$. Following this, the ion fluxes at the reactor wall can be expressed as ${\varGamma }_{\mathrm{L},\mathrm{i}}={n}_{\mathrm{s}}{u}_{\mathrm{B}}$ and $\varGamma '_{\mathrm{L},\mathrm{i}}=n'_{\mathrm{i}}{u}_{\mathrm{i}}$, respectively. In this study, plasma discharge assumes a low-pressure condition where the ionization mean free path is significantly larger than the reactor scale. Under these conditions, the spatial distribution of plasma is determined by global ionization across the entire reactor volume rather than local ionization. Thus, the ion flux supplied by bulk plasma ionization is not locally nonuniform. These conditions approximate the bulk plasma in a 0-dimensional form, where the physical characteristics of ions reaching ${A}_{\mathrm{L}}$ and $A'_{\mathrm{L}}$ are determined by the global ionization properties within the reactor.

To simplify equation (3) and applying the low-pressure limit where ${\lambda }_{\mathrm{i}}\gg$ reactor scale, the particle balance equation can be approximated as $\iiint {\nu }_{\mathrm{i}\mathrm{z}}n\left(x,y,z\right)\mathrm{d}V\cong {\nu }_{\mathrm{i}\mathrm{z}}{n}_{0}{V}_{0}$, and equation (3) can thus be described as:

Here, ${n}_{0}$ and ${V}_{0}$ represent the bulk electron density and reactor volume, respectively. The volume ionization frequency can be expressed in the Arrhenius form as $\nu_{\mathrm{i}\mathrm{z}}= n_{\mathrm{g}}K_{\mathrm{i}\mathrm{z}0}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\varepsilon_{\mathrm{i}\mathrm{z}}/T_{\mathrm{e}}\right)$ [18]. To modify equation (4) into a form that includes the effective loss area, we define $h\equiv {n}_{\mathrm{s}}$/${n}_{0}$ and $h'\equiv n'_{\mathrm{i}}$/${n}_{0}$. Using this, we express the effective plasma loss areas as ${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}=h{\cdot A}_{\mathrm{L}}$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}=h'\cdot {A}_{\mathrm{L}}$. Applying this to equation (4) yields the following equation:

In general, obtaining a general solution for ${T}_{\mathrm{e}}$ can be challenging when the last term ${u}_{\mathrm{i}}A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ is present in equation (5). To avoid this, there is a need to normalize the parameters ${u}_{\mathrm{i}}$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$. Firstly, ${u}_{\mathrm{i}}$ can be normalized as ${N}_{u}\equiv {u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$. Here, ${N}_{u}$ is mathematically normalized to the Bohm velocity but does not inherently depend on the electron temperature. Secondly, $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ can be normalized as ${N}_{A}\equiv A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$. The total effective loss area ${A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ of the plasma reactor is always conserved, so this area can be defined as:

Assuming that θ is a real number within the range of 0–1.0, ${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ can be represented as:

By using ${N}_{u}={u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ and ${N}_{A}=A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$, the last term of equation (5) can be included in ${\left(e{T}_{\mathrm{e}}/{M}_{\mathrm{i}}\right)}^{\frac{1}{2}}{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$, and the particle balance equation can be expressed as:

If we define constants $\alpha$, $\beta$, and $\gamma$ as $\alpha \equiv {V}_{0}{{n}_{\mathrm{g}}K}_{\mathrm{i}\mathrm{z}0}$, $\beta \equiv e/{M}_{\mathrm{i}}$, $\gamma \equiv {A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}\left(1+{N}_{u}{N}_{A}\right)$, equation (5) can be expressed as follows:

Using $\mu \equiv \beta \cdot {\left(\gamma /\alpha \right)}^{2}$, we can rewrite equation (10) as follows:

When solving equation (11) for the electron temperature ${T}_{\mathrm{e}}$, the following result is obtained:

Equation (12) is a function of the Lambert W function [22]. Using the definitions of $\alpha$, $\beta$, and $\gamma$, equation (12) can be expressed as follows:

To simplify the solution of equation (13), it is necessary to define ${\nu }_{\mathrm{i}\mathrm{z}0}\equiv {{n}_{\mathrm{g}}K}_{\mathrm{i}\mathrm{z}0}$, $u_{\mathrm{i}\mathrm{z}}\equiv\left(e\varepsilon_{\mathrm{i}\mathrm{z}}/M_{\mathrm{i}}\right)^{1/2}$, ${d}_{\mathrm{i}\mathrm{z}}\equiv {u}_{\mathrm{i}\mathrm{z}}/{\nu }_{\mathrm{i}\mathrm{z}0}$, and ${d}_{\mathrm{e}\mathrm{f}\mathrm{f}}\equiv {V}_{0}/{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$. Using this, we can obtain the following expression:

The electron temperature can be determined through four parameters for gas state ($\varepsilon_{\mathrm{i}\mathrm{z}}$, ${K}_{\mathrm{i}\mathrm{z}0}$, ${n}_{\mathrm{g}}$, ${M}_{\mathrm{i}}$), three parameters for the reactor (${A}_{\mathrm{L}}$, $A'_{\mathrm{L}}$, ${V}_{0}$), and four parameters for the plasma (${u}_{\mathrm{i}}$, ${n}_{\mathrm{s}}$, $n'_{\mathrm{i}}$, ${n}_{0}$). Using equation (14), it is also possible to calculate the change in electron temperature with respect to ${N}_{A}$ and ${N}_{u}$.

Here, $\psi$ is defined as $\psi \equiv \mathrm{i}\sqrt{2}{(d}_{\mathrm{i}\mathrm{z}}/{d}_{\mathrm{e}\mathrm{f}\mathrm{f}})\left(1+{N}_{u}{N}_{A}\right)$. In the case of the fully bounded plasma, where $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}=0$ (i.e., $\theta =0$) is satisfied, equation (14) is approximated as:

Equation (17) represents the analytic solution for the electron temperature obtained from the particle balance equation of the fully bounded plasma. The electron temperature characteristics of the locally bounded plasma need to be compared to those of the conventional fully bounded plasma. If we define the electron temperature based on the conditions of the locally bounded plasma as ${T}_{\mathrm{e}}\left(\mathrm{L}\right)$, and the electron temperature under the conditions of the fully bounded plasma as ${T}_{\mathrm{e}}\left(\mathrm{F}\right)$, then we can express the normalized electron temperature as:

In representing the change in the normalized electron temperature ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ as a function of ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ (figure 3(a)), argon gas at 20 mTorr pressure, 400 K temperature, a chamber volume of $V_0=\left(0.2\right)^3\mathrm{\ m}^3$, an effective area of $A_{\mathrm{T}}= 6\times\left(0.2\right)^2\mathrm{\ m}^2$, and $h=0.5$, and $h'=0.5$ [18] were used. Under the conditions $0 < A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$, the normalized electron temperature ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ increases as the ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ increases (figure 3(a)). As $n'_{\mathrm{i}}{u}_{\mathrm{i}}A'_{\mathrm{L}}$ increases in equation (4), the plasma volume ionization should equilibrate with wall loss. Since the plasma volume ionization frequency is proportional to $\nu_{\mathrm{i}\mathrm{z}}\propto\mathrm{e}\mathrm{x}\mathrm{p}\left(-\varepsilon_{\mathrm{i}\mathrm{z}}/T_{\mathrm{e}}\right)$, an increase in ${u}_{\mathrm{i}}$ should lead to an increase in ${T}_{\mathrm{e}}$ as well. Under the conditions of ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}=1$, where the plasma loss characteristics of the unbounded and bounded areas become identical, ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ does not depend on changes in the $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$.

Figure  3.  Normalized electron temperature ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ trends according to the (a) normalized ion velocity ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$, (b) normalized unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$.

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When the variation of the normalized electron temperature ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ with increasing $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ is considered (figure 3(b)), the plasma loss per unit area of the unbounded walls is less than that of the bounded walls when ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ < 1. As a result, the total plasma loss of the reactor decreases as $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ increases. The volume ionization frequency inside the reactor should be balanced by the plasma wall loss, and $\nu_{\mathrm{i}\mathrm{z}}\propto\mathrm{e}\mathrm{x}\mathrm{p}\left(-\varepsilon_{\mathrm{i}\mathrm{z}}/T_{\mathrm{e}}\right)$ should be reduced. To achieve this, ${T}_{\mathrm{e}}$ should decrease (figure 3(b)). When ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ = 1, the plasma loss per unit area of the unbounded walls becomes equal to the plasma loss per unit area of the bounded walls. Since the total plasma loss area ${A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ is always constant, there is no change in the electron temperature even when the area of $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ changes. When ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ > 1, the plasma loss per unit area of the unbounded walls is greater than that of the bounded walls, and the total plasma loss inside the reactor increases with $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$. The plasma volume ionization frequency within the reactor should remain in equilibrium with plasma wall loss, so $\nu_{\mathrm{i}\mathrm{z}}\propto\mathrm{e}\mathrm{x}\mathrm{p}\left(-\varepsilon_{\mathrm{i}\mathrm{z}}/T_{\mathrm{e}}\right)$ also increases. To achieve this, ${T}_{\mathrm{e}}$ should also increase.

Figure 4 represents the variation in the normalized electron temperature with respect to ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$. According to the 2-dimensional plot results for ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$, when both ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}\to 0$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}\to 1$ are satisfied, ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ is minimized. In this case, particle losses ${n}_{\mathrm{s}}{u}_{\mathrm{B}}{A}_{\mathrm{L}}+n'_{\mathrm{i}}{u}_{\mathrm{i}}A'_{\mathrm{L}}$ through both the bounded area ${A}_{\mathrm{L}}$ and unbounded area $A'_{\mathrm{L}}$ are minimized, and plasma volume ionization frequency is maximally reduced. Numerically, under conditions of ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}\approx 0.2$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}\approx 0.6$, the normalized electron temperature satisfies ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)\approx 0.92$. This indicates an approximately 8% reduction in electron temperature compared to fully bounded plasma.

Figure  4.  Normalization electron temperature ${T}_{\mathrm{e}}\left(\mathrm{L}\right)/{T}_{\mathrm{e}}\left(\mathrm{F}\right)$ trends according to the normalized ion velocity ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$, normalized unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$.

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4.   Energy balance equation in the locally bounded plasma

The RF power (${P}_{\mathrm{a}\mathrm{b}\mathrm{s}}$) delivered to the plasma is always equal to the sum of the volume energy loss (${P}_{\mathrm{c}}$) in the plasma and the kinetic energy loss of electrons and ions (${P}_{\mathrm{e}}+{P}_{\mathrm{i}}$) at the reactor walls. This can be expressed as:

Under the locally bounded plasma conditions, equation (19) can be specified as follows for ${P}_{\mathrm{c}}$, ${P}_{\mathrm{e}}$, and ${P}_{\mathrm{i}}$.

Here, ${\varepsilon }_{\mathrm{c}}$ represents the plasma energy loss due to collisions, and ${\varepsilon }_{\mathrm{e}}$ and ${\varepsilon }_{\mathrm{i}}$ represent the kinetic energy loss of electrons and ions to the reactor boundaries. In the case of plasma discharge that includes an unbounded area, it is necessary to separately consider the energy loss of electrons and ions passing through this boundary surface, with this energy loss denoted as $\varepsilon' _{\mathrm{e}}$, and $\varepsilon' _{\mathrm{i}}$, respectively. ${\varGamma }_{\mathrm{L},\mathrm{e}}$, $\varGamma'_{\mathrm{L},\mathrm{e}}$, and ${\varGamma }_{\mathrm{L},\mathrm{i}}$, $\varGamma '_{\mathrm{L},\mathrm{i}}$ in equations (21) and (22) represent the electron and ion fluxes reaching the bounded area (${A}_{\mathrm{L}}$) and unbounded area ($A'_{\mathrm{L}}$), respectively. In plasmas with heterogeneous boundaries, maintaining the quasi-neutrality condition necessitates satisfying conditions ${\varGamma }_{\mathrm{L},\mathrm{e}}={\varGamma }_{\mathrm{L},\mathrm{i}}$ and $\varGamma' _{\mathrm{L},\mathrm{e}}=\varGamma' _{\mathrm{L},\mathrm{i}}$. Assuming a low-pressure limit satisfying the ${\lambda }_{\mathrm{i}}\gg$ reactor scale, the plasma energy loss ${P}_{\mathrm{c}}$, ${P}_{\mathrm{e}}$, and ${P}_{\mathrm{i}}$ can be simplified as:

Through equations (23) and (24), the RF power delivered to the plasma can be formulated as:

Applying $h={n}_{\mathrm{s}}$/${n}_{0}$, ${h'}=n'_{\mathrm{i}}$/${n}_{0}$, and ${A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}=h{\cdot A}_{\mathrm{L}}$, $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}= {h'}\cdot A'_{\mathrm{L}}$ to equation (25), we can determine the bulk electron density ${n}_{0}$ as follows:

By applying ${N}_{u}={u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ and ${N}_{A}=A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ to equation (26) and rearranging it, we obtain the energy balance equation for the locally bounded plasma as follows:

The electron density can be determined through four parameters regarding the gas state ($\varepsilon_{\mathrm{i}\mathrm{z}}$, ${K}_{\mathrm{i}\mathrm{z}0}$, ${n}_{\mathrm{g}}$, ${M}_{\mathrm{i}}$), three parameters regarding the chamber (${A}_{\mathrm{L}}$, $A'_{\mathrm{L}}$, ${V}_{0}$), seven parameters regarding the plasma $\left(u\mathrm{_i},\ u_{\mathrm{B}},\ \varepsilon_{\mathrm{c}},\ \varepsilon_{\mathrm{e}},\ \varepsilon_{\mathrm{i}},\ \varepsilon'_{\mathrm{e}},\ \varepsilon'_{\mathrm{i}}\right)$, and one parameter regarding RF power (${P}_{\mathrm{a}\mathrm{b}\mathrm{s}}$). By utilizing equation (27), changes in electron density with respect to ${N}_{A}$ and ${N}_{u}$ can also be calculated.

By using the condition $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}=0$ and equation (2), we obtain the classical energy balance equation for the fully bounded plasma as:

Defining the electron density based on the locally bounded plasma conditions as $n\mathrm{_e}\left(\mathrm{L}\right)$ and the electron density based on the fully bounded plasma conditions as $n_{\mathrm{e}}\left(\mathrm{F}\right)$, the normalized electron density can be formulated as:

We used argon gas at 20 mTorr pressure, 400 K temperature, chamber volume ${V}_{0}={\left(0.2\right)}^{3}\;{\mathrm{m}}^{3}$, ${A}_{\mathrm{T}}=6\times {\left(0.2\right)}^{2}\;{\mathrm{m}}^{2}$, $h=0.5$, ${h'}=0.5$, ${P}_{\mathrm{a}\mathrm{b}\mathrm{s}}=500\;\mathrm{W}$, ${T}_{\mathrm{e}}=2\;\mathrm{e}\mathrm{V}$, ${V}_{\mathrm{s}}=100\;\mathrm{V}$, ${\varepsilon }_{\mathrm{e}}=2{T}_{\mathrm{e}}$, ${\varepsilon }_{\mathrm{i}}={V}_{\mathrm{s}}+(1/2){T}_{\mathrm{e}}$, and $\varepsilon '_{\mathrm{e}}=2{T}_{\mathrm{e}}$ to analyze equation (31). The collisional energy loss $\varepsilon\mathrm{_c}$ of bulk plasma was calculated using [23]:

The variation of the normalized electron density ${n}_{\mathrm{e}}\left(\mathrm{L}\right)/{n}_{\mathrm{e}}\left(\mathrm{F}\right)$ with respect to ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ (figure 5(a)) indicates that the normalized electron density decreases as ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ increases. This can be interpreted as an increase in ion loss through the unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$. Conversely, there is an increasing trend in normalized plasma density as $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ increases. Physically, ions and electrons passing through the unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ do not experience strong sheath voltage acceleration. This means that $\varepsilon '_{\mathrm{e}}+\varepsilon'_{\mathrm{i}}< {\varepsilon }_{\mathrm{e}}+{\varepsilon }_{\mathrm{i}}$ should always be satisfied. Hence, as $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ increases, total plasma wall loss decreases, and the electron density of the bulk plasma increases.

Figure  5.  Normalized electron density $n_{\mathrm{e}}\left(\mathrm{L}\right)/n_{\mathrm{e}}\left(\mathrm{F}\right)$ trends according to the (a) normalized ion velocity ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$, (b) normalized unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$.

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The variation of the normalized electron density ${n}_{\mathrm{e}}\left(\mathrm{L}\right)/{n}_{\mathrm{e}}\left(\mathrm{F}\right)$ with respect to $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ (figure 5(b)) shows that the normalized electron density rapidly increases as $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ increases. This is due to the decrease in total plasma wall loss. Additionally, it appears that ${n}_{\mathrm{e}}\left(\mathrm{L}\right)/{n}_{\mathrm{e}}\left(\mathrm{F}\right)$ decreases as the normalized ion velocity ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ increases from 0 to 1.5. This can be interpreted as an increase in ion loss through the unbounded area.

The variation of the normalized electron density with respect to ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$ (figure 6) shows that the electron density reaches its maximum when ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}\to 0$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}\to 1$. Under these conditions, $(\varepsilon' _{\mathrm{e}}+\varepsilon '_{\mathrm{i}}){u}_{\mathrm{i}}A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}\to 0$ and $\left({\varepsilon }_{\mathrm{e}}+{\varepsilon }_{\mathrm{i}}\right){u}_{\mathrm{B}}{A}_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}\to 0$ are satisfied in equations (26), resulting in an increase in electron density. Numerically, when ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}\approx 0.2$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}\approx 0.6$, the electron density increases by approximately 38% (${n}_{\mathrm{e}}\left(L\right)/{n}_{\mathrm{e}}\left(F\right)\approx 1.38$). The variation in the heterogeneous boundary conditions of the reactor is a key factor in altering plasma density, and this is directly linked to changes in the unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$.

Figure  6.  Normalized electron density $n_{\mathrm{e}}\left(\mathrm{L}\right)/n_{\mathrm{e}}\left(\mathrm{F}\right)$ according to the normalized ion velocity ${u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$, normalized unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}/{A}_{\mathrm{T},\mathrm{e}\mathrm{f}\mathrm{f}}$.

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5.   Minimum ion escape velocity at the open reactor boundary

We derived the particle and energy balance equations for the locally bounded plasma through equations (14) and (27). To predict global electron temperatures and densities in the reactor using these equations, it is necessary to calculate the ion velocity $u\mathrm{_i}$ passing through the reactor’s open boundary. The electron flux and ion flux passing through the unbounded area $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$ must always be equal (i.e., $\varGamma '_{\mathrm{L},\mathrm{e}}=\varGamma '_{\mathrm{L},\mathrm{i}}$), giving us:

Here, $n'_{\mathrm{e}}$ and $n'_{\mathrm{i}}$ represent the electron density and ion density at the open boundary. The position of the open boundary is defined as the origin $(x=0)$. When the change in electric potential from the bulk plasma to the open boundary is denoted as $\delta \varPhi$, the ion velocity at the open boundary can be expressed as:

Here, ${u}_{\mathrm{i}\mathrm{b}}$ represents the initial ion velocity in the bulk plasma, and its value is typically assumed to be negligibly small. Substituting equation (33) into equation (34), we can determine $n'_{\mathrm{i}}/n'_{\mathrm{e}}$ as:

Considering ${m}_{\mathrm{e}}\ll {M}_{\mathrm{i}}$, the ion density passing through the open boundary will always be larger than the electron density. Thus, it would not be possible to assume $n'_{\mathrm{e}}=n'_{\mathrm{i}}$ at the open boundary of the plasma reactor. Furthermore, since there is no sheath structure around the open boundary, it is also not possible to assume that the conventional Bohm criterion ${u}_{\mathrm{i}}\geqslant {u}_{\mathrm{B}}$ exists at the open boundary. In conclusion, there is a need to redefine the velocity of ions around the open plasma boundary. To achieve this, it is essential to solve the Poisson equation that includes the open boundary of the plasma.

Here, ${e\varepsilon }'_{\mathrm{i}}$ represents the ion energy passing through the open boundary $x=0$, given by ${e\varepsilon }'_{\mathrm{i}}=(1/2){{M}_{\mathrm{i}}u}_{\mathrm{i}}^{2}$. If we define the electric potential and electric field as $\varPhi \left(x=0\right)\equiv 0$ and $\mathrm{d}\varPhi (x=0)/\mathrm{d}x\equiv {E}_{0}$, equation (36) can be expressed as [3]:

After Taylor expansion for $\mathrm{exp}(\varPhi /{T}_{\mathrm{e}})$ and ${\left(1-\varPhi /{\varepsilon }'_{\mathrm{i}}\right)}^{\frac{1}{2}}$, we obtain:

Each term in equation (38) can be rearranged as follows:

When the ions reach the open boundary, $0 < \varPhi$ is satisfied, and the condition $n'_{\mathrm{e}} < n'_{\mathrm{i}}$ is confirmed by equation (35). This verifies that the left-hand side of equation (39) is always positive. Consequently, this makes the right-hand side of equation (39) also positive, which can then be represented as follows:

Here, ${\lambda }_{\mathrm{D}\mathrm{e}}$ is the electron Debye length given by ${\lambda }_{\mathrm{D}\mathrm{e}}={\left({\varepsilon }_{0}{T}_{\mathrm{e}}/en'_{\mathrm{e}}\right)}^{1/2}$. Using the ion plasma frequency ${\omega }_{\mathrm{p}\mathrm{i}}={\left({e}^{2}n'_{\mathrm{i}}/{\varepsilon }_{0}{M}_{\mathrm{i}}\right)}^{1/2}$, we can determine the minimum ion escape velocity as:

The key factors determining the characteristics of equation (41) are ${E}_{0}$ and $\varPhi /{\lambda }_{\mathrm{D}\mathrm{e}}$. In relation to this, two approaches are possible. The first is when the electric field at the open boundary shares the electrical characteristics of the surrounding closed boundaries. It is well known that electric boundary conditions are required at the sheath edge to smoothly connect the electric field distributions of the presheath region and the non-neutral ionic sheath [24, 25]. The magnitude of this electric field is known as ${E}_{\mathrm{s}}={T}_{\mathrm{e}}/{\lambda }_{\mathrm{D}\mathrm{e}}$, which is derived from the equilibrium between plasma electrons and the electric field at the plasma-sheath interface. Assuming that the width of the open boundary applied in the plasma reactor is sufficiently narrow so that the electric field ${E}_{\mathrm{s}}$ formed by the sheath edge along the bounded wall is equally shared in the open boundary region, equation (41) can be expressed as follows:

To ensure the validity of the Taylor expansion of equation (37), the conditions $\varPhi \ll {T}_{\mathrm{e}}$ must be satisfied. Then, the minimum escape velocity of ions is simplified as:

By rearranging equation (43) using ${N}_{u}={u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$, we obtain:

For equation (44) to converge to the classical limit ${N}_{u}$ = 1, the following condition must be satisfied.

Equation (45) always holds at the plasma edge where the condition $\varPhi \ll {T}_{\mathrm{e}}$ is satisfied. Therefore, the assumed sheath edge electric field ${E}_{\mathrm{s}}={T}_{\mathrm{e}}/{\lambda }_{\mathrm{D}\mathrm{e}}$ in equation (42) allows equation (41) to satisfy the Bohm criterion.

The second approach is when the electric field satisfies ${E}_{0}^{2}\ll {\left(\varPhi /{\lambda }_{\mathrm{D}\mathrm{e}}\right)}^{2}$. In this approximation, equation (41) can be approximated as follows:

Here, this minimum ion escape velocity must be greater than the product of the ion plasma frequency and the electron Debye length. Physically, the ion’s displacement during its oscillation period must be equal to or greater than the electron Debye length. By rearranging equation (46) using ${N}_{u}={u}_{\mathrm{i}}$/${u}_{\mathrm{B}}$, we obtain:

The normalized ion escape velocity ${N}_{u}$ passing through the open boundary is determined by ${\left(n'_{\mathrm{i}}/n'_{\mathrm{e}}\right)}^{1/2}$. Considering the condition $n'_{\mathrm{e}}\leqslant n'_{\mathrm{i}}$ at the plasma edge, equation (47) does not approximate the classical limit ${N}_{u}$ = 1. The condition ${E}_{0}^{2}\ll {\left(\varPhi /{\lambda }_{\mathrm{D}\mathrm{e}}\right)}^{2}$ used in deriving equation (47) minimizes the electric field at the open boundary. Consequently, ions passing through the open boundary do not accelerate sufficiently. Therefore, ions entering the open boundary must have velocities exceeding ${\left(n'_{\mathrm{i}}/n'_{\mathrm{e}}\right)}^{1/2}{u}_{\mathrm{B}}$ to satisfy the flux balance $\varGamma '_{\mathrm{L},\mathrm{e}}=\varGamma '_{\mathrm{L},\mathrm{i}}$. This mathematical criterion arises from the flux balance required to keep the bulk plasma neutral. Finally, solutions to the particle balance equation (equation (14)) and the energy balance equation (equation (27)) for a locally bounded plasma can be determined using equations (44) and (47).

6.   Conclusion and future applications

In this study, we developed new particle and energy balance equations that are applicable to locally-bounded plasmas. Using these equations, we studied changes in electron temperature and density in response to the unbounded plasma loss area and ion loss velocity. The basis of the newly developed particle balance equation is the equilibrium between the plasma loss parameters (${u}_{\mathrm{i}}$, $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$) and plasma volume ionization ${\nu }_{\mathrm{i}\mathrm{z}}\propto \mathrm{e}\mathrm{x}\mathrm{p}\left(-{\varepsilon}_{\mathrm{i}\mathrm{z}}/{T}_{\mathrm{e}}\right)$. From this equilibrium, we analyzed the impact of the plasma loss parameters (${u}_{\mathrm{i}}$, $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$) on the electron temperature. In the newly developed energy balance equation, the key consideration is the equilibrium between the RF power supplied to the plasma and the kinetic energy loss through the heterogeneous boundary. This heterogeneous boundary changes the characteristics of plasma wall loss, and directly affects plasma density variations. The most important input parameter for the particle and energy balance equations is the ion escape velocity through the open reactor boundary. To determine this, we developed new electric potential equations that connect the bulk plasma with the open boundary, and mathematically proposed a minimum escape velocity for ions. When ${{E}_{0}}^{2}\ll {\left(\varPhi /{\lambda }_{\mathrm{D}\mathrm{e}}\right)}^{2}$, we showed that the minimum velocity of ions escaping through the open boundary is ${u}_{\mathrm{i}}\geqslant {\omega }_{\mathrm{p}\mathrm{i}}{\lambda }_{\mathrm{D}\mathrm{e}}$. This allows us to define the plasma wall loss characteristics of a plasma reactor that includes a heterogeneous boundary.

The uncertainty of core variables (${u}_{\mathrm{i}}$ and $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$) significantly impacts the calculation of electron temperature and electron density. Typically, the plasma potential can vary depending on the characteristics of the plasma source, potentially influencing the ion escape velocity through $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$. Additionally, the structure of the reactor’s open boundary can distort ion paths and ion escape velocities. These hidden variables primarily affect ${u}_{\mathrm{i}}$. In conclusion, the uncertainty of ${u}_{\mathrm{i}}$ is expected to be greater than that of $A'_{\mathrm{L},\mathrm{e}\mathrm{f}\mathrm{f}}$, which will directly influence the accuracy of predictions for electron temperature and electron density.

One important assumption in this study is the low-pressure discharge condition satisfying “${\lambda }_{\mathrm{i}}\gg$ reactor scale”. This condition ensures uniform volume ionization in the reactor. However, as the discharge pressure increases such that conditions reach “${\lambda }_{\mathrm{i}}$ < reactor scale”, localized plasma is generated due to increased electron-neutral gas collisions. In such cases, equations (14) and (27) are inadequate for predicting electron temperature and electron density within the reactor. To validate the applicability of equations (14) and (27) under high-pressure discharge conditions, experimental verification of the theory is necessary. In this regard, measuring the escape velocity of ions with respect to plasma discharge pressure will be the most crucial verification process going forward. This validation will reveal how unbounded structures such as plasma confinement rings and gas baffles affect electron temperature and density.

In developing semiconductor equipment, the loss area of the locally bounded plasma can be a key factor for controlling the electron density and electron temperature. Typically, the electron density and electron temperature of the plasma directly impacts parameters such as etch selectivity, etch rate, and etch wall profile angle [26–30]. To control these factors, it is necessary to optimize process and equipment parameters, including RF frequency, RF power, gas pressure, and etch gases. Considering the unbounded plasma loss area as the primary design parameter for plasma reactor performance allows for more advanced plasma equipment capable of controlling electron density and temperature to be developed.